IEEE - Aerospace and Electronic Systems - January 2020 - 35

function (PAF) [3], [4]. This explains the name given to
this approach. When the pulse duration tp is short relative
to the PRI, namely tp < Tr =2 then the PAF is given by






xM T ðt; nÞ ¼ jx ðt; nÞj  sinðMTr pnÞ 
r
 M sinðT pnÞ 

(2)

r

where x ðt; nÞ is the Woodward's ambiguity function [4]
of a single pulse, t is delay, and n will represent Doppler
shift (together with fD ). The Doppler resolution is given
by the cut of (2) at zero delay. Setting t ¼ 0 in (2) yields






xM T ð0; nÞ ¼ jx ð0; nÞj  sinðMTr pnÞ :
r
 M sinðT pnÞ 

(3)

r

One of the properties of the AF is that its zero-delay
cut is independent of any phase or frequency modulation,
but depends only on the magnitude of the pulse's complex
envelopeuðtÞ,

Z

 tp =2


2
jx ð0; nÞj ¼ 
juðtÞj expðj2pntÞ dt:

 Àtp =2

(4)

In our example the pulse is phase coded, but its amplitude is constant, which results in
Á
 À
sin tp p n 
:
jxð0; nÞj ¼ 
tp p n 

(5)

Figure 1.
Coherent I and Q sampling of a Doppler shifted pulse train.

JANUARY 2020

Inserting (5) in (3) yields the expected Doppler
response of a coherent train of M identical phase coded
(or frequency modulated) pulses,
Á
 À


 
xMT ð0; nÞ ¼ sin tp p n 
r
 t pn 
p



 sinðMT r pnÞ 


 MsinðT pnÞ :
r

(6)

Using a decibel scale, Figure 3 displays (6), for the
case of M ¼ 32 ; Tr =tp ¼ 5 ) MT r =tp ¼ 160. For
such a case, the first term on the right-hand side (r.h.s.) of
(6) hardly influences the Doppler response. Its effect is
seen by the attenuation of 0.6 dB of the recurrent lobes at
f MT r ¼ Æ 32. Note that the Doppler resolution (defined
as the separation between the mainlobe's peak and the first
null) is Dn ¼ Df ¼ 1=ðM Tr Þ ¼ 1=CPI.
Nonuniform weight windows, such as Hamming,
Dolph-Chebyshev, [5], [6] can help reduce the sidelobes
of the Doppler response, with a penalty of wider mainlobe.
To avoid transmitting pulses of different amplitudes, the
weight window appears only at the receiver side, causing
mismatch, hence the SNR loss. Typical loss is about
1.5 dB (e.g., in the Hamming case). Figure 4 displays the
normalized Doppler response when a Hamming window
is used. Note the reduced Doppler sidelobes and the doubling of the Doppler resolution.
Figures 3 and 4 show that the Doppler resolution
achieved by the processor outlined in Figure 2 is either
Dn ¼ 1=ðMT r Þ, without interpulse amplitude weight window, or Dn ¼ 2=ðMT r Þwhen a Hamming weight window
is used. To learn the two-dimensional range-Doppler
response, we need to define a more specific waveform. To
keep the example simple, we will add Barker 13 binary
phase-coding to the previous example. With this choice,
the duration of a code element is tb ¼ tp =13. We will also
add Doppler shift. Figure 5 displays the noise-free
phase evolution along six (out of 32) received pulses.
From a phase change during one PRI, Df ¼ 0:5893 rad,
we can derive the normalized dimensionless Doppler
shift
ntb ¼ 0:5893=ð130pÞ ¼ 0:0014; nTr ¼ 0:0938,
and finally nMT r ¼ 3. The last normalization is the
one used in the delay-Doppler responses shown in
Figures 6 and 7.

IEEE A&E SYSTEMS MAGAZINE

35
IEEE - Aerospace and Electronic Systems - January 2020

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