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0.10 Pulse shaping and receive filtering  (Page 11/13)

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F - 1 { W ( f ) e - j 2 π f T d } = w ( t - T d ) .

Combining these two transform pairs yields

F - 1 { ( W ( f ) e j 2 π f T d ) * } = w * ( - ( t - T d ) ) = w * ( T d - t ) .

Thus, when g ( t ) is real,

F - 1 { k ( G ( f ) e j 2 π f τ ) * } = k g * ( τ - t ) = k g ( τ - t ) .

Observe that this filter has the following characteristics:

  • This filter results in the maximum signal-to-noise ratio of v 2 ( t ) / P w at the time instant t = τ for a noise signal with a flat power spectral density.
  • Because the impulse response of this filter is a scaled time reversal of the pulse shape p ( t ) , it is said to be “matched”to the pulse shape, and is called a “matched filter.”
  • The shape of the magnitude spectrum of the matched filter H R ( f ) is the same as the magnitude spectrum G ( f ) .
  • The shape of the magnitude spectrum of G ( f ) is the same as the shape of the frequency response of thepulse shape P ( f ) for a broadband m ( k T ) , as in "Spectrum of the Pulse: Spectrum of the Signal" .
  • The matched filter for any filter with an even symmetric (about some t ) time-limited impulse response is a delayed replica of that filter.The minimum delay is the upper limit of the time-limited range of the impulse response.

The following code allows hands-on exploration of this theoretical result. The pulse shape is defined by the variable ps (the default is the sinc function srrc(L,0,M) for L=10 ). The receive filter is analogously defined by recfilt . As usual, the symbol alphabet is easily specified by the pam subroutine, and the system operates at an oversampling rate M . The noise is specified in n , and the ratio of the powers is output as powv/poww . Observe that, for any pulse shape, the ratio of the powers is maximized when the receive filteris the same as the pulse shape (the fliplr command carries out the time reversal). This holds no matterwhat the noise, no matter what the symbol alphabet, and no matter what the pulse shape.

N=2^15; m=pam(N,2,1);                    % 2-PAM signal of length N M=10; mup=zeros(1,N*M); mup(1:M:N*M)=m;  % oversample by ML=10; ps=srrc(L,0,M);                    % define pulse shape ps=ps/sqrt(sum(ps.^2));                  % and normalizen=0.5*randn(size(mup));                  % noise g=filter(ps,1,mup);                      % convolve ps with datarecfilt=srrc(L,0,M);                     % receive filter H sub R recfilt=recfilt/sqrt(sum(recfilt.^2));   % normalize the pulse shapev=filter(fliplr(recfilt),1,g);           % matched filter with data w=filter(fliplr(recfilt),1,n);           % matched filter with noisevdownsamp=v(1:M:N*M);                    % downsample to symbol rate wdownsamp=w(1:M:N*M);                    % downsample to symbol ratepowv=pow(vdownsamp);                     % power in downsampled v poww=pow(wdownsamp);                     % power in downsampled wpowv/poww                                % ratio
matchfilt.m test of SNR maximization (download file)

In general, when the noise power spectral density is flat (i.e.,  P n ( f ) = η ), the output of the matched filter may be realized by correlating the input to the matched filterwith the pulse shape p ( t ) . To see this, recall that the output is described bythe convolution
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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