ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN2I ARBOR THE RESPONSE OF A PANORAMIC RECEIVER TO CW AIND PULSE SIGNALS Tecmhical Report No. 5 Electronic Defense Group Department of Electrical Engineering By: IH. W. Batten R. A. Jorgensen A. B. Mlacnee W. W. Peterson Approved by: H. W. -Alch Jr H. W. Welch, Jr. Project M970 TASK ORDER NO. EDG-3 CONTRACT NO. DA-36-039 sc-15358 SIGNAL CORPS, DEPARTMIENT OF THE ARMIY DEPARTMET OF ARMY PROJECT NO. 3-99-04-042 SIGNAL CORPS PROJECT NO. 29-194B-0 June, 1952

TABLE OF CO T 1TABLE OF CONTENITS Page LIST OF ILLUSTRATIONS iv ABSTIVCTix 1. IINRODUCTION1 2. DESCRIPTION OF TIE PROBLEMI 2.1 Assumptions 2.2 Definition of Parameters 2. Factors Describing the Response 7 3. 5T1hORETICAL ANALYSIS 11 3.1 Single-Tuned Circuit 11 5.2 The Gaussian Case 13 3.5 TIJO General Formulas 22 4. SOLUTIONS BY DIF!FE;RENTIAL ANATjZLYZER 25 4.1 Statement of the Problem 25 4.2 Method of Solution 27 4.2.1 Differential Analyzer Setup 27 4.2.2 Parameter Values 29 4.3 Discussion of Solutions 31 4.3.1 Qualitative Results 32 4.5.2 Quantitative Results 36 5. COiMPARISON OF SOLUTIONS 48 35.1 Single-Tuned Circuit by Tho Methods 48 5.2 The Gaussian Case and Differential Analyzer Solutions 48 6. SUI1ARY 62 ACTX-IOW SLI DGEvENT 63 APPEIDIX A - Derivation of Response to a Single-Tuned Circuit 64 APPENDIX B - Derivation of the Response of a Gaussian Filter 68 APPENDIX C - Curves for the Gaussian Case 7 APPEiNDIX D - Derivation of the General Relations Among Factors Describing Response 91 APPENjEDIXI IE - DiLff:erential Analyzer Procedure 99 E.1 Analyzer Setup 99 E.2 Differential Analyzer Solutions 104 E.2.1 Run Procedures 104 E.2.2 Extraction and Processing of Differential Analyzer Output Data 105 ii

E.3 Discussion of Errors 109 E.3.1 Machine Errors 109 E.5.2 Data Processing Errors 109 APPENDIX F - Differential Analyzer Data 112 F.1 Examples of Differential Analyzer Solutions 112 F.2 Curves for the Factors Ao, B, and W 117 BIBLIOGRAPHY 129 LIST OF SYMBOLS 152 DISTRIBUTION LIST 153 iii

LIST OF ILLUSTRAftTIONS Fig. INo. Title Page 2.1 Block Di-aagral of Idealized Superheterodyne 4 2.2a Time-Frequency Diagram Before MIixer 6 2.2b Time-Frequency Diagram After Mixer 6 2.3a Time-Frequency Diagralm NIormalized to Bandwidth 8 2.5b Time-Frequency Diagranm Nolralized to Sweep-Rate 8 2.4L The Response of a Scanning Receiver to a Series of Pulses at a Fixed Frequency 10 5.1 DiagrLa of Single-Tuned Circuit Filter 12 5.2 Response of Single-Tuned Circuit to Signal of Linearly Varying Frequency (Theoretical Curves) -.7 14 5.5 Response of Single-Tuned Circuit to Signal of Linearly Varying Frequency (Theoretical Curves) b = 0.1 15 5.4 The Relative fAplitude of the Response for the Gaussian Case 19 355 The Output Pulse Width for the Gaussian Case as a Function of Sweep-Rate 20 5.6 The Apparent Bandwidth of a Gaussian Filter as a Function of Sweep-Rate 21 3.7 Definition of Apparent Bandwidth 23 4.1 Block Diagram of the Problem 25 4.2 A-C Equivalent Circuit of Idealized I-F Amplifier to be Represented by Differential Analyzer 28 4.3 Block Diagram of Differential Analyzer Setup for the Solution of Eq. 4.8 30 4.4 Time-Frequency Diagram Showing Problem Parameters 32 14.5 Responses for Various Bandwidths (2 Circuits, d = 0 ) 33 4.6 Responses for Various Bandwidths iv

Responses for 1, 2, and. ' Circuits 55 4.<b 5Responses for Various Input Pulse Tidths 57 4.9 PXResponses of 1 and 2 Circuits for Various Pulse Positions 58 4.10 Relative Amplitude Response of a Panoramiic iReceiver as a Function of Sweep-Rate (Data from Differential Analyzer) for bd = 00 40 4.11 Relative Auplitude Response of a Panoramic Receiver as a Function of Sweep-Rate and Pulse Length for Two Circuits, Pulse Centered on Passband (Data from Differential Analyzer) 41 4.12 Frequency Swept by Panoramic Receiver during Output Pulse as a Function of Sweep-Rate. (2 Circuits, Pulse Centered on Passband) (Data from Differential Analyzer) 45 4.13 Frequency Swept by Panoramic Receiver During Output Pulse as a Function of Sweep-Rate. Pulse Centered on Piassand For bd == (Data from Differential Analyzer) 44 4.14 Intermediate Plot Response of a Panoramic Receiver as a Function of Pulse Center in Frequency. For bd = 2jr, 2 circuits 45 4.15 Apparent Bandwidth as a Function of Sweep-Rate (Data from Differential Analyzer) 46 5.1 Response of Single-Tuned Circuit to a Signal of Linearly Varying Frequency For _ ----1.2, Signal Begiininng at / t = - 00 49 5.2 Response of Single-Tuned Circuit to a Signal of Linearly Varying Frequency For b -= 7, Signal Beginning at it = - 5 50 5.3 Response of Single-Tuned Circuit to a Signal of Linearly Varying Frequency For -f =.1, Signal Beginning at / t = 1 51 5.4 Relative Amplitude for a CUW Input Signal 52 v

5.5 Relative AAmplitude for the Gaussian Case and Differential Analyzer Solutions (bd = 2x) 5 5.6 Relative Amplitude for the Gaussian Case and Differential Analyzer Solutions (bd = 3/2) 54 5.7 Output Pulse Width for a CW Input Signal 55 5.8 Output Pulse Width for the Gaussian Case and Differential Analyzer Solutions (bd = 2) 55 5.9 Output Pulse Width as a Function of Sweep-Rate for Gaussian Case and Differential Analyzer Data (bd = /2) 57 5.10 Apparent Bandwidth for the Gaussian Case and the Differential Analyzer Solutions (bd = 27t) 58 5.11 Apparent Bancdidthl for the Gaussian Case and the Differential Analyzer Solutions (bd = /2) 59 C. 1 Relative Aitpli tude as a Function of Input Pulse Width for the Gaussian C as) 7 C 2 Output Pulse Width as a Function of Tnput Pulse Widthlc for the Gaussian Case 77 C.5 Apparent Bandwidthc ' as a Function of Input Pulse Wlidth for the Gaussian Case 78 C.4 Relative Aml, itude as a Function of Sweep-Rate for the Gaussian Case 79 C.5 Output Pulse WWidth as a Function of Swecp-Rate for the Gaussian Case 80 C.6 Apparent Bandwtidth as a Function of Sweep-Rate for the Gaussian Case 81 C.7 Relative AmlLitude as a Function of Bandwidth for the Gaussian Case 82 C.8 Output Pulse Width as a Function of Bandwidth for the Gaussian Case 83 C.9 Apparent Bandwidth as a Function of Filter Bandwidth for the Gaussian Case 84 C. 10 Relative Ar;mpli.tude as a Function of Bandwi.dth for the Gaussian Case 85 C.ll Output Pulse,Wid.tn as a Function of Filter Bandwidth for the Gaussian Case 86 vi

C.12 Apparent Bandwrridth as a Function of Filter BandCtidth for the Gaussian Case 87 C.15 Relative Amplitude as a Function of Input Pulse Width for the Gaussian Case 88 C.114 Output Pulse Width as a Function of Input Pulse WTidth for the Gaussian Case 89 C.15 Apparent Bandwidth as a Function of Input Pulse WIidtih for the Gaussian Case 90 D.1 Definition of Apparent Bandwidth 95 L.1 Single-Tuned Filter and Its Analogue 101 E.2 Differential Analyzer Setup of Input Function Generator 103 E. Typi cal Results of a Run as Observed at the Differential Analyzer Output 106 E.~4 Intenrediate Plot Riesponse of a. Panoramic Receiver as a Function of Pulse Center in Frequency. For bd = 2r, 2 circuits 108 3.5 The Apparent Bandwidth of a Two Circuit Filter as a Function of Sweep-Rate (Data from Differential Analyzer) bd = 27c 110 3.6 Relative kAmplitude as a Function of Sweep-Rate and Pulse Len,-gth for Two Circuits, Pulse Centered on Passband 111 F.1 Responses for Various LNumbers of Circuits and Input Pulse Positions 115 F.2 Responses for Various Bandwidths (1 Circuit, d = ) 114 F.3 Rsesponses for Various Bandwidths (4 Circuits, d = 0 ) 115 F.]4a Responses for Slightly Different Length Pulses (5 Circuits) 116 F.4b Responses for Slightly Different Length Pulses (2 Circuits) 116 F.5 Responses for Different NTumbers of Circuits 118 F.6 Relative Amplitude as a Function of Sweep-RatePuls e Centered on Passband (Data from Differential Analyzer) bd = 1 119 F.7 Relative Amplitude as a Function of Sweep-Rate, Pulse Centered on Passband (Data from Differential Analyzer) bd = 2 120 F. 8 Relative,Amplitude as a Function of STweep-Rate, Pulse Centered on Passband (Data from Differential Analyzer) bd =- t/2 121 vii.

F.9 Relative Armplitude as a Function of Sweep-Rate,Pulse Centered on Passband (Data from Differential Analyzer) bd = t 122 F.10 Relative Amplitude as a Function of Sweep-Rate,Pulse Centered on Passband (Data from Differential Analyzer) bd = 2t 123 F.11 Output Pulse W;idth as a Function of Sweep-RatePulse Centered on Passband (Data frorm Differential Analyzer) bd = 1 12r F.12 Output Pulse Width as a Function of Sweep-Rate, Pulse Centered on Passband (Data flrom Differential Analyzer) bd = 2 125 F.15 Output Pulse Width as a Function of Sweep-Rate,Pulse Centered on Passband (Data from Differential Analyzer) bd = C 126 F.14 Output Pulse Width as a Function of Sweep-Rate,Pulse Centered on Passband (Data from Diifferential Analyzer) bd = 2 e 127 F.15 Frequency Swept by Panorsmic Receiver During Output Pulse as a Function of Sweep-Rate, bd = 0 128 Lr ZL 1

ABSTRACT An analysis of the response of a panoramic receiver to cw and pulse signals is presented. The receiver's response is studied quantitatively as a function of the parameters: signal pulse length and frequency, receiver bandwidth, sweep-rate, and type of i-f amplifier. The effect of these parameters on the relative output amplitude, output pulse width, and apparent bandwidth is emphasized. Some general relations are derived. Two specific cases are considered. An electronic differential analyzer is used to study the response of a receiver with a single-tuned i-f amplifier to pulses having rectangular envelopes. Theoretically the response of a receiver with a Gaussian shaped i-f passband to pulses having Gaussian envelopes is derived. This answer is given in closed form. The agreement between these two cases justifies application of the Gaussian case to most practical design problems. Many curves are presented to aid the design engineer. ix

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN THE RESPONSE OF A PANORAMIC RECEIVER TO CW AND PULSE SIGNALS 1. INTRODUCTION The study of the response of a linear resonant system to a sinusoidal driving function having a linear variation of frequency with time is pertinent in various fields of engineering. This problem is encountered when an engine is accelerated uniformnly through a critical frequency.l The same situation occurs in the analysis of records of ocean waves by means of vibration galvanometers.2 A panoramic superheterodyne receiver also presents this problem; and this is the problem studied in this report. An analogous second problem is the response of a system whose resonant frequency varies linearly with time to a fixed frequency sinusoidal signal. This problem is encountered in various types of spectrum analyzers and in panoramic radio receivers.p For the high-Q or very much underdamped system, the two problems prove to be essentially equivalent.2 4 As indicated by the bibliography, there is a considerable amount of published work on these problems. Previous theoretical work has been confined to a single-tuned circuit or its mechanical analogue with constant amplitude driving functions. Microwave spectrum analyzers or panoramic radio receivers usually represent resonant systems having many degrees of freedom; and with the growing importance of pulse modulated communication and radar systems, the response to pulsed driving functions is important. This report is concerned with 1Lewis, F.M., Ref. 2 2Barber, N.F. and Ursel, F., Ref. 3 5lilliams, E.M., Ref. 4; Barlow, H.M. and Cullen, A.L., Ref. 5; Montgomery, C.G., Ref. 6; Marlic, W.E., Ref. 11; Thomasson, D.W., Ref. 12 4Hok, G., Ref. 1

- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN this more general case of the response of a resonant system to a constant amplitude or pulse modulated sinusoidal dcriving function having a linear variation of frequency with time. Throughout this report the problem is stated and discussed in tetrs of a panoramic radio receiver, but the results obtained here are directly applicable to other enLJi neering problems. C

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 2. DESCRIPTION OF THE PROBLEM 2.1 As sumptions The receivers of this investigation are idealizations of conventional superheterodyne receivers. Diagrcm4i atically, they are shown in Fig. 2.1. The function of the mixer is to convert an incoming signal of fixed instantaneous frequency to one with an instantaneous frequency changing linearly with time. It is assumed that the envelope of the incoming signal is not distorted by the mixer. The filter of Fig. 2.1 merely selects the desired frequencies, and the dertector operates on the output of the filter to obtain the envelope. These assumptions reduce the problenm to that of obtaininr the response of a filter to a particular fm signal.1 Several types of filters and a variety of input signals are considered in this report. Two filters are examined theoretically: the one a single resonate circuit, the other a filter with a Gaussian amplitude response and a linear phase response curve. Both of these cases are examined with a cw input signal to the mixer. In addition, sinusoidal pulses with square envelopes (see Fig. 2.1) are studied in conjunction with the single circuit filter, and sinusoidal pulses with Gaussian envelopes are treated as input signals to the mixer with the Gaussian filter. By means of a differential analyzer filters with one, two and four r:.y-nchronously aligned single-tuned stages are examined with cwr and pulse i.nLput signals to the mixer. The pulses studied all have square envelopes in this investigation. 1Under sore circumistayces the response of a trf receiver which sweeps in frequency by chang-ing the resonant elements is approximately the same as that of a receiver in which the resonant elements are fixed and the input signal sweeps in frequency. ( Hok, G., Ref. 1) The analysis in the report is appropriate in such cases. (See also Barber, N.F. and Ursel, F., Ref. 5) ---— 3 _ --- —

I )( 10 I (D ya) U, r I SIGNAL INPUT TO RECEIVER ES- TRANSMITTER FM SIGNAL FILTER OUTPUT DETECTOR OUTPUT Ift FIG. 2.1 BLOCK DIAGRAM OF IDEALIZED SUPERHETERODYNE RECEIVER

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2.2 Definition of Parameters The same notation is used throughout this report and wherever it is convenient, formulas and results are presented in normalize. and dimensionless form. This facilitates the comparison of the several cases discussed. The following para-meters are used in describing the signal and the filter. The signal is assumed to have passed through the mixer, so that the filter has a constant resonant frequency and the signal frequency is varying. a -- center frequency of the filter (radians per second) a -- frequency of signal at the time t = 0 (radians per second) b -- bandwidth of filterl (radians per second) c -- time of center of input pulse (seconds) d -- input pulse widthl (seconds) s -- sweep-rate of signal (radians per second) These paraeters are illustrated on a ti;e-Ifrequency diagrami in Fig. 2.2. Note that the second definiton of "a" zamounts to stating thl-at the time origin is taken as te time hen the signal seeps (or woul see) through the center frequency of the:ilter. The results can be presented in terms of d.imensionless variables by nozrmalizing with respect to one of the paramleters. The effect is to reduce by one the numlber of parameters involved. In this report the normalization is usually with respect to bandwidtth. Frequencies are then in bandwidth units, -e.g., b and times in reciprocal bandwidth units, t/l/b or bt. The appearance Thie bandwidths and the widths of input and output pulses are generally measured between points where the amplitude drops to 0.70'7 of its lmaximum value. This convention is adopted in this report e::cept in,the treatment of the Gaussian case, twhere all widths are measured to the e-/4 points. This simplifies tl}e formulas without seriously affecting the accuracy of the results, since e-l/ f is only 9% larger than 0.707, the width so defined is about lS' smaller than the width between the 0.707 Ioints. No adjustment has been made for this 'dif ference in this report. 5

Z9-OZ-9 '1>i1 Z-ES-V OZ6-1N t d t=c () RECEIVER PASS BAND FIG. 2.2a TIME-FREQUENCY DIAGRAM BEFORE MIXER tO t=c +=c FILTER PASS BAND SLOPE =S FIG. 2.2b TIME-FREQUENCY DIAGRAM AFTER MIXER 6

1. - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN of the time-frequency diagram in two possible sets of normalized coordinates is shown in Fig. 2.3. 2.5 Factors Describing the Response The general nature of the response of a panoramic receiver is quite evident. The amrplitude of the voltage output as a function of time is zero sufficiently early; it reaches some peak value and again approaches zero. The peak value for a pulse input depends upon the value of sc, the difference between the filter frequency and the signal frequency at the center of the pulse. The peak value is a maximum for sc approximately zero and approaches zero for large values of sc. The most important features of the response are: (1) its peak amplitude when sc is zero, (2) the approximate width of the response in time, and (3) the width of the I)eak amplitude curve plotted as a function of sc. Three dimensionless measures of these features are defined below; these quantities are used to compare the cases discussed in this report. The Relative Anplitude A With a given input signal suppose the output voltage of the filter is g(t). Let g0 denote the steady state output voltage when the input to the filter is a cw signal with the center frequency of the filter and the same peak voltage as that of the given signal. Then the relative amplitude A is defined as: A max g(t) (2.1) The value of A for sc = 0 is denoted by A0, which is also referred to as relative amplitude. A0 is a measure of the effect of changes in bandwidth, sweep-rate, and pulse width on the amplitude of the response for a given type of filter. I 7 I

Z~9 -z -g G 1>18 2- 2 - t OL6 - l bt bt = bt=bc H Hr -J U-F b2 SLOPE = - S 5 II bd SC b I -BLI — ~~- Ih —~~ --- — ~~- -~-~ —~ IIIC~ I eC~~~ rC - - a -31~ra I ~ F-L a b b FIG. 2.3a TIME- FREQUENCY DIAGRAM NORMALIZED TO BANDWIDTH t/s= t/s = 0 I C SLOPE = I FIG. 2.3b is TIME- FREQUENCY DIAGRAM NORMALIZED TO SWEEP-RATE 8

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The Output Pulse Width W Suppose to is the width of the output pulse in seconds (between halfpower pointsl). This width can conveniently be described in dimensionless fonrm by sto W= st (2.2) b W is the number of receiver bandwidths swept through by t&he signal in the duration of the output pulse, Thus, if the output of the detector is presented on an oscilloscope with the abscissa calibrated in frequency, is the width of the output pulse in cycles per second. The Apparent BandwTidtcl.t B Sulppose for a!given f ilter anld. a given type pulse input the relative allplitude A, as a function of sc, drops to 0.707 Ao at sc2 and scl. Then the apparent bandwidth of the receiver for pulses is by definition sc2 - scl B =. (2.3) b Essentially, a signal within bB cycles per second of the receiver frequency is received with little attenuation. Both W4 and B are im,.ortant factors in the discussion of the resolution of a receiver. It was pointed out that if the output of the panoramic receiver is presented on an oscilloscope calibrated in frequency, the width of the pip on bW on the oscilloscope would be 2c cycles per second; the resolution could hardly be expected to be much better than this width. It was also pointed out that a I 1See footnote p. 5 9

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN signal within 2 cycles per second of the receiver frequency is received with little attenuation, and thus appears on an oscillograph at the receiver frequency. This discrepancy limits resolution to about bB (see Fig. 2.4). ~ ~ ~ ~~~~~~~~~~~~~~~~~i J FIG. 2.4 THE RESPONSE OF A SCANNING RECEIVER TO A SERIES OF PULSES AT A FIXED FREQUENCY

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3. TSHORELTICAL ANALYSIS. 1 Single-Tuned. Circuit The special case of the single-tuned circuit filter, shown diagrrammatically in Fig. 3.1, is examined theoretically in this section for input pulses which have square envelopes. The assumed input current pulse to the filter is the real part of exp(jat + j 2 t), or c - t < c +2 i(t) = (5.1) d d (3.1) 0, for t < c - d or t > c + d; 2 2 then the real part of the response represents the output. In this form, a is the pulse frequency at zero time in radians per second, s is the sweep rate in radians per second per second, d is the pulse width in seconds, and c is the center of the pulse in time. The determination of the voltage that appears across the resonant circuit for this input current pulse is carried out in Appendix A. The result in complex form is given by e(t) - exp(jat + j 2 t2) G(y) (5.2) where — r - n + jo/~ t 2RC =/Jand G is a function related to the error function of a complex variable and can be calculated using tables for that function. The envelope for this voltage is given by 1i

Z9c-ZI- 9 -1 1 9-~S-V 06- W i(t) i0 - 1 L R {C e(t) FIG. 3.1 DIAGRAM OF SINGLE-TUNED CIRCUIT FILTER 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN = G/(y) (I-3) The case of d = o (cw inputs), can be found. elsewhere in the literature.1 The extension to.pulses brings in two new parameters: the normalized pulse width rs d and the normalized pulse position /-c. Fortunately, the increase in complexity is not as severe as the two additional parameters suggest since the envelope at any time after (c + d) is simply given by t e(t) = e(c + e)- R (5.4) d Therelore the starting time (c - 2) can be taken as the only parameter, and the envelope of any pulse is readily deterwmined from the response to the pulse which starts at the same time and continues indefinitely. Nevertheless, the complete analytic study of the normalized response with two parameters (normalized bandwidth and starting time) is tedious and, since the results can easily be obtained from a differential analyzer, only select cases have been carried out numerically Two sets of calculated curves for particular bandwidths with the starting time as parameter are given in Figs. 3.2 and 3.3; the curves are normalized to sweeprate. A comparison of these curves with the results obtained from the differential analyzer is made in Section 5.1 (see Figs. 5.1, 5.2 and 5.3). 3.2 The Gaussian Case In this section the response of a filter with a Gaussian transfer function to a cw signal and to pulses with Gaussian envelopes is discussed. 1lok, Gunnar, Ref. 1; Lewis, F. M., Ref. 2; Barber, N.F. and Ursel, F., Uef. 5 15

CM OJ I 0 frf 1 -r rt tO I 0 0 1,0 4 -( 5.5 1~- br =...... i]..........~07 J8s: bR t X,9 X FT 1 -- -- -- -- -- -- -- — i --- ^ -- - y ^ ^ - -- -- -- -- -- --, ^ tf ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a~~~~~~~~~~~~~~~~~~~~r-Tzz~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CN -5 -5 -4 -3 -2 -I 0 I 2 3 4 5 2 FIG. 3.2 RESPONSE OF SINGLE-TUNED CIRCUIT TO SIGNAL OF LINEARLY VARYING FREQUENCY (THEORETICAL CURVES)

I. 1.1 4 -0i) Y- I U FIG. 3.3 RESPONSE OF SINGLE-TUNED CIRCUIT TOSIGNAL OF LINEARLY VARYING FREQUENCY \ (THEORETICAL CURVES) b b~- = 0.1 ]....!!, -/,f t i i~,', /:~ c I II B" I l -6 -5 -4 -3 -2 0 I / 2 V T 3 4 5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The most important reason for considering this case is that a closed form answer can be obtained. The Gaussian filter is not physically realizable; however, if the time delay is neglected, the transfer function of n single-tuned circuits all at the same frequency approaches the Gaussian function as n becomes large.l As is pointed out in Section 5.2, the envelope of the response for this hypothetical filter differs very little from that of several synchronous singletuned circuits; therefore, a study of this case gives insight into the problem. The transfer function assumed is I() 1 -e{- (C. - })2 -1/4 The center frequency of the filter is "a", and the bandwidth between e points is b 2 Note that the phase delay is completely neglected here. The introduction of a linear phase delay would not significantly change the answers. The signal assumed for the cw case is, st2 f(t) = cos [at + -2 (3.6) and for the pulse case is, 2 f(t) = exp[- (-L2 ) ] cos[at + 2 ] * (357) -1/4 The center-time of the pulse is c, and the pulse width between e points is d. The answer is derived first for the pulse case, and the cw case is obtained from it by letting d approach infinity. See Section 4.1, p. 26 See Footnote p. 5 16

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The analysis is straightforwaard. The signal function is transformed to the co - plane, multiplied by the transfer function H(wc), and transformed back to the t-plane. An outline of this manipulation is given in Appendix B. The envelope of the output for the pulse case is l s(e -_ 2 2 g(t) = A0 exp { [ b ]2 2 b)2} (3.8) where Ao, B, W, and - are functions of s, b, and d as follows: A0 b () AO 4[( _ + b2) 2 + 4s2]1/4 (39) B 1 B = b/ 2 + b s2d2 (5.10) / I *b2. 2 2 = b A02B (3.11) 2+ b +2.2s tm = cFT +b s- (5.12) 7- + b2 + 82s 2 The envelope response in the cw case reduces to g(t) = Ao exp{ [s } (3.13) where /A b~ b Ao = b + 42)14 (5.14) w b4 4s2 A (5.15) A0' I J L 17

ENGINEERING RESEARCH NSTITUTE UNIVERSITY OF MICHIGAN - The definitions of A0,. and B given here are consistent with those in Section 2.2: Ao is the relative amplitude, W is the output pulse width, and B is the apparent bandwidth. For the Gaussian case graphs of Ao,., and B as functions of b are given in Figs, 5,4, 5.5, and 3.6. A0 is defined so that it has the value one for an infinite pulse and zero sweep-rate, Ao is affected little by sweeping until s 4i. is of the order 1 + (bd)2, and drops off rapidly for higher sweep-rates, By definition the output pulse width W is one for a cw signal and zero 2s sweep-rate. The curves are never below tW ---.; this is the output pulse width corr:esponding to tlhe impulse rxesponse of the filter (bd = 0). For low sweeprates the output pulse width is between Lth,.e value for the cw signal and that for the impulse response. For high sweep-rates the output pulse is essentially the impulse response of the filter and is independent of bd. The apparent bandwidth B is defined so that it is unity when the sweeprate is very low and the pulses very long, For short pulses B is greater than one even for zero sweeep-rate. As the -sweep-rate increases above - 1 - (a 'v b2d2 s the curve rises sharply and approaches B = - bd asymptotically (see Fig. Mlo:re curves and. further discussaion are included in Appendix C to show the dependence of Ao, W^ and B on all the parameters, In Section 5.2, Ao, W, and B as computed from the differential analyzer data are compared with Ao, W, and B for the Gaussian case, The reader should keep in mind that A0, W. and B are expressed in dimensionless form,

1.O V) 0.8 0.6 0.4 0.2 0.1.0 10.0 s b2

WF FIG. 3.5 10 THE OUTPUT PULSE WIDTH FOR THE GAUSSIAN- _ CASE AS A FUNCTION OF SWEEP-RATE fl g 1.0 ii-i --- —-.. — — ^^ Z7,,, ----_____ - ~^ /oy^"""""- — /, L"/ bd c 1 1 L-do * bd= 2 bd = 0 0.1____ —.01 0.1 1.0 S T?~ CQ LO 0 I ro I < 0 10 10

100 10.01.1 1.0 S bt 10

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ).) 3 rtO Gengral Formulas In Section 352 it was found that for the Gaussian case with constant emplitude input signal, Ao2 = 1, (3.16) and with pulse input signals, A 2 B ib = Ao2 B to = 1 o sd - (3.17) It is natural to ask whether these formulas are more general. The answer is that they can be proved with very broad hypotheses. The general validity of (3.16) and (3.17) depends on a different definition of bandwidths and pulse widths. The width of a pulse is defined as its total energy divided by its maximum power. The bandwidth of a filter with transfer function H(u)) is defined by 00 _ fX IH(u) HI(X) dc b max H(-) H( max H(X) H(w) i I L This is lnown as the "noise bandwidth."1 A similar definition, in tenns of the energy of pulses, is given for apparent bandwidth B.2 Equations 3.16 and 3.17 are correct if the widths of bands and pulses are calculated in the above manner. Usually these widths differ little from the 3 db widths of curves. They coincide for square pulses, The ratio of noise bandwidth to 3 db bandwidth for synchronous single-tuned amplifiers is 1.57, 1.12, 1.15, 1.06 for one, two, four, and an infinite number of stages respectively.1 Therefore, the formulas are still Wallman, Ho and Valley, G. E., Ref. 9, p. 169; Lawson, J. L. and Uhlenbeck, G. E., Roef. 10, pp. -76-177 2See Appendix D. 22

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN approximately true for 3 db bandwidths and pulse widths. The details of the derivations are given in Appendix D. In deriving Eq 3.16, the signal f(t) = fo cos at + s-] is assumed to be applied to an arbitrary filter with a finite bandwidth. The energy of the output pulse is calculated using Fourier transforms and Parseval's Theorem, and the result is interpreted using the energy-type definition of bandwidth and pulse width. The first step in deriving Eq 3.17 is to define apparent bandwidth. This is done as follows: Given any signal of finite energy, a family of signals can be constructed; each member of the family is formed by shifting the original signal in frequency. Let fa (t) denote the signal which is shifted from the original by a radians per second. For each input signal fa (t) the energy Eo of the output pulse is calculated. Eo is a function of a which approaches zero for large values of a (see Fig. 3.7). Then the apparent bandwidth B is defined as the width of the curve of E as a function of a, divided by the filter Eo a Figure 3.7 25

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN bandw'.idth, 'whlere the width of the curve is determined by dividing the area under the curve by its maximum height. Thus, 00 B, =!/- da B 1 Co.. (5.18) b maxl " Equation 5.17 follows from this definition of B when the energy of the output pulse is calculated using Parseval's Theorem (see Appendix D). It seems worth repeating that the derivations above assume an arbitrary filter of finite bandwidth and an arbitrary type of pulse. It is noted in Section 3.2 that for fast sweep-rates, the apparent s bandwidth B, plotted on a log log graph, has the asymptote B*= bd * b2 This is true for an arbitrary filter. A proof using Eq 3.17 is given in Appendix D. By Eq 3.17, B d= (5.19) to Ao2 It is shown in the appendix that the response approaches the impulse response of the filter as the sweep-rate becomes large, and thus to is the width of the response to an impulse, while Ao is related to the strength of the equivalent impulse. Then the behavior of B for large sweep-rates follows from Eq 3.19. 24

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 4. SOLUTIONS BY DIFFERENTIAL ANALYZER 4.1 Statement of the Problem The physical problem is to observe the response of a bandpass filter to pulse modulated sinusoidal signal whose frequency varies linearly with time. A block diagram of the apparatus required is shown in Fig. 4.1. PULSED fm SIGNAL F FILTER a RECORDER GENERATOR (c, d, and s) a, b, type Fig. 4.1 Block diagram of differential analyzer. This is equivalent to solving the ordinary differential equation ny dy F[d..n' '., Y, t] = f(t) (4.1) dt where f(t) is the time representation of the pulsed frequency-modulated signal. Since only linear filters with fixed parameters are considered in this report, the left side of Eq 4.1 is an-ordinary linear differential form with constant coefficients; the order of the equation, n, depends on the complexity of the filter considered. A differential analyzer is well suited both to the solution of Eq 4.1 and to the generation of the driving function f(t) (input pulse), through the solution of an auxiliary differential equation.1 1Busl, V. and Caldweell, S.H, ef.!;a Razziri, J.., 2Randall, R.H., and Russell, F.A., Ref. 14: Macnee, A.B., Ref. 15 25

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The driving function is f(t) = a d f(t) = 0 if t< c - Z, f(t) = cos[at + st2 if c - d t < and (4.2) f(t) = if t>c + 2 Thus f(t) is a sinusoidal pulse with a square envelope. The pulse has a duration of d seconds and is centered at c seconds; the instantmaeous ingular frequency of the signal is = a + st (4.3) where s is the sweep-rate in radians per second per second. The filters studied are of the sort encountered in the simplest intermediate-frequency amplifiers, a synchronous single-tuned amlplifier.1 Each filter has maximum response to sinusoidal steady-state signals at "a" radians per second, and the 3 decibel bandwidth is "b" radians per second. Filters consisting of one, two, and four single-tuned circuits are investigated. The use of one single-tuned circuit is of special interest since this case can be calculated without too much difficulty.2 This gives a check on the analyzer solutions.3 The steady-state amplitude response of the n stage filter is n 2 2 2 H(c) [(J { [(2) - (4.4) For large n 1 1 (2) n_ 1 I E ~n(2), (4.5) lWallnan, H. and Valley, G. E., Jr., Ref. 9 2See Section 3.1 3See Section 5.1 4loc. cit. p. 172 ______ _-k

-- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN and limit JI(c) = limit 1 +[ - a2 2 ] 7 nL-0 P n —l =n -O L o --- ' n ~n2 W2 - a2 2 = 2 b J ~ (4. 6) Furer, or equences nr resonnce, 2, nd erore Further, for frequencies near resonance, 1 + w ~ 2, and therefore 1 li it Hm e as n-00 G - 2 Qn2 [ ' - a] -b J. (4.7) A filter having this Gaussian amprlitude response curve can be handled by analytic means if the input pulses are assuimed to have a Gaussian rather than a square envelope curve.1 The differential analyzer data for synachronous single-tuned amplifier gives a measure of the usefulness of this analytic result.2 4.2 Method of Solution 4.2.1 Differential Analyzer Setup An alternating current equivalent circuit of the intermediate frequency filter-amplifier studied is shown in Fig. 4.2. The vacuum tubes are assumed to operate in a linear fashion. The input and output capacities as well as the plate resistance rp are lumped into the appropriate single-tuned interstage circuits which all have identical element values. The grid-plate capacities are neglected. This filter amnplifier is de scribed by the equations del el 1 + el dt C - t + — + e dt at R L Cde e + f1 f dt dt R L C d + E + L f e4 dt dt 1" = il(t) = f(t), =- g-rue. - ine2. = - W )'1 -- iSee Section 3.2 2See Section 5.2 27

CU In -J r() I Or (I, 0 0> i, = f(t) i2= -gme, 13=-g9me2 i4= -gme3 IN) co FIG. 4.2 A-C EQUIVALENT CIRCUIT OF IDEALIZED I-F AMPLIFIER TO BE REPRESENTED BY DIFFERENTIAL ANALYZER

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A block diagram of the differential analyzer setup for the solution of Eq 4.8 is shown in Fig. 4.3. Each block labeled "circuit" in Fig. 4.3 solves one of the second order differential equations of Eq 4.8. The details of the interconnection of the differential analyzer components within these blocks as well as the function generator block are given in Appendix E. Since all the circuits are tuned to a common resonant frequency and all voltages and currents are at zero at the start of every solution, there are just two parameters for each circuit box. For the jth circuit the amplitude of the driving function ej is controlled by a potentiometer, Pij, and the bandwidth bo is controlled by a potentiometer, Pr. In this study the bandwidths of all circuits are identical; the band1 b = bo (2) 1. (4.9) As indicated in Fig. 4.3, a four channel recorder permitted simultaneous recording of the input pulse f(t) and the responses of one, two and four singletuned circuits. The input potentiometers were generally adjusted to give approximately full scale deflection of the recorders at the time of maximum response. 4.2.2 Parameter Values The function generator output was a constant amplitude sinusoidal signal that varied linearly in frequency from 10.25 to 20.19 radians per second in a period of 140 seconds. Therefore, the sweep-rate of the driving signal was always.0710 radians per second per second. The length of the input pulse d and the time of the center of the input pulse c were controlled by d closing a key connecting the input signal to the first circuit box, at c - 2 seconds; d seconds later an electronic interval timer disconnected the input I 1oc. cit. p. 172 29

INPUT CIRCUIT CIRCUIT CIRCUIT CIRCUIT f (t e e FUNCTION f() o. No. 2 e2 No. I o.2 N3 o. 4 GENERATOR 5 P L P P, Pr 2, Pr Pi, Pr Pi4, Pr o KKEY AND INTERVAL ~INTERVAL ~FOUR CHANNEL RECORDER TIMER, d FIG. 4.3 BLOCK DIAGRAM OF DIFFERENTIAL ANALYZER SETUP FOR THE SOLUTION OF EQ. 4.8 Cl N I rC -0 CD I 0!CY n 4 if

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN sial. T fruenc of he inpu sinal equaled the frequency of the filter a = 14.18 radians per second, 55.4 seconds after the start of each sweep. Table 4.1 summarizes the ranges of parameter values. It will be noted that the Q's of the tuned circuit, b, range from 11.28 to 567.2. TABLE 4.1 RANGES OF PARAM.ETER VALUES a b c d s 4.-3 Discussion of Solutions 14.18 radians per second 0.025 to 1.257 radians per second - 55.4 to +70 seconds 2.65 to 125.7 seconds.0710 radians per second per second For convenience in the operation of the differential analyzer, all solutions were run with a fixed filter resonant frequency "a" and a fixed input signal sweep-rate s.1 The differential analyzer solutions were run to observe the effect of varying the four remaining parameters: (1) the center time of the input pulse c, (2) the length of the input pulse d, (3) the bandwidth of the filter b, and (4) the number of single-tuned circuits. These parameters are displayed on the time-frequency daiagram of Fig. 4.4. Some of the solutions as observed at the output of the differential analyzer are discussed here. Additional solutions and discussion are found in Appendix F. A comparison between the solutions measured on the differential analyzer and the solutions calculated analytically is given in Section 5. 1See Appendix E for a discussion of this and for details of the differential analyzer operating proceedure. J J L - -- -- -- 31 - -- — '

I 11I -- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN o ____-1 _b 4 -0o a frequency C I Ce __,!e ^ I I I Ksd --- Fitg. )i. - TiLme-Frequency Diagraml Sho-ing Problem Paraueters 3,.l1 Qualitative Results A. Varying Filter Parameters The effect of varying the filter parameters b and n is showmn by Figs. 4.5 to 4.7. Figure 4.5 shows the response of a two circuit filter to an extremely long input signal as the filter bandwidth is varied. The four output traces illustrate an important effect, For a fixed sweep-rate there is a filter bandwidth which leads to the minimum output Vgb pulse width in seconds, -. From the data shomwn the optimum bandwidth appears to be about 0.34 radians per second. Figure 4.6 shows the effect of varying the bandwidth of a one circuit filter; the input signal is a pulse 10.6 seconds long with a center frequency equal to the filter center frequency. The two important effects observable here are: (1) the reduction in the relative amplitude A0 as the filter bandwidth is 32

m.. Y 1~ I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~k - 4 i:_i-l_ I__ir- i ~~~~~~~~~~~~~~~ i i. i i II: r- r. L1-1-J1 -7 l~L~ f: I:' _i.-.-I- -. 1-J J i:_[=:~~7_t L-. ~ ~ L — T I- - I I 11.1 L - I - --— I - 4 --- I t L i~ i i i 1. I!:1 /- - 'I ':!!! 1,,!:.:.;.)! - t i 1.:'1I- -: - u:LtL-::t:- - — f ':: -t:::::-.- I —:. 1 ':Il '- - - - -- --— f_:J..::l:-: vLI-. I, i ~ ~ ~~~~~~I il: I - II I- 1 -,- - / I 7777 LiI z: (30NVNOS3) 0 = 4 < or- -, - \ ' ' i \ 7 \ \ -. \ \ -7 SaN033S S OILO' = S, = p SIlnod11 z SHIGIMONVt SnOlIVA aOJ S3SNOdS38 V't7 '91

M-970 B -P3 -5 RKL 5 - - 52, - IlI~l;lIL:~r-il — E —.21/:;/ - / - I^ ---71 -—. -- -^ —! - / / - -f ti ----- ---- - — I —.-/ -=1j —, —, I..,,,_ FIG. 4.6 RESPONSES FOR VARIOUS BANDWIDTHS - - I -- -i:.- -- -- - --- \ - - t -:.., V -.A I CIRCUIT C = 0 d = 10.6 S =.0710 - _;__i _: = — - I1-_-_ —.,f..f f Jf -; 1 _____j_, — I- J:: -:::::::::: -' -: /: _ ---' — - -—:.-i --- —--?- ~ 71! —, ttt.- it I-tfi It-fi-Ttt-!t-~t'#J~ J ----—: ----- ' --- — -- — t — -—!I ---— i --- — -. -, - -- -.;- -.:: 0 ---l-= — L-\-I'-i- 1ttn:t~ \\f~~l-fttftt- ~t I\ t ] L-T 0Mfl --- -- -—: 1 —! i l_:i0 --- _ - v A 9 '. I HE-~\\ \-A:.::::::::::::77F '- ~ lid- -—::.t777.: ':Ifi-,:; --- -llid Ar = 96':/>f-:= - l S -l-X-lt.Xliftlf..niJ~~~~~~lt, U l~~~iX-f ff __ I,-'tI ilsl iAl 31 8!1' - ~~~~~~~~: 3 - -- \;- -- -\ — -~\ --- —: g^ ^- = O:A.S:- - ---— ' t = 0 (RESONANCE) 5 SECONDS I.._ 1 --- — l \ --- - I --- -Tt=- --- '-/ --- —! --- -/- - ---,-...,, -I -:: --::! --- -:::: i - - -j./. -, A- - i - L — _~ —i II -i i -- 1_:i. k- i _ \.._ k _ ---. ', _ - _ _ __ 1 7k 74. l'-.- t _ \ _ = 34

ZS-I-S I- d f — 'd- - O06-W FIG. 4.7 RESPONSES FOR I, 2 AND 4 CIRCUITS b =.31 c 5 d = 20 s.0710 5 SECONDS -—:-t' _____,.,_} --— / ---' ---/-7 __l.2___2__ _.t_ -f —IJ-{-f- - — f-f8 - I CIRCUIT II F t = 0 (RESONANCE) 2 CIRICU1IT /1 ~J_ A~ — I r-~~~~~~~~~~~I YA" 7iE~ 4 CIRCUITS 55

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN reduced (A0 drops from 1 to.59 as the bandwidth b is reduced from 1.51 to.075), and (2) the "spreading" of the output pulse as the filter bandwidth is reduced. The spreading of the output signal for large bandwidths observed in the previous figure does not occur here since for large bandwidth the input pulse width determines the output pulse width directly. Figure 4).7 shows the response of 1, 2 and 4 circuit filters, all having the same bandrwidth, to an input pulse having a center frequency somewhat above the resonant frequency of the filters. There is an increase in the delay of the output pulse relative to the input pulse as the number of circuits is increased, but otherwise increasing the number of circuits has little effect on -the relative amplitude and output pulse width. Note the envelope of the output pulse tends toiwards a Gaussian shape as the number of circuits increases. B. Varying Signal Parameters The effect of varying the signal duration d and center frequency (a - sc) is shown in Figs. 4.8 and 4.9 respectively. As one would expect, there is a reduction in the relative amplitude of the output if either (a) the input pulse duration is reduced, or (b) the center frequency of the input pulse differs appreciably from the filter center frequency. Figure 4.8 shows the effect on the output of varying d. Figure 4.9 illustrates the change in filter response as c is varied. Each output pulse is the result of a separate differential analyzer solution. The various output tapes w7ere cut and pasted together in the proper sequence to form the figure. 4.3.2 Quantitative Results Figures 4.5 to 4.9 indicate qualitatively the effect on the filter output of varying the various signal and filter parameters. A quantitative measure of the filter response is furnished by the factors introduced in Section 2.5: the output relative amplitude Ao, the output pulse width W, and the apparent bandwidth B. More than four hundred solutions of the I 36

ZS-o~-b 'lIN ~-~d-V OL6-W FIG. 4.8 RESPONSES FOR VARIOUS INPUT PULSE WIDTHS b =.97 C = 0 S =.0710 2 CIRCUITS 5 SECONDS I ____ - 37

In I, In Ir rT FIG. 4.9 RESPONSES OF I AND 2 CIRCUITS FOR VARIOUS PULSE POSITIONS d = 5 s =.0710 5 SECONDS 0 (RESONANCE) I Iw_ I o CO 46P; 1 EZ. Z E

I - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN type shown in Figs. 4.5 to 4.9 were run on the differential analyzer to determine the dependence of these factors on the circuit and signal parameters. Some typical curves of Ao, W and B are discussed below. Additional curves and discussion are included in Appendix F. Relative Amplitude A The relative amplitude is the peak amplitude of the output pulse normalized to the steady-state amplitude of the output for a cw input signal at the bandcenter frequency. Ao is the value of A which corresponds to bandcenter pulses (c = 0). Fig. 4.10 shows the variation of the relative amplitude versus sweep-rate for a cw receiver input. Curves for one, two and four circuit filters are plotted. For every case, the response is unity for very low sweep-rates. For very high sweep-rates, the responses drop off rapidly. The response of the single circuit filter is considerably greater than that of the other filters. For all filters measured the response has dropped 3 decibels at - 1. bThe relative amplitude as a function of the input pulse length and the sweep-rate are shown in Fig. 4.11. The cw response curve is replotted in this figure for comparison. If the pulses are short, relative to the reciprocal of the bandwidth in cycles per second, (bd < 2)), the relative amplitude response is less than the response for the cw case for very low sweep-rates. At high sweep-rates the response to pulses is approximately bounded by the cw response curve, but in every case there is a range of sweep-rates over which the response to a short pulse exceeds the response to a cw signal. For very high sweep-rates one would expect the response curves for all pulse lengths to approach the cw response curve. Output Pulse Width Ij W is the width of the output pulse in time (measured between 3 decibel points) relative to the time it takes the input 39

Ao 1.0.8 FIG. 4.10 4__ _____ x;. L______ __ RELATIVE AMPLITUDE RESPONSE OF A PANORAMIC x7 a '". [RECEIVER AS A FUNCTION OF SWEEP-RATE _.,, __ FOR bd= __ (DATA FROM DIFFERENTIAL ANALYZER) __ 1...t_..I t _ _ _ _ t__... _ _ _..... _.__ _ I CIRCUIT 2CIRCU 4CIRCL ITS 1 i4. I TS )ITS| 10 LO I N I C, 0) ao I r I 3E 0.4.2 0.(I 1I.1 IU 70 S b2

Ao 1.0.8 FIG. 4.11 RELATIVE AMPLITUDE RESPONSE OF A PANORAMIC RECEIVER AS A FUNCTION OF SWEEP-RATE AND PULSE LENGTH FOR TWO CIRCUITS, PULSE CENTERED ON PASSBAND (DATA FROM DIFFERENTIAL ANALYZER) 4.6.2 O L-.01.1 10 50 s b2

-- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - signal to sweep over the filter passband. Data have 'been obtained.(ivin_ the dependence of W on the signal and circuit parameters. The curves of Figs. 4.12 and 4.15 are typical of the results obtained. There is an increase of W as the sweep-rate is raised. For large sweep-rates the width of the output pulse to approaches a constant determined by the filter impulse response. Then W = t b becomes linear in s. The length of the input signal d and the number of circuits n have little influence on W for high sweep-rates. An exception occurs for the case of one circuit; W is about one-half the value for the two and four circuit cases. The beat phenomena which are observed in the output for the one circuit case have the effect of appreciably reducing the output pulse length as measured between 3 db points.1 It should be pointed out that this beating phenomenon can give rise to some other anomalous results.2 Apparent Bandwidth B As indicated qualitatively in Fig. 4.9, the relative amplitude of the output pulse is a function of the difference between the center frequency of the input pulse and the filter bandcenter, sc. The peak response drops off as the magnitude of sc is increased. A typical series of SC plots of relative amplitude A versus - is shown in Fig. 4.14. The center value of the curves shown is Ao. The apparent bandwidth, B, is defined as the distance Ao between the points on these curves at which A -=. Each curve of Fig. 4.14 determines one value of B. Plots of B as a function of sweep-rate with the number of circuits and the length of the input pulse as parameters are given in Fig. 4.15. For very low sweep-rates B has a horizontal asymptote. For very large sweep-rates B has I1-Se Fig. '.. Appendix F,,ee Appendix I 42

S- 6-C5 IN; L.-~9-V OZ6 - WI 100 10 W 1.0 0.1 L 0.1 1.0 10 S b2 40 FIG. 4.12 FREQUENCY SWEPT BY PANORAMIC RECEIVER DURING OUTPUT PULSE AS A FUNCTION OF SWEEP-RATE. (2 CIRCUITS, PULSE CENTERED ON PASSBAND) (DATA FROM DIFFERENTIAL ANALYZER)

i7 (83ZAIVNV I VIIN3y83JdJ l WOdJ viva) -- = pq aO CNVtSSSVd NO C3d31JN33 3Sind '3iVd-d33MS JO NOIlONnld V SV ~17 '91id Zq s 01 0'1 fI 0 I'0 / I I3tJ 10 I It 11 I l l l _1 = < < t In I t1<e i I S Xv/ ) L H /I 1-,,1 1 S I 1 S- 15 s I j I -" —^/ _ _ — _ _ _ _ _,/- - - --- _ _ --- - -/ - /7'- _ — -- / ^- ---— ~~~~~~~ siinoblIO ~, sunodio fr ----- -~~~~~~~~~~~~~ 0'1 M 01 001 M-970 A-G3-9 RKL 5-12-52

1.2 1.1 1.0.9.8.7 A.6.5.4.3.2 I0 0o 1.r 2 I II I I I I I I I RESPONSE FUNCTION FIG. 4.14 INTERMEDIATE PLOT OF A PANORAMIC RECEIVER AS A OF PULSE CENTER IN FREQUENCY FOR bd = 27r, 2 CIRCUITS _____ I___ I__ __ IX 1L p, 3 CI - - — ", - b = 0.31422 = b = 0.6283 = b = 1.257 b b_ 0.0450 0.180 0.719 r x _ _ X~~~~~~~~~~~ -4 -3 -2 -I 0 I 2 3 4 SC b

100 50 40 30 20 FIG. 4.15 APPARENT BANDWIDTH AS A FUNCTION OF SWEEP-RATE (DIFFERENTIAL ANALYZER DATA) a) T (D 0> <0 a: oI t CD ~o I 0 CO m S 10 ONE CIRCUIT FILTER TWO CIRCUIT FILTER FOUR CIRCUIT FILTER f- B 10, ON"

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN an asrlptote given by (b2 bd). As a result the curves of Fig. 4.15 are reasonably reliable even though only three points were obtained. for iriost curves. Since each point on this plot represents about seven differential analyzer solutions, about two huLndred solutions were required to obtain the dcata plotted.. The number of circuits is seen to be of little importance, but the length of the input pulse has a major effect. For low sweep-rates narrowing the pulse increases the applarlent bandwidth while for high sweep-rates the reverse is true. See Section 5.5

I - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 5. COMPARISON OF SOLUTIONS 5.1 -Single-timed. Circuit by' Two Methods The response of sinle-btuned c-ircuit-s to pulses with square envelopes has been discussed in earlier sections of this report.l- Both the theoretical and differential analyzer curves for certain cases were presented. The results of the. tw.o investigations are compared: here. Figure 5.1 illustrates curves obtained by tlhese two techniques for very broad pulses. Figures 5.2 and 5.3 are similar ex:ce-pt that thle p:)ulses started at frequencies near the center of the passband of the filter. The bancdwidths for the three figures are different A parlt of the discrepancy of Fig. 5.3 probably is due to the slight difference in starting tine of the two curves. These three figures and Figs. 5.4 and 5.5 indicate a very good agreenrlent between the numerical calculations and the differential maalyzer solutions. 5.2 The Gaussian Case and. Differerntial Analyzer Solutions The relative cmlplitude Ac, output pulse width W, and apparent bandwidth B have been calculated both for the Gaussian case (Section 3.2) and from the differential analyzer solutions for one, two and four circuit synchronous single-tuned filters (Section 4.3). Eight typical f ri-lies of curves co.mparing these data are given in Figs. 5.4 to 5.11. The relative arplitude for e a ci i input Fig. 5.4. The curves for two circuits, four circuits, and the Gaussian case differ very little; the curve for the one circuit filter is higher at fast sweep-rates, but not by more than a factor of two. The relative amplitude curves for pulses (Figs. 5., 1Section 3.1 and Section 4 L 48

1.5 1.0 4 -a) -p-.5 0 FIG. 5.1 RESPONSE OF SINGLE-TUNED CIRCUIT TO A SIGNAL OF LINEARLY VARYING FREQUENCY FOR 8b =.2, SIGNAL BEGINNING AT, t = -00 '\ THEORETICAL CURVE \ --- DIFFERENTIAL ANALYZER CURVE ___ \\__ I j - - — V _- ia - 1 - -^ ^^^ — - - - -^/Ijj ~...... I I I I ~ 1~I I _ //' ~~ -5 -4 -3 -2 -I 0 2 3 2t 4 5 6

l I -J I cL I o.5 1.0 4-,.,..5 FIG. 5.2 RESPONSE OF SINGLE-TUNED CIRCUIT TO A |___ ______ SIGNAL OF LINEARLY VARYING FREQUENCY FOR.7, SIGNAL BEGINNING AT 't = -3 -I - - THEORETICAL CURVE t — -- -- -t ---.DIFFERENTIAL ANALYZER CURVE, f 1 1- S 1 5 E - 1 -____________-....... — -.- -1 0 5 -4 -3 -2 -I 0 2 3 t 4 5 6

4 -a) 1") \ > I0.8 --- -I ------ FIG. 5.3 l - - -- - --- RESPONSE OF SINGLE-TUNED CIRCUIT TO t.7 ____ SIGNAL OF LINEARLY VARYING FREQUENCY _ _ _ / \ _ FOR.1, SIGNAL BEGINNING AT, t = I./ L ---- THEORETICAL CURVE.6__ __ or Jt/ - \V. ----- DIFFERENTIAL ANALYZER CURVE.5 __ _- -_ __ T Li.4 04 I \1\ ~ * ~<.2_ __ _ / _ / _ _ _ _./.1l~ ~ ~ ~ ~ ~~~i _ I! Q~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, I I 0 I 2 3 2 t 4 5 6

N (14 I -J I to (D cr 0 t... I ft II 4 1.2 FIG. 5.4 RELATIVE AMPLITUDE FOR I 1.0 isXS- - -X DIFFERENTIAL ANALYZER DATA I. I CIRCUIT 2 CIRCUITS 4 CIRCUITS A CW INPUT SIGNAL X THEORETICAL { I CIRCUIT GAUSSIAN 0.8 A o \Ji No 0 1.6 -, --- —-- F 0 Iq 11 i ~o0 0.02 0.1 1.0 S b2 10.0 70.0

a. ff o 0 2 gE 1.1 O. A, OT. I \? rN bd =27r O - - - __ "-"^*"^**~i --- ~~PULSE CENTERED ~____ _ i lS ^_ ~ m —.. —m___ _ _ _ _ ___ ON PASSBAND 8 6 N N N S Is K xx 0 FIG. 5.5 RELATIVE AMPLITUDE FOR THE GAUSSIAN CASE ).2 AND DIFFERENTIAL ANALYZER SOLUTIONS - GAUSSIAN I CIRCUIT -X- 2 CIRCUITS 4 CIRCUITS O, s _.01 0.1 1.0 7.0 S b2

CM rO I I to O0 0 7 r 70 70.0 S1l C Ao 0 0: 0 OL0 0.1 1.0 10.0 S bt

70.0 FIG. 5.7 I IIII[IIIIII IZiiIII II OUTPUT PULSE WIDTH FOR A CW INPUT SIGNAL DIFFERENTIAL ANALYZER DATA THEORETICAL I CIRCUIT 2 CIRCUITS 4 CIRCUITS x -I -. _, i I I CIRCUIT GAUSS IAN /, e J1 /, // / /0 / ( / // W 10.0 I. 1.0 0. A - _ I —_ ___ XgX t i I m,.. I 1.0 10.0 50.0 s bv

Zg-91-9 Mld 6t-~9-V OL6-Ni 6.0 O — -- bd:=27r PULSE GENTERED ON PASSBAND Itto t W 0.O 025 0.10 I.0 s G. 5. FIG. 5.8 OUTPUT PULSE WIDTH FOR THE GAUSSIAN CASE AND DIFFERENTIAL ANALYZER SOLUTIONS — I- GAUSS IAN -- I CIRCUIT.-. ->- 2 CIRCUITS ' — 4 CIRCUITS 56

-Zl-9 Mld 9S-~9-V OL6-W 100.0 - bd ' r PULSE CENTERED ON PASSBAND /I, I Cl to.0 - - -- - -...- _X ___ 0. C.0~o -- 7-_ _ - — _|___ ___-b FIG. 5.9 OUTPUT PULSE WIDTHAS A FUNCTIONOFSWEEP*RATE FOR GAUSSIAN CASE AND DIFFERENTIAL ANALYZER DATA o -o 7_-_-_............ _. _. --- FOR GAUSSIAN CASE AND DIFFERENTIAL ANALYZER DATA 57

N UC I 0. IIt 80.0 B 10.0 r I I I I I 1 I I IIl 91 FIG. 5.10 APPARENT BANDWIDTH FOR THE GAUSSIAN CASE ANDTHE DIFFERENTIAL ANALYZER SOLUTIONS - GAUSSIAN ~ — I CIRCUIT -— x — 2 CIRCUITS -- 4 CIRCUITS bd= 2t -__I_ _ _._ — _ Jol, / j1 CO I I ----- ------- ----- I I --- -I -I -I -- II I I I(.0 / I 1 I I I 1 1 1 6 1 1/ l ll l l lI I I /A 7 1.01 r.o S b2

80.0 I '( [ FIG. 5.11 APPARENT BANDWIDTH FOR THE GAUSSIAN CASE AND THE DIFFERENTIAL ANALYZER SOLUTIONS - GAUSSIAN --— o I CIRCUIT bd - bd = 75 --— x ---- 2 CIRCUITS -0- 4C -4 CIRCUITS I ~~~~~~~~~~I I I m=fX f II I~~~~~~~ f ~~~~~~~~~~~~~~~~~~, i A I i 1111=ix/I / 7 10 -/-0 E J S S a S L/ r~x.... I I - 1 I.( ( 3..I 1.0 10.0 70.0 b2 b2

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN and 5.6) are more irregular. Tlhe curves still have the some general features, 1howeverC1 and they differ by no more -than a factor of two. Figure 5.7 shows the output Iulse wzidth for a constanst amplitude signal. Again the curves for two and four circuits and the Gaussian case are very close. The curves for the one circuit filter are lower, but not by more than a factor of three. The output pulse rJidthl curves for a pulse input (Figs. 5.8 and 5.9) are all of t"-le sane general shape and. ae wJithlin a factor of three of eac' other, 1 with- the curve for a one c-ircuit f ilter d:ivergJing the most. The curves for B in Figs. 5.10 and 5.!! are within forty per cent of each other fo fast swreep-rates and- within a factor of two and one-half for slow swee!r-rates. The curves for a particular value of bd have 1the sane asy-,ptote for fast sweep-rates (see Section 3.3) ancd the asL.yptotes are parallel for slow, sweeprates. The limiting values of B for slow sweep-rates can be obtained theoretically for the single-tuned circuit as Twell as for the Gaussian case, and these are col-mpared in Table 5.1. TABLE 5.1 APPAPENT BADTWIDTH FOR ZERLO SW1,nEEP-RAITE -...... bd — C i 2T 2 1 circuit 5.6 1.8 9 1.17 Gaussian Filter 1.62 1.18 1.05 Generally thle one circuit filter gives sorewhat greater relative apvlitude, a little narrower outpu.t pulse, and a slightl2y Iarger apparent ban(d.icdth ltan any of the other filters considered 6o

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - Certain features of th;le receiver response in thie diff erential a4nalyszer solutions differ consid-erably from the solution of the Gaussian case. For example, the tine of masximtir response can hardly be expected to agree, since the Gaussian filter is assumed to have no phase delay. Also, the output pulse shape for the Gaussian case is always Gaussian, while a wide variety of shapes ap-pear in the differential analyzer solutions, as observed in Section 4.3. The solution of the Gaussian case gives an understanding of the nature of the response of a panoramic receiver. M;oreover, the Gaussian case is quantitatively consistant enough with the other cases studied to be used in many design problems involving peak amplitude, output pulse width, apparent bandwlcidth, and resolution. 61

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 6. SUIlLTRY The response of a panora-nic receiver to cw signals and pulses has been investigated- in this report. The receivers studied were idealizec superheterodyn:es; several types of i-f amplifier filters were considered. In many cases the analysis applies to a trf panoranic receiver. Theoretical studies were made of the following cases: a) Single-tuned i-f filter with cw and square envelope pulses as as input signals, and b) Gaussian shaped i-f passband with cw and Gaussian envelope pulses as input signals. Differential analyzer solutions were made for i-f filters of one, two, and four singfle-tuned circuits with cw and square envelope pulses as input signals. The character of the response is found to depend in a minor way upon th'e type of i-f filter used. General formulas presented in Section 5.5 hold regardless of the type of filter or the type of input pulse. Thus one anticipates that the important features of the response are little altered with other types of bandpass filters. Only the envelope of the response is studied in this report. The pertinent response factors investigated iwere the relative amplitude, the output pulse width, and the apparent bandwidth. These factors largely determine the character of the response. They are functions of the receiver sweep-rate, the i-f bandwidth, and the signal pulse width. The Gaussian case has been presented in considerable detail in this study. Analytically, the response for this case is very satisfying. The answer is riv-n in closed form,a and it is simple Jwhen expcressed in terms of the i 62

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN important factors: relative amplitude, output pulse width, and effective bandwidth. More significant is the fact that the Gauspian case is fairly representative and can be used with suitable caution in the design of panoramic equipment. The formulas and the many curves in this report will enable the engineer to optimize certain design features of a panoramic receiver subject to the application requirements. ACKNOWLEDGEMENT The authors wish to thank Mr. R. Bradley, Mr. R. Lyons, and Mrs. C. Martin for assistance in the preparation of the text. The authors are also indebted to members of the engineering staff at Sperry Gyroscope Company for their work on the Gaussian filter.

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX Al Derivation of Response to a Single-Tuned Circuit Let jst2 d d exp at + 2 for c 2c t < c + 2 (A.1) i(t) = 0, for t < c - - and t > c + 2C 2 2 be the input signal current to a passive lumped-constant network and let i designate its Fourier transform. Similarly, let e(t) be the voltage produced at a given output and e its Fourier transform. Assume the circuit is quiescent prior to the application of the signal. Then, d e == z(p) = f exp[jaTi + 2sT exp(-pT) dT (A.2) c - 2 where z(p) is the appropriate transfer impedance. For a passive network z(p) is a rational function and can be expanded in partial fractions as follows. z = h(p) = g(Pk) h(p) = - (A.3) ki 1 ht (Pk)(P-Pk) where pk designates a root of h(p) = 0, and multiple roots are excluded.2 Then equation A.2 becomes d c + Z = -- -.k (Pk)=- ( exp [jaT + ]exp(-pT)dT. (A.4) e =-I k =1 h (Pk)(P-Pk) d c - 2 1The proof given here follows that of Gunnar Hok, Ref. 1. 2The extension to multiple roots is not difficult. 64

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Taking the inverse transform and interchanging the order of integra tion, e(t) n F g(Pk) k = 1 h' (Pk) d c + f exp [jaT + jT] 2 j0o exp p(t - T) dpdT. 2irj (p - Pk) d c - 2 2 -jco (A.5) By the inversion theorem of operational calculus, if the real part of Pk < 0, 1 jo -io -jG0 0 for t < T exp p(t - T) dp = (P - Pk) (A.6) exp pk(t - T) for t > T, and therefore the expression for the voltage becomes, e(t) n h' (Pk) = ht 1 k = 1 " I exp {Pkt + (ja - pk)T + jsl. dT 2 (A.7) c - d 2 d d for c - < t < c +-. 2 - 2 The upper limit of the integration t is replaced by d d! c -L for t> c + 2 Now define, Y4(T) = (ja - Pk) + ji s T. (A.8 Then the exponent of the integrand in Eq A.7 can be written Pkt + (ja - Pk)T + jT2 = 2 2 - 4 (T) 2 2 'k (t) + ij 2 2 + j [at + s2 and the voltage expression assumes the form 65

- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN n e(t) = ~ k = 1 (Pk) exp[at + 2j G(T) (A.9) where G(o)1 = rk 2(t) - j exp [j 2 Wk(t) J exp (c- 2) 2 d) IX. (A. 10) The error functio of a complex variable deined The error function of a complex variable defined by v/2i has been tabulated and can be used to evaluate G(yr).2 For the case of the single tuned circuit shown in formula for the voltage reduces as follows: x + iy 0 2 e e 2 dz Fig. 3.1, the general z() = - I1 + 1 + pC RL-o p' RLC + pL + R Then, g(P) h(p) h'(p) g(p) h t (s)) = RLp, = RLCp2 + Lp + R, = 2RLCp + L, and RLp 1_ 1 2RLCp + L 2C 1 + 1 ' 2RCp The roots of h(p) = 0 are 1 d Mc d G( o) = 0 for t<c - 2, and for t>c + 2 the upper limit of the integral is to be replaced by Yk(c + 2). 2 2"Tables of Integrals Associated with the Error Function of a Complex Variable," Hastings, C. and Mercum, J., RAD-284, Douglas.Aircraft, August, 1948. 66

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1P P, 1 + i1 1 2 2RC LC L2RCJ = - 12RC + ja = 2RC - - 2 + ja, 2a Then, if - >> 1, g(pl) h' (pi) g(p2) h' (P2) 1 2C By Eq A.9 the voltage is e(t) C1 exp (jat + t2 2 [G(Yl) + G(Y2. (A.11) Also, it is readily shown that lim a - O G(Y2) = 0, so that for high frequency the voltage response is given approximately by e(t) -- C 2s exp (jat + jst2 2 G( Y1) (A. 12) where Y1 b + jr t. 2 Is (A.13) 67

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX B Derivation of the Response of a Gaussian Filter In this appendix the formulae are derived for the response of a filter with a Gaussian shaped transfer function to a signal which is changing linearly in frequency and has either a constant amplitude or a Gaussian shaped envelope. Assume the filter transfer function is (c) -1 ex [ (oa)2, (B.1) b2 and the input signal is (the real part of) f(t) = exp [ (t2 — + at) -(t-c2 (B.2) 2 d2 The -rocedure is to find the Fourier transform F(() of f(t), multiply it by H()), and transform back to the t-plane. The filter response is the real part of the resulting function g(t). The calculation is simplified by using the convolution formula: 1 D H() F() eJt F = O f( ) h(t-%) d (B.3) 00 -C y 00 where h(t) is the Fourier transforms of H()). 12 Two preliminary remarks will make the derivation go smoothly. In the first place, the envelope of the real part of a complex function of time is just Titchmlarsh, E. C., "Introduction to the Theory of Fourier Integrals", Oxford University Press, 1937, p. 51. 2The use of complex functions for the signal f(t) and the impulse response h(t) is justified if the response of the filter is negligible at zero frequency. 68

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN the absolute value of the function. This can be seen as follows: Let Z(t) be any complex function of t. It can be written, Z(t) = Iz(t) exp (jG(t)), where e(t) is the argument of Z(t). The real part of Z(t) is then IZ(t)1 cos @(t), and the envelope of this is z(t). Secondly, in computing the Fourier transforms, use will be made of the following formula: co v2 exp ( - ut2 + vt) dt = - exp - (B.4) -OD uu 4u The integration is along the real axis in the t-plane, and u and v are complex numbers, with the real part of u positive. This formula can be derived as follows: fexp [-ut2 + vt dt = exp [ ] / exp [ - u [t v2 dt. 4uJ 2uJ I - -O — co Letting Z = u(t - 2u), G0c 2 v2 co 2 exp (- ut + vt) dt = exp exp(- Z = exp -00 O -00 Note that the path of integration in the Z-plane is not along the real axis, but along a line which may be oblique to the real axis. From the requirement that the real part of u be positive, it can be shown that the path of integration in the Z-plane makes no more than a 45~ angle with the real axis. With 00 this restriction the integral exp(- z2) dZ is independent of the angle of the -00 path and thus equal to/i, which is given by integration along the real axis. Now the calculation of g(t) can be carried out. Before use is made of (B.3), h(t) must be calculated. 69

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1 Go ~1(t) = 2_ _ OD II(e) exp (jct) d) '-aio =7- 00 exp [ jet. ( - a)2 b2 1 r-a2 c - da = exp b2 JJ C p [ 2 +(jt + ) b2 which yields, after application of (B.4) and simplification, h(t) = - -]ex [jat - b2t The expressions for f(t) and h(t) can now be substituted in (B.3): (B.5) 00 g(t) = / — 0 (x) 1h(t-X) d 1 - 1 00 ep-[ + ja - -d. b( c cxp + ja(t-))2 X d2 )l =b [ _b2t2 = b_ xp -d2 4 + jat]f exp [- 2 (i + b2 - ) + ) 2 d L d~ '4 2 d and using (B.4) again; g(t) b 2LA + L - js d2 4 2 ex2 b2t2 a exp - +d jat +. (B.6) As has been remarked, the can be obtained by taking only the real part of the Jg(t) = b required answer is the absolute value of g(t), which the absolute value of the first factor and keeping exponent. 2c t t 1 2 b c2 b2t2 + 2 d2 4 exp -- 2 1 d2 4 2- 2 + 4} 4 J4 4 (B.7) f-% 2 [(1 b2 7'o

ENGINEERING RESEARCH INSTITUTE - UNIVERSITY OF MICHIGAN The exponent of (B.7) can be put into the followin forma-:..~[ 4 2. ] t - _....| (B 8) - 2[4 b2,22 4 11. _ d2 [(+2b2)4+ 2] [ - +b2 2d] + 21 b2 +2 s22 (B.8) Referring to the definitons of A, W, and B in section 2.3, and recalling 1 that in the Gaussian case the width of a curve is taken to the e 4 points, it is clear that A = 27 77b (B.9) bd2 1 4s B 2 + b + s2d2, and (B.o0) ( + b2 ) 2 + 42 s d2s b2 t 2 2 s2dE (Bll) The time of maximum response is given by 4,. 4+ b2 d2+ t,-, = c -+b2 -.(B.12) - *b 2 + s2d2 Now jg(t) can be written 2 Fo-r a cn inJ.put t'Jie si-ial is e-? i[ st" t ] n 'an:.e aoutr-u't can 'be obtainedl from (B.7) by takinf thle limit as d 4 a- oachles infini ty. limr b b2 2 t2 d-00 O g(t)j [ + 4s2> ep { b' + (s2} * (B.7') I._ 71.

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN In the notation of (B.9) to (B.13), A0 = b B.9') [b4 + 4s2] w = 1 (b4 + 4 =2) 1 b2 A02. and b-~(b +4s2) = --—, ~na Ao k (B.1l') lim d-Oo |(t) o {w2 [b]} W2 b2j -oXP ^ ^{-^k-Jh (B.13') 72

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENiD IX C Curves for the Gaussian Case Thle curves in this appendix together with th1ose of Section 3.2 nSake up a complete set of curves of Ao, W, and B for the Gaussian case. They are listed in Table C.1. TABLE C.1 Curves for the Gaussian Case I. Norrmlalization with Respect to Bandwidth A. Sweep-Rate as Abscissa, Pulse Width as Parameter 3.4 Ao 3.5 r 3.6 B B. Pulse W-Tidth as Abscissa, Sweep-R-ate as Pa-rameter C.1 Ao C.2 W C., B II. i\Norm-alization with Respect to Pulse WIidth A. Sweep-Rate as Abscissa, Bandwidth as Paramleter C. 4 Ao C. 5 bdW C.6 bdB B. Bandwidth as Abscissa, Sweep-Rate as Parameter C.7 Ao C.8 bdW C.9 bdB III. Normalization with Respect to Sweep- Rae A. Bandwidth as Abscissa, Pulse Widnt as Parm-eter C. 10 Ao C.11 / TW b C.12 B 1 iThese numbers are fi-gure nulbers. 75

I -- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - B. -Pulse Width as Abscissa, Bandwidth as Parameter C.15 Ao b c.i4 ib_ w y^ c.-15 bB The output pulse width W and aplparent bandwidth B are no:;rm-alized wit h respect to bandwidth by definiton. IWhere they are presented with a different normaalization, they are multiplied by the factor which will corr- ect their normalization. Sumxose one wishes to know how apparent bandwidth varies as a function of pulse width for a fixed sweep-rate. INolralization w-,ith resecct to sweep-rate should be chosen; then the apparent bandwidth in radians per second bB -f bB s is - or -— * —. Sincc- is fixed, apparent bancdidth in radians -er 2is or 2o it ans a0er second is proportional to B, and its dependence upon nulse width is shown in Fig. C.15. A brief discussion of the dependence of Ao, T, and B on sweep)-rate and pulse width- w;as given in Section 5.2. Several features are brought out by the curves in this appendix. For any given sweep-ratc, there is a pulse owidth which leads to a minimum apparent bandwidith It is given by sd2 = 2. This can be seen in Figs. C.5 and C.15. For a fix:ed w sweep-;ate and for pulses such thlat st2 > 2, there is a bandwidth which gi.ves a minimum output pulse width (see Fig. C.11). For long pulses (sd2 > 10), the minimum occurs very nearly where b2 2s. In many of the gra-hs there is an unattainable region. For elxmple, in Fig. C.7, the relative cm-plitude is seen to be bounded by the curves for sd - 0. Thus, if bd < 1, AO must be less than 0. 5 regardless of the sweeprate. In Fig. C.5, t;he curve for c = 0 is the envelope of the family - J L

I - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN no curve is found below it. In Fig. 5.6, the envelope is found to be B = /1 +. B is never less than this, and it has this value only when b2 sd2 = 2. The envelope is also shown in Fi,. C.14; it corresponds to the minima in Fig. C.ll. I i 75

N (D I -J (0 cr (9 I rcc I 4 I.0 0.8 Ao 0.6 -.Q C~>' 0.4 0.2 0 0.1 1.0 10.0 bd 50.0

-9Z-9 "1>8 ZI-~99-V OL6-1N 10.0 1.0 W 0.1 0.01 1 0.2 1 10 bd 100 FIG. C.2 OUTPUT PULSE WIDTH AS A FUNCTION OF INPUT PULSE WIDTH FOR THE GAUSSIAN CASE 77

0'001 3SVO NVISSnV9 3H1 80d HICIM 3S-Ind indNI JO NOIlONfnI V SV H.LOIMNV lN38VddV ~' '91.d pq 0'01 O' IO'0 cYOI 0.1 _ N - -- ~0_ _ __ 0 1 = 1. _ 0 - - i --- -,' _ - - - -----..... _ - - - -- 0 0 1 /.i --.'-'^- -.,__ ------ ~ 0001 M-970 A-G3-18 RKL 5-26-52

FIG. C.4 A 0.6 1.0 10.0 100.0 1000.0 sd2

z9 ez-9 - 1 18t 9Z - 0 - OL6 - Vs 100.0 (bd)W 10.0 sd2 FIG. C.5 OUTPUT PULSE WIDTH AS A FUNCTION OF SWEEP-RATE FOR THE GAUSSIAN CASE

IOO.C (bd)B FIG. C.6 APPARENT BANDWIDTH AS A FUNCTION OF ------ SWEEP-RATE FOR THE GAUSSIAN CASE - _ ____ ___67S -- __^ - -a-=-- -= --- - --- bd = 277 bd = 4r bd = - - - - - - _ _ ~ ~ ~ ~~ ~ ~ ~ ~~ ~~_M_ _ _ _ _ _ _ _ _ _ I 10.0 OD I 1 I bd o / / llb `O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1.0 L0.2 1.0 10.0 100.0 sdt

1.2 cM 10 CJ 10 N 0 f-J 0 o I?e M7 0 S~ Ao 0.6 O3 N) 0.I21.010II I I II - I 0.2 1.0 10.0 100.0 bd

'S-ZZ-S "IME ISZ-~9 -V OLZ6- (bd)W | — -- _ __ — - - - 1 tE m1 oft s —LOCUS OF MINIMA sd' = 0.316 1,0 %, __ sd 0.1 0.1 0.2 1.0 10.0 100.0 bd FIG. C.8 OUTPUT PULSE WIDTH AS A FUNCTION OF BANDWIDTH FOR THE GAUSSIAN CASE 8,

100.0 (bd) B 10.0 L) I CM r 0 ro 0a CD -p 1.0 L 0.2 1.0 10.0 100.0 b,:

1.2 1.0 0.8 Ao 0.6 FIG. C.1O RELATIVE AMPLITUDE AS A FUNCTION OF BANDWIDTH FOR THE GAUSSIAN CASE!! 0,o o: o1 Ifsd 10.0. Ad =3.0 'd=.44 d d1.0 I ---/Ty -- I01" ^ ^/ / ^"vsd 0.3 sd XO.I l-.~~~~~~~~~~d0.3 -0.1l 0000p,-~~~~~~~~~~~~~~~~~~~~~~~~~~~ 00010~~~~~~~~~~~~~~~~~~ 0o k J1 0.4 0.2 n 0.2 0.2 0.3 0.4 0.6 0.8 1.0 2.0 3.0 4.0 6.0 8.0 10.0 b,

ZS-Z-9 1 X 8Z- ~9-V OZ6- W 100.0 10.0 b 1.0 - 0.1 0.1 1.0 b,i 10.0 86

i-~-9 1"N8I 02- ~9-V 016- w 1000.0 100.0 VJsd 0loo OR 0.02 ____,/'d = 40 OR 0.05 - /S d =10 OR 0.20 Sd = 4 OR 0.50 __ < _ sd = 0. 2._ 10.0 1.0 0.2 0.2 1.0 10.0 100.0 b,4S FIG. C.12 APPARENT BANDWIDTH AS A FUNCTION OF FILTER BANDWIDTH FOR THE GAUSSIAN CASE

1.2 1.0 0.8 Ao 0.6 FIG. C.13 CM 0) I -i o, 4 500.0 oz CO 0.4 0.2 0 0.01 0.1 1.0 10.0 100.0 Isd

ZS-Z-9 'INN 6Z - 9-V O6-W 100.0 10.0 b 1.0 0.1 0.1 1.0 Vsd 10.0 89

Vw In I re) I to I le (D 0: 0) 0) I,1E 100.0 -B 10.0 FIG. C.15 APPARENT BANDWIDTH AS A FUNCTION OF INPUT PULSE WIDTH FOR THE GAUSSIAN CASE bh:!0.0!====^^ -— ~.b - _,,..... ~ = 1.78. _ ^. ^ _ _ _ _ _________L, ^. ^ ^______ _ -- _ — _______ _ _~~~~~~~~~~~~~~~~~~~~~~~ In 0.04 0.04 0.1 1.0 -/sd 10.0 50.0

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPEINDIX D Derivation of the General Relations JAnong Factors Describing Response The success of the derivation of Eq 5.16 and Eq 35.17 depends on a different definition of the width of a curve. Suppose f(x) is a function which approaches zero when jxj becomes large. Then the width w of the graph of f(x) is defined here as -f Jf(x)2 dx rm ax |f(x) 2 For example, the width of a pulse is defined as its total energy divided by its maximum ipower, and the bandwzidth of a filter with transfer function H(c)) is b= 00. (D.2 ma H(I() H(o)) d max H(cD) HC This is knorn as the "hoise bandwidth.1' Suppose a scanning signal of constant amplitude, f(t) = fo exp[jat + st2] (D.3) 2 is applied to a filter with transfer function H(c). Then the output signal is given by 00 g(t) =J F()) H(c) et dc) -00 where F(c)) is the Fourier transfonrm of f(t). By Parseval's Theorem,2 the energy 1Walhlman, II. and Valley, G.E., Ref. 9, p. 169 Lawson, J.L. and Uhlenbeck, G.E., Ref. 10, p. 101 sTitchaarsh, An Introduction to the Theory of Fourier Integrals, Oxford University Press, 1937, p. 50 y-i

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN in the output pulse is, I [g(t)] -00o dt = 2rfJ -C0 F(()) H(cO) F~ IW(-T dw. (D.4) But, _ r0_ F(o=) v -f -CO j (a-os)2 - jet 2s f(t) e dt = f cOs (D.5) so that, F(W) iF() -- -- and from (D.U), s7 rfX [g(t)] dt -00 2-tfo2 CO -- 0 II(ow) II(c)) dD (D.6) NTow all that remains is to interpret these intecgrals using the definition given above for the width of a curve. As was pointed out above, the energy of a pulse is the product of pulse width and maximum power, so 00 f [g(t)] 2 dt = to (maximura power of output pulse) (D.7) where to is the width of the output pulse. The maximum of 2itI(n) H() is the ma;rimumlu powier gain of the filter, so by Eq D.2, 00 I H( )) II(a)- dm = 2 '(maxirlmum power gain of filter). (D.8) Substituting in Eq D.6, ~, bf - to'(maximu;r power of output pulse) = -0.(.lmaximum power gain of filter) s (D.9) 92

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2 fo is the power of the input signal, and by the definition of relative amplitude A, 0 2 (maximum power of output pulse) (D.lO) ^I ^ ^^^(D.10) (input signal power)-(maximum power gain of filter) Substituting this in Eq D.9 gives A = Ao2 * W = 1 (D1.) 0 b b which is Eq 5.16. The difficult part of proving Eq 3.17 is defining "apparent bandwidth." Suppose the input signal f(t) to the filter is a pulse of arbitrary shape and frequency, and suppose its transform is F(a)). If thle pulse is shifted in frequency by a radians, the transform of the resulting pulse fa (t) is F(c + a). The response of the filter to each of tilhe family of pulses resulting from shifting f(t) in frequency is a function of a. In particular, the energy of the output pulse is a function of a which approaches zero for large Ia. The apparent banldwidth B is defined here as the wicdth of the gratph of energy of the output pulse as a function of a, divided by the filter ban'dwidth (see Fig. D.l). I- aa. a Di 3~~~~~~~~ Figure. 1 93

-- ENGINEERING RESEARCH INSTITUTE ' UNIVERSITY OF MICHIGAN There is no loss of generality in assuming that the output pulse energy is a max:rinur for a = 0. The output pulse energyS as a function of a is, by Parseval s T1heorem, 200 -CD F(w -+ a) Il(w) F(w + a ) c', (D.12) and byr Eq D.1, 2 it B = b F((w + a) I (cw) FI()w + a ) H(cw 7 dw d.a (D. 13) 00 2<t -Go F(w) 1(wX) H(w) F()) (W (cow Changing the E. _D order of JO intoegration in the numlerator rfrn i11(w) I-J F(w + a ) F(wa.T da cow (D. 1 4) b 2<t -Go00 F,( w) 11(03) P17wY' iTl(wY 6; w But, Go — co so that 1 FP(w + a) FEj +a) ca H(co) 11^7i d-w 2 itG 0 - A- 00 -G -G -co F.( a F(a ) da, J-G '-0 F(a ) FP(-Tc da 2 —J -00 2<O (D. 15) F() (w) P(w) F (w) cho and all that remains is to ierpret t hre thr ee integrals. By (D.2), 00 2jf Il(o) 11(w) o(w) (w = b-(mrax power geain of' filter ) -CO (D. 16) 9'4

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN and by Parseval's Theorem again, O0 J F( a) F(a ) da = d-(max power of input pulse). (D.17) -00 O0 2TJ F(c) H(o) F() HTs- d = to(maximum power of output pulse for -00o "centered" input pulse), (D.18) where d and to are the widths of the input and output pulses respectively. Substituting in Eq D.15 gives B d'(max power of input pulse).b(max power gain of filter) b-to'(max power of output pulse for "centered" input pulse) By the definition of relative amplitude Ao, 2 ( max power of output pulse for "centered" input pulse) 0 (max power of input pulse)(max power gain of filter) Substituting (D.20) in (D.19) gives 2 to A B - = 1 (D.21) which is Eq 5.17. As an application of (D.21), a proof is given here that the straight lines B* = bd'8 are asymptotes for the apparent bandwidth as the sweep-rate b becomes large. Since the curves are plotted on log-log graph paper, thle condition for B* to be an asymptote for B is lim B* lim = 1. (D.22) s- CO B By Eq D.21, B = to A2 0 1 -- 95- -_ --- —— 9 --- J

L ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Substituting this for B and bd -* for B*, the condition (D.22) becomes b lim sd toAo2 = lim stoAo = 1.(. s-0 b d s- b(D.23) S — *. CO b d S -a 0 b Let M(t) denote the envelope of the input signal. to choose the time origin so that M(t) is a maximum when t achieved by substituting t + c for t. The frequency of the t = 0 is then sc + a, and the input signal is It is convenient = 0; this is input signal when 2 f(t) = M(t) est + j(sc + a)t (D. 24) Since o00 1 - v -0o F(o( +a) et e-ja t cO 4m iC O F( +a) e( +a )t d( + a) = e-a t f(t), a translation in frequency in general is equivalent to multiplying f(t) by e-ja t. This can be achieved by changing the parameter c in (D.24). Thus, the "centered pulse" is the input signal which has the value for the parameter c which maximizes the output power. The output signal is, by the convolution formula1 00 g(t,c) = f(T ) h(t- T) dT -0 2 = M( r ) expjS ---- + j(se + a)rh(t -T ) dr, -CD 2 where h(t) is the transform of H(w), the filter transfer function. Titchmarsh, loc.cit., p. 51 (D.25) 96

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN As in Eq D.20, Ao2 (max power of output ulse for centered input pulse) (max power of input pulse)'(max power gain of filter) max I g(t,c)l 2 (D.26) max |f(t)| 2 * max 2it H(w) H(() and lim sto Ao2 s-C00 ---- b slim [to max /~ g(t,c)j 2 b a- x [I(t)] 2 J OiD 20 H(S) () b max [M (t)] 2 * max 2 T H1(w) H1(w) max sli to - I- g(t,c) 2 s -CO (D.27) max [M1(t)]2 * max 2t H((w) II(cw) Referring to Eq D.25, 1irs-C O' s -GO, g(t,c)l2 lim O& S — o- 00 - 00 2 M(r T ) exp.j. L + j(a + sc) 2 2 h(t- T ) dT. Now,1 jexp { js -aO = (+ j), S.s (D. 28) and from this it can be showna that lim 1 5 s —O (1 + j) it 0 iS( T + C2 f GO F(T ) dr = F(-c). (D. 29) 1Dwight, H.B. Tables of Integrals and Other latheratical Data, Eq. 859.5, Macmillan, New York, 19 47 97

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN It follows that limo g(t) = (1 + j) j/ exp{-P } M(-c) hi(t - c), S -SGCo2 (D.30) and the response, in the limit, is the same as the response to an impulse. This observation makes possible the calculation of the limiting value of to. By (D.1) and using Parseval's Theorem, 0o CG lirn == IiSd-t 2 I(C) f[h c 2 anct lim t-_ 1 Y,., I C -l (o - and -o s —oCO max h(t)I | nax l iI(t) - b max H(cw) H(-)7 max I h(t) I2 lim t s -Co (D.31) Substituting (D.30) and (D.31) into (D.27) gives lim sto Ao2 _ 1 + j4 2 max |IM(-c)2 max Ih(t - c)12 b max H() H(() s-X)O b b max | M(t) 12 max 2it H(w) H(w) max h(t)| 2, o' lim stoAo2 s- (O - 1 (D.32) which is the required condition, Eq D.25. 98

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENIDIX E DCifferent, la1 na ly-zer oce(i.ure IE.I Analyzer Setup The mathematical problem is to study solutions of the equations del cdt 01 1 + - +1 i eI dt R L = i(t), Cd2 + 2+ 1f e2 dt C2- ++ e2 dt dt R L j2 + l - / eJ dt dt R L = - grel1 (E.1) = - — 2 C- -- + 1 e dt dt R L ' = - smle3, where, i(t) =0 for t < c - d, 2' st2 i(t) = cos (at + 2-) 2 for c - < t < c + 2, and (E.2) i(t) =0 I for t > c + d 2 * The system of equations E.I is of a particularly sirmple type in that the solution of each differential equation becomes the forcing function for the succeeding equation; otherwise all four differential equations are identical. The electronic differential analyzer is an analog~ue machine epiloy in. voltages as the dependent variables and time as the independent variable. It consists of a number of high-gain amnlifiers which, through suitable feedback connections, can be made to perform the operations of addition and. integration 99

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN of these variables. The interconnection of analyzer units to solve any one of equations E.1 is given in Fig. E1.. In this figure the ground connection is commion to all elements, although it is omitted in the drawing. All voltages are measured with respect to ground. The equation solved by this setup is ce = -. ej - 200 ej dt + Pij. i(j) (.3) dt 250 2000 which is identical with any of Eq E.1 provided, 1 = -Pr RC ~ 250 and L = a2 = 200. (B.4) LC bo is the circuit bandwidth in radians per second and "a" is the circuit centerfrequency, which for these solutions was held constant at 10 /17 radians per second. Fig. E.la represents the contents of any one of the blocks labelled "circuit" in Fig. 4.5. A sinusoidal signal with frequency proportional to time was generated by solving the differential equation, d2f d- + (10 + st)2 f = (E.5) dt2 with the initial conditions nd~ = 1.5, fo = 0 df 100

I fd I fd i(t) 100 K - Ioe(t) I ej(t) ~~c~~~~ I c ~~ejit II 2000 ii i - 200 le dt- 2ejl L._ ____________ (a) FIG. E.Il SINGLE-TUNED FILTER AND ITS ANALOGUE 250K Pij 10 K (b)

-- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN For small s, one would expect the solution of this equation to be dt 1 t.5 cos 1Ot + st (2]6) dt [ 2 J v / When Eq E.5 is solved on the differential analyzer, it is observed that the amplitude of the first derivative increases as 0 < st < 10. This increase can be reduced to negligible proportions by modifying the equation solved to d2f+ E-df + (10 + st)2 f = 0. (E.') dt2 dt The differential analyzer circuit for the solution of Eq E.5 is shown in Fig. E.2. The variable coefficient of Eq E. 5 is generated by the two ganged potentiometers 1-l and 1A-2 driven by a synchronous motor at 3 PPM. The circuit 14 shown solves the equation d2 + d [10 + f = 0. (E.7) dt2 dt I t it 10 e t 1 Since O =l 40' = 1T; and Eq E.7 is the same as Eq E.5' with s = 1l radian per second per second. The amplitude of the first derivative, which is used as the output signal, is observed to remain constant to within five per cent when E = 5 x 10. Thus the signal used is i(t) = = 15 cos [Ot + volts (8) dt 2,J volts(8 for 0 < t < 140 seconds. 10

o / r --- L -F I G. -E ID j - 400 f ( 2 2 f _ 7 _ + IM IM I —VVVV, V --- — -W - SYNCHRONOUS MOTOR 3/14RPIM IOK FIG. E.2 DIFFERENTIAL ANALYZER SETUP OF INPUT FUNCTION GENERATOR!( ^ — - ~ - - - I-^DIFFERENTIAL ANALYZER SETUP OF INPUT FUNCTION GENERATOR

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The interconnection of the input function generator and the various circuits to solve equation E.1 is given in Fig. 4.3. The block labelled key and interval timer consists of a switch which is closed manually at c - seconds 2 and an electronic interval timer which opens the circuit d seconds later. The four channel recorder is made up of two dual Brush recorders. E.2 Differential Analyzer Solutions E.2.1 Run Proceedures For all differential analyzer solutions the input signal sweep-rate s, the filter center frequency a, and the input signal amplitude are held constant. In each run four quantities have to be specified: (1) the type of filter (number of single-tuned circuits n), (2) the bandwidth of the filter b, (3) the length of the input pulse in time d, and (4) the time of the center of the input pulse c. Table E.1 shows a typical series of runs. These were made to study the effect of varying the tilne c and pulse length d on the response of a two circuit filter. The bandwidth of the filter can conveniently be normalized in terms of the reciprocal of the pulse width. This series is for bd = 2r. (E.9) Table E.1 Typical Series of Runs 2 circuits bd = 2x s =.0710 rad/sec2 d in seconds 5 10 20 - 25 - 20 - 15 1 - 10 - - 10 c in seconds - 5 - 5 - 5 0 0 0 + 5 + 5 + 5 + 10 + 10 + 10 + 25 + 20 + 15 o104

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - In accordance with Eq 4.10, the bandwidth of the individual circuits is b 3. 15( o.636 - d (E.10) and from Eq E. 4, the damping potentiometer settings are given by p = 7781c divisions. (E.ll) r d For each run the time of the start of the pulse must be determined relative to the start of the run. For every run the time t = 0 of Fig. 4.4 occurred 55.3 seconds after the run started. If c = -15 seconds and d = 20 seconds, the signal is connected to the filter input 55.3 -[15 + 20] = 30.3 seconds after the run begins. The signal is disconnected from the filter input by the electronics interval timer 20 seconds later. The proper time to connect the signal is determined by observing the Brush recorder tape, which is driven by a synchronous motor at a rate of 5 millimeters per second. The input pulse is always recorded on one of the four output channels so that a check on this time is available. E.2.2 Extraction and Processing of Differential Analyzer Output Data A data tape as recorded at the output of the differential analyzer is shown in Fig. E.3. Two data are extracted from each tape as indicated in the figure: the maximum amplitude of the pulse envelope A' and the length of the output pulse in time to. The time to is measured between the points at which the envelope amplitude is A'/ /~. Throughout this study a comparison of the behavior of various filters was made in terms of three quantities: A, W, and B.1 See Section 2.3 for a more detailed discussion. 105

Z9;-6-9 '1>18 Zl —SdtV OL6-Wn 111. VW/4:1/ ---u-I /-T / ' —7- 1 I -7V7j F TE 1-7 113/- III — I-I --- It -1] --- -7 --- ~ -~ i 7~~~~~~~~~~~~~~~~~~~~~~~~~A ---- ----- -~~~~~~~~~~~~~-t_ Ij I:~f — I - Ii 4A~~~~~~~~~i:I1 i --- — -- A.* -----— rt --- t-V!VI.-; - ^y~ --- - - ** —;m-111.-Wmm:31/<miffi-j 17.1/11/1. -1-I-1'-1 t — 1 -= m\:yt Wx I=\: \ i::1::1'f\-1:z-:\: \z=n-E —.\ - - 1= (a) INPUT AND OUTPUT PULSES FOR A TWO CIRCUIT FILTER d = 20, bd = 27-r and c = (b) STEADY STATE RESPONSE TAPE FIG. E.3 TYPICAL RESULTS OF A RUN AS OBSERVED AT THE DIFFERENTIAL ANALYZER OUTPUT jo6

- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN A is the peak amplitude of the output pulse relative to the steadystate output amplitude of the filter when a sinusoidal signal is applied having the frequency a. The steady-state response is a function of the filter bandwidth, and therefore three steady-state runs have to be made for the series indicated in Table E.1. The differential analyzer output for one such steadystate run is shown in Fig. E.Ab. The steady-state amplitude A ' is determined from this tape, and then the normalized amplitude is given by A' A = Ao (E.12) The output pulse width W is the width of the output pulse in frequency relative to the bandwidth of the filter b. Since the frequency of the input pulse depends linearly on time as given by Eq 4.3, the width of the output pulse in frequency is sto, and sto W = b (E.13) The determination of the apparent bandwidth B of the filter circuit is a somewhat more involved process. Two steps are required: (1) Plots of the normalized arlolitude A versus C- are made, (sc is the difference between the center frequency of the input pulse and the center frequency of the filter). The data taken from the runs indicated in Table E.1 yield the three plots shown in Fig. E.4. The normalized response is a maximum when the pulse outer frequency equals thle filter frequency and drops off: at higher and low7er frequencies. The apparent bandcwidth B can be measured directly from the curves of Fig. E. 4; it is the width of these curves between the points at which response is 70.7 per 107

1.2 1.0.9 RESPONSE FUNCTION FIG. E.4 INTERMEDIATE PLOT OF A PANORAMIC RECEIVER AS A OF PULSE CENTER IN FREQUENCY FOR bd = 2rr, 2 CIRCUITS.... J ' I I.. I I... II II I I III I CN 0 I I 0 o 0 I I I X, I I I I - - I~~~~~~~~~~~~ X V-x X b = 0.3142 - = b = 0.6283 -S b = 1.257 -J = 0.0450 0.180 0.719 (I x A in A 0 Co /q x.7 r I 1 1S 1~~~~1.0I. I I _I_ I_ _.3 __._..___..2..i111iiiiiiiiiiiiiiii. r_ r,__ F_ 7F.._._._ 0 ----- -- --- -- -- -- --- -- -- -- -- --- -- -- ---- -4 -3 -2 -I 0 I 2 3 4 b

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN cent of the response at sc = 0. The three values of B measured from the intermediate plots of Fig. E.4 are plotted against the nominalized sweep-rate s in b2 Fig. E.5. E.5 Discussion of Errors E.3.1 Machine Errors The circuit resistor and condenser values are correct to within + 0.1 of their nominal values. Trimmer resistances were used to provide fine adjustment of the resonant frequencies of the individual circuit loops. These frequencies are adjusted to be within + 0.1*9 of one another and of the input signal frequency, when taking steady-state tapes. The potentiometers which were used to control bandwidth and amplitude are 10 revolution Helipots marked with 1000 divisions. These potentiometer settings are correct within one division. The errors which have the greatest effect on the data taken were those which occurred in making bandwidth settings, particularly at the smaller values of b. The values of bo are accurate to within +.01 radians per second; thus the percentage error ranges from -.01* for bo = 1.257 radians per second to + 15o for b = 0.07854 radians per second. The pulse length and the center of pulses in time are correct within + 0.3 second, except for the centered pulse group of runs, in which the tolerance was ~ 1 second. E.3.2 Data Processing Errors Readings from the tapes could be made within one-fourth division out of twenty. This is ~ 1.25% at full scale, and within ~+ 5/ at one-fourth of the full deflection. Readings at less than onefourth full deflection were usually avoided. Some idea of the overall reproducibility of the data may be gained by comparing plots from two sets of data for the same parameters, taken at different times, as shown in Fig. E.6. 109

0 T 9 8 7 -------------------- -- -- -- ------------------------------------ -- - - I B 3 2 0.02 O.I S b2 FIG. E.5 THE APPARENT BANDWIDTH OF A TWO CIRCUIT FILTER AS A FUNCTION OF SWEEP-R (DATA FROM DIFFERENTIAL ANALYZER) bd = 2-r Cal U) (D y I,D _! -J I 0 N1.0?ATE

1.2 FIG. E.6 ---- --- _ RELATIVE AMPLITUDE AS A FUNCTION 0 SWEEP-RATE AND PULSE LENGTH FOR T 1.o0 -- - CIRCUITS, PULSE CENTERED ON PASSBAI bd = 2-r I - I______ II___ ' ___ III FIRST SERIES - SOLID CURVES -— "X x ls SECOND SERIES- DASHED CURVES I. I _ _ _ __ _ _ _ 1 1bd: 1 0.01 0. 1.0 10.0 b2 F WO JD 50.0

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPINDIX F Differential Analyzer Data F.1 Examples of Differential Analyzer Solutions Section 4.5 includes a discussion of several typical solutions taken from the differential analyzer. Additional runs showing the form of the responses and certain anomalous effects are included in this appendix. Figure F.1 shows how the response to pulses changes with the pulse position for one, two, and four circuits. The middle pulse is centered on the filter passband and the other two pulses are each about midway between the center of the passband and the 3 db point. The tendency of the envelope of the response to approach a Gaussian shape with an increase in the number of circuits is evident in this figure. Figure F.2 gives the response of one circuit with various bandwidths to a cw signal. The output pulse width is clearly a minimum for some intermediate bandwidth. The undulatory nature of the response that predominates in single circuit filters and narrow passbands is evident here. Figure F.5 shows a similar set of curves for the four circuit case. These responses are delayed in time and depressed in amplitude by comparison with the single circuit solutions of the same bandwidth. The output pulse width is also seen to be longer for the greater number of circuits, and the undulatory character has been suppressed. Figure F.4 shows an interesting anomalous effect that can be attributed to the beat phenomena. Figure F.4a indicates that a change of 25% in the input pulse length can cause the output pulse width to triple. Figure F.4b shows a similar situation for the two circuit case, The solutions in Fig. F.5 112

FIG. F.I RESPONSES FOR VARIOUS NUMBERS OF CIRCUITS & INPUT PULSE POSITIONS b = 1.26 d = 5 s =.0710 INPUT SIGNAL 5 SECONDS 1-< ---- ~- 1 I CIRCUIT ~4 t=0 (RESONANCE) 2 CIRCUITS LA 4 CIRCUITS, 113

-- r —4 OIL ' = S ( = p linotio I SHIGaIMNV8 SnfOIVtA 0.i S3SNOdS38 Z'J l91d

FIG. F.3 RESPONSES FOR VARIOUS BANDWIDTHS 4 CIRCUITS d = s =.0710 - r'V-'V - ' -^tft ^.v. l i At-w _ — ^ — -t — 4- 1 —Y — g —A — t \ A\c a_ M -: En~~~~~~ =-1 H. is Il, t = 0 (RESONANCE)

?g-g-g ) 9 -I r a. (D - (3 1 CD) LLOCZ 2 0 w I (jLL cf F W Z r1( C ) LLJ Zn w Ow cLLL cn:: Cl L I C ~d - OL6 - W 0 0). - of.0 (C (D (0 Io at Vo 116

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN are also included because they show an anomalous effect. Note that the beat phenomenon is much more pronounced in the two circuit solution that in the single circuit case. By the fourth circuit the ripple has disappeared and the envelope approximates the Gaussian shape. F.2 Curves for the Factors A, B, and W Most of the solutions colmpiled from the differential analyzer are summarized in this section. The objective is to present fairly complete data on the relative amplitude, effective bandwidth, and output pulse width as a function of the number of circuits, pulse width, bandwidth and tihe sweep-rate. Table F.1 summarizes the curves of Ao, B, and W obtained from the differential analyzer solutions. Since examples of each type of curve have been discussed previously, the compiled results are included without cormment. 1Table F.1 Location of Curves of Ao, B, and W for Differential Analyzer Solutions Type of Filter 1, 2, and 4 Circuits 2 Circuits bd 1 2 2 i 2 ( C _ 2, and CO 2 2 AO F.6 F.7 F.8 F.9 F.10 4.10 4.11 F.11 F.12 4.13 F.13 F.14 F.15 4.12 B 4.15_ lNuubers in table are figure numbers. 117

ZS-L-S 'INN 1l-2d- oZG-N-W RESPONSES FOR FIG.. F.5 DIFFERENT NUMBERS OF CIRCUITS b =.38 d = o S -.0710 INPUT PULSE 5 SECONDS I _ 4 CIRCUITS >^ ~ ~ ~ ~ ~ ~~ ~~~~~~~~~~~~~~~~~~"- U ' t = 0 (RESONANCE) O rrl ~I IIre 1 i f i i A A AM A L ---h-"1 Ilk I I li A A A-A A=A V-V=: I CIRCUIT

1.2 1.0 0.8 Ao 0.6 —.J i-u 'O 0.4 0.2 0 0.2 S v

Ao 0.6 0 L 0.1 1.0 10.0 S bv 50.0

1.2 H 1.0 0.8 Ao 0.6 0.4 0.2 0 al s bz

Ci I I _1 y - ro 0!) X 1.2 1.0 Q8 Ao 0.6 C\J U, (D 0 I Ko (D I 0 I Pr 0.4 0.2 0.02 0.I 1.0 S ve 10.0

1.21 1).80_ Noma.,~ Now C AO ( 0.6 1 CIRCUIT-' 2 CIRCUITS 4 CIRCUITS bd = 27r -J 0)4 FIG. F 0 RELATIVE AMPLITUDE AS A FUNCTION OF SWEEP-RATE PULSE CENTERED ON PASSBAND (DATA FROM DIFFERENTIAL ANALYZER) 1I 1 5,. i.C )i 0.1 1.0 50.0 s

Zg-6-9 "-11 SG~-29-V OL6-W 100.0 10.0 W 1.0 4 CIRCUITS -- - --- - -- ------- --- - -- - 2CIRCUITS bd l I — I I I- -- - - ~~~~ --- ---- -~~T- I CIRCUIT f~_I _ - - V _./______^_ - - _______j XOOF X --- Ix-1EL- _f_ --- --!-_1 __ _-...-. _ --- --- - --—....j... — i --- —------- --- - — ___ - - - '..... or 0.I C >.1 1.0 10.0 30.0 s *p FIG. F.II OUTPUT PULSE WIDTH AS A FUNCTION OF SWEEP-RATE PULSE CENTERED ON PASSBAND (DATA FROM DIFFERENTIAL ANALYZER) 124

-01-9 "l)lN 9~-~9-V OZ6-W 100.0 4 CIRCUITS Y// ^2 CIRCUITS bd 2 10.0 - - - --- -- - _ _ m _ -_ /__ - |~ I CIRCUIT _~____, _ ii -------- -- L_ __ --— i-i- - 1.0 - - 0.1 0.1 1.0 10.0 b2 FIG. F.12 OUTPUT PULSE WIDTH AS A FUNCTION OF SWEEP-RATE PULSE CENTERED ON PASSBAND (DATA FROM DIFFERENTIAL ANALYZER) --------- -^ ^ — ^ - _ _ _ --------- -- - - - _ - _ ----- - -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ '~~~~~~~~~~b '~~~~~~FG. OUPTPLE IT SA UCINOFSEPRT PUSXETRD NPSBN /-?. DAAFOMDFEENILANLZR 30.0 125

29-6-9 "I> S9~-~9-V OL6-1W Z- 01 -9 'lIN L~-~9-V OL6-W W.03 0.1 1.0 s b2 FIG. F.13 OUTPUT PULSE WIDTH AS A FUNCTION OF SWEEP-RATE PULSE CENTERED ON PASSBAND (DATA FROM DIFFERENTIAL ANALYZER) 10.0 126

G- 11-9 'I>I 82-~9-V OL6-W 100.0 10.0 W, 4 CIRCUITS. 2 CIRCUITS --------- -- -- - _ _ _ --------- -- - ^._ _ I C IRC U IT bd = 2r I I I S: I _=1 11<X < 1.0 0.1 C ).01 0.1 1.0 3.0 S b2 FIG. F.14 OUTPUT PULSE WIDTH AS A FUNCTION OF SWEEP-RATE PULSE CENTERED ON PASSBAND (DATA FROM DIFFERENTIAL ANALYZER) 127

w 10 2 CIRCUITS — _____ __ _ _ _ _ —__ _ _ - - _ __4 CIRCUITS x I CIRCUIT I 1 1 1 1 1 11 1 1 f FI[G. F.15 FREQUENCY SWEPT BY PANORAMIC RECEIVER DURING OUTPUT PULSE AS A FUNCTION OF SWEEP-RATE bd = o0 --------- ^ ~/ ---:=::=^==^=:^ __^^f - - -______p/ 7 _7 _ __/ --- - - --— y ^^^ ---- -- ---—.,x ~o.ou. - --- -----— / -^ - - -/ - - -----— ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t -- — ^~~~~ ~~, --- ^z:::;:^_::::::/./ ^^ ^^ FIG. F.15" ^^ ^FEQEC SETBYPNRAI ECIE DRN.^^^*_S ^ UTUTPUSEASA UNTIN F WEP-AT I 0.1 I 10 60 SC b

BIBLIOGRAPHY 1. Hok, G., "Response of Linear Resonant Systems to Excitation of a Frequency Varying Linearly with Time," Journal of Applied Physics, Vol. 19, pp. 242 -250; March 1948. It is pointed out in Appendix A that the derivation in that section is essentially contained in Hok's paper. In addition, Hok gives a general form for the response relation when the sweep is not necessarily linear, derives an apparent bandwidth, and presents suitable curves so that the output phase as well as amplitude for the single resonant circuit can be *calculated in the case of cw input and a linear sweep. Many interesting curves are included. 2. Lewis, F.M., "Vibration During Acceleration Through a Critical Speed," Trans. Am. Soc. Mech. Eng., APM-54-24, pp. 253-261; 1932. This paper presents a solution of the fundamentally similar mechanical problem of "running a system having a single degree of freedom and linear damping through its critical speed from rest at a uniform acceleration." 3. Barber, N.F. and Ursell, F., "The Response of a Resonant System to a Gliding Tone," London Philosophical Magazine, Vol. 39, PP. 345-361; May 1948. This study is similar to the work of Hok and that of Lewis. The same integral is derived for the response and discussed in some detail. The discussion of the case of varying the natural period of the system rather than the frequency of the signal is carried farther in this paper. Interesting diagrams are presented and good experimental agreement is shown. 4. Williams, E.M., "Radio-Frequency Spectrum Analyzers," Proc. I.R.E., Vol. 34, pp. 18-22; January 1946. This paper is an experimental study of resolution and the bandwidth corresponding to optimum resolution. The author concludes that the best resolution defined as the frequency between 3 db points of the output is 1.3/ s/2g and that this resolution is achieved for a bandwidth of S s/4 Cps. 5. Barlow, H.M. and Cullen, A.L., Microwave Measurements, pp. 520-332; 1950. In the section on microwave spectrum analysis, the authors discuss the response of a fixed resonant system to a signal of varying frequency as it applies to microwave spectrum analyzers. For resolution defined as the frequency between 3 db points of the output, the optimum Q is given as Qpt = t, i e., b = qopt ~~ 129

ZG;-6-9 7)18 9-29-Vt OL6 -IN 6. Montgomery, C.G., Editor, Technique of Microwave Measurements, Vol. 11, MIT Radiation Laboratory Series, p- 4 0855; 197. Chapter 7, "The Measurement of Frequency Spectrum and Pulse Shape," discusses the principles and design of spectrum analyzers, giving descriptions and circuit diagrams of representative spectrum analyzers. The author also discusses the response of a Gaussian filter to signals of constant amaplitude and linearly varying frequency in connection with the generation of pulses (Section 4.6). 7. Marique, J., "Response of a Circuit to a Linear-Frequency-Sweep Voltage, " Onde Elect., Vol. 31, pp. 3135-15; July 1951. (in French) The universal response curves of Hok are transformed into a set of curves which can be applied more readily in practice. These are based on circuit bandwidth corresponding to a 3 db fall in response, and show response as a function of sweep-rate. As this increases the current maximum decreases and occurs later, while the passband (3 db below the maximum) increases. 8. Hatton, W.L., "Simplified FM Transient Response," Technical-Report No. 196, MIT; April 1951. This paper derives the response of a single tuned circuit when the input current makes a sudden jump in frequency. The paper examines the instantaneous output frequency, the rise time, and the maximum overshoot. There are also earlier papers by D.A. Bell, H. Salinger, and C.C. Eaglesfield which treat the special problem of transient response with a sudden change in frequency. 9. Walaran, II. and Valley, G.E., Vacuum Tube mplifiers, Vol. 18, MIT Radiation Laboratory Series. 10. Lawson, J.L. and Uhlenbeck, G.E., Threshold Signals, Vol. 24, MIT Radiation Laboratory Series. 11. Moulic, W.E., "Panoramic Principles," Electronic Industries, Vol. III, No. 7, pp. 86-88, 206; July 1944. A description is given of the basic components in the panoramic system and applications to industrial and other problems are suggested. 12. Thomasson, D.W., "Panoramic Display - Design Considerations," Electronic Engineering, Vol. 21, No. 257, pp. 259-261; July 1949. Basic design principles of panoramic display are presented. General limitations are pointed out and several applications are suggested. 13. Bush, V. and Cald.well, S.H., "A New Type of Differential Analyzer," Journal of the Franklin Inst., Vol. 240, pp. 255-525; October 1945. 130

14. RaTgazzini, J.i., E andll, R.II. and Russell, F.A., "Analysis of Problems in Dy,,amics by Electronic Circuits," Proc. I.RE., Vol. 55, pp. 444-452; MayrCJ 1947. 15. IMacnee A.B., "An -Electronic Differential Analyzer," rtoc. I.R.E. Vol. 7, pp. 1315-1321-; November 191.9.. 16. Salinger, II., "On the Theory of Frequency Analysis by MIeans of a Searching Tone," Elektrische Itachrichten Technik, Vol. 6, pp. 2935-02; August 1929 (in Ge rmian) This paper describes an automatic spectjrumL analyzer; the response of lowpass filter to signal sweepin, linearly in time is presented. The author considers an "ideal" filter with a rectangular amplitude response curve and a linear phase response curve. He concludes that sweeping will not seriously affect the speectrLum analyzer output provided __. > ~4- ^y2j 15]1

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN iLIST OF SYMBOLS a, tile instantaneous frequency in radians per second of the filter input signal at time, t = 0. a, the center frequency of the filter: in radians per second. A(relative amplitude), the peak response to a pulse per unit of centered cw signal with the sane input amplitude (see Section 2.3). Ao(relative amplitude), the value of A for centered pulses (see Section 2.3). b, the bandwvidth of the filter in radi-as per second. B = ( 2- cl). (effective bandwidth), the rance of frequencies of the input b pulse per unit bandwidth such that the response is at least 0.707 of the maximum (see Section 2.3). c, the center of the input pulse in time (seconds). cl, C2, the centers in time of the input pulse such that t:e peak response is 0.707 of its maximum value. d, the input pulse width in time (seconds). s, the sweep-rate in radians per second per second. sc, the distance in radians per second from the filter to the center of the input pulse. t, the time in seconds. tmn the time of maximum response in seconds. to, the width of the output pulse in seconds. WT =.t. (output pulse width), the width of the output pulse in frequency per b unit of bandwidth (see Section 3.2)..I, the frequency in radians per second. Additional symbols of less significance are defined where they are used in the texts 152

DISTRIBUTION LIST I co py M. Keiser Chief-, CounterUrmeasures Branch Evans Signal Laboratory BeLmar, New Jersey 75 copies Transportation Officer, SCEL Evans Signal Laboratory Buildinc No. 42 Belniar, New Jersey i copy FOR - Signal Property Officer Inspect at Destination File ITNo. 2505-PIH-51-91 (1443) W. G. Dow, Professor Dept. of Electrical EPngineering University of Michigan Ann Arbor, Michigan IIT.. Welch, Jr. Engineering Research Institute University o-f Mlichig an Ann Arbor, Michigan 1 copy 7 copies 1 co-,yr 1 co-py Electronic University Ann Arbor, De'fonse GrouL-) Project P il e of Michigan Miclhigan Document R-oori Willowj Run Research Center University of Michigan Ann Arbor, lvichigan Enginreer ing Rsearchl Institute P'roject File Universi-t of Michig!an Ann Arblor, M,1ichigian k 01_3

UNIVERSITY OF MICHIGAN 3 9015 02523 0338111111 3 9015 02523 0338