606 Journal of Radars Vol. 3
One of the most widely used pulse
compression waveforms is LFM waveform
[3]
or
the chirp waveform. The frequency of LFM
linearly increases (up-chirp) or decreases (down-
chirp) with time. Therefore, the waveform is
feasible to be achieved by the analog circuits.
The mathematical equation for LFM is:
()
()
2
c
rect exp exp 2
t
jkt j ft
T
ππ
⎛⎞
⎟
⎜
⎟
⎜
⎟
⎜
⎝⎠
(7)
where
k is the rate of frequency increase or the
chirp rate. The instantaneous frequency of the
baseband waveform is a function of time:
()
t =
kt
()
rect / .tT
In the beginning of the pulse,
()
/2 ( )/2.
TkT−=−
At the end of the pulse,
)
/2 ( )/2.
TkT=
As a result, the bandwidth of
LFM is
kT. The time-bandwidth product of
LFM is
kT
2
. According to Eq. (6), the LFM
waveform can simply increase its chirp rate
k in
order to obtain the high range resolution and
keep a wide pulse width at the same time.
The ambiguity function is used to evaluate
how the returned pulse is distorted due to the
receiver matched filter:
()
() ( )
()
*
dd
,exp2d
sts t j ft tχτ τ π
∞
−∞
=−
∫
(8)
When the Doppler frequency is not
considered, the ambiguity function reduces to
the autocorrelation of
s(t), i.e.
()
() ( )
*
,0 dsts t tχτ τ
∞
−∞
=−
∫
(9)
Fig. 3 shows an LFM waveform with a 3
μs
pulse width and 64 MHz bandwidth. The auto-
correlation of the waveform produces a sharp
mainlobe with serious sidelobes. The Peak to
SideLobe Ratio (PSLR) of LFM is
13.2 dB.
−
These sidelobes are highly responsible for
blocking nearby weak reflections from small
targets or for blurring SAR images.
One traditional way to suppress the
sidelobes is using windowing procedures. Harris
[4]
provides an extensive list of windows and their
properties. Coming along with the sidelobe
suppression, the windowing procedure brings a
side effect of increased mainlobe width, which
also degrades the range resolution in target
detection. Compared with LFM, the stepped
frequency waveform
[5]
increases (or decreases) its
frequency discretely by a fixed frequency incre-
ment (or decrement) each time. The Frequency
Jump Burst (FJB)
[6]
divides LFM into multiple
sub-pulses in the time domain, and transmits
each separately. Consequently, these discrete
versions of LFM perform similarly as the
original LFM waveform in range detection.
Aiming at suppressing range sidelobes, some
researchers have selected nonlinear frequency
modulated waveforms. Griffiths
[7]
generates
nonlinear frequency waveform using equation:
() () ()
12
st s t s t=+
(10)
A central linear FM component
()
1
st
and
the additional higher FM rate portion
()
2
st
are plotted in Fig. 4. The PSLR of his proposed
waveform is
50.4 dB
−
, which is much lower than
the traditional LFM waveform. Instead of the
piecewise nonlinear FM waveform, Witte
[8]
further studies a continuous nonlinear FM wave-
form to suppress the sidelobes to below
70 dB.
−
Fig. 3 Baseband LFM waveform and its autocorrelation