Comments on "Optimum Bandwidth of a Low-Pass Filterfor Detection of a Pulse in Nonstationary Noise"'

Abstract

In a recent correspondence1 a calculation of the optimum

bandwidth of a low-pass RC filter for the detection of a pulse signal

in nonstationary noise was presented. The purpose of this

correspondence is: 1) to point out additional references to the workwhich has been conducted in the stationary noise case, and 2) to

present an interesting alternate derivation of the expected output

noise power for the nonstationary noise case.

px(to) = E[x(to)]

VW(t,T) = Q(t)6(t - t)

Vx(to) = E [X(to)XT(to)]

it can be shown5 that the time-varying variance of x(t) canbe determined from the differential equation

Vx (t) = FVx(t) + VxF'T + BQ(t)BT (2)

In his recent correspondence, Mason' initiates hisdiscussion with a reference to the work reported bySchwartz2 for the stationary noise case. A detaileddiscussion of the RC low-pass filter analyzed by Schwartzcan be found in a thesis by Fine,3 who investigated therelative performance of the RC low-pass filter and anideally matched filter for the detection of a rectangularpulse in stationary white Gaussian noise. The investigationmade by Fine was extended by Carlock,4 who considered alarger class of filters and several signal pulse shapes. Thefilters considered were the 1-stage RC filter, the 2-stage RCfilter, the choke-input filter, the bridges-T filter, the2nd-order Butterworth filter, the 2nd-order Chebyshevfilter, and the 2nd-order catenary filter. The pulse shapesconsidered were rectangular, triangular, trapezoidal, andsine squared. The peak normalized signal-to-noise powerratio was computed for each combination of filter andinput pulse shape and the results were presented graphicallyas a function of the bandwidth-pulselength product.

Mason continues his presentation with the derivation ofthe equation for the time-varying expected power of thenonstationary noise using a two-dimensional powerspectrum equation. An interesting alternate derivation canbe made by solving a simple differential equation describingthe propagation of noise variance through a linear system.

For a dynamic system described by the state variableequation

x =Fx +Bw (1)

where Q(t) is time varying when w(t) is a nonstationaryrandom process.

For an RC low-pass filter, the Laplace transfer functioncan be written

X(s) = H(s) = +W(S) s c

c RC.

The corresponding state variable equation can be written

x(t) + WCx(t) = (OcW(t)

(3)where

or

x(t) =- ccx(t) + (cw(t) (4)

where for this problem w(t) is the white Gaussian randomnoise process with variance

VW(t,r) = Q(t)6(t -t)

and x(t) is the output noise process from the RC low-passfilter. Associating the coefficients of (4) with those in (1),we see that

F=-c

B = coc.

where the prior mean and variance of w(t) and x(t) areassumed to be known as

iw(t)=E[w(t)] =0

Manuscript received March 1, 1971.

lC. A. Mason, IEEE Trans. Aerospace and Electronic Systems,vol. AES-7, pp. 213-216, January 1971.

2M. Schwartz, Information Transmission, Modulation and Noise.New York: McGraw-Hill, 1959, pp. 287-291.

3A. M. Fine, "Optimum filters for pulsed signals in noise,"M.E.E. thesis, Polytechnic Institute of Brooklyn, Brooklyn, N.Y.,1957.

4G. W. Carlock, "Approximating the matched filter for pulsesignals in noise," M.S.E.E. thesis, University of Texas at Arlington,1968.

Thus, the time-varying variance equation (2) can be written

x(t) = -2cocVx(t) + WCc2Q(t) (5)and has the solution

Vx(t) = e-2woctVx(O) + f Q2wc(t- )c'Q(T)dT. (6)

For the nonstationary noise considered by Mason,

Q(t) =.e-2at for all t> O .2

5A. P. Sage and J. L. Melsa, Estimation Theory with Applicationsto Communications and Control. New York: McGraw-Hill, 1971, p.52.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS JULY 1971710

Thus, (6) becomes (assuming Vx(O) = 0)

Jt

Vx(t) = e-2wc(t - T) WJC2 T e -2a'rdro~~~~~~

=Cwc2 Te-2wtf e2(0c- a)-rddrX2T

= -- - [ e- 2at - e- 2wct (7)

Equation (7) corresponds to the results obtained byMason' in his equation (12).

GAYLORD W. CARLOCKConvair Aerospace Div.General Dynamics Corp.Fort Worth, Tex. 76101

Radar Performance on Targets With Range Acceleration

AbstractRange acceleration introduces a coupling between range, Doppler,and range acceleration, with potentially severe consequences on

radar performance. The following analysis of the effect shows that it

is serious only when the radar uses a constant-carrier pulse or a

waveform with small time-bandwidth product.

1. Introduction

As the duration of a radar signal is increased, a capabilityin range rate resolution is obtained, then in rangeacceleration, and so forth [1]. Although the theory forrange rate resolution is well developed, this is not so forrange acceleration, which introduces a coupling between theparameters [1], [2]. In the following we determine thepractical significance of the effect for matched filter orcorrelation radar.

Since measurement and resolution performance of aradar depend on the form of the receiver response to thetarget return, the best method of arriving at anunderstanding of the radar capability in range, Doppler, andrange acceleration is to study the receiver response to apoint target. This response is obtained by expanding theround-trip delay of the signal into a series, thus introducingthe parameters of Doppler, range acceleration, and so forth.In the many applications where the time-bandwidthproduct of the waveform does not exceed about 10 percentof the ratio of speed of light and target range rate, themotion distortions of the complex modulation function canbe neglected, and the complex modulation function can be

assumed simply delayed. The complex envelope of thereceiver response then has the form [1]

Xo(,r,zy yf) = j j(t)p*(t -i-) exp [ j27r(r' + 2-yf-r)t]

. exp (j2ryt2)dt. (1)

Here we have introduced the mismatch between theparameters of the received signal and those of the filter,Tr-ro -Tf, P-P0 -Pf, and y- yo -3yf. The behavior ofthis response will be analyzed subsequently.

11. Analysis of the Coupling Effect

Since the only parameter associated with t2 in (1) is therange acceleration mismatch y, only a filter matched inrange acceleration will give a peak response. This meansthat there is no problem in measuring or resolving in rangeacceleration (only its effect on the Doppler measurement).Hence, it is only necessary to study the response for a filtermatched in range acceleration. With y = 0 in (1), thisresponse becomes

Xo (r-,O,°'yf) = , S(Ot)g*(t - r)00

- exp [ j27r(v + 2yfyr)t] dt. (2)

The form of the exponenetial shows that a change inm Is canbe offset by an appropriate change in 2,yf-. The parametersz', yf, and X thus are coupled in the sense that different setscan produce identical forms of the exponential; i.e., theyare ambiguously coupled.

For an analysis of the coupling effect, we note that inthe absence of the term 2yf-r, the response of (2) would bethe point-target response in delay and Doppler associatedwith the waveform ,u(t), commonly called the ambiguityfunction. The value that the ambiguity function has on thedelay axis, v = 0, will now simply appear on the line v +2zfr = 0, or v = -2fr. Excluding the linear FM signal forthe moment, we can represent the main lobe of theambiguity function of any waveform by an ellipse whoseaxes are oriented along the coordinate axes, and whosewidth along the r and z axes is on the order of 1/B and lIT,respectively [1 ] . The parameter yf shears the ellipse off ther axis, as indicated in Fig. 1.

The dimensions of the sheared ellipse can be obtained asfollows. Its width along the Doppler axis remainsunchanged at 1T, as can be verified by observing thatletting X = 0 in (2) eliminates the effects of yf. Todetermine the width along the delay axis, we calculate theambiguity function

X(7,v) = f ,i (t)MuS* (t - r) exp (q27Twt)dt-10Manuscript received October 1, 1970; revised March 1, 1971.

CORRESPONDENCE 711