Page 269 - Fundamentals of Radar Signal Processing
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proper matched filtering achieves a signal processing gain of G  = βτ compared
                                                                                          sp
               to a simple pulse of the same Rayleigh resolution.
                     Unlike the simple pulse case, the matched filter output for the LFM pulse
               exhibits  a  sidelobe  structure. Figure 4.26  expands  the  central  portion  of Fig.
               4.25b, showing the distinctly sinc-like mainlobe and first few sidelobes. This
               should  not  be  surprising:  the  waveform  spectrum X(F)  (Fig.  4.24b)  is
               approximately  a  rectangle  of  width β  Hz.  Consequently,  the  spectrum  of  the

                                                2
               matched filter output, |X(F)| , will also be approximately a rectangle of width β.
               The  time-domain  output  of  the  matched  filter  is  therefore  expected  to  be
               approximately a sinc function with a Rayleigh resolution of 1/β seconds.



































               FIGURE 4.26   Expanded view of central portion of Fig. 4.25b.



                     To  summarize,  the  LFM  waveform  enables  separate  control  of  pulse
               energy (through its duration) and range resolution (through its swept bandwidth).
               The possibility of pulse compression is created by the use of matched filters.
               The output of the matched filter is not a replica of the transmitted waveform x(t),
               but  of  its  autocorrelation  function s (t).  Therefore,  if  a  waveform  can  be
                                                             x
               designed that has a long duration but a narrowly concentrated autocorrelation,

               both  good  range  resolution  and  good  energy  can  be  obtained  simultaneously.
               This in turn is accomplished by modulating a long pulse to spread its bandwidth
               beyond the usual 1/τ. Since the spectrum of the autocorrelation function is just
               the squared magnitude of the waveform spectrum, a spectrum spread over β Hz
               will tend to produce a filter output with most of its energy concentrated in a
               mainlobe  of  about  1/β  seconds  duration.  The  linear  FM  pulse  is  the  first
               example of such a waveform, but phase coded waveforms will provide more

               examples of this approach.
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