Page 301 - Fundamentals of Radar Signal Processing

(4.133)

The total pulse length is τ = Nτ . Individual subpulses are often referred to as
c
chips. Phase coded waveforms are divided into biphase codes and polyphase
codes. A biphase code has only two possible choices for the phase state ϕ ,
n
typically 0 and π; a polyphase code has more than two phase states. There are
several common subcategories of each. Figure 4.1c was an example of a
biphase-coded waveform.

The matched filter output for a phase coded pulse is derived in detail in
Levanon and Mozeson (2004); the result is now summarized. Denote the
sequence of complex amplitudes of the individual pulse chips x (t) of Eq.
n
(4.133) as {A } = {exp[jϕ ]}. Express the time variable t in terms of the chip
n
n
duration τ and an offset η, as t = kτ + η, 0 ≤ η < τ . The matched filter output,
c
c
c
which is just the autocorrelation of x(t), is

(4.134)

where s [k] is the discrete autocorrelation of the complex amplitude sequence
A
{A }. Equation (4.134) shows that s (t) takes on the value s [k] at t = kτ , and is
n
c
A
x
linearly interpolated (in the complex plane) between adjacent samples. Thus,
the matched filter output can be determined by computing the autocorrelation of
the amplitude sequence and interpolating between those values. One
consequence of this result and the fact that the {A } have unit magnitude is that
n
the peak value of the autocorrelation will always be s (0) = N.
x

4.10.1 Biphase Codes
The most common biphase codes in radar are the Barker codes. Barker codes
are a specific set of biphase sequences that have a maximum sidelobe magnitude
of 1 at the matched filter output and therefore attain an N : 1 ratio of the peak to
the highest sidelobe. A low-frequency Barker coded waveform for N = 13 is
shown in Fig. 4.43. The phase switches are visible at τ = 5τ , 7τ , 9τ , 10τ , 11 τ ,
c
c
c
c
c
and 12τ . Because there are only two phase states, the waveform is often
c
represented by a diagram such as the one shown in Fig. 4.44, using either “+”
and “–” symbols as shown, or +1 and –1 symbols. Note that biphase codes do
not necessarily change phase state at every subpulse transition.