Efficient Pulse-Doppler Processing and Ambiguity Functions...
Transcript of Efficient Pulse-Doppler Processing and Ambiguity Functions...
Efficient Pulse-Doppler Processing and Ambiguity
Functions of Nonuniform Coherent Pulse Trains
Shahzada B. Rasool and Mark R. Bell
School of Electrical and Computer Engineering
Purdue University, West Lafayette, Indiana 47907
Email: {srasool, mrb}@purdue.edu
Abstract—We propose a DFT based pulse Doppler processingreceiver for staggered pulse trains. The proposed receiver is asimple extension of traditional DFT based coherent pulse trainprocessing. We show that P DFT processors are required toprocess the staggered train of pulses as a coherent signal, where P
is the number of available pulse positions in each pulse repetitioninterval (PRI). Thus the complexity of the processing hardwareonly increases linearly with the number of available positions.We also look at the distribution of ambiguity volume aroundthe delay-Doppler map by varying the pulse positions and theselection of pulse shapes.
I. INTRODUCTION
A uniformly spaced coherent train of pulses is commonly
used in radar systems for improving Doppler resolution. In
traditional coherent pulse radars, a basic pulse with good
autocorrelation properties is chosen and transmitted period-
ically at a certain pulse repetition frequency (PRF). The
received echoes are then processed coherently. In principle,
transmitting same signal periodically and then processing the
returns coherently introduces large ambiguities in the matched
filter delay-Doppler response, or the pulse train ambiguity
function, which occur at multiples of pulse repetition interval
along the delay axis and at multiples of PRF along Doppler
axis [1]. This necessitates a design choice to be made since
decreasing PRF would result in longer delay range but will
impact Doppler resolution. Pulse Doppler radars are classified
as low or high PRF radars, depending on the choice made
during the system design.
Different parameters of the pulse train can be varied to op-
timize the ambiguity function and trade resolution properties.
Common variations, leaving aside the amplitude weighting,
include phase coding of individual pulses or employment of
diverse pulses to decrease correlation [1], [2]. One simple
technique that has not been used as commonly is to stagger
the pulses in the pulse train. Pulse staggering is not a new
idea [3], [4] and it is known that pulse staggering can ‘break
up’ the ‘bed of nails’ delay-Doppler response of uniform pulse
trains. It was shown in [4] that in a train of N pulses, proper
pulse staggering can result in an average ambiguity of 1/N2
for all major sidelobes off Doppler axis. An algorithm based
on uniform staggering to obtain these levels was also given.
It is interesting to note that if pulse staggering is restricted
to integer multiples of pulse duration, we can use a ‘Costas
type staggering’ that will also reduce the ambiguities to an
average level of 1/N2. In [5], random staggering of pulses
is considered and average ambiguity function based on the
ensemble of possible pulse sequences is studied. It is found
that ambiguity peaks can be smeared by choosing discrete
staggering with a uniform probability density.
One major reason that pulse staggering is not used widely
to control ambiguities may be the lack of a computationally
efficient receiver for staggered pulses. Uniformly spaced co-
herent pulse train lends itself very well to DFT processing.
Introducing pulse staggering increases the processing com-
plexity and simple DFT based Doppler filtering becomes
difficult. In this work, we show a computationally efficient
approach to generating a bank of Doppler matched filters for
staggered pulse trains. By limiting the staggering to multiples
of pulse duration, we show that the complexity of DFT based
processing increases linearly with the number of possible
staggered positions.
We emphasize here that pulse staggering is different from
PRF staggering. PRF staggering is commonly used in radars to
compensate for blind speeds and redistributing ambiguities [2].
Usually in PRF staggering, a burst is sent and processed at a
constant PRF so that DFT processing can be used. In staggered
pulse trains, each pulse is displaced from its nominal position
in a uniformly spaced pulse train.
II. MATHEMATICAL MODEL AND AMBIGUITY FUNCTION
ANALYSIS
A coherent pulse train signal consisting of N pulses can be
expressed in the following general form:
s(t) =N−1∑
n=0
sn(t − nTr − pnT ), (1)
where sn(t) is the complex envelope of n-th transmitted pulse,
Tr is the nominal pulse repetition interval (PRI), and T is the
PPM offset duration quantization. In this notation, pn adds
an offset relative to the n-th pulse position to allow for pulse
staggering for desired autocorrelation properties. We assume
that sn(t) = 0 ∀ t /∈ [0, T ]. Defining P = max0≤n≤N−1(pn),we require that Tr ≥ 2PT . This makes sure that only one
pulse is transmitted in each PRI along with nonoverlapping
cross ambiguity function contributions for the pulse train.
Letting pn = 0 and sn(t) = s0(t) ∀n, we have a uniformly
spaced pulse train. In this paper, we investigate the ambiguity
properties of pulse trains with staggered pulses in each PRI.
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Tr
T
p0 p1 p2 pN−1· · · p2 t
Fig. 1. A rectangular pulse train with PPM modulation.
Figure 1 shows the structure of a typical staggered pulse train
of rectangular pulses.
We define the ambiguity function of a narrowband signal
s(t) as
χ(τ, ν) =
∫
R
s(t)s∗(t − τ)ej2πνtdt. (2)
Likewise, the cross ambiguity function is defined as
χn,m(τ, ν) =
∫
R
sn(t)s∗m(t − τ)ej2πνtdt. (3)
Substituting (1) in (2) and rearranging the terms
χ(τ, ν) =
N−1∑
n=0
ej2πν(nTr+pnT )χn,n(τ, ν)
+
N−1∑
k=1
N−1−k∑
n=0
ej2πν(nTr+pnT )χn,n+k(τ2, ν)
+N−1∑
k=1
N−1−k∑
n=0
ej2πν[(n+k)Tr+pn+kT ]χn+k,n(τ3, ν), (4)
where τ2 = τ+kTr+(pn+k−pn)T and τ3 = τ−kTr−(pn+k−pn)T . The first summation is the auto ambiguity function
of the component pulses in the pulse train weighted by an
exponential factor. The second and third summations arise
from the cross ambiguity of component pulses. As is clearly
seen from (4), pulse staggering not only affects weighting of
the main lobes along Doppler axis but also shifts the cross
ambiguity functions along the delay axis.
If Tr ≥ 2PT , and sn(t) = 0 ∀t /∈ [0, T ], then the cross
ambiguity and auto ambiguity functions are nonoverlapping.
Then the response for |τ | ≤ T is solely determined by the
first summation in (4) and the other summed terms define the
response at PRF multiples along the delay axis. We can care-
fully choose the pulse staggering such that the peaks of cross
ambiguity functions do not coincide, thus giving an average
low value around τ = ±kTr, k = 1, . . . , N−1. In the absence
of any pulse staggering, as in uniformly spaced coherent pulse
train, the cross ambiguity peaks combine to give a larger
response around τ = ±kTr. Thus pulse staggering can be
used for effective ambiguity reduction along the delay axis.
For example, if {pn}N−1n=0 is chosen as a Costas sequence,
then we shall have good cross correlation properties along
the entire delay axis, in the sense that, for all shifts greater
than pulse duration T , there will be only one overlap out of
N − 1 possible overlaps for a uniformly spaced pulse train.
Thus pulse staggering is an effective technique for removing
ambiguities from the delay axis. Since we cannot eliminate
ambiguity due to basic limitations imposed by the ambiguity
function volume constraint, the removed volume has to appear
somewhere else. As will be shown later, the ambiguities show
up along the Doppler direction.
A. Firing sequences
The staggered pulse train (1) can also be viewed as a pulse
position modulated (PPM) signal, where in each PRI, we trans-
mit a pulse at a certain position. We can construct a PPM firing
sequence represented by N×P array of amplitude coefficients.
For each n-th row, there is only one nonzero coefficient in the
p-th column indicating that the p-th PPM position is occupied
in the n-th pulse. There are many possibilities for selecting
the firing sequences in this fashion. The firing sequence (along
with individual pulses) will determine the ambiguity function
properties. As mentioned earlier, a Costas firing order where
positions are determined according to a Costas sequence will
result in minimum delay axis sidelobes.
III. DFT PROCESSING OF STAGGERED PULSE TRAIN
The success of pulse Doppler radar may be attributed to
an efficient processing of the coherent pulse train which is
based on a DFT implementation of the matched filter for the
pulse train. Since pulse staggering can be effectively used to
redistribute the volume around the delay Doppler plane, one
may ask why pulse staggering is not as commonly used in
radars. Perhaps, this is due to a lack of simple processing
structure as compared to a uniformly spaced pulse train.
Nonuniform spacing does not lend itself easily to simple
DFT processing. In this section, we present a simple DFT
implementation of the matched filter for a staggered pulse
train.
We show that if, for all relevant Doppler shifts ν, the pulse
duration T is small enough such that ej2πνT ≈ 1, which is
true for most radar systems, then there is a simple extension
of traditional DFT processing that can be employed to process
the echoes. The proposed solution requires P DFT processors,
where P is the number of available staggered positions.
A. Proposed DFT processing
Within a scaling factor, the received echo from a point
target for n-th pulse, after demodulation, can be written as
sn(t)ej2πfdt, where fd is the Doppler frequency. To perform
coherent processing for a particular delay, we must collect Nsamples at that delay from last N pulses1. However, due to
pulse staggering, samples do not fall after Tr as in a traditional
pulse train. To solve this problem, we propose the following
solution.
Without loss of generality, assume that the first pulse is
sent at the first position, i.e. p0 = 0. Because the n-th pulse
can be sent at any of P positions, we must account for that
additional degree of freedom when collecting samples from
last N pulses. Therefore, we construct a three dimensional
1We use the word samples loosely here. The samples are actually matchedfilter outputs for n-th pulse.
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N − 1
n
0
Delay
First Pulse
Pposi
tions
N th pulse
n th pulse
Pco
mple
xsa
mple
s
Fig. 2. A conceptual diagram of the accumulation of I and Q samples fromthe matched filter at various delays from N position modulated pulses.
array of I and Q samples: First dimension is the delay, second
dimension is the pulse number and the third is its position.
For a certain delay, we consider all P possible positions and
collect P samples for each pulse corresponding to that delay.
Thus for every delay, we have N×P samples. Note that out of
P samples for nth pulse, we know which sample corresponds
to the transmitted pulse. We set the remaining P − 1 samples
to zero.
Just as in traditional DFT processing, each row of the 3-
dimensional array includes all samples from one pulse in order
of increasing delays. Each column include all the samples
received from the delay associated with that column during
last N pulses. Figure 2 shows a conceptual diagram for the
accumulation of NP matched filter samples for every delay.
N×P double samples (matched filter outputs) of a column,
corresponding to a certain delay, are processed simultaneously
in a bank of DFT processors. For P possible pulse positions,
we need P DFT processors. Each DFT processor is fed with
N samples that are Tr units apart. The p-th DFT processor
output is
Gp(fm) =1
N
N−1∑
n=0
fp(tn)ej2πfmtn p = 1, . . . , P, (5)
where fp(tn) is the sampled value at t = nTr + pT for the
n-th pulse transmitted at position p, tn = nTr, and fm =m
NTr
, m = 0, 1, . . . N − 1. Note that of the P samples, all
but one sample corresponding to nth pulse is nonzero. That is
fp(tn) is nonzero iff n-th pulse was transmitted at position p.
Example: A pulse was sent at position number 3 for pulse
number 2 (Figure 1). Now DFT processor 1, which is being
input all samples due to pulse position number 1 will have its
second input (due to pulse number 2) set to zero. Indeed, all
DFT processors will have their second input set to zero except
DFT processor 3 which is accounting for position 3.
The outputs of P DFT processors are then combined in the
following manner.
G(fm) =P−1∑
p=0
ej2πfmp T Gp(fm). (6)
The exponential factor takes care of the additional delay asso-
ciated with p-th pulse position. Equation (6) yields an output
resembling the ambiguity function and it can be considered as
a matched filter to the PPM pulse train.
Length-N DFT Filter Bank
Delay (M-1)T
Length-N DFT Filter Bank
Delay (M-2)T
Length-N DFT Filter Bank
P (f)
P (f)
P (f)
tn = nTr
tn = nTr
tn = nTr
a∗
n1
a∗
n2
a∗
nM
H̃(1)1
H̃(1)2
H̃(1)N
H̃(2)1
H̃(2)2
H̃(2)N
H̃(M)N
H̃(M)
1
H̃(M)
2
H̃1
H̃2
H̃N
Fig. 3. Coherent processing of a multiphase signal.
B. Staggered pulse train as a multiphase uniform pulse train
In this section, we show how an efficient Doppler processing
for staggered trains follows naturally from Doppler processing
of a multiphase uniform pulse trains. Let s(t) be a complex
baseband uniform pulse train of the form
s(t) =
N−1∑
n=0
M−1∑
m=0
anmp(t − nTr − mT ), (7)
where anm are (generally complex) amplitude weighting co-
efficients, Tr is the interpulse period, and p(t) is a basic pulse
of duration T . We refer to this signal as a multiphase pulse
train (M phases in each pulse).
It is well known that in the presence of wide-sense station-
ary additive noise with PSD Snn(f), the filter that maximizes
the output at time t = NTr has the transfer function
H(f) =S∗(f)e−j2πfNTr
Snn(f), (8)
where S(f) is the Fourier transform of multiphase signal s(t).Substituting (7) in (8), we have
H(f) =P ∗(f)
Snn(f)e−j2πfNTr .
N−1∑
n=0
M−1∑
m=0
a∗nmej2πfnTrej2πfmT
=P ∗(f)
Snn(f)e−j2πfNTr
[M−1∑
m=0
ej2πfmT
N−1∑
n=0
a∗nmej2πfnTr
]
.
This shows that we can process each of the M pulse “phase”
as a separate uniform pulse train and add the results to get
the complete matched filter response for the multiphase pulse
train. Figure 3 shows the corresponding receiver structure.
The multiphase signal (7) is a generalization of a staggered
pulse train. Specifically, for each n = 0, . . . , N −1, we set all
but one anm equal to zero. The resulting discretely staggered
pulse train can also be viewed as a pulse train based on pulse
position modulation (PPM).
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C. Ambiguity function and DFT processor response
We show that the signal ambiguity function and proposed
DFT processing have same response at sampled coordinates
along the delay-Doppler plane. We will show that how equa-
tions (5) and (6) follow naturally, if the transmitted signal is
given by (1). For ease of exposition, we analyze the ambiguity
function of the transmitted signal for zero delay.
Leaving out the scaling factor, the ambiguity function of (1)
at zero relative delay reduces to
χ(0, ν) =N−1∑
n=0
∫
R
sn(t − nTr − pnT )s∗n(t − nTr − pnT )
· ej2πνtdt
=N−1∑
n=0
ej2πν(nTr+pnT )
∫
R
sn(t)s∗n(t)ej2πνtdt
=
N−1∑
n=0
ej2πν(nTr+pnT )χn,n(0, ν).
Recall that pn ∈ {0, 1, . . . , P − 1}, where P may be less
than N . The above integral can be expanded as follows:
χ(0, ν) =
N−1∑
n=0
ej2πν(nTr+pnT )[fp0(tn) + fp2
(tn)+
· · · + fpP−1(tn)] (9)
=
N−1∑
n=0
ej2πν(nTr+pnT )fp(tn). (10)
The notation fpk(tn) means that the matched filter output
for n-th pulse is taken at relative delay (corresponding to
transmitted pulse position) pkT, k ∈ {0, 1, . . . , P − 1}i.e. fpk
(tn) is matched filter output at t = nTr + pkT . Since
we know the position of n-th transmitted pulse already, we
set all the P samples to zero except the one corresponding to
true relative pulse position pn. Equation (9) can be re-written
as
χ(0, ν) =N−1∑
n=0
ej2πν(nTr+pnT )P−1∑
k=0
fpk(tn).
Exchanging the two summations yields
χ(0, ν) =P−1∑
k=0
N−1∑
n=0
ej2πν(nTr+pnT )fpk(tn).
Taking into account that fpk(tn) = 0 except for pk = pn, we
can write
χ(0, ν) =P−1∑
k=0
N−1∑
n=0
ej2πνnTrej2πνpkT fpn(tn)
=P−1∑
k=0
ej2πνpkT
N−1∑
n=0
ej2πνnTrfpn(tn).
Fig. 4. Partial ambiguity surface for a uniformly spaced LFM pulse train of8 pulses and bandwidth time product of 20.
Again noting that Tr is the sampling interval here and defining
νm = mNTr
, m = 0, 1, . . . , N − 1, we can write
χ(0, νm) =P−1∑
k=0
ej2π m
NTrpkT
N−1∑
n=0
ej2π mn
N fpn(tn), (11)
which is the same as Equation (6).
IV. EXAMPLES AND DISCUSSION
In this section we compare the ambiguity surfaces of
staggered pulse trains with a uniformly spaced pulse train.
We highlight the gains and shortcomings of pulse staggering.
For reference purposes, Figure 4 shows the ambiguity
surface for a uniformly spaced LFM pulse train with N = 8and BT = 20. Only the partial ambiguity surface is shown.
In particular, recurrent delay axis sidelobes are not shown
here. For comparison, Figure 5 shows the same AF region
when pulse train is also position modulated, in this case by
using a length 13 Costas sequence. The implication is that
a uniform pulse spacing results in higher sidelobes along
the delay axis for τ = ±mTr, m = 1, . . . , N − 1. The
pulse staggering reduces these higher sidelobes and smears
the contribution of cross ambiguity terms. In particular, for
Costas pulse staggering, there is minimum overlap from cross
terms and this results in a low level pedestal around multiples
of PRI as shown in Figure 5.
Figure 6 shows the AF surface for Costas staggered rectan-
gular pulse train. As expected, we have lower sidelobes along
delay axis and somewhat higher sidelobes along Doppler axis
as compared to a uniformly spaced pulse train. Obviously,
the pulse staggering introduces Doppler ambiguity. It is also
interesting to look at staggered pulse train of Costas pulses. In
Figure 7, we plot the ambiguity surface for a doubly Costas
pulse train of 13 pulses. Since Costas pulses already have good
autocorrelation properties, introducing pulse staggering even
more suppresses the delay axis sidelobes, resulting in a large
ambiguous region along doppler axis as shown in Figure 7.
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(a) Tr = 26T . (b) Tr = 26T .
(c) Tr = 13T . (d) Tr = 13T .
Fig. 6. Partial ambiguity surface figures for rectangular pulse train with pulse positions that satisfy the Costas property in time. N = 13, P = 13.
A. Doppler axis Response
As we have shown, pulse staggering can be used to reduce
delay axis sidelobes considerably. In this section, we investi-
gate how pulse staggering affects the Doppler resolution and
ambiguities. From the ambiguity function analysis of staggered
pulse train, we have
χ(0, ν) =N−1∑
n=0
ej2πν(nTr+pnT )χn,n(0, ν). (12)
Note that if pn = 0, and sn(t) = s0(t) i.e., a uniformly spaced
pulse train, then
|χ(0, ν)| =sin πνNTr
sin πνTr
|χ0,0(0, ν)|. (13)
If we choose pn = n, a stepped staggering, then
|χ(0, ν)| =sinπνN(Tr + T )
sin πν(Tr + T )|χ0,0(0, ν)|. (14)
Usually, Tr >> T , so the ambiguity function shall be very
similar to uniformly spaced pulse train in this case.
To obtain the best possible Doppler axis response, we can
formulate the problem as follows. Minimize (12) as a function
of pn for all ν ≥ 1NTr
. This in itself is an ill-posed problem.
The best we can do is to choose pn such that we have
acceptably lower sidelobes for relevant Dopplers. However,
even that imposition does not help in solving the problem.
Ultimately, its a trade-off between the delay axis sidelobes and
Doppler axis sidelobes; best delay axis response will require
a Costas type staggering and orthogonal signals while best
overall Doppler axis response needs uniform staggering.
V. CONCLUSIONS
We have proposed an efficient pulse Doppler processing
receiver for staggered pulse trains. The proposed receiver is a
simple extension of traditional FFT based coherent pulse train
processing. We showed that P FFT processors are required
to process the staggered train of pulses as a coherent signal,
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(a)
(b)
Fig. 5. Partial AF of Costas position modulated LFM pulse train. BT =
20, Tr = 26T, N = 8, P = 13.
where P is the number of available pulse positions in each
PRI. Thus, compared to the traditional uniformly spaced pulse
train processing, the complexity of the processing hardware
only increases linearly with the number of available positions.
We also looked at the distribution of ambiguity volume around
the delay-Doppler map by varying the pulse positions and the
selection of pulse shapes.
Processing the staggered pulse train with P degrees of free-
dom with the proposed algorithm thus requires O(PN log N)arithmetic operations. We mention here that most efficient
algorithms for calculation of DFT of nonuniformly sampled
data also require O(N log N) arithmetic operations, but with
a larger, precision-dependent, (and dimension-dependent) con-
stant [6]. These algorithms first calculate interpolation coef-
ficients for exponential functions, using some form of least
squares, and then calculate regular FFT on the oversampled
grid. The solution proposed here has the same order of
complexity and uses more convenient and widely used FFT
banks.
(a)
−1
−0.5
0
0.5
1
0
5
10
15
0
0.2
0.4
0.6
0.8
1
τ/T
νNTr
|χ(τ
,ν)|
(b)
Fig. 7. Partial ambiguity surface for a doubly Costas pulse train. The trans-mitted pulse train is a frequency modulated Costas signal. The pulse staggeringalso satisfy the Costas property in time. Tr = 26T, P = 13, N = 13.
ACKNOWLEDGMENT
This work has been funded by the Air Force Office of
Scientific Research (AFOSR) MURI “Adaptive Waveform
Design for Full Spectral Dominance.”
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[5] M. Kaveh and G. R. Cooper, “Average ambiguity function for a randomlystaggered pulse sequence,” IEEE Trans. Aerosp. Electron. Syst., pp. 410–413, May 1976.
[6] L. Greengard and J.-Y. Lee, “Accelerating the nonuniform fast fouriertransform,” SIAM Review, vol. 46, no. 3, pp. 443–454, 2004.
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