Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 47

Example 2.3: The chirp signal

Consider an example of the chirp signal with duration T = 600 μ s, and chirp
parameter K =0 . 0001 in units of (MHz) 2 . Figure 2.32 shows the real and
imaginary parts of p ( t )= e jπKt 2 and the autocorrelation R ( τ ) . Note how
sharp the autocorrelation is, compared to the pulse duration. Figure 2.33
shows the autocorrelation of the chirp together with the autocorrelation of
a rectangular pulse of duration 600 μ s. The height of p ( t ) is scaled such that
the autocorrelation has a peak value of unity.
In this example the bandwidth is F m = KT = 60 kHz. The magnitude
of the Fourier transform P ( jω ) is shown in Fig. 2.34 for 0 ≤ f ≤ 500 kHz
(with passband normalized to have a maximum of 0 dB). Note that the
passband extends from 0 to about F m / 2 = 30 kHz, as expected from the
theory. The time-bandwidth product in this example is

F m T =36 .

So the pulse compression ratio is F m T/ 2=18 . This means that the duration
of the main lobe of the autocorrelation plot is about 18 times smaller than
the pulse width of 600 μ s.

As a practical matter, it is desirable to use the real causal signal

cos( πKt 2 )0

≤ t ≤ T

p ( t )= (2 . 74)

0 otherwise

asthechirp. Inthiscasetheautocorrelationcontinuestobeverysharp,asin
the above example. This is demonstrated in Fig. 2.35, which shows the pulse
and the autocorrelation ( p ( t ) is scaled such that the autocorrelation has a peak
value of unity). In this example we once again have chosen T = 600 μ s, and chirp
parameter K =0 . 0001 (MHz) 2 . Note, however, that the autocorrelation plot in
Fig. 2.35 is even sharper than the one in Fig. 2.32. This is intuitively expected
from the fact that the bandwidth is wider, as seen from the plot of

| P ( j 2 πf ) | .

To see why the bandwidth is higher, write

cos( πKt 2 )=0 . 5 e jπKt 2 +0 . 5 e −jπKt 2 .

The first term has a frequency sweep from 0 to KT, and the second term has
a sweep from 0 to − KT. So there are dominant frequency components in the

range 60 to 60 kHz in this case. In the previous example

of a complex chirp it was only from

− KT to KT ,thatis, −

− 30 to 30 kHz.