Weighting simulations in the time domain were considered to shape the time-compressed pulse waveform. The digital input radar data were 32 bit I,Q, and simulated data from a point target, imaged by a Seasat-A type system. Weighting functions tested include stepped-amplitude distributions, (with 1 through 5 steps), and the cosine-squared plus pedestal distribution. Effects treated include mainlobe ratio, signal to noise ratio, and nearest sidelobe suppression

Dicenzo, A.

English

NASA Technical Reports Server

N79-11270
EFFECT OF WEIGHTING ON TIME SIDELOBE SUPPRESSION*
ALAN DI CENZO
- JET PROPULSION LABORATORY
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91103
-•-CEDING PAGE BLANK KQT. F&$-jffl
SUMMARY
Weighting is a well known technique for shaping the compress*.i puls= wavefonr
it. radar processing. Usually, weighting is applied to the transfer function
»md has the effect of sacrificing the time nainlobe width (resolution) in
exchange f T decreasing the height of the neighboring siielobes. This paper
reports on simulations of weighting in the time dom-.iu, as used to shape the
ti're-compressed pulse wavefomi. The digital input radar data is 32 bit
I,Q, and simulates data from a point target as imaged by a Seasat-A type
system. Weighting functions tested include stepped-amplitude distributions,
(with 1 through 5 steps), anc the cosine-squared plus prdestal distribution.
Effects treated include mtdnlobe broadening, peak energy reduction, the
intigr&ted sidelobe ratio, signal to noise ratio, and nearest sidelc/be
suppression.
1.0 INTRODUCTION
Pulse compression in radar is commonly accomplished through matched filtering.
A radar point target return s(t) is correlated against a T-shifted version
of its conjugate s*(t) over a time interval T, to yield an output at tine T:
/•T/a
J-T/2
S(T) =J s(t)s*(t - T) dt. (1)
-T/2
* This paper presents the results of one phase of research carried out at the
Jet Propulsion Laboratory, California Institute of Technology, under Contract
No. HAS7-100, sponsored by the National Aeronautics and Space Administration.
V-8-1
https://ntrs.nasa.gov/search.jsp?R=19790003099 2020-03-22T02:21:53+00:00Z
The estimate S(T) yields a prediction of the return from a target at the
•location specified by T. The function |S(T)| usually has multiple peaks, j
and thus creates detection ambiguities when there is large signal dynamic
range. For example, if s(t) is a linear FM signal, S(T) approximates e sine ;
function with high 1-13.2 db) sidelobes. . .
The relation (l) may also be expressed in the frequency domain. Set
s (t) = s*(-t).
Then (l) becomes
/•T/2
s(t) sr
•J-T/2
s®s r (T) .
(T - t) dt (2)
If capital letters denote fourier transforms, then (2) Implies
S(w' = £(w) • S(w). (3)r
The inverse transform of F(w) is of course S(T). In (3) we can multiply
S (w) by a weighting function W(w) to obtain a new output S :
S^w) = S(w) • Sr(w) ••W(w). (li) j
•|
.]
The inverse transform, s..(r)» is now only an approximaticn to S(T)- The goal of i
frequency domain weighting is to select a function W in (It) such that s.. (T) is * j
a good local approximation to §(T), but has lower sidelobes.
It is known that weighting of the tine domain signal can produce similar
effects, particularly if the signal it- linear FM. In time weighting the
weighting function is multiplied with the return signal
' V-8-2
s^ t) = s(t) wft)
or with the reference function s'Ut)
sj(t) = s*(t) w(t), |
prior to filtering. li
!iii
Tliis paper reports on simulations o*" time-weighting, performed OP. digits". ft
SAR data of the type expected from satellites such as SEASAT-A. j]
ii
It is shown that use of a cosir.e-squared-plus-pedestal tine-domain weighting '!
scheme can result in sidelobe suppression down to -36 db, with mair.lobe '[
broadening of less thsn 30$ for the simulated data.
Section 2 gives a brief description of the frequency domain weighting rpprosch,
which is tb'in used to motivate the ideas on time weighting presented in
section 3. Finally, section 3 contain;; th? numerical results extrapolated
from the simulations.
2.0 FREQUENCY DOMAIN WEIGHTING
Let the signal givpn by eq. (5) be receivad at the radar receiver at tine t.
<
exp j J (fot + Y- M f or -T <_ t <_ T
(
 ° ' (5).
0 elsewhere.
The matched filter response at time T is
/•-* , ,-*). (
 e
3
^ •""> .-JU"
•'-
'dt. (6)
T
Ij
.T-T *i. . t 1 /. o \ i
V-8-3
By elementary calculation, we have
-Jf T sin (bit - |-
8(0- Me °._L__JL (T)
Usually, the range of interest is such that T is small compared to T, so that
the approximation
S(T) = 2Te~
JfoT sin (bTT)
bTr (8)
a
= 2T sine (bTr) x (phase factor)
ic valid. Ignoring the phase factor, we see that S(T) peaks at T = 0. There-
after, sidelobes 13.2 db below the peak intensity occur near T = ± 3w/2bT.
These sidelobes can mask weaker peaks corresponding to neighboring targets,
as shown in Figure 1.
By slightly mismatching the filter response, (and hence sacrificing signal to
noise ratio and nainlobe width), the sidelobe levels of the filter output can
be decreased. The process is most easily understood when mismatching is
Fig. 1. Sidrlobes Masking Weak Peaks
V-3-A
•••v.
:uv:ot:ii\H:-.l o«l l\v weinht.iii« ,in t i n - 1'ivnuoncy iU<r.j:«ln [*•]. I tf l : ' • « r
tho oulim'. 01* a in;iti-l><»i) f i l l - i - r . rtn fonrior U-iiiisform i:-. r.ivon l\v
a(w) = I - \p i-.)::nwt) v i l .
l
c<
I We n:'-:-. x> no t '(w) = 0 for |w| > IV':'). t\v tlio l'o',irii:r f .h i t ' t . t.hi'or.-m Uu% rur.x-t '.on
r"fa-
ni\ l\nirii?i ' t r.'r.i:M\'!-m t ' . ivoti l\v
oi-t- 11 it'. !>n inti>-,i-x' ;i!-..i :i i :: iirln l.i':ir.v 1 .
1
I
Tim.'- :'• ( I ) i:' t I'.o iM ' i f - . h in l i:i:.t i - tu- . l : ' i l t . or .niUnil w i ' . h p.-'.ii'i-.i o.'!*.^:'. :'.i;','i'!'iti:i>-v>i
ill :l t.ii'.n: v' 1'tV-ot, of ' i j / l * . Till- ivllv-:: i lVi? : - .>-: i l i^ . i by :i ''•i.'t.or .M' :i /'.'. in t hi>
v-.-i-.'-o wlu'i'o .-. = A. llio :•. hi.' riiiu'! i . M i ill (.'5), t i n ' i i : ' - ( w ^ i.-. :i rci't.i-.n.-.'.U' r i i i i t - t . :.-r.
(i-xot-pt. IVM' n -.•hi'.'t. rsu'toi- ihu: t .»< Uu' i'li:i:u- .'t't'r.t't, ) . T.'iK x uj*. n = .1 , v,v ,<bt.!iin
fivm (S):
14
8
!$
r
^
'*
V-8-5
/
.i.
\v-
\Here B is proportional to T, r>y virtue of (8)). When the three sine functions
are nlded, a new "quasi-sinc" function results with lower sidelobes. (See fig-
ure 2).
Thus weighting by W(w) = [1 + a cos 2nw/B] can lower the time sidelobes. In
the radar case, we vould actually be confronted with a superposition c-f sine
functions at the unweighted filter output. Each sine function vould be centered
on its own resolution element. However, by the linearity of the fourier trans-
form operation, one weighting function applied to the superposition spectrum
serves to suppress the sidelobe? of ali sine functions til. once.
3.0 TIME DOMAIN WEIGHTING FOR TIME SIDEr.OBE SUPPRESSION
Time domain weighting is accomplished by multiplying the return signal or time
reference function by a properly designed weighting function. Since the weights
can easily be incorporated as part of the time doinain reference function, the
technique is a promising one for application to real-time spacecraft 3a board
processors, ihe effect of time weighting on time output can often be easily
determined [l],
.1
TJj
•'<!
For example, let the radar signal s(t) be the linear FM signal given in (5).
Then, in a rough manner of speaking, there is a one-to-one correspondence between
time and frequency. That is, each time interval At corresponds to a frequency
interval aw in the spectrum. Hence a time weighting function W(t) would have
about the same effect as the frequency weighting function W(t = w). Thus a first
order estimate of the effects of time weighting by W(t) may be obtained through
— = SUPERPOSITION OF
SINC FUNCTIONS
2. Superposition of Sine Function and Scaled Echo
V-8-b
I
frequency weighting with W(w). More precisely, it is known that for the
linear FM signal (5), the filter time output g(t) resulting from a tine
weighting w(t), satisfies
ls(t)| = J-57 W(T) exp {Jbtrl dt1C- (13)
For further details, see [l] p. l8Y.
In SAR processing, the azimuth aperture is generated by u sensor moving linearly
with constant velocity over the aperture length. Thus, to a reasonable
approximation, the azimuth matched filter function is a linear FM chirp, and the
above time weighting approximations (e.g. (13)) aoply. However, the a:-.imuth chirp
is a sampled (discrete) function, t'.nd the results mo smnewhut different.
Therefore, to discover the effects of weighting vjth discrete time functions,
actual simulations were performed. The weightings were applied along the azimuth
direction of simulated ?KASAT-A SAK data. Kach datT inpr.t point war. quantised
to U bits, as was each reference and weighting coefficient. The ground spacing
between adjacent data input points was around ') meters. A total of -iC96 input
points was used to process a b-look output data pouit. Ihus each look required
filtering of 10214 points, or 102'( separate weighting coefficients. The
spacing between output data points wan around 16 meters. The simulated input •
data was designed to represent the return from a point target so that the effects
of weighting could be easily observed. The output image was ]6 (range) by 512
(azimuth) points.
The weighting functions selected for simulation were i.he stepped amplitude
distributions and the cosine - squared plvu; pedestal, function [3], ['*]» [5].
The stepped i'tnplitude distributions are given in table 1 for 1 through 5 steps.
(Gee also figure 3)- These distributions were chosen for their optima]ity in
antenna pattern adjustment, and their ease of imploiaentat.ion [3]. Koto that
T is proportional to N = 10f?-'i, the number of weights-.
V-8-7
TABLE 1
STEPPED AMPLITUDE TIME WEIGHTING FUNCTIONS
1
0.
0.
0.
0.
5
35
25
300
• *
0.
0.
0.
0.
.
5
35
25
225
• •
0.
c.
0.
.
.
30
25
235
0.
0.
. • • •
• * * *
. •••
25
170 0.070
1
1
1
1
1
0.
0.
0.
0.
.
5t
625
78
72
• • •
• • •
0.350
0.5^
0.5't 0.36 0.18
3.
 £
It is defined
w(t) = H
l.o
b,T
J_
°N
ULl
for
T
(14)
-\ '
where
H = pedestal height (0 <_ H < l)
T = pulse timewidth.
•
For the simulation, we select H = 0.05 to approximate Hamming weighting, which
produces the lowest sidelobes attainable ,/ith this type weighting in the
frequency domain [5]. The resulting weighting is similar to the 'j'ayior
approximation to the physically unrealizable optimum Dolph - Chebyshev weighting
[2], [6], and represents a practical approach for digital processing.
Table i gives the total energy in decibels in the 3 center lines about the peak.
(By "line" is meant a range line of 16 data points centered around the azimuth
peak. See figure l».)
TABLE 2
PEAK ENERGY AND INTEGRATED SIDELOBE RATIO
Weighting Technique Total Energy in 3 Center Lines Peak Energy
Rectangular 103.109 100.067
2 step amplitude 103. '4 58 99.66?
3 step amplitude 103. 51^ 99-^15
»4 step amplitude 103. 51*0 99.359
5 step amplitude 103.552 99.362
cos".+ pedestal 103. ''78 98. 7 W
Note: Total energy in 512 lines is 203.671 db for each type of weighting.
t
V-S-9 f,
00
o
o
o
0
o
o
o
RANGE
16
PEAK
o
0
o
o
o
0
° - AZIMUTH
O
o
Fig.
Since the total energy in each 16 x 512 image was normalized to 103.671 db,
the energy in the center lines reveals in a rough way how mu^h energy is left
over for sidelobe generation. This data can of course be used to compute the
"integrated sidelobe ratio" in decibels, defined by
j;
( itotal energy outside muinlobe; — : : :—rtotal energy in mainiobe
(3 center lines)
The results are summarized in table 3, for both U and 32 bit quantisation, and
give a fair estimate of the 2-dinensional integrated sidelobe ratio.
TABLE 3
INTEGRATED SIDELOBE RATIO, DB FOR VARIOUS WEIGHTINGS
Weighting
Rectangular
(no weighting)
2 - step amplitude
3 - step amplitude
b - step amplitude
5 - step amplitude
2
cos + pedestal
Integrated Sidelobe Ratio
1) bits
- 12.1 db
- 13.3 db
- 114.5 db
- 15.6 db
- 15.7 db
- 17.3 db
32 bits
- 12.8 db
- Il4.3 db
- 16.1 db
- 17.8 db
- 18.0 db
- 21.7 db
V-8-10
I .
I * , - . - • ! ' —«=.— '- r, *-7^\J-
'.'able C also gives an indiotvtion or loss in signal to noise ratio in terms of
peak energy, i'ince ea>%h weighted output has the sume total energy, Uie p«*:ik
onors.'.' in the second column iiulicut.es the loss in PNK rathor sioi'uj-iitojy. Thf?
Ki'ojttest loy.; (3.319 db) relative to tiie unweighted case oecurr. with eo.">' +
Table -i is iuu?vl to derive the nninlobo broaiicninp i*esultins t'ro::i vt»i|r.hti:ic,. A:1.
>
o\pi»»:tcd, the \.ort-.t, broadonint; it- seen to occur with cos" + pod vci»:htin>:, w ' o l i
o n l y H S vlh ::uppre:ir..ion -l6 meter:; from the potik. Tu I l i i ?
i .-•• .!e;;:'.'t>"ie»i to .\0 motors.
the < ilb ro •o
The value:- in TtiMo !i can he used to derive t'!\v.-ly vr-^'iso- valuer for the
••«.•:•.,* Int . ion achieved w i t h each woit:htinf: scheme, in the -following s.annor. Tho
output. i M l t e r t'l.tiction i^ mode.1Jed na a sine Ainotiou wli ioh is plf.sel.y track'^d
:!! l.he ratif'o of intoivst by a quadrat ic . Ur-in;: La^ran.-u'1 a intorpolat i . -n p> iy-
i i o n - ' a l , w i t h the peak a:ut .? nearest nol«shbors a;; input ' point:'. , the dii-.taree to
the J «il% na in lobo threshold is calcuhited. The results -\re shown in table ?.
TA'^LK •'.
inirn\::;v-ro:i AT NKAHK-.'-T NKJ^H'.VK, IM«
\\oif.htinj*. Te
:? step aopl i tu>ie
3 step -uiiplitndo
•i step :oplitUvie
') step air.plitnde
^
 tOvus" •*• pedestal
m i n
a
- m a x
o
6
V-8-ll
TABLE 5
RESOLUTION ACHIEVED WITH WEIGHTING
Weighting
Rectangular
(No weighting)
2 - step amplitude
3 - step amplitude
'» - step.amplitude
5 - step amplitude
?
Cos + pedestal
Resolution
2'. m
27.2 m
28.8 m
28.8 m
28.8 m
30.1* m
Ki'.ially, table 6 gives the side Lobe suppression act willy achieved with each
weighting scheme. Each technique generated 2 sidelobes of generally different
heights; hence the min and max suppressions are both tabulated. The best
•.•>
suppression is achieved by cot;" + pedestal weighting, with a -3o db meusure-
;:ient j:t the highest sidelobe. (The sideiobe asymmetry results from tho
true peak occurring borween two data output points).
TABLE 6
NEAREST SIDKLOBE SUPPRESSION, DH
Weighting Tech:iique
Koctangular
2 step amplitude
3 step amp] itudo.
l» step amplitude
5 :itep amplitude
o
Cos" + pedestal
Sidelode Suppression, db
min uax
20 -2!»
22 . JS
26 29
30 3?
29 33-
V-8-12
U'eii:ht i:i»i in tho tine do.i::iin i'or Lino si dejobe- suppression pivr.ent.s an attru
t Lvo al ternative to frequency domain weight ing, especially if Mno-.i.-n.-i i-t
^
anirnvil i : prices t- hiji 2o T.O be ncooaplishod. The :;imul.'itiQnr. show tlint. i-r>p' •»
ivJo.jlul weir.htin t; achieved 36 db sidelobe suppression with n Sf.'K J. ':-.*•. oT u:n
l.!i:in i.5,vlb, and resolution loss of u'nnmd ?5I*.
The -rithor would like to t.h:uik Dr. i\ Wu Tor valuable' srl'tvara u:-d ji:>.:i::::;.ir>
,-!:-.,i sij.!»o H. !He:vso.n, W. Arena, and V. Tyr^e for helpful siiiv:estionf. Ii 'v.-il
aid wuj furnishon by B. L'arkan.
V-8-13
REFERENCES
[l] Cook, C. and Bernfeld, M. , Radar Signals, nn Introduction to Theory and
Application, Academic Press, llev York, 19^ 7, chapter 7.
[2] Dolph, C. L., "A Current Distribution for Broadside Arrays Which Optimizes
the Relationship Between Beam Width and Sidelobe Level," Proc. IRE, vol. 3
pp. 335-3')3, June,
[3] Murakami, T. , Optimum Waveform Study for Coherent Pulse Doppler, RCA /inal
Report, prepared for Office of Ilaval Research, Contract Ncnr i»£!»9(0r )(x),
Feb. >8, 1965, AD6M391.
[1*1 Hash, R. J. , "Stepped Amplitude Distributions," IEEE Trans., vol. AP-12,
pp. 515-516, July, 196U.
[5] Skoinick, N. I., ed. . Radar. Handbook. McGraw-Hill, 1970.
[6] Taylor, T. T. , "Design of Line Source Antennas for Narrow Beamwidth and
Low Sidelobes," IRE Trans., vol. AP-3, pp. 16-28, January, 1955-
V-8-U


Effect of weighting on time sidelobe suppression

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