ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR SIGNAL DETECTION WITH A PANORAMIC RECEIVER Technical Report. No. 38 Electronic Defense Group Department of Electrical Engineering By: W. W. Peterson Approved by:..... T. G. Birdsall A. B. Macnee Project 2262 TASK ORDER NO. EDG-3 CONTRACT NO. DA-36-039 sc-63203 SIGNAL CORPS, DEPARTMENT OF THE ARMY DEPARTMENT OF ARMY PROJECT NO. 3-99-04-042 SIGNAL CORPS PROJECT NO. 194B June, 1955

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv I. INTRODUCTION 1 II. A SIMPLE CASE 2 III. SIGNAL-TO-NOISE RATIO IN A PANORAMIC RECEIVER 7 IV. PANORAMIC RECEIVER DESIGNED TO COVER A FIXED BAND 14 APPENDIX A 21 APPENDIX B 23 ii

LIST OF ILLUSTRATIONS Figure No. Page 1 Normalized Output Signal-to-Noise Power Ratio VS Normalized Filter Bandwidth 6 2 Block Diagram of Idealized Panoramic Receiver 7 3 Normalized Signal-to-Noise Ratio VS Normalized Bandwidth, Normalized Sweep Rate as a Parameter 11 4 Normalized Signal-to-Noise Ratio VS Normalized Sweep Rate, Normalized Bandwidth as a Parameter, Using No Post-Detection Filter 12 5 Normalized Signal-to-Noise Ratio VS. Normalized Sweep Rate; Normalized Bandwidth as a Parameter, Using Ideal Post-Detection Filter 13 6 Normalized Signal-to-Noise Ratio VS. Normalized Bandwidth, Normalized Apparent Bandwidth as a Parameter 15 7 Optimum Video Bandwidth VS. I. F. Bandwidth 18 8 Signal-Noise Ratio VS. Video Mismatch Ratio, Optimum Video to I. F. Bandwidth as a Parameter 19 B.1 Block Diagram of Panoramic Receiver 23 iii

ABSTRACT Formulas for the signal-to-noise ratio at the output of a particular panoramic receiver are calculated. The receiver considered consists of a linearly sweeping local oscillator, an ideal mixer, an IF amplifier with a Gaussian passband, a square law detector, and a video amplifier with a Gaussian passband. For a given signal pulse duration the optimum IF bandwidth, video bandwidth, and sweep rate are determined. The results of this derivation are applied to compare a wideband non-scanning receiver with a narrow band receiver scanning at a rapid enough rate to make it effectively a wideband receiver over the same frequency range. If the optimum video filter is used, both receivers have very nearly the same output signal-to-noise ratio; but if the video filter is not matched, the fast-scanning narrow-band receiver is superior. iv

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN SIGNAL DETECTION WITH A PANORAMIC RECEIVER I. INTRODUCTION In this report formulas for signal-to-noise ratio are calculated for a odel of panoramic receiver. The receiver model consists of a linearly sweeping ocal oscillator, an ideal mixer, an IF amplifier with Gaussian passband, a square 1w detector, and a video amplifier with a Gaussian passband. This model was hosen because an explicit formula for signal-to-noise ratio can be obtained for b- a rather remarkable fact in view of the complexity of the system. Although the analysis is made only for the case of pulses with Gaussian iaped envelopes and filters with Gaussian passbands, the results should be ndicative of what will occur in the more general case. Certainly the noise )wer will be little different. It was shown in Technical Report No. 31 that the iussian case is quantitatively consistent enough with the other cases studied to > used in many design problems involving peak amplitude, output pulse width, and pparent bandwidth. These are the features of the receiver output which are,rtinent here. Ideally, one would like to know probability of detection and probability' false alarm as a function of signal, noise, and receiver parameters. For a (stem as complicated as the panoramic receiver, this appears to be out of the.estion at this time. Signal-to-noise ratio has been used successfully for many zars as a guide for designing linear systems for detection. I. W. Batten, R. A. Jorgensen, A. B. Macnee, and W. W. Peterson, "The Response of. Panoramic Receiver to CW and Pulse Signals" Technical Report No. 3, Electronic )efense Group, University of Michigan, June, 1952.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Signal-to-noise ratio is defined as the ratio of peak signal power in t absence of noise to the average noise power, at the receiver output. Since the noise and the signal are uncorrelated, the average voltage output with signal plus noise equals the sum of the output with signal only and the average output with noise alone.l Thus the peak signal output power measures the change in outp voltage, on the average, caused by the signal. The noise power measures the average amplitude of noise fluctuations and hence signal-to-noise ratio is a measure of the amplitude of the change in voltage output caused by the signal, relative to the amplitude of noise fluctuations. II. A SIMPLE CASE In this section, signal-to-noise ratio is calculated at the output of a Gaussian filter when white noise plus a Gaussian pulse are passed through the filter. This problem is relatively simple and serves as a good introduction to the more complicated panoramic receiver problem. It is well known that the filter which maximizes signal-to-noise ratio has a transfer function proportional to the conjugate of the spectrum of the 2 pulse. It is the purpose of this section to consider the effect of matching the filter bandwidth improperly to the pulses. This is strictly true only with a square law detector. 2 See, for example, Part II, pp. 12 and 20 of Technical Report No. 13 by W. W. Peterson, and T. G. Birdsall, "The Theory of Signal Detectability"; Part I," The General Theory", June, 1953; Part II,"Applications with Gaussian noise," July, 1953. If the transfer function is proportional to the conjugate of the pulse spectrum, then the impulse response is proportional to the pulse reversed in time, and conversely. _ ~~~~~~~~~~~2

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Assume that the pulse is (the real part of) f(t) = (d ~) exp. jat -2 ) (2.1) here a is the center frequency in radians per second, and d is the pulse duration n seconds between times when the amplitude is exp (-1/4). The constant factor is chosen to make the pulse have energy E, i.e., S-O f(t) f-t dt = E. (2.2) hie Fourier transform (or spectrum) of this pulse is2 F() f (t) exp (-jot)dt = 7E) fexp (j(a.J)t - 2 dt d -00 1/2 E\,v/ d) exp(- - (C -a2) (2.3) 2 ie pulse spectrum, and hence the optimum filter, has bandwidth d radians per bcond between exp (-1/4) points. Consider a filter with pass band of the ume shape and center frequency but of arbitrary bandwidth, i.e., a filter with.ansfer function 2 2=, -i b2 iere b is the bandwidth in radians per second between exp(-l/4) points. The,sponse of this filter to the pulse given by equation (2.1) is found by multiying the pulse spectrum by the filter transfer function to get the spectrum the output, -and then transforming this back to the time domain, i.e., the his assumes a normalized load of one ohm, a convenient and non-restrictive ssumption. s integral, and a number of others appearing in this report, can be evaluated ith the aid of equation B.4, page 69 of Technical Report No. 3: If U and V are omplex numbers, and if the real part of U is positive, then exp (-Ut2 + Vt)dt =7ij exp' (V2/4U). (25) - 0

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN output pulse g(t) is g(t) = F(m) H(w) exp (-Jwt) dt = (dg O [4 )b2 E,/_..1/2 - a00 d2 xp2 jmt) dm = (,I/ 2> f 0exp [- + ] a)+ t} d (26) 1/2 _ =exp - 4+ ) exp (jat) / E d2 \~ /b2d2+4.x. The maximum power of the output pulse, i.e., the maximum of g(t) fi occurs when t = O. It has the value S [g(t) ~T]T]E b2d2 Smax d' b2d2 + 4(2.7) If the noise is assumed white with spectral power density of No watts per cycle per second (or No- watts per radian per second) at the input of the 2n. filter, then at the output the noise spectrum is, by equation (A.7) of Appendix A Po(W):2 * 2 *. 2H(W) and the total noise power is N = 2 I H(a)H(W) dw 2 OD0 NO= OC exp b (u-2 ) d (2.8) No b /4 J/ The signal-to-noise ratio is found from equations (2.7) and (2.8) to be 2E 4bd (2.9) N No' b2d2+ 4

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN This function is graphed in Figure 1. The maximum signal-to-noise ratio achieved when bd = 2, and it has the value 2E/No. The asymptotes are lines slope one. A filter designed for ten microsecond pulses has one-tenth the |timum bandwidth for one microsecond pulses, and hence, from Fig. 1, the signal-noise ratio would be one-tenth of its maximum value if this filter rather than e optimum filter were used with one microsecond pulses. The signal-to-noise tio would be reduced by a factor of 100 if a filter designed for ten microsecond ses is used with one-tenth microsecond pulses. Likewise, using a filter signed for one-tenth microsecond pulses would result in a signal-to-noise ratio 1/100 of the maximum value when used with ten microsecond pulses. Clearly, filter bandwidth must be matched to the signal if the receiver sensitivity is important consideration. This is true also in the case of a panoramic receiver. This analysis assumed for simplicity that the pulse and the filter are lussian. The question arises as to whether it is safe to accept these results approximately true for other types of signals and filters. A partial answer this question is possible. In the first place it can be shown that the ymptotes are always parallel to 450 lines. In the second place, if an optimum pe filter is used, the maximum signal-to-noise ratio is 2E/N0.2 If the curves ways have the same maximum and parallel asymptotes, they are probably very milar. Curves for two other cases are shown on Fig. 1 for comparison. Note at in the case of a square pulse and a single-tuned circuit filter, the filter not of the optimum type for the pulse. Et is true in general that the signal-to-noise ratio at the output of a filter which is optimum for a single signal is 2E/No. See Technical Report No. 13, Part II, p 67. (In the first printing a factor 1/2 was omitted in equation (5.4) and therefore the output was found to be E/No rather than the correct result BE/No ). Bee footnote 1 above. 5

-6 -.,,,,.,.,I~ ~ ~~ ~~~~~ /'N ~II -o -9'o~~~~~~~~~~o.4'~~~~~~~~~~~~. \\ \,%~ 2 -35 =~~~ ~~~~~~~~~- /'N ~~~~~~~~~~~~~~ ~,-6,'// 0."z 12 2 -. -- __ _ - - - -1246 8 10 C~ 9~~~~~~~~~~~~~~~~ ~'1~-la// I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' /1 N~oee/N /'/I // -/'" I 4, -15./ /I /* 4,,, -18 -21.01 2 4 6 8.1 2 4 6 8 I. 2 4 6 8 1.0 2 4 6 8,00 b/bOPT. FIG. I. NORMALIZED OUTPUT SIGNAL-TO-NOISE POWER RATIO VS. NORMALIZED FILTER BANDWIDTH. O GAUSSIAN PULSE GAUSSIAN BANDPASS (GAUSSIAN IMPULSE RESPONSE) G~ SQUARE PULSE R-C FILTER (EXPONENTIAL IMPULSE RESPONSE) (Q SQUARE PULSE sin x BANDPASS (SQUARE IMPULSE RESPONSE) x

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN III. SIGNAL-TO-NOISE RATIO IN A PANORAMIC RECEIVER In this section formulas and graphs are given for the signal-to-noise tio at the output of a panoramic receiver. A block diagram of the receiver'del is shown in Figure 2. SIGNAL INPUT TO RECEIVER FM SIGNAL ~TRANSMITTER -B~lMIXER FILTER FILTER OUTPUT DETECTOR OUTPUT VIDEO " k,- DETECTOR FILTER FIG 2. BLOCK DIAGRAM OF IDEALIZED PANORAMIC RECEIVER CW signals and pulses with Gaussian shaped envelopes are considered. ite Gaussian noise is assumed. The signal-to-noise ratio is calculated for this re complicated problem in much the same manner as in the previous section. rmulas in closed form are obtained for the signal-to-noise ratio as a function pulse width, sweep rate, IF bandwidth and video bandwidth. The derivation of the formula for signal-to-noise ratio is rather hgthy and not enlightening, and therefore it is placed'in Appendix B. The gnals are assumed to have the form f(t) E exp j(st + t - C) (321) the output of the ideal mixer, where the signal has a linearly varying frequency. L signalat the output of the IF filter is found with the aid of Technical port No. 3. The square law detector is assumed to have at its output the square the, envelope of the input. The response of the video filter to this function

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN is found also with the aid of Technical Report No. 3. The noise spectrum is assumed white at the input to the IF filter. At the output its spectrum is proportional to the power gain function of the IF filter. The spectrum after detection is found through the use of autocorrelation functions, and the power gain of the video filter is taken into account in finding the total noise power at the output of the receiver. The signal-to-noise ratio is found by dividing the peak signal power at the output by the average output noise power.1 The resulting expression for signal-to-noise ratio is S E2 16Ao 4W2/ 4 SC 2 l6A0W1 13 +22exp (- (5 N No2 d2(8s2+ b2P2W2) exp b (32) where Ao relative amplitude; the peak IF amplitude when the receiver is not sweeping and tuned to the pulse divided by the amplitude when the input is CW -1/4 b bandwidth of the IF (radians/sec between e points) 13 bandwidth of the video (radians/sec between e'1/4 points) C the time difference between the occurance of the center of the puli and the instant that the receiver is tuned to the pulse frequency (seconds) d input pulse duration (seconds) E energy of the pulse No noise power per cycle per second S sweep rate of the receiver (radians/ sec) W normalized IF output pulse duration; the number of IF bandwidths swept through during the output pulse. See Appendix B. 8

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN It will usually be possible to adjust the video bandwidth P, and therelore the optimum video bandwidth deserves consideration. By differentiating (3.2) ith respect to P, it can be shown that the signal-to-noise ratio has its maximum orl b (b2 1/2 (3)3) 8s2 f this value of P is substituted in equation (3.2), the result simplifies to E2 1 2 exp[- 4(b)2 (3.4) = N02 B2 bbW [b N) y using the expressions for AO, W, and B given in Technical Report No. 3, the ollowing alternate forms can be derived from equations (3.3) and (3.4): bB/8 opt.: (3.5) opt*. ()opt. video E.exp Q ()) (3.6) filter No2 4+b2 b ) quations (3.3) through (3.6) are derived in Appendix B. The effective bandwidthB,can be increased indefinitely by increasing he sweep rate. Hence for large enough sweep rates S, the expression 4 + b2d2 _ 4B2 (3.7) s negative, and the optimum video bandwidth is imaginary. If P = j7, then the ptimum video filter has a transfer function of the following type: H(c) = -exp (+722) (3.8) See Appendix B.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN - Such a filter is unrealizable, of course. It is not entirely unreasonable to consider such a filter; however, a situation very similar to this occurs in the problem of detecting signals in radar clutter, and Urkowitzl has shown that signa: to-noise ratio can be improved by using an approximation to a similar unrealizable filter. That case will not be considered in any detail in this report, however. Since the ideal filter is unrealizable in certain ranges of the variable b, d, and s, signal-to-noise ratio with no video filter at all will be considered in these ranges. The expression for signal-to-noise ratio with no video filter can be found by letting 3 approach infinity in equation (3.2). That equation ther reduces to S_ E2 16A4 4 c2 N 0.exp - (-)) (3.9) N 2 b2d2 B2 b The dividing line between real P and imaginary P occurs where P becomes infinite and 4 + b2d2 4B2 = 0 (3.10) If this equation is satisfied, the ideal video filter is no filter at all. Signal-to-noise ratio is plotted in Figures 3, 4 and 5 to show its dependence upon sweep-rate and IF bandwidth. In these figures the pulse is always assumed centered on the passband, i.e., C is assumed to be zero. Equation (3.6) shows the dependence of signal-to-noise ratio on IF bandwidth and effective bandwidth when an optimum video filter is used. The corresponding equation for the case of no video filter, obtained by eliminating Ao in favor of B from equation (3-9) is (S) no video filter = exp ( (3.11) No2 4B2+4+b2d2 B2 b Harry Urkowitz, "Filters for Detection of Small Radar Signals in Clutter", Journal of Applied Physics, Vol. 24 No. 8, August 1953. 10

FIG. 3. NORMALIZED SIGNAL-TO-NOISE RATIO VS. NORMALIZED BANDWIDTH, NORMALIZED SWEEP RATE AS A PARAMETER. IDEAL POST-DETECTION FILTER IS "-'..IDEAL POST-DETECTION FILTER IS GAUSSIAN INVERSE GAUSSIAN - USING NO POST-DETECTION FILTER - USING IDEAL POST-DETECTION FILTER — USING IDEAL POST-DETECTION FILTER -6 db sdb:.31 -9db -3 db -6 db -24db 0.1 10 NORMALIZED BANDWIDTH bd, IN RADIANS 15~,,~ db?o —.!,,'' \ 0 _-18db /7/'~//~~~~~~~~~~00

bd = 2 bd = /2 bd ir bd =?,r/4 -3db bd = 27r 0~~~~~~~~~~~~~~~~~ -6db bd 7r/8 bd 47 -9db -12db -15db -18db -21db.01 02.04 06.08.10.2 A.6.8 I 2 4 6 80K) 20 40 60 80 100 200 400 600800000 sd2 FIG. 4. NORMALIZED SIGNAL-TO-NOISE RATIO VS. NORMALIZED SWEEP RATE, NORMALIZED BANDWIDTH AS A PARAMETER, USING NO POST-DETECTION FILTER.

0 bd = /2 bd bd = 17r/4 -3db D bd~ir/8 -- - bd 47r A -9db -12db -15db, FIG. 5. NORMALIZED SIGNAL-TO-NOISE RATIO VS. NORMALIZED SWEEP RATE, NORMALIZED BANDWIDTH AS A PARAMETER, USING IDEAL POST-DETECTION FILTER. IDEAL POST-DETECTION FILTER IS.GAUSSIAN _ IDEAL POST-DETECTION FILTER IS INVERSE GAUSSIAN

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN IV. PANORAMIC RECEIVER DESIGNED TO COVER A FIXED BAND Suppose that a receiver is required to be essentially wide open to pull of duration d that may occur anywhere in a certain band. The effective bandwidt] of a panoramic receiver is bB radians per second, and hence it is desired to adji the receiver parameters so that bB always equals the band to be covered. Figure 6 is a plot of the signal-to-noise ratio as a function of IF bandwidth, bd, where the sweep rate has been adjusted along with the IF bandwidth in order to hold th( effective bandwidth bdB constant. Consider a receiver required to receive pulses that last ten microsecox and may be anywhere in a 1.6 megacycle band. This effective bandwidth could be accomplished either by having a 1.6 megacycle IF bandwidth, or by having a narrower IF bandwidth and by scanning rapidly. For this hypothetical case, bB i. 2X x 1.6 = 10 megaradians per second, and dbB is therefore 100. It can be seen from Figure 6 that for bdB equal to 100, the maximum signal-to-noise ratio occurs for bd - 100 and s = 0. As the IF bandwidth is decreased to bd = 14, the point at which the optimum video bandwidth is infinite, the signal-to-noise ratio dropE only.04 db. Beyond that point, the signal-to-noise ratio begins to drop rapid] with the (unrealizable) ideal filter, and even more rapidly with no filter at alJ The receiver could thus be operated with nearly maximum signal-to-noise ratio wit IF bandwidth anywhere in the range from b = 100 to b =,16 me to 226 kcd d to226kc. At b - 100 the sweep rate would be zero, while at b 14.2 the sweep rate d s d would be 98.51 radians per second per second, or 156 kilomegacycles per second pe d2 second. 1 Table I summarizes this and several similar cases.

IK-2-11 W3r egi-es-is agaz FIG. 6. NORMALIZED SIGNAL-TO-NOISE RATIO VS. NORMALIZED BANDWIDTH, NORMALIZED APPARENT BANDWIDTH AS A PARAMETER. IDEAL POST-DETECTION FILTER IS I' — IDEAL POST-DETECTION FILTER IS GAUSSIAN INVERSE GAUSSIAN -— USING NO POST-DETECTION FILTER USING IDEAL POST-DETECTION FILTER -— USING IDEAL POST-DETECTION FILTER 8db, 0 db -; 10...-1- IOdB= 3'e ~Bdb=4 / I ~. / ~ ~ ~ ~ ~ ~, // -2ldb ~ 0 Bdb -24d b \irrI YBb B =0 3db,.,, // I X I I I II~~~~~~~~~~~~~d I 0 d -5 //~~~/ /',,,,,,/ /" 00, 4.0..0,, -26db I/'~~~~~~~~~~~~~~ //-//''/".'" / Bd 20Bdb -90dbA 0.1 10 00 NORMALIZED BANDWIDTH bd, IN ADIANS -12d~~ ~~~~~~~~~~~~~~~~~~~~~~~~~Bb = 0 100'\ _,. ~~~~~~~~~~~~~~~.-, 4e~~ / -24~~~~~~~~~~~ d -27 ~ ~ ~ ~ ~ / db\

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE I...TABLE. For 10 usec Pulse Bdb sd2 bd (N) diff Sweep Rate IF Bandwidth Video Bandwidth 0 19.90 0 318 Kc/s 45.4 Kc/s 20.o85 db 19.286 6.485 31.6 KMc/s2 103 Kc/s ---—. — 0 99.98 0 1.6 Mc/s 45.3 Kc/s 100.042 db 98.51 14.212 156 KMc/s2.226 Mc/s 0 M 1/2 M>100 4343 db ( 8 M 2M M \ b2d2 -4/ The collected equations of interest for Bdb rad/sec are the following: (bd)2 = (Bdb)2 4- (sd2)2 (4.1) The video bandwidth, 1, is found from (3.5) to be. 44 22 (Bdb)2 b (4.2) - ~.25bd 4 +bbd (Bdb For s = 0, the IF bandwidth, b, is determined by (bd)2 = (Bdb)2 4 (4.3) and the video bandwidth $, is 1/2 B.,= (. b. (4.4) (Bdb)2 - 8 If the receiver is used for bdB >> 4, then (Bdb) radians/sec (4.6) d and = b = - radians/sec (4.7) Bdb d when the sweep rate is increased so that no video filter is necessary, equation 1/2 (3.10) yields d /(d) +, (4.5)

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN nd f or Bdb > 4, 2(Bdb) b = radians/sec (4.8) ummarizing, equations (4.1) through (4.5) apply whenever (bd)4 > 4(4 + s2d4) (4.9) o that P is real, and for large effective bandwidth, the same signal-to-noise an be obtained by adjustment of sweep rate, IF and video bandwidths from slow sweep bd = (Bdb) and Ad =V8 o bd = 2(Bdb) and wide open video6 ote that the fast-scanning narrow band and the slow-scanning or non-scanning roadband receivers have nearly equal signal-to-noise ratio only if the optimum ideo filter is used. If the video filter is not matched, the fast-scanning arrow-band receiver is superior. Because the video filter is so important when a slow scanning broadband eceiver is used to cover a fixed band, the equation for mismatched video bandidth is determined from the general signal-to-noise ratio Eq 3.2. I/2 2 J/2'(..Nh (No0) -45 4; 7 7 />? 1( > )2 [2 +1] 2+ P opt) b / + he last factor in (4.10), the video mismatch factor, has a maximum value of one t P = Popt, and depends on the parameter (Popt/b), the ratio of optimum video D IF bandwidth. In Fig. 7 the optimum video bandwidth, Boptd, necessary to zhieve an apparent bandwidth Bdb is plotted as a function of the IF bandwidthlbd. ie video mismatch factor is plotted in Fig. 8. From Fig. 8 it is evident that b is better to have the video too wide than too narrow, and no video at all 3 -co) is less than a decibel below optimum if the optimum video is as wide as 17

GG-IZ.-; R13P tf 1-f9l-toV Z9Z so 60 T T I 40 - (PTb) 30 _ 20 /' 80 Ca., | I Bd. A <|]|/\ / 82 C _ T dbi l 24 z:Jzz IL: Izi r 0 40 1.0 2 3 4 6 8 10 20 30 40 60 8C' —'bd FIG. 7. OPTIMUM VIDEO BANDWIDTH VS. I. BANDWIDTH. APPARENT BANDWIDTH AS A PARAMETER. 18

0 0.1 _- I 8.0.0 1 / 1 0.1 1N A 10 100 -1.01 0 () 10 100 -18PT FIG. 8. SIGNAL-NOISE RATIO VS. VIDEO MISMATCH RATIO, OPTIMUM VIDEO TO I. F BANDWIDTH AS A PARAMETER.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN the IF. However, a computation of the sweep rate shows that sd2 is only slighti; less than Bdb for Bdb > 10. At the other extreme, if sd2 is required to be less than one, then it can be shown with a little juggling that the optimum-video-to-IF ratio will be slightly greater than for Bdb > 10, and the video match is important. Practical considerations may favor either the fast-scanning wide video or the non-scanning matched video type receiver. No, the noise power per unit bandwidth at the input of the receiver, was considered constant in the analysis made in this report. This quantity No includes both the noise coming into the receiver and the noise produced in the receiver itself, and thus includes, for example, local oscillator noise and the noise produced in the IF amplifier. As bandwidth and sweep-rate are varied, the noise per unit bandwidth will not remair constant. Broadband IF amplifiers generally have higher noise figures than narrow-band amplifiers. It may not be possible to build a fast-scanning local oscillator with low noise level in its output. These factors must be taken into consideration, and thus the optimum sweep rate IF bandwidth and video bandwidth for a receiver is certainly a function of the state of the art,among other things. For 1opt5 = 1 b d2 = Bdb- __2. Bdb (Bdb)2 so that Bdb - 2<sd2 < Bdb - 20

-ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APPENDIX A Fourier transforms are commonly used in several different forms, and with each form the constant multipliers are different. The form used in this report is F(w) = f(t)e dt (A.1) rhere F(o) is the Fourier transform of f(t). The pertinent formulas are collected Ln this appendix for reference. With this form of Fourier transform, the inverse transform is o jcwt f(t) =i fF(o)e d. (A.2) Parseval's theorem takes the form jf(t) fmdt = F() w d' (A -Co -00 Lnd it can be shown that if H(co) is the transform of h(t), 1 C 1 j H(wo)F(0)e da=. ( X)h(t - )d. (A.4) _ - — _o Thus if a filter has impulse response h(t), the response to a signal f(t).s the transform of H(u)F(w) multiplied byv.l The energy spectrum of a signal f(t) is P(w) = F(w) j-)' (A.5) Lnd the total energy is f0 ft ( P(c)dC (A)6) f(t) f(t)= | F()F) dm= f P(~)d~. (A.6) -cO -0CO f the signal'(t) is passed thrpugh a filter with impulse response h(t), or transer function E(c), the spectrum at the output is 21

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN PO(D) -= [Vx H()F(co)][ ^/H(cD)F(w) 2AH(Y) W7 F(c)F(To P (o) = 2xH(W)H(m)Pin(c) (A.7) Oin where Pi (X) is the spectrum of the input signal. Equations (A.6) and (A.7) holdE talso if PO () and P in() are interpreted as the power spectrum of noise. The autocorrelation function p( T ) for stationary noise is proportional to the transform of the spectrum.. That is, p() K7 -2 J P(cm)e dc, (A.8) where P(w) is the noise spectrum. The value of K depends upon the choice of the form of the Fourier transform. The constant can be determined from the fact that p(O) must be the total noise power, and hence C0 p(O) = S P(w)d It follows that K = / 2 and the auto correlation is the transform of the spectrum, multiplied by iJ2, i.e., -c0 22

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APPENDIX B In this appendix the expression for signal-to-noise ratio in a panoramic eceiver is derived. The receiver model is shown in block. diagrat form in igure B.1. IF with Square Gaussian Input Ideal Gaussian Law Video Output Mixer Passband Detector Amplifier Scanning Local Oscillator Fig. B.1 BLOCK DIAGRAM OF PANORAMIC RECEIVER "he signals assumed are pulses with Gaussian shaped envelopes. At the output of'he mixer they have linearly varying frequency, and at this point are assumed to Lave the form 2 (t-e)2 f(t) - exp [J( t -t t) -2 (B.1),he constant factor was chosen to make the signhal have energy E, i.e.,, cD E = / f(t) ftdt, (B.2),he noise is assumed to be white stationary Gaussian noise with a power of No watts er cycle of bandwidth. The transfer function of the IF is assumed to be (to) = exp [-, (B.),nd the transfer function of the video filter is G(oO = 1Q exp [- (2] (Bi) 23

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN First the maximum signal power is found with the aid of The Response of, Panoramic Receiver to CW and Pulse Signals. Then the noise power is calculated, and finally the expression is given for the ratio. The signal function and the transfer function of the IF amplifier are the same except for a constant factor as those assumed in The Response of a Panoramic Receiver to CW and Pulse Signalsl, and hence the expression for the envelope of ti signal at the output of the IF amplifier can be taken directly from that report. Jg(t)j = A I 1 s(t _ t)t ] 2. l (c ) 3 (.) A -.. + b +; B A0 o |( +mb2 + i +s2] 1/4 B = -4 +b2+ 2.d2 (B.7) t = c d, (B.9) + b + s2d2 b d2 s2 g(t) = E 2 p B2 s(t(t) 2B. o. o i., bia2 b. op. cit., page 1.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN This is a Gaussian pulse, and the video filter is Gaussian also; thereore the formulas (B.5) through (B.9) can be applied again. This time the sweep ate, s, is zero, the center frequency, a, is zero, the filter bandwidth is P, he center time of the pulse, c, is tm, and the pulse width d is s here is also the constant factor E A2 exp 2 sb (B.1) o be carried along. Only the maximum response is required, and that is the facor, (B.11), times Ao evaluated for these special values of sweep rate, filter andwidth, and pulse width. From the expression (B.6) for Ao we obtain 2E 2 2 SCl(.. S I d A0 exp ( b/8s (B.12) b2W2 he peak signal power is the square of this voltage. The noise spectrum at the input is assumed to be constant with a power f No watts per cycle. If this is thought of as spread over all positive and egative frequencies (as is convenient in working with Fourier transforms), then ne-half of the power should be associated with the negative frequencies and onealf with the positive frequencies. Also, since there are 2n radians per cycle, he density per cycle must be divided by 2: to give power density per radian. ence the power density at the IF input is P (D) = N No (B.13) in 2 2" N o - -~ y equation (A.7), the spectru at the output of the IF filter is 25

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN No ex2p1r 2 -a,2 2% H(&) F Pi (OD) = 0 exp b By equation (A.9), the autocorrelation function of the noise is/ 2S tin the Fourier transform of the noise spectrum, and hence the autocorrelation at the input to the detector is / times the transform of (B.14), or ri (tr) Nb exp [ar b22 ] (B.15) rin (r) A formula for the autocorrelation at the output of a square law detector is presented in Threshold Signals1. In that book, the noise is represented in th form n(t) = x(t) cos wt + y(t) sin It (B.16) where n(t) is the noise voltage, w the center frequency of the noise spectrum, an x(t) and y(t) are assumed to be uncorrelated and to have the same autocorrelation function p(t )'. The autocorrelation function of this noise n(t) is r(- ) = average {n(t) n(t +r)} = average {{x(t) cos cDT + y(t) sin CT] [x(t + r ) cos u(t + r )+ y(t + r) sin w(t +r)]} = average { x(t) x (t +r) cos ot cos o (t+T) + x(t)y(t +r) cos wot sin CD(t +r) + y(t)x(t+ T) sin wt cos D(t +r) +y(t)y(t + r) sin wct sin co(t + T) (B.17) = Iaverage {x(t)x(t ++r)}] cos Wt cos cW(t +r) + + [average {y(t)y(t +r)}]l sin cot sin wa(t +rT) 1 J. A. Lawson, G. E. Uhlenbeck, "Threshold Signals," McGraw-Hill,1950. 26

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN t this point the other terms drop out because x(t) and y(t) are uncorrelated. he expressions in brackets in (B.16) are the autocorrelation of x(t) and of y(t),.e., p(r). Thus r(T ) = p(r ) [ cos t cosc(t +r) + sin at sin X (t + r) = p(r) cos cT (B.18) hus p(r ) is the envelope of r(t ). Since r(r ) is given in complex form, the hvelope is the absolute value, Ir( T ). The autocorrelation function at the output of the detector as given by!wson and Uhlenbeck is rout [ ()]. (B.19) rout() 4N2 + 4 i[r n (T)]2 out ( [( ie first term in this expression is the dc component of the output, and hence may * dropped. Thus by equation (A.9),the noise spectrum is the Fourier transform of Le seeond term divided byJ2, or N() L f 4 ri (-) 2exp[ -cor] dr 2it in N2b2 =n 2]=o0 _ exp - - JDr dr 16n2 b2 exp 2 = v exp [ ]b2 (B.2) (A.7), the spectrum at the output of the video filter is obtained by multiply27

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN ing (B.20) by 2% G(w)G(c), where G(w) is the transfer function of the video filt, The noise power output of the receiver is the integral of the noise spectrum: co 8 F L b2j b +exp ex (B22) N N2 b2 E _W 2 1+ 2F 2 N 82 _ exp (B.21) b2 8 s = E2 16A0o W24 SC'.N ep (B.22) o. 2 2 b2 2W22 by taking the limit as the video bandwidth P becomes infinite. If the numerator and denominator of (B.22) are divided by,p2, the equation takes the form 2b2 02 2 S E2,ep. 4 soC (B.23) o d2.2 and as P approaches infinity, the limiting signal-to-noise ratio is video filter 2 2 — No8 b dkB.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN abs'~E2 16A04w2 J SC 21 + 2b 2 2SS\ E ~6~4w -B 2 /82 + b2JW2 0 = 1/2 P + 2b P [ (4p3 + 4b2p)(8S2 + b2W2)2 L (882 + b2P2W2 )2 (-4 2b+ 2)2(8s2 + b2 2)(2b2pW2) 0o (p2 + b2)(8s2 + b2p2W2).(P4 + 2b2 2)(b2W2) 8 2 +8.bp4W2 + 8b2s2 + b4 2W2 _4b2W2 2b4 0 = 8sb2 + B.b 25 P2 882 his can also be written in the following form: P = b (B.26) /b -- 8s2 The signal-to-noise ratio with the optimum filter can be found by subtituting the expression (B.26) for P in equation (B.22). It is somewhat simpler o substitute for 1/2 from (B.25) into (B.23). 29

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN b4W2 r_ E 16A. * e xp - - --- A2 882 / No b 4W2 /1 +W -b b4W2- 1.2 4ex{2 J 4 sc B 48 } d2 8s2 b4W2 ) -1 b) b24 4s2 By equation (3.11) of Technical Report No. 3, b A2 B (B.28) Substituting for W in (B.27) and (B.26) yields ()i,E2 2 exp ( _ ) (B.29) a m optit bmum b -1 (B3o) a~s 1 op. cit., p.l. 30

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN From equations (3.9) and (3.10) of Technical Report No. 3, d2 = 4 + 1 + 2d (32) b2d2 b2 b_ 2 4B + 4 + b(B~33) bd 22 2( 2 4 bdd 4 + 22 A4 0 is expression can be used to eliminate A from (B.24), (B.29) and (B.30). 0 Lus (ES = B2 _ 16 exp (B.34) bd (S )nE2 4 exp{ SC (B} 35) video 22 N2 + bd B~ flter + + b d dBV opt 4 + b2d(B36),, _. 1d1 L4 4 1Op. Cit. P. 1 31

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