Efficient Pulse-Doppler Processing and Ambiguity Functions ...nehorai/MURI/publications/Bell_Radar.pdf · autocorrelation properties is chosen and transmitted period-ically at a

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.”

REFERENCES

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[2] N. Levanon and E. Mozeson, Radar Signals. John Wiley & Sons, 2004.[3] A. Rihaczek, “Radar resolution properties of pulse trains,” Proceedings

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environment,” Master’s thesis, Massachusetts Inst. of Tech., Camridge,MA., Jun. 1962.

[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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