radio-frequency interference among linear-fm radars

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-21, NO. 3, AUGUST 1979

Radio-Frequency Interference Among Linear-FM RadarsJ. J. GERALD McCUE, FELLOW, IEEE

Abstract-The problem of radio-frequency interference (RFI) be-tween radars using linear-FM pulses is examined. For a very large classof cases, the RFI is the same as if the FM were removed; for the casesnot in that category, it is shown that the peak response of an unweightedreceiver can be closely approximated by a hand calculation. The effectsof cosine-on-a-pedestal weighting, in either the frequency or the .timedomain, are then considered. Exact expressions for the RFI are de-veloped, and it is shown that, if one is satisfied with knowing the peakof the response, the effect of the weighting can be well approximatedby some quite simple expressions.

Key Words-Linear-FM radars, weighted, unweighted, frequencydomain, time domain, mutual RF interference, exact expressions, ap-proximations.

NOMENCLATUREa Ratio of FM rate of interfering pulse to FM rate of

receiver.b FM rate; b = tB/T.B FM bandwidth.C(x) The Fresnel cosine integral.E(t) Unit-amplitude pulse waveform.f A frequency; sometimes, in particular, the center

frequency of the receiving filter.fo Center frequency of transmitted pulse.F(x) The complex Fresnel integral.N Normalizing factor for receiving filter.p (fo - f)/B, the same as v/B, a normalized measure

of frequency difference.P Duration of transmitted pulse.S(x) The Fresnel sine integral.t Time.T I)ration of pulse, or of transmitted pulse to which

receiver is matched.v Center frequency of pulse minus center frequency

of receiver: v = fo -f.X1, X2 Variables defined by (22).y(t) Response of a (noiseless) receiver.YF(t) Response with weighting in the frequency domain.YT(t) Response with weighting in the time domain.Y(t) Envelope of y(t).Z1, Z2 Variables defined by (3).(f Complex spectrum (Fourier transform) of E(t).e ) 12 Energy spectrum of E(t)

v Frequency offset f- fo, where f is any frequency.

Manuscript received October 2, 1978; revised April 16, 1979. Thiswork was sponsored by the Department of the Army. The views andconclusions contained in this document are those of the contractor andshould not be intrepreted as necessarily representing the official policies,either expressed or implied, of the United States Government.

The author is with Lincoln Laboratory, Massachusetts Institute ofTechnology, Lexington, MA 02173. (617) 862-5500.

pa

1T

A weighting parameter; see (25) and (30).1 -p.Used in two senses: 1) the variable of integration ina convolution; 2) tIT, a normalized measure of time.Phase spectrum of pulse.

I. INTRODUCTIONTHE proliferation of radars that use wideband linear-FM

pulses is increasing the demand for calculations of radio-frequency interference (RFI) between such radars. Methodsof approximate calculation have been given by several authors[1] -[7]. Some useful things remain to be said. The presentpaper reviews the basic relations, gives some closely approxi-mate methods of calculating RFI, and then considers the ef-fects of weighting in the receiver, developing expressions forthe power response of a receiver with cosine-on-a-pedestalweighting (including Hamming and cosine-squared) in thefrequency or the time domain, when the received pulse may bemismatched in center frequency, bandwidth, duration, and FMrate; a computer program embodying these expressions isavailable [8], but there are rapid approximations adequate formost needs.

II. THE ENERGY SPECTRUMBasic to interference is the spectrum of the radiation. The

real part of

E(t) = rectT exp [iir(2fot + bt2)] (1)

represents a pulse starting at -T/2, having a rectangular enve-lope, unit amplitude, frequency fo + bt, duration T, and band-width b T. Its energy spectrum is 1 e (f) 2, where if) isthe Fourier transform of E(t):

JT/2cE(f) =

-T/2exp {iir [2(fo-f)t + bt2] } dt. (2)

Introducing z such that the expression in square bracketsequals (z2/2) - (fo - f)2/b, one finds that, for b > 0,

1 z2e(f) = exp [-irv2/b] exp (hrz2/2) dz

where v = f - fo is the offset from the center frequency and,

0018-9375/79/0800-0228$00.75 i 1979 IEEE

228

McCUE: RFI AMONG LINEAR-FM RADARS

with ib T=B,

(3)

The function F(a) = foc exp (iirz2/2)dz is the complexFresnel integral, with real and imaginary parts C(ce) and S(a).From the definition, it is an odd function of a, and F (ia) =iF4*(a), where * denotes "conjugate;" consequently,

6(f)= exp [TiIrp2T/B]28

*{C(Z2)-C(Zl)± i[S(Z2)--S(Z1)]}

-1.0 -0.8 -0.6 -0.4 -0.2

(4)

is valid for all b # 0, provided the upper signs are used whenb > 0 (frequency rising during the pulse) and the lower whenb <0 (frequency dropping). Thus the energy spectrum is

TI e(f) 12 = {[C(Z2)-C(Z1) 2 + [S(z2)-S(zl)] 2}. (5)

2B

Because changing the sign of I in (3) merely interchanges z1and-Z2, the energy spectrum is symmetric about the centerfrequency fo. Equation (5) has been known for many years[9], [10], but its interpretation by means of Cornu's spiral isinsufficiently mentioned. The spiral [11]-[13], Fig. 1, is aplot of S(a) against (c(), with a marked off along the curve;in the third quadrant, et is negative. QZ2) -C(zl), which ap-pears in (5), is the difference of abscissas on the spiral, andS(Z2) - S (z1) is the difference of ordinates. Therefore, e(f) Iis represented by a straight line joining Z2 to z1. The arclength Z2-Zl is for us a constant, namely V2T; it can bethought of as a thin tubing that slides along the spiral as wechange f. The reader can verify that when f = fo, so thatv = 0, the center of the tubing is at the origin, whereas whenf is at one limit of the nominal frequency sweep, so thats= ±B/2, an end of the tubing is at the origin. He can go on to

visualize the way the variables combine to give the spectrumits familiar form, exemplified in Fig. 2: a central plateau withundulations that are small when TB is large, a 6-dB width B,and a height TIB.

When there is no FM, b = 0, and the Fourier transform (2)leads merely to

sin (rvT)e(f) = T irvT

C (z)

I-1-.0Fig. 1. Cornu's spiral.

m

NI

wa.

wzw

FREQUENCY OFFSET v (MHz)Fig. 2. Energy spectrum of a rectangular 5-,us 10-MHz linear-FM

pulse. It is symmetrical about the vertical axis. The simple line ispart of the envelope of the spectrum of a constant-frequency pulsewith the same center frequency.

(6)

The peak of the energy spectrum in this case is at T2, in con-trast with TIB for the general level of the plateau when b = 0;FM lowers the spectral peak by a factor 1/TB.A valuable relation between these two classes of spectrums

is found by using the asymptotic expansions [141,I10 1.3---.(4m -1)

h(z) = Xz + F, ( 1) 7Z1r22lIfZ m=1 7rZ(7TZ2)2m

1.3 ...(4m + 1)g(Z) = ° + F, ( 1)m XZ1Z22

in the relations [14]

C(z)=-1 + h(z) sin (_Z2 )-g(z)Cos ( 2Z2)

S(z) = 1--h(z) cos (_z2 -g(z) sin (_z2)

(8a)

(8b)

(7a)In (7), the error caused by truncation is less than the first termthat is dropped, and for the skirt region of a chirp spectrum,

(7b) much good can be done with (7) by dropping the whole series.Suppose that 1v I 1.5 B, and let TB = 32. Then I Z2 I 8,

229

p IZ1, 2 .-- N/2-T-B +

-B 2-


and 1z1 is larger; g(z) < 10-4, and simplifying h(z) to I/lrzintroduces an error of less than 25 parts per million. In excel-lent approximation, therefore,

C(z) = -+-sin _Z2)irz 2 /

1S(z) = --frz

Cos Z2)

for TB > 30 and P> 1.5 B. Using these forms in (4) yields

1 e+i7TB/4e(f) - [2v sin irTv ± iB cos irTv]

27T v2 -B2/4

from which follows

sin2 (irTv) + (B2/4V2) cos2 (rrTv)

(irTV)2(1-B2/4V2 )2

(9a)

sionally correct, which is important when comparing the ef-fects of a given pulse on different receivers; the chosen factorgives the filter unity gain near the center of the passband.1

From (1) and (12), the convolution

r

y(t) = E(r)h(t -- r) dr(00

9b) giving the response of the second radar to a pulse from thefirst is [15], [16]

y(t) =Tj exp [i2r(fiot + bT2/2)]

(10)

(11)

regardless of the sign of b. Whenever 2Tv is an odd integer, thisapproximation has a local maximum. The denominator showsthat when v > 1.5 B, the values near these maximums aregiven with an error of less than 1 dB by (6). That is, whenTB > 30 and v I> 1.5 B, the envelope of the energy spec-trum of a rectangular linear-FM pulse is within 1 dB of theenvelope of the energy spectrum of a rectangular pulse withthe same duration and no FM.

We have completed the first step in showing that for a verylarge class of linear-FM interference problems, the FM can beignored. The second step, an examination of the phase spec-trum, is taken in Section IV, after discussing the effect of fre-quency offset.

III. EFFECT OF A FREQUENCY OFFSETIt is useful to consider an idealization of interference, as-

suming a pulse envelope that is rectangular and disregardingany spurious transmissions ("hash"). Rectangular pulses arelikely to be good models of real pulses, because high-powerradar transmitters commonly operate in saturation. Trape-zoidal pulses have been treated by Newhouse [4]. We assumethat the receiver is linear, so that the pulse interference, thereceiver noise, and any spurious transmissions there may be,can be considered separately.

In the simplest case of interference between two linear-FMradars, their waveforms would be alike, except for a differencein center frequency. This case is of practical concern, becauseit permits a cluster of radars (e.g., for missile defense) to useidentical software. Consider a receiver matched to a waveformlike (1), except with center frequencyf Its matched filter hasthe impulse response

h(t) = B rect ( T ) exp [itn(2ft -bt2)] (12)

where the normalizing factor s,B77 makes the equation dimen-

* exp {i2rr[f(t-r)-b(t-r)2/21} dr (13)where 'rl is the greater of -T/2 and t -T/2, and r2 is thelesser of t + T/2 and T/2. A delay in the filter has been disre-garded. There follow, for t < T,

sin [7rTB(I -IT I )(p±)] exp [ir(fo +f)t] (14)

y(t) 12-=TB( ±- exp [ ))

sin [irTB(p ± r)(I- I )1 12

7rTB(p±T)(l -IT J)(14b)

where p = (fo -f)/B is the frequency offset in bandwidthsand r, which is distinct from the variable of integration in(13), is tIT, so that it runs from -1 to +1; the upper andlower signs are for b > 0 and b < 0, respectively.

Instead of treating the frequency offset p as a parameter,we can view y as a function of two variables, r and p, renor-malize it to unity at the origin, and call it X (r, p). ThenI X (r, p) 12, or sometimes just I X (r, p) l, is called the ambi-guity function of the linear-FM signal. For p < 1, it has beendiscussed extensively [15], [16], because it gives the responseof a linear-FM receiver to a pulse to which it is matched, ex-cept for a Doppler effect that can be represented as a constantshift in frequency [17]

Inspection of (14b) shows that, for Ip < 1, there is a timer = p or T = -p at which the denominator vanishes and thefactor in brackets is unity. If p l, and consequently this valueof T, is small, the maximum of Iy(t) 12 occurs close to thistime. The influence of the frequency offset on the time ofpeak response is the well-known "range-Doppler coupling;" inan RFI context, we are interested in the height of the peakrather than its time of occurrence, and our interest is notlimited to Ip I< 1. The height of the peak can readily befound from (14b) when its time of occurrence T' is known.That can be estimated as follows. To simplify the discussion,

1 The transfer function H of a filter that matches a spectmm e(f) isNE*(f), where N is arbitrary in magnitude and should be a frequency,in order to make the transfer function dimensionless. The conditionI H(fo) = 1 implies N-1 = e(fo) 1. From (6), this N is T-1 whenb = 0. When b s 0, (5) applies; at v = 0, the factor in braces is close to2, and e(fo) - ITB, so that H(fo) I is near unity ifN = rBi7

230

e(f) 12 - T2


TIME-TO-DURATION RATIO (T)(a)

-0.5 0 0.5

TIME-TO-DURATION RATIO (r)(c)

C',

U)

Li'

C,)zI0a.

U)

Li'

TIME-TO-DURATION RATIO (r)(b)

1.0-10 -0.5 0 Q5

TIME-TO-DURATION RATIO (T)(d)

Fig. 3. Response of a linear-FM receiver for TB = 100 to a pulse that matches it, except for a frequency offset. In all

graphs of frequency response in this article, the receiver is for an upward-sweeping pulse and, except when the con-

trary is stated, the pulse to which it is responding is upward-sweeping. (a) No frequency offset; the pulse matchesthe receiver. (b) The frequency offset p, in bandwidths, is 0.60. (c) Like (b), except that p = 1.0. (d) Like (b), exceptthat p = 2.5.

consider an upward sweep and a positive frequency offset.The locus of maximum Iy 12 is found by equating the time

derivative of y to zero. Let 7rTB(p + rXl + r) = x. Then,forthe cases chosen and p + r 0, the condition for vanishingderivative is

tan x = x + (p + ,)2.

Replacing tan x by the first three terms of its power series,which is legitimate ifx < rr/2, and consequently if 2TB(p +T)-(1 + r) < 1, gives the approximate locus of r'as some root of

(irTB)2(p+r)(1 +T)3 2(rTB)4(p+T)3(1 +±T)53 15

By inspection of (15), one sees that for p > 0, T' is negativeand that p + T is small and positive. With this guidance, one

can use a trial value of X and see how nearly (15) is satisfied forthe given p. By this means, 'r' can be located with sufficientprecision in two or three tries; using it in (14) gives IyIeak2in good approximation. An alternative procedure is simply to

plot a few evaluations of (14) for Xr near -p.The reason these procedures work, for Ip I < 1, but

fail when p is appreciably larger, is evident in Fig. 3. For

mLiw

C,)z0a.C/)

:

:

Li.

ULi

231


a~-30-

-40-

~. -50-_ \ T~~~~~~~~~~~B= 100

-60-

-70TB= 1000

-80 _

-90-o0 3 4

FREQUENCY OFFSET p (bandwidths)

Fig. 4. Normalized peak response Y Ipeakj2/TB as a function ofnormalized frequency offset p, for the cases TB = 100 and TB=1000.

Ip 1< 1, Iy 12 varies slowly near its peak; for p - 2, y 12fluctuates rapidly, as in Fig. 3 (d). For these larger values ofp, such that the frequency offset appreciably exceeds thebandwidth, one can replace (14) by the envelope equation

1I y(r) 12 =

TBrr2(p + T,)2I|T I< l. (16)

The pulse envelope, exemplified in Fig. 3(d), is a sloping mesawith its maximum near T = ± 1, whichever is opposite in signto p. This method estimates ly Imax2 to within 0.1 dB downto about p = 1.5 when TB = 100, and to even smaller p whenTB is larger. For p slightly in excess of unity, it is wise tocalculate ly lpeak2 by each method in the range of its validity,and then to interpolate graphically. Fig. 4 was obtainedby precise calculation of the waveform defined by (14b),using a large computer. Note from (14) that for Ip < 1,ly lpeak2/TB is nearly independent of TB, and that forp 1> 1, IjY lpeak2/TB varies like (TB)2.An implication of Fig. 3 (d) is that, if two signals have the

same FM rate, a receiver that will compress one of them maynot compress the other. The explanation is found in the phasespectrum.

IV. PHASE CONSIDERATIONS

From (4), the phase spectrum of a rectangular pulse withlinear FM is

7rTv2 S(Z2 )-S(Z1)~p(f) = ± arctLan (7B C(Z2)-C(z1) (17)

The lower signs are to be used when the FM sweeps downward(b < 0). The arctangent is not to be limited to its principalvalue, because q5 needs to vary through a range of 2fr in orderto represent all differences in phase between two complexFresnel integrals.

If TB is large and v < B/2, then z1 and Z2 are near theeyes of the Cornu spiral, Fig. 1, and the argument of the arc-tangent, which is the slope of the line from z1 to Z2, is neverfar from unity. The arctangent is nearly constant, and thephase departs little from a quadratic dependence on frequency.

V0

I

w

a.a '

nr

FREQUENCY OFFSET v (MHz)(a)

200.o 200.2 200.4FREQUENCY OFFSET v (MHz)

(b)Fig. 5. The phase spectrum of a lO-js 100-MHz upward-sweeping

linear-FM pulse in a 0.4-MHz band 2 bandwidths above the centerfrequency of the pulse, as given by (a) the quadratic term in (17)and (b) all of (17).

But if we let v, the departure from center frequency, approachand then pass the edge of the nominal band, the line joiningz1 to Z2 wobbles increasingly and then begins to revolve;the behavior of the arctangent term destroys the nearlyquadratic relationship. The in-band behavior of the phase hasbeen amply described [16], but the out-of-band behavior hasreceived little attention. It can be calculated from (17) or fromthe approximation

7r BPf±f)_- TB ± arctan - cot (irTv)

4 L2v(18)

which emerges when the approximations (9) are used in (4).When TB = 1000, (18) is correct within less than 0.1 percentwhen v > B. Some sample results appear in Figs. 5 and 6.

I-1

232


0

.-

LId()I)4I

D.-cr

C-)ILJU)

FREQUENCY OFFSET v (MHz)

Fig. 6. Like Fig. 5(b), but 10 bandwidths above the center frequencyof the pulse.

They show that, as v increases, the phase spectrum approachesthat of a constant-frequency pulse, which (6) shows to changeabruptly by 7r when v is a multiple of T-1.

The Fourier relation has uniqueness: each transformablefunction has just one transform. If the part of the spectrum inthe passband of a chirp receiver does not have phase that isapproximately quadratic in frequency, the waveform in thereceiver cannot approximate linear FM, and it will not undergocompression.

Section II demonstrated that when TB> 30 and v > 1.5 B,the energy spectrum is nearly the same for a chirp as for a con-stant-frequency pulse with the same duration. Now, it hasbeen demonstrated that phase considerations rule out com-pression if v much exceeds B/2. Consequently, if the nominalfrequency sweep of a pulse lies wholly outside the nominalpassband of the receiver, with a little margin to spare, then,even though the frequency offset may be the only mismatchbetween pulse and receiver, we can calculate the receiver re-sponse by ignoring the FM. The needed margin will be dis-cussed after a method of calculating the response to a con-stant-frequency pulse has been established.

V. GENERAL CASE OF UNWEIGHTED RECEIVERNOT MATCHED TO PULSE

Consider now a pulse that may be mismatched to the re-ceiver, not only in center frequency, but also in bandwidth,duration, and FM rate. As before, let f, b, and T represent thecenter frequency, the FM rate, and the duration of the pulsefor which the receiver is matched, and let fo, ab, and P applyin like manner to the pulse that is actually received; designatethe receiver bandwidth b T by B. Then the extended equiva-lent of (13) is

/()P/2y(t)= T- exp {iri[2for +abr2 I }

/-t* rect t exp {iir[2f(t-r)-b(t-,r)2 }dr.

(19)This time, the quadratic terms in the integrand do not cancel;the consequence is that y (t) has to be expressed, not trigono-metrically, as in (14), but by means of Fresnel integrals. Also,if P and T are different, the convolution needs to be done inthree segments, with some dependence on whether P < T orP> T. The result is

lBy(t) v 2(a- )bT exp [i7r(2ft - bt2)]

* exp [-irr(v+ bt)2/(a-I)bI

IX2* exp (iIrz2/2) dz

X1

when

I y(t) 12 = 2( 1)bT C(X2 -C(X) 2

+ [S(X2)-S(X1)] 2}

(20)

(21)

where X1 and X2 have the following values for the first andlast segments of the response:

/+2TB [a-i]X1 = p±r +-(P/T)

a-I L2

/+2TB aa-iva-i L 2

+2TB aa-1aX1 =

a-I p±arTI2

X2 = ' i +r +-(PIT)

(22a)

-(1 +P/T)I2 r.-I 1 -P/TI /2

(22b)

(22c)

I 1-P/T 1/2 <-r< (1 + P/T)/2.(22d)

233


As before, the upper signs are for an upward-sweeping pulse,and the lower ones apply to a downsweep. Recall that when Xis imaginary, C(iW) = iC(W) and S(iW) = -iS(W). For P # T,there is an inner segment,- II PIT /2 < T < 1-PIT 1/2,for which X1 and X2 are given by (22a) and (22d) whenP< T, but by (22c) and (22b) when P >T.

Define X such that y(r) = X(r, p, a, P/I7) exp [iir(fo + f)T] .Then X 12 or X l, after renormalization to make X(O, 0, 0,0) I = 1, is the cross-ambiguity function of the actual signaland the signal to which the receiver is matched. It has beendiscussed by Cohen [5], who gives some interesting three-dimensional plots. Note the combination (a - I)b, whichappears explicitly in (21) and pervades (22) implicitly, as±(a - 1)B/T. If b $ 0, (a - I)b is the difference in FMrate of pulse and receiver filter. Its importance was recognizedby Cook and Bernfeld [15] and by Cohen [3]; Cook [7]named it the FM mismatch factor. In (21) and (22), the valuesa = 1 and b = 0 are forbidden. For a = 1, the analysis inSection III applies, though it must be elaborated ifP T [3].The case b = 0 is that of a receiver for constant-frequencypulses; to apply (21) and (22) to this case, one needs only asimple reciprocity relation, as follows.

Let Radar j use a waveform E3(t). The impulse response ofits matched filter is hj(t) = NjEj*(-t), where Nj is a normaliz-ing factor; in our case, Ni is oBy/Tj,or Tj-1, depending onwhether the radar uses FM or not. If Radar 1 receives a pulsefrom Radar 2, the receiver response is

Y21(t) = E2(r)ATIEl *(r --t) dr.

By the same token,

Yl 2-t)= f E1 (u -t)N2E2 *(u) du.

Therefore,

Y2 1 (t) = (N1/N2 )Yl2 *(-t).

If Radar 1 is nonFM and Radar 2 is FM, y12(t) is given by(20) with a = 0 and, for an FM pulse in the constant-frequencyradar, we have

z -200.w

cr. W 40 0llF W _ 1 0 S X

-60 T

-1.0 -0.5 0 0.5 1 0TIME-TO-DURATION RATIO (r)

Fig. 7. Response of a receiver for 10-js 10-MHz pulses to a 10-Ms5-MHz pulse with the same center frequency; TB = 100, P = T, a =0.50. p = 0.

pulse equals that of the receiver, as in Fig. 7. That figure andothers that follow were produced by a large digital computerprogrammed for (21). They are offered as useful illustrationsof what happens, but the evaluation of RFI will seldom, ifever, demand such a brute-strength treatment; the computercan usually be replaced by the back of an envelope.

If an appreciable portion of the pulse spectrum lies withinthe passband of the receiver, X1 and X2 can be large in magni-tude and opposite in sign. Fig. 1 shows that, if their absolutevalues are as large as 3, they lie near the two eyes of the spiral,so that the quantity in braces in (21) has a value near 2, andonly fluctuates a little until r changes by enough to makeX1 or X2 move out of the coiled part of the spiral. The re-spohse, therefore, has a plateau; from (21), one sees that thelevel of the plateau is simply a- I -1. Its duration, be-tween 6-dB points, is set by the values of T that lie in theproper time bands and cause vanishing ofX1 and X2.

As an example, consider the response of a 20-,s 10-MHzreceiver to a 10-,s 20-MHz pulse, whose center frequency is5 MHz above that of the receiver. We have T = 20, P = 10,B = 10, a=4, p =-. In -0.75.<rS -0.25, (22) gives

TIY21(t) 12 = -T2 IY12(-t) 12IT B2

(23) 20Xi = [r-1 /4]

20X2 = - [4T + 2].

The remaining case, a constant-frequency pulse interferingin a constant-frequency receiver, lies outside the theme of thisarticle. For the sake of completeness, however, it is covered inthe Appendix.

Returning to (21), one sees that it is similar in form to (5),which describes the energy spectrum of a linear-FM pulse.However, the arguments z of the Fresnel integrals in (5) aresuch that the spectrum is symmetric about v= 0, whereas thereceiver response is seen from (22) to be asymmetric aboutt = 0 unless v = 0; that is, unless the center frequency of the

For r > -0.50 in this interval, X2 > 0 and XA1 <-10/V, sothere is a plateau. The vanishing ofX2 at -r = -0.50, when X1 isnear the lower eye of the spiral (Fig. 1), marks a point 6 dBbelow the plateau. In the central segnent, -0.25 < T < 0.25,both X's stay near the eyes of the spiral; that segment is allplateau. Use of (22) shows that X2 is large and X1 = 0 whenr = 0.25, which is, therefore, the other 6-dB point boundingthe plateau. Thus the salient features of Fig. 8 can be deter-mined accurately without a computer; the response is a

234


20

00o1-

w

U) -4

z -20g

U)

n

uJ -640

-60

-801-. T-0.5 O 0.5

TIME-TO-DURATION RATIO (T)

Fig. 8. Response of a receiver for 20-,s 10-MHz pulses to a 10-ps20-MHz pulse 5 MHz above the center frequency of the receiver;TB= 200,P= T/2,a = 4,p = 0.50.

flat-tapped pulse whose level is 14- I- , which is -5 dB,and whose duration in normalized time is 0.25 + 0.50 = 0.75.

Another example is offered by a recently designed bistaticradar for a space application. It has a 0.32-s 10-MHz pulse,and the receiver is matched to the return from a target withradial velocity 1.5 km/s, which produces a Doppler shift [17]of one part in 105. The pulse traveling directly from transmit-ter to receiver is Doppler-shifted negligibly; for it, 1 - a10-5, and the plateau of the receiver response, Ia - I1-1,is 105, or 50 dB, whereas the response if the receiver matchedthe pulse would be TB, which is 65 dB; the Doppler mismatchlowers the response by 15 dB.

In cases like this where the only mismatch is in the FM rate,(22) shows that the duration of the plateau, in normalizedtime T, is (1 - a) if a < 1 and (a - 1)/a if a > 1.2 For theradar just cited, the duration of the pulse received directly is(1 - a)T, or 3.2 ,us, between 6-dB points; the receiver-matching echo from the target is compressed to B-1, which is0.1 ,us, between 4-dB points.

It is evident in (22) that these rules for plateau height andduration are not valid when a - 1l > TB because, in suchcases, the X's cannot be large in magnitude and opposite insign, so there is no plateau.

The situation la - 1 l1 > TB has other special interest. In-tuition persuades us that a "very small" mismatch in FM rates,i.e., a "very small" but nonzero value for I a -1 1, would not

2When a = -1, i.e., when the pulse is matched except that it"chirps the wrong way," the rule exaggerates the duration of theplateau by about 5 percent (Fig. 9). The reason is that the value ofX that makes X1 = 0 in (22) also makes X2 = 0, so instead of being 6dB below the plateau, y 12 for this time is zero. For a's slightly dif-ferent from -1, the validity of the rule can be assessed by calculatingthe value of one X at the time when the other vanishes, and seeing howfar out on the spiral it is.

co

U)z0C,)

C-)

ia

20 *T 1

-20

-40-- -

-60 A]

- -80 --1.0 -0.5 0 0.5


Fig. 9. Response of a receiver for 10-pus 10-MHz upward-sweepingpulses to a 10-Ms 10-MHz downward-sweeping pulse centered inreceiver passband; TB = l00, P = T, a = 1, p = 0.

seriously invalidate (14). We can agree to accept as smallenough, for example, the error that is caused by a differencein FM rates that produces a phase discrepancy of 1 radian dur-ing the course of the pulse. The differential FM rate is (a -I )brad/s2, so during the pulse the phase error builds up to ±(a-l)bT2, whose absolute value is Ia -1 BT. For an error notexceeding a radian, we must restrict a so that I a -I h < TB.When the Doppler effect is small enough so that this conditionis met, we can use (14) and a nonzero p, instead of the morecomplicated (21). On the other hand, if one already has acomputer program for (21), it is unnecessary to have one for(14); the program can be used when a = I (no mismatch in FMrate) by plugging in a = 1 +7±, where 0 < l I< TB-1.

Cornu's spiral is not of much help when the two Xs arelarge in magnitude and the same in sign. In such cases, how-ever, the asymptotic relations (9) can be called upon. From (7)and (8), it can be verified that, for 1-dB accuracy in ly 12, it issufficient if XI I > 1 and I X2 I> 1, for all t under considera-tion. Using (9) in (21) gives

1 I 1 2Iy 12 = + _-

2ir2la-1I X12 X22 X1X2

cos [ (X22 -XI2)

A bound on y 12 is the envelope

r 1±21y 12 = 2l I + - I.

27r21-- lXli IlX21J(24)

Consider a 10-,s constant-frequency pulse 25 MHz belowthe center frequency of a receiver for 10-,us 10-MHz upward-

235

1 .0 1.0


m

w

(I)

z

0

0.

w

cr

w

w

-60 w-r710.-lO -0.5 O 0.5

TIME-TO-DURATION RATIO (r)

Fig. 10. Response of a receiver for 10-,s 10-MHz pulses to a 10-,sconstant-frequency pulse 25 MHz below the center frequency of thereceiver; TB = l00, P = T, a = 0, p = - 2.5.

sweeping pulses. From (22), for -T < t < 0,

-25+t -25±+tX, =2-i5 X2=N i + i(t+5)

where i= X/7 and the unit of time is the ,us. At t = -10 s

these reduce to X1 = 30iV2 and X2 = 30iV2.; (24) yieldsY(_10) 12 = _39.5 dB. Similarly, Y(0) 12 = _37.6 dB andY(l0) 12 = -36.0 dB. These are seen to be in good accord

with the accurately calculated Fig. 10.Comparing Figs. 10 and 3 (d), one sees that at their centers,

the responses have the same value. The envelopes tilt oppo-sitely because in Fig. 3 (d), the center frequency of the pulseis above that of the receiver, whereas in Fig. 10 the reverse istrue. One response tilts more than the other because when theFM pulse starts, its frequency is closer to the passband of thereceiver than it is at the end. This comparison solidifiesa conclusion reached in Sections II and IV, namely, that theresponse of an FM receiver to a pulse that lies outside itsnominal passband does not depend much on whether thepulse is chirped or not.

We can go a step further, and replace the chirp receiver withone that is matched to constant frequency. The response can

be calculated from (19) by using b = 0 and B = lIT; the resultis given in the Appendix. The maximum of the response isgiven by (7rvT) 2, where T is the duration of the pulse towhich the receiver is matched. The receiver in Fig. 3 (d)matched the 10-,s 10-MHz pulse, except for having a centerfrequency 25 MHz lower. The closest approach of the pulseto the center of the receiver passband was 20 MHz. Replacethat scene with a receiver for 10-,us, constant-frequency pulses,and such a pulse 20 MHz above the nominal frequency of thefilter. The maximum of the response is (7r-20-I0)-2 - -56.0

dB. If this unchirped pulse matched the receiver, the maxi-mum response would be 0 dB; the 20 MHz of mismatch re-duces the response by 56.0 dB. A mismatch of 30 MHz wouldcause a reduction of ( r-30-10)-2 = 59.5 dB. In Fig. 3 (d),the response of the FM receiver to the FM pulse runs from-33.5 to -40.5 dB, whereas if the pulse matched the receiver,as in Fig. 3(a), the response would be TB, which is +20 dB.The reduction due to the 25 MHz mismatch runs, therefore,from 53.5 dB at one end to 60.5 dB at the other. Replacingthis FM situation with the totally constant-frequency oneoverestimates the reduction by 2.5 dB at the beginning of theresponse and underestimates it by 1.0 dB at the end. At thebeginning, the FM-pulse frequency is only 1.5 bandwidthsfrom the edge of the nominal passband; for larger frequencyoffsets, the discrepancy between the real situation and theconstant-frequency substitute diminishes.

VI. A SAMPLE CALCULATION OF RFI

As an illustration of the application of what has beendiscussed above, consider a radar receiver matched to a 10-,us10-MHz upsweep. We need to estimate the level of RFI from aradar, 20 km away, that emits a 10-lus 20-MHz upsweep at aradiated level of 2.0 MW; this radar has a center frequency10 MHz higher than that of the receiver. Fig. 11 is applicable,but we do not have to use it.

At the outset, we need the transmission loss between theantennas [ 18] and the noise temperature of the receiver. Sup-pose that they are respectively 150 dB and 360 K. Begin theRFI calculation by supposing that the transmitted pulsematched the receiver. The emitted energy is 2.0 MW X 10 ,us =20 J, or +13 dB-J. The energy picked up at the receiver is150 dB less, or -137 dB-J. The receiver noise power per hertz,in a one-sided spectrum, is 1.38 X 10-23 J/K X 360 K =5.0 X 10-21 J, or -203 dB-J. The interference-to-noise ratiofor our fictitious (matching) pulse would be -137 dB-J + 203dB-J = 66 dB.

The response of the receiver to a unit-amplitude pulse thatmatched it (Fig. 3 (a)) would be TB, or 20 dB. To a unit-amplitude pulse like that of the interfering radar, which hastwice the sweep rate (a = 2.0) and some frequency overlap,the maximum response would be a - h1; in this case,0 dB. Therefore, the mismatch reduces the receiver responseby 20 dB, and the expected interference-to-noise ratio is 66-20 = 46 dB.

It is possible to raise the frequency of the interfering radarby 15 MHz, and we are asked how much good that would do.The center-frequency offset is now 2.5 receiver bandwidths,but the closest frequency in the pulse is only 1.5 bandwidthsfrom the center of the receiver passband, This is a marginalsituation for applying the constant-frequency approximation,but invoke ain imagined receiver for a 10-ps constant-frequencypulse, with the same noise temperature as that of the real re-ceiver. If the interfering pulse had constant frequency, and ifit matched the receiver, the interference-to-noise ratio wouldbe 66 dB, as above. The 25-MHz offset in frequency would re-duce the response by a factor (ir 10-25)2, which is 58 dB.For moral support, compare Figs. 3(a) and 12; at midpulse,they indeed do differ by 58 dB. Raising the center frequency

236


m

wC')z0a-u)U

>U

Ut

-0.5 0 0.5TIME-TO-DURATION RATIO (T)

Fig. 11. Response of a receiver for 10-gs 10-MHz pulses to a 10-s20-MHz pulse with center frequency 10 MHz above that of thereceiver; TB = 100, P = T, a = 2, p = 1.0.

of the interfering radar, as contemplated, would reduce thelevel of interference to 66 -58 = 8 dB at midpulse. One islikely to want the maximum response rather than the responseat midpulse. It is associated with the minimum offset in fre-quency, which is 15 MHz from the center of the passband.Using (7rTv)2 would lead us to conclude that the maximum is(25/15)2, or 4.5 dB, above the midpulse level. The correctfigure, readable from Fig. 12, is 8 dB; we have used the con-stant-frequency substitution when the frequency offset is toosmall for it to give really good results.

To get a closer approximation in this intermediate regionbetween frequency overlap and large offset, we can resort to(24), which (7) shows to be valid when 7rX2 > 1. For v = 25MHz and t = -9.5 ps, which is near the high end of the re-sponse, we have, from (22), X1 = 14.8, X2 = 15.6; (24) givesI Y 12 = -30.6 dB. With t = -9.9 ,ps the same procedure givesY 12 = -30.1 dB, which is higher; therefore t = -9.9 ps is

not too near the start of the pulse, and we can take the highend of the response to be at -30 dB, which is 9 dB above themidpulse level, and 50 dB below the response to a matchedpulse, so the peak interference-to-noise level resulting from thenew center frequency would be 16 dB.

This example has illustrated three approaches. When thefrequency offset v is small (roughly speaking, when the pulselies partly or wholly in the receiver passband), the maximumresponse occurs in a time when the Xs are large and haveopposite sign; if - a I-1 < TB, there is a plateau and itslevel is I I -a I-1; but, ifIl -a I-1 > TB, there is a peakwhose level is TB. When the frequency offset exceeds a coupleof receiver bandwidths, the X's are large and have the samesign; we can ignore the FM, and the frequency offset causesthe response to be lower by (irvT)2 than the response to a

pulse matching the receiver. There is an intermediate situa-tion such that the pulse, or part of it, lies on the filter skirt

U

U)z0a.(I)U

Ur

iiiU

U-20m

-1.0 -0.5 0 0.5TIME-TO-DURATION RATIO (T)

1.0

Fig. 12. Response of a receiver for 10-,us 10-MHz pulses to a 10-ps20-MHz pulse with center frequency 25 MHz above that of thereceiver. TB = 100, P = T, a = 2, p = 2.5.

near the edge of the passband. If the Xs are both large enoughfor the asymptotic approximations to be valid, one can apply(24), as in the example. Finally, in the time interval in whichthe maximum lies, it may not be true that both Xs are largeenough to justify the asymptotic approximations (9). Aninstance occurs if we use v = 15 in the example above. Fort = -9.5 ps, that results in X1 = V7/2 and X2 = NE. These lieon the open part of the Comu spiral, meaning that the rapidfluctuations evident in Fig. 12 do not occur, and that there-fore (21) can be evaluated adequately by means of thespiral.The result is --8.1 dB. By plotting the response for a smallnumber of values of t in this neighborhood, one finds that thisvalue is close to the maximum. The upshot is that, for valuesof v where the simple rules of thumb involving Ia1-1 and(irvT)-2 fail, the asymptotic relations are often justified, andwhen they are not, the Comu spiral comes to the rescue.

VII. COMMENTS AND REFINEMENTS

In using the methods described above, attention must fallstrongly on the receiver, and on the pulse duration for which itis designed. For example, when calculating the interferencelevel that a matched pulse would cause, one must match thepulse to the receiver, even though that may mean using a ficti-tious transmitted energy, based on a pulse duration (7) differ-ent from that (P) actually used by the transmitter. The reasonis that the analysis permits us to calculate how the response tothe actual pulse compares to the response to a pulse thatmatches the receiver. Likewise, when using the constant-frequency shortcut, one needs (7rvT-2, not (7rvP)-2, becauseour reference level (unity, in the constant-frequency case) isthe peak response to a pulse that matches the receiver; thespectral density sampled by the receiver is the same for the fic-titious pulse as for the real pulse.

237


In general, each of the procedures described is, to somedegree, inexact by an amount that will usually be small com-pared with the uncertainty in knowledge of the transmissionloss. For a clean, ideal spectrum and a filter matched ideally toa rectangular pulse, (20)-(22) are exact. There are two prin-cipal contributions to ly 12: the skirt spectrum of the pulseseen through the passband of the filter, and the spectral plateauseen through the skirt of the filter. If the only mismatch is afrequency offset, these contributions to the calculated valueare equal. Increasing a bandwidth does not make much dif-ference; it makes a plateau broader but lower. The two con-tributions are not equal if the pulse is longer or shorter thanthat for which the receiver is matched, because changing thepulse duration changes the level of the spectral plateau with-out significantly altering the spectral skirts. Replacing an FMpulse with a constant-frequency pulse, without altering theduration, leaves the spectral skirt unchanged and also leavesunchanged the total energy in the plateau region of the spec-trum, but the contraction of the plateau into a peak concen-trates energy near the nominal frequency of the pulse. Whenspectrum and filter conform to the ideal assumed in the anal-ysis, the analysis takes all these effects into account. (For ex-ample, the item just mentioned is reflected in the slightlylower average power exhibited in Fig. 10, as compared withFig. 12.) However, these considerations provide a basis for re-fining the estimate of interference-to-noise ratio, if it seemsworthwhile to do so, in cases where the filter or the spectrumis known to depart from the ideal. A practical receiver mayhave IF or RF selectivity sharp enough to reduce acceptanceof the spectral plateau of the interfering pulse. Also, especiallyin very-wideband radars, it may be that the spectral skirt ofthe pulse is attenuated by the RF properties of the transmitterand its antenna. Of course, one must be on guard against thepresence of "hash," spectral components unrelated to the ideal.

Wideband (e.g., 250-MHz) pulses are sometimes processedin receivers that map time into frequency by using a linearlyswept local oscillator: range, within a chosen interval selectedby a gate, is read on a frequency analyzer. The bandwidthneeded may be as little as the reciprocal of the pulse durationT. Noise is less than matched-receiver noise by the ratio of thisbandwidth to the pulse bandwidth B, so it can be less by afactor TB. However, if we have a unit-amplitude pulse and aunity-gain receiver, the peak signal power at the output isunity, whereas in a matched receiver it is TB. In regard to sig-nal-to-noise ratio, therefore, the time-bandwidth exchange re-

ceiver performs about as well as a matched one. Interferencefrom an out-of-band pulse is a chaotic waveform; the receivertreats it like noise. The RFI calculations above can thereforebe applied to a time-bandwidth-exchange receiver, with thedifference that the interfering pulse may be shut out, entirelyor in part, by the range gate.

In the realm of practice, a chirp receiver nearly always has afilter that is deliberately mismatched to the desired pulse, forthe sake of reducing the time sidelobes evident in Fig. 3 (a).Such mismatching is called "weighting;" the rest of this articlewill be concerned with its effects.

VIII. WEIGHTING

Prior to investigating the effects of weighting on RFI, someprefatory remarks about the technique will establish the basicideas and equations. Attention will be restricted to one class ofweighting function, the cosine on either a pedestal or an end-less platform; the much-used Hamming weighting is a particu-lar case.

It is apparent from (14) that successive sidelobes of y(t),the voltage waveform underlying Fig. 3 (a), alternate in phase.Therefore an interference effect can be set up, by superimpos-ing three waveforms of that kind, one a little advanced in timeand one a little retarded. A delay 1/B is chosen, because that isthe total width (very nearly) of the large sidelobes, for which T

is small. No choice of delay can result in destructive interfer-ence at all sidelobes, because the zeros of (14) are not spacedevenly in r, as can be seen in Fig. 3(a). The desired effect canbe achieved by multiplying the spectrum of the matched-filteroutput by a weighting function WF(t) such that

WF(t) = p + a cos 2ir(T-f)/B (25)

where ¢ is the frequency offset from f, the center of the re-ceiver passband; p and a are constants, and p + a = 1. If thesignal spectrum prior to weighting is S(t), upon multiplicationby (25) it becomnes

SF() = pS(¢) + (a/2)S(¢) ei2 X ( -f )/B

+ (a/2)S(¢)e i2 r(-f)/B (26)

Taking the Fourier transform of both sides gives the waveform:

YF(t) = py(t) + (a/2)y(t - l/B)ei21rf/B

+ (ul2)y(t + I/B)e-i2 Tf/B (27)

This kind of weighting, therefore, can be accomplished bysumming phaseshifted outputs of a tapped delay line (a trans-versal filter) [ 15 ], [ 19] . It is an example of"weighting in thefrequency domain," which means shaping the spectrum insome prescribed manner. The weights (/2 of the advanced andretarded components must be chosen judiciously, to producea large amount of cancellation of the major sidelobes withouttoo greatly depressing or widening the main lobe. A much-usedchoice is a = 0.46, p = 0.54; this is called Hamming weighting.The function (25) is a cosine curve on a platform whose heightis p -a. It is arguable that the true Hamming weighting func-tion in the frequency domain is a single cosine hump on apedestal with vertical sides [191. Such is achieved by adding to(25) the stipulation WF(t) = 0 for I> B/2. If applied to theunweighted waveform (14a), the implied (unrealizable) sharp-cutoff filter would make the envelope seen in Fig. 3 undulate.The impulse response of the filter is (sin 7rBt)/irt, and thepeaks of the resulting ripples would be spaced by approximately2/B. We see from (27) that 2/B is twice the delay and advanceintroduced by the weighting function (25), and when the

238


OD1-

wU)z0a.U)w

c-w

w

-1.0 -0.5 0 0.5 1.0TIME-TO-DURATION RATIO (T)

Fig. 13. Response of a receiver for TB = 100, with pseudo-Hammingweighting in the frequency domain, to a pulse with matching dura-tion, bandwidth, and center frequency. The peak lies at 14.7 dB.

sharp-cutoff filter is applied, the ripples it puts on the retardedand advanced components in (27) almost exactly fill in theripples it puts on the p-weighted term; the sharp cutoff wouldhave hardly any effect. Because (27) is so much easier to eval-uate than (26) transformed with limits of ±= +B/2 on the in-tegration, we will use only the bandlimiting provided by thechirp-matching filter in conjunction with (25); when p = 0.54,we will call the operation pseudo-Hamming weighting.

By using (14a) in (27), one finds

quency offset from the center frequency f of the receiver byan amount pB (which may be zero).

For a radar receiving its own pulse with no Doppler shift(p = 0), the result of pseudo-Hamming weighting in frequencyis illustrated by comparing Figs. 13 and 3(a). The weighting re-duces the peak response to about 5.4 - dB below TB, but italso reduces the noise; the loss in signal-to-noise ratio is about1.3 dB [15]. The largest sidelobes are 37 dB below the mainpeak. For large time-bandwidth products (e.g., 1000) thelargest sidelobes are the third ones, and these are 42.7 dBbelow the main peak. Apart from generating precursor andpost-cursor waves at about -65-dB peak, bandlimiting to get"true" Hamming weighting would slightly (-1 dB) perturbsome of the near sidelobes, would lower some of the very farsidelobes, and would cause the pairs of sidelobes at = 1(which come from the terminaton of overlap) to merge intosingle ones with slower rise time [20].

Weighting can also be applied in the time domain. A com-monly used weighting function [211, [22] is

p + a cos (27rt/T),WT(y) =

O,

I t < T/2

I t I > T/2.(30)

It is cosine-on-a-pedestal weighting and, if p = 0.54, this, too,is called Hamming weighting; in fact, it is the original Ham-ming weighting [21].

When (30) weights a receiver3 that is otherwise matched toa linear-FM pulse with length T, the counterpart of (13) is

YT(t) =r2 ei7r(2foT+bT2)[p + or cos 2T(t -r)]

. eiT(2f(t-T)-b(t-T)21 dr (31)

(28)

The upper signs apply when the receiver matches an upsweptpulse.

If we designate the moduli of the three tenns in the braces

asK,L,M,we find

I yF(t) 12=K2+ L2 ±+M + 2K(L + M) cos (irp)

-F 2LM cos (2irp) (29)

as the response of a receiver, pseudo-Hamming weighted in fre-quency, to a linear-FM pulse having the duration and band-width that the receiver is designed for, but with a center fre-

where, as before, r1 is the greater of-T/2 and t- T/2, while r2is the lesser of t + T2 and T/2. The part multiplied by p is justthe integrand in (13). In the part multiplied by a, the cosine

3 Hamming weighting was introduced [211 to weight the data in a

time series. Early application to radar [221 involved range-gating thereceived signal and weighting it with (30). Present technology is con-

ducive to using unweighted pulses, but matching the receiver to a

weighted pulse [161. If the ambiguity functions resulting from thesetwo procedures are called Y2(r, P) and Y1(T, p) 1, it is straight-forward to prove that Y2(r, p) = Y1(-r, -p) 1. If the pulse islinear FM, then Y2(r, P) = Y1(r, p) I; both procedures give the same

waveform.

YF(t) = exp [i1r(fO +f)tI | p sin [TB(p ±+r)(I -I T I )]

ue-iTP sin [7rTB(p ± r T I/TB)(1- i--1/TB I )]2 T7rT (p ± rT 1/TB)

ueiTP sin [7rTB(p ± r± l/TBXl -I r + 1/TB I )]

2 ir'fTJ.(p ± r 1/TB)

239


z -20

0

0-

cn(IJw

_40w

w

w

-0.5 0 0.5 1.0


Fig. 14. Like Fig. 13, except that the weighting is Hamming in thetime domain.

can be put in exponential form and merged with the other ex-

ponentials. After suppression of a carrier exp [inr(fo + f)t],

sin [7rTB(p ± r)(1 - T )] ae1iTrYT(K =P ±p + ) 2

sin [lTTB(p ±+ T- I/TB)(Il- i )] ae~-iirrTB(p ±+ -)/TB) 2

sin [irTB(p + T + l/TB)(l ITiI)](32)

TVT_B(p ±T + l/TB)

The notation here is the same as in (14); observe that r is tIT,and is not the variable of integration used in (3 1).

Introduce 4¢(T) and ;(T) as the moduli of the second andthird terms on the right of (32), so that YT(t) = K + eil r +

,e-i7rT; then

YT(t) 2=K2 + I)2 +4,2 +2K(Q+ i)cos7tr

+ 2(DIb cos 2trT. (33)

Such an equation characterizes an interference, or beat,phenomenon. Hamming weighting selects p and a such that,with only slight degradation of signal-to-noise (compared withthe unweighted case) the beating results in low sidelobes. Fig.14 gives an example, for comparison with Figs. 3(a) and 13.

It is often stated that one uses weighting in one domain to

reduce sidelobes in the other domain. Though the statementdoes have wide validity, it is not applicable to linear FM.When the time-bandwidth product exceeds about 170, thelargest sidelobes in the time domain, after Hamming weighting,are about the same when the weighting is in the time do-main as when it is in the frequency domain. Moreover, whenTB < 170, the advantage is with the time-domain weighting[201. The selection of the domain for cosine-on-a-pedestal

weighting of linear-FM pulses can be based on the means ofimplementation; filters using surface acoustic waves are wellsuited to weighting in the time domain.

IX. EFFECT OF WEIGHTING ON RFI

If an interfering pulse matches the receiver in bandwidthand duration, but not in center frequency, then (28), (29),(32), and (33) apply. Some evaluations of (29) and (33) bymeans of a computer are graphed in Figs. 15-17; it is of in-terest to compare these with the corresponding parts of Fig. 3.

In calculations of RFI, interest focuses on the maximumvalue of the response. That can be obtained without a com-

puter. The first step is to calculate the level of the interferenceif weighting were absent, i.e., if p = 1, a = 0. It is apparentfrom (14) or (28) that, if Ip < 1 and there is no weighting,the response peaks at a normalized time T' close to Tp, the signdepending on the direction of the chirp, and that y(T') is thenVfTY (1 - ' I). Simple manipulation of (28), using the ap-proximation ' = Tp and the stipulation p I > I/TB, leads to

YF(T) a sin 27r IT' i1-/TBy(r) 2ir(I -|IT )

(34)

where the upper sign is for an upward chirp. Similarly, from(32)

YT(T) asin27r Ir'I=P +

y(r') 2it(1 -|IT |)(35)

The squares of these differ by having a factor cos(7r/TB) in thecross term; unless TB is small (say <10) the result is effectivelythe same no matter which domain is used. Even for the stressingcase p = 1.0, (34) and (35) calculate the reduction as closelyas Figs. 3(c) and 16 can be read. After suppression by theweighting, response at T' may no longer be the maximum re-sponse, but it will be a useful approximation to the maximum.Figs. 3(c) and 16 illustrate the phenomenon, which can onlyoccur when p is near unity. As a precaution, one can setI p = 1 or 1.1 and calculate the height of the broad hump bythe method that follows.

When the frequency offset exceeds the bandwidth, i.e.,when p I > 1, the maximum of the unweighted response isclose to one of the ends, as in Fig. 3(d), but the maximum ofthe weighted response may be elsewhere (Fig. 17). For per-spicuity, (28) and (32) can be recast into the restricted forms

YF(t) sin [7rTB(p + r)(I + r)] ueiwrp

N/TB 7rTB(p +T) 2

sin [irTB(p + r)(1 + -r) + 7r(p + 2r) + ir/TB]7rTB(p + r) + ir

ae -irp

2

sin [7rTB(p + T)(I ± r) - 7f(p + 2T) + ir/TB]irTB(p + r)- ir

(28')

240


20

m1-

wU)z0

a-w

w

w

0- -

-40 I-I-F -fw 1

.. .

-60 _ 1

-80 _ I -_1r _n n, nzF% I


(a)

V - -.- -O.-O0.5I -

wUf)z -20--

0w-

W-40~---1 -~

-80-1.0 -0.5 0 0.5 1.0

TIME-TO-DURATION RATIO (T)(b)

Fig. 15. Like Fig. 3(b) (TB = 100, p = 0.6), but with weighting: (a)pseudo-Hamming in the frequency domain; (b) Hamming in thetime domain.

YT(t) sin [7rTB(p + r)(1 + r)]

N ~T_B ~p

7rTB(p + r)

ae-i7 sin [nTB(p + r)(1 + r) + -r]2 irTB(p+r)+ir

aeT sin [7rTB(p + r)(1 + r) - rr]2 7rTB(p+r)--ir

(32')

The restrictions, which serve to simplify the scene, are that the

sweep be upward (b > 0), and that i- < -1/TB; if p > 1, the

maximum of the response will lie at a value of r complyingwith the second restriction.

m

wcnz0

U1)w

w

CL

w

cn.


'nw11)zI0a-Uf)w

w0LU

-10 -0.5 0 0.5 1.0TIME-TO-DURATION RATIO (r)

(b)Fig. 16. Like Fig. 15, except that the center-frequency mismatch is

1.0 bandwidth. Compare with Fig. 3(c).

Use the abbreviations irTB(p + r)(1 + r) = a and 7r7TB (p +r)= ,. Then, under the restrictions,

YT sin a ae-iT sin (a + irr)(T- 2(3+±7r)

uei7r sin (a -7rr)2(o -7r)

The second and third terms can be expressed with the com-mon denominator 2,3(1 -ir2/12); replacing it by 2,B introduceserror of less than 10 percent if p > 1.2 and -0.9 < r whenTB = 10, and, for larger TB, the requirements on p and 7 are

T X | s * | W w * w r r 241

-u.D U.0 .U-1.,u


m'0

wcnz0a.(I)wa:

w

wwa:


(a)

m

wC,-z -200a.U)wa:a:W -40-

0wa:

-10 -0.5 0 0.5 10

TIME-TO-DURATION RATIO (-r)(b)

Fig. 17. Like Figs. 15 and 16, except that the center-frequency mis-match is 2.5 bandwidths. Compare with Fig. 3(d).

even less inhibitive. When the numerator is multiplied out,these same requirements guarantee that two terms can be ne-

glected, and one finds that

TBYT 12 - [(p - a COS2 irT)2 sin2 a

132

+ a2 sin4 irn cos2 ax].

The largest value of the factor in brackets occurs when sin a =

1, which happens frequently, and cos27rr is near 0, which hap-pens at r = ±1 At such time, YT 12 --p2 Iy 12. At times not

much different from that (depending on how large p and TBare), cos2a = 1 and IYT 12 = a2 ly 12. Near r = -1 or 7r = 0,the sin4irr is negligible and IYT 12 = (p - a)2 Iy 12. The rib-bon-like double arch in Figs. 16(b) and 17(b) is therefore theform of the response whenever a chirp receiver with cosine-on-a-pedestal weighting in the time domain responds to a pulsethat matches it, except for being out-of-band. For Hammingweighting, the suppression factor due to the weighting variesfrom 5.4 dB at Ir = i to 21.9 dB at the middle and the ends.At the top of each arch, the width of the ribbon is given byp2_ a2.

The form of IYF 12, on the other hand, depends on theoffset p. When that is a half-integer, there is a resemblance tothe time-weighted case, though with an important difference.If p = m/2, where m is an odd integer greater than 2, (28')becomes

YF sin -a a cos (a +rr/TB + 27rr)-vlT 2 _I

cos (a + ir/TB - 217r'T)+ - _

which, in the approximation delineated in connection with YT,yields

12 TB 2 2 2 22iTTcos2 ot].IYF 12=- [p2 sin2 a ± a2 Cos2 rcs ]132

(36)

At maximum, IYF 12 = p21y 12. That is, the effect of theweighting is to suppress the envelope uniformly by p2, whichfor Hamming weighting is 5.4 dB. The term in a2 gives a four-fold arch as in Fig. 17(a); the summits of the arches at T- =

are the same as when weighting is in the time domain.When p is an integer, (28') reduces to

YF sin a a sin (a±+ r/TB + 2irr)NoT__ 1 - 2L 1+ir

+ sin (a + 7r/TB 2T)+ -7r

which, under the same approximations, yields

IYF 2 /I y 12 = (p-a cos 2irT)2. (37)

At the ends and middle of the pulse, the suppression is (p - a)2which, for Hamming weighting, is 22 dB. However, whenr = 1, the ratio is unity; in those parts of the pulse, the

weighting does not suppress the RFI at all. The effect is il-lustrated by Figs. 3(c) and 16(a), even though p is only 1 inthese. When the offset is two or more bandwidths, however,the larger hump is more nearly symmetrical, both humps are

242


lower, and their heights are more nearly alike, because theenvelope of the unweighted response is lower and less tilted.Compare Figs. 3(c) and 3(d); and also 16(b) and 17(b).

When the frequency offset is neither an integer nor a half-integer number of bandwidths, the envelope of the frequency-weighted response is intermediate between those in Figs. 16(a)and 17(a); as p goes from an integer to a half-integer value, thedouble hump turns into a gentler undulation and then be-comes monotonic.

Putting these findings together, we can formulate a simplerecipe for determining the effect of cosine-on-a-pedestalweighting on RFI when the interfering chirp matches the un,weighted receiver, except for lying outside the passband. Un-less the weighting is in the frequency domain and the center-frequency offset is a half-integer number of bandwidths,calculate the frequency offset at a time 0.25 or 0.75 of theway through the transmitted pulse, using whichever gives thesmaller result. Using that offset, calculate, as described in Sec-tions V and VI, the level of interference with weighting absent;call this level L. If the weighting is in the time domain, thepeak level of interference will be p2L. For weighting in thefrequency domain and p = v/B an integer, the peak interfer-ence is L. If p is a half-integer, the interference, at all parts ofthe pulse, is p2 less than it would be with no weighting, andthe peak is at one end, as in Fig. 17(a). If p is neither an in-teger nor a half-integer, calculate for the nearest integer andhalf-integer; the two results will bracket the interference level.

It is interesting that unless the center-frequency offsetclosely approaches or exceeds the bandwidth, the peak RFI isindependent of whether the weighting is done in the time orthe frequency domain; but if the offset equals or exceeds thebandwidth, time-domain weighting suppresses RFI more thanfrequency weighting does.Now generalize, by removing the restrictions, that the

pulse have the duration, the bandwidth, and the sweep rate forwhich the receiver is intended. One must then work from (20)and (22) instead of (14a). For weighting in the frequency do-main, using (27) leads to

I B .1YF(t) = e P|p[F(X2)-F(X1)]

2(a -- 1)bTI

+ -[F(X2 ) F(X1-)] exp {-iT[ (ab/B2)2

-2(v + abt)/B] /(a- 1)} Jr 2 F(X2 +)2

-F(X1+)] exp {-hrT[(ab/B2)

+ 2(v + abt)/B]/(a -1)}j (38)

where -= -r [(v2/b) + 2(fo - af) t + abt2 ]/(a - 1). Thearguments X of the p-weighted Fresnel integrals are defined by(22); the arguments X- and XI are obtained from these by

substituting r- 1/TB and r + 1/TB, respectively, for t. Thereceiver response yF(t) 12 is given by

2(a- l)bTB IYF(t)12

U2=p2[(C2-C1 )2 + (S2-S1)2] +-[(C2--C1-)2

4

a2+ (S2--S-)21 - [(C2+-C1+)24

+ (S2 +-Si +)2 ] + pa {[ (C2 -C1)(C2--CC)+ (S2 -S1)(S2--S )I Cos 0-

+ [ (C2 -C1)(S2--S1)-(C2- -C1-)(S2-Sj)] sin 0-} + pa{ [ (C2 -C1 )(C2+ -C1)+)+ (S2 -Sl)(S2+-S1 +)] COS a+ + [(C2 -Cl )(S2 +

-S1+)-(C2+ -C1+)(S2 -SO)] sin 0+}

2±-+ {[(C2 -Cg)(C2 +-Cl+)±(+ -Sl-)2S+] o 1+2[(2-1) + S1iXS

(39)

where C1 and S2' stand for the Fresnel integrals C(X1) andS(X21), and so on, while

v+abto =47r

(a -1 )B7rab 0

0- = .-_B _

(a- J)2 2

rrab 00+= + -.

(a-1)B2 2

To obtain the output YT(t) of a receiver weighted in thetime domain, one can work from (31) after replacing b with abin the first exponential. The result is

B

Y ) 2(- e)bT [hr(2ft-bt2)]p[F(X2)-F(X1)] exp [-i7r(v + bt)2/(a - I)b]

+- ei27rt/T [F(X2') - F(X1 ')] exp [-i7T(v'2

± bt)2/(a 1)b] -e-i2lrt/T[F(X2")2

-F(Xl")] exp [-iir(v" + bt)2/(a - 1)bI (40)

in which X' and X" are obtained from (22) by putting, re-spectively, p' = p 1/TB and p" = p +1/TB in place of p.

243

(C2' Cl')(S2 Sl-A sin 0}


For the time-weighted response IyT(t) 12, we have

2(a-1 )bT IYT(012B

a2=p2[(C2-C1)2 + (S2-S1)2] -[ (C2 -C1')2

4

+2±(S2 -Sl')2 ] +- [(C211-C1t)2+(S2t-SI')2]4

m

cnwC,)z0a-(nw

w

wC)llJJ

*1 COS t-+ [ (C2 -Cl )(2 -Sl')-(C2' -Cl')(S2-S1l)] sin t-j + pa {[ (C2 -Cl)(C2 " -C1 ")

+ (S2 -S1)(S2 ' -S1 )] COS t + [(C2 -C1)(S2"

a2-S1 )-(C2 "-C1")(S2 -S1)] sin t+ +-

2

-(SC ") COS + I (C2'-S1C)(S2-S} )

- W2"-Cl "0(2' -Sl')] sint} (41)

where C2' and S1 stand for the Fresnel integrals C(X2') andS(X1), and so on, while

v+abt= (a-l)bT

1r t(a_

(a-1)bT2 2

1rt+- ~+_(a-l)bT2 2

t here being the same as 0 above.Fortran programs for evaluating IYF(t) 12 and IyT(t)12

have been published [8] ; for estimating the peak response,however, rapid approximations can be found. In what follows,statements will be simplified by assuming that (a - 1)/b >0,so that the Fresnel integrals are real. The restriction is mademerely for verbal convenience; it can be dropped without af-fecting any of the conclusions.

In a wide range of cases, X1 and X2 in (22) are not muchchanged by the substitutions r ± 1/TB for T, which yield thesuperscripted Fresnel integrals in (39). Ignoring the modifica-tions produced by the superscripts causes (39) to collapse to

IYF(t) 12 - y(t) 12 {p2 + 2pa cos (ac) cos (6/2)

± a2 COS2 (0/2)}

where 0 is defined below (39), and e = 7t/(a - I)TB. Similarly,

iYT 12 - y(t)- 12 [p2 + 2pu cos e cos (0/2)

+ a2 cos2 (0/2)] .

These approximations are valid where the unweighted responseis nearly level. They may also be valid when X1 and X2 have

ma)w(I)z0a.C,)wcr.irw

ww


-20 _

-40 - I-20 -1-- -I1\ H --

-60 _ [-

-IA) U~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-80 -__ r [_1,n _n C; n n_U.t u

TIME-TO- DURATION(b)

RATIO (r)1.0

Fig. 18. Like Fig. 8, but with weighting: (a) pseudo-Hamming in thefrequency domain; (b) Hamming in the time domain.

opposite sign but are not large enough to be near the eye ofthe spiral. The approximations are alike except for cos aeversus cos e; these can differ only slightly, and, unless a is near1, both cosines are near 1 and can be ignored, in which situa-tion we have

IYF(t) 12 = y(t) 12 [p +± cos (0/2)] 2

and the same for IYT(t) 12. If the frequency offset v is lessthan the pulse bandwidth, there is a time when v + abt = 0,which makes cos (0/2) = 1. This means that cosine-on-a-pedestal weighting rounds off a plateau, but does not lowerthe maximum of the response. Fig. 18 shows the effect. Anexception is any constant-frequency pulse; for it, 0 = -4irp, so

244

-1 .0


that YF 12 and YT 12 are given by ly 12 [p + acos 27rp] 2; theweighting lowers the response by a constant factor when per-turbing r by ± 1/TB has negligible effect on the Fresnel in-tegrals in (39) or (41), unless the pulse is at the center of thepassband, in which case the plateau (or, for short pulses, thepeak) is not changed by the weighting.

Other cases that yield to approximation are those in whichthe frequency offset is large enough to put the whole pulsewell outside the passband of the receiver. Sections II and IVdemonstrated that then the spectrum in the passband iseffectively independent of T and B and FM rate. In suchcases, therefore, we can replace the real pulse by one that hasthe same frequency offset, but has the duration and band-width that the receiver was designed for. The methods devel-oped at the beginning of this section then apply. Fig. 19 isobtained from (39) and (41), using the same receiver as in Fig.17, but using a pulse that is mismatched to the receiver in T,B, TB, and FM rate ab. The details of the responses are differ-ent, but if all we need is the response peak, we can get it bykeeping the frequency offset, and replacing the waveform withthat for which the receiver is intended. The response to that isgiven in Fig. 17. The peak levels indicated by Figs. 17 and 19are nearly the same. If the frequency offset were higher, thedifferences in level would be even smaller. We already knowhow to find the peak levels of Fig. 17 on the back of an enve-lope, and for the situation in Fig. 19, the same simple handcalculation can replace (39) or (41).

X. SUMMARYThe problem of RFI by a linear-FM pulse in a receiver in-

tended for a different linear-FM pulse divides mathematicallyinto two classes. If the FM rates of the two pulses are thesame, the receiver response evolves from a function related to(sin x)/x or, when the receiver is weighted, a number of suchfunctions. When the FM rates are not the same, the response isexpressed in Fresnel integrals, for which the Comu spiral is auseful graphical aid. In this article, the second situation hasbeen treated in full generality. For the equal FM rates, it hasbeen sufficient to take up the case of a pulse and a receiverthat differ only in center frequency. This situation has beenstudied intensively [1] , [15] , [17] because it is that of aradar whose pulse has experienced a constant Doppler shift.What may be new is that, in those RFI situations where thefrequency offset is larger than about 1.5 pulse bandwidths,the receiver response is nearly the same as if there were no FM,because the phase spectrum of the pulse as seen by the receiveris nearly the same as if the pulse had constant frequency, andthe same is true of the energy spectrum. Cosine-on-a-pedestal(for example, Hamming) weighting has an effect that can easilybe taken into account; the details depend on whether theweighting is done in the time or the frequency domain.

For the more general case of unequal FM rates, an exactexpression for the response of an unweighted receiver has beengiven (21), with examples of the waveforms (Figs. 7-12). Theshape and level of the response can be deduced, however, with-out tedious calculation. Cosine-on-a-pedestal weighting, whichresults in the complicated expressions (39) and (41), modifiesthe peak response by an amount that may be easy to deter-

m

wC,)z0a.C,)wcna:

ww

wa:-

20 - -

-0

-20

III!~~~~~~~~~~~~~~~~~~~-60C.

-80 ~ ~ L

-1.0 -0.5 0 0.5 1.0TIME-TO-DURATION RATIO (T.)

(a)

wC,)z0a.C,)wa:a:w

0w

nr

-1.0 -0.5 0 0.5 1.0TIME-TO-DURATION RATIO (r)

(b)Fig. 19. Response of a 10-,is 10-MHz receiver to a 5-,us 15-MHz pulse

whose center frequency is 25 MHz above that of the receiver; TB =100, P = T/2, a = 3, p = 2.5. Weighting (a) pseudo-Hamming in thefrequency domain; (b) Hamming in the time domain. The peaklevels are comparable with those in Fig. 17, where the only mis-match is the frequency offset.

mine; in some cases, the modification is nil. If the pulse liesoutside the receiver passband, the procedure developed forfinding the peak response when the FM rates are equal is alsoapplicable when they differ.

The reference level for all of the discussion is the peak levelthat would result if the received pulse had unit amplitude andno FM, and passed through a noiseless, matched receiver withunity gain at midband. A method for relating the RFI level toreceiver noise is given in Section VI.

245


APPENDIXRESPONSE OF A CONSTANT-FREQUENCY RECEIVER

TO A CONSTANT-FREQUENCY PULSE

For a receiver intended for constant-frequency pulses, theFM rate b in (19) is zero, and the normalizing factor V7B-Tis replaced by T- (footnote on (12)). In place of (21) and(22), one has [23]

2T+P

sin2 rV _- It]

(7rvT)2

T±P IT--P2 2

T-Pt T+P

2 2

sin2 rrvP

(rrvT)2if<

I y(t) 12 = sin2 vTsin2irvT , ifP>T(iTvT)2

_T-P< t<I-2 2

The duration of each oscillatory segment is P or T, whicheveris smaller; the duration of the constant segment is I T-- P 1.

REFERENCES

[11 R. L. Mitchell and A. W. Rihaczek, "Matched-filter responses ofthe linear FM waveform," IEEE Trans. Aerosp. Electron. Syst.,vol. AES4, pp. 417-432, May 1968.

[21 J. Elefant, P. Schiffres, and J. L. Perry, "Spurious sidebands of alinear signal," IEEE Trans. Electromagn. Compat., vol. EMC-7,pp. 375-387, Dec. 1965.

[31 S. A. Cohen, "Generalized response of a linear FM pulse com-pression matched filter," IEEE Trans. Aerosp. Electron. Syst.vol. AES-6, pp. 708-712, Sept. 1970.

[4] P. D. Newhouse, "Simplify EMC design," Microwaves, vol. 9,pp. 59-62, May 1970.

[5] S. A. Cohen, "Cross-ambiguity function for a linear FM pulsecompression radar," IEEE Thans. Electromagn. Compat., vol.EMC-14, pp. 85-91, Aug. 1972.

[61 P. D. Newhouse, "Bounds on the spectrum of a CHIRP pulse,"IEEE Trans. Electromagn. Compat., vol. EMC-15, pp. 27-33,Feb. 1973.

[7] C. E. Cook, "Linear FM signal formats for beacon and com-munication systems," IEEE 7rans. Aerosp. Electron. Syst., vol.AES-10, pp. 471-478, July 1974.

[81 J. J. G. McCue and C. W. Edwards, "Compatibility betweenlinear-FM radars: Two programs," 1976 Microwave EngineersHandbook, Microwave J., Dedham, MA, 1976.

[9] C. E. Cook, "Pulse compression-key to more efficient radartransmission," Proc. IRE, vol. 48, pp. 310-316, Mar. 1960.

[10] J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim,"The theory and design of Chirp radars," Bell Syst. Tech. J.,vol. 39, pp. 745-808, July 1960.

[11] A. Cornu, "Methode nouvelle pour la discussion des problemesde diffraction dans le cas d'une onde cylindrique," Journal dePhysique, series 1, vol. 3, pp. 5-15, 44-52, 1874.

[121 F. A. Jenkins and H. E. White, Fundamentals of Optics. NewYork: McGraw-Hill, 1957.

[13] E. Jahnke and F. Emde, Tables of Functions. New York:Dover 1945.

[14] M. Abramowitz and I. A. Stegun, Handbook of MathematicalFunctions. Washington: U. S. Government Printing Office,1964.

[15] C. E. Cook and M. Bernfeld, Radar Signals. New York: Aca-demic, 1967.

[16] A. W. Rihaczek, Principles of High-Resolution Radar. NewYork: McGraw-Hill, 1969.

[17] H. 0. Ramp and E. R. Wingrove, "Performance degradation oflinear FM-pulse-compression systems due to the Doppler effect,"Proc. IRE, vol. 49, pp. 1693-1694, Nov. 1961.

[18] P. L. Rice, A. G. Longley, K. A. Norton, and A. P. Barsis,"Transmission loss predictions for troposheric communicationcircuits," National Bureau of Standards Tech. Note No. 101,vol. I. Washington, DC: U.S. Government Printing Office,1965 (revised 1967); AD-687820.

[19] M. Skolnik, Ed., Radar Handbook. New York: McGraw-Hill,1970 (Article 20.7).

[201 J. J. G. McCue, "Time-domain vs. frequency-domain weightingof linear-FM pulses," Proc. IEEE, to be published.

[211 R. B. Blackman and J. W. Tukey, The Measurement of PowerSpectra. New York: Dover, 1959 (reprinted from Bell Syst.Tech. J., 1958).

[22] C. L. Temes, "Sidelobe suppression in a range-channel pulse-compression radar," IRE Trans. Military Electron. MIL-6, pp.162-169, Apr. 1962.

[23] J. J. G. McCue, "Radio-frequency interference from linear-FMpulses," Lincoln Laboratory, M.I.T., Lexington, MA, Tech.Rep. 512, July 1974; AD-A003677.

246

radio-frequency interference among linear-fm radars

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