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The transmit and receive filter designs rely on the assumption that all other parts of the system are working well.For instance, the modulation and demodulation blocks have been removed from [link] , and the assumption is that they are perfect:the receiver knows the correct frequency and phase of the carrier. Similarly, the downsamplingblock has been removed, and the assumption is that this is implemented so that the decision deviceis a fully synchronized sampler and quantizer. Chapter [link] examines methods of satisfying these synchronization needs, but for now, they areassumed to be met. In addition, the channel is assumed benign.

Spectrum of the pulse: spectrum of the signal

Probably the major reason that the design of the pulse shape is important is because the shape of thespectrum of the pulse shape dictates the spectrum of the whole transmission.To see this, suppose that the discrete-time message sequence w ( k T ) is turned into the analog pulse train

w a ( t ) = k w ( k T ) δ ( t - k T ) = w ( k T ) t = k T 0 t k T

as it enters the pulse shaping filter. The response of the filter, with impulse response p ( t ) , is the convolution

x ( t ) = w a ( t ) * p ( t ) ,

as suggested by [link] . Since the Fourier transform of a convolution is theproduct of the Fourier transforms (from [link] ), it follows that

X ( f ) = W a ( f ) P ( f ) .

Though W a ( f ) is unknown, this shows that X ( f ) can have no energy at frequencies where P ( f ) vanishes. Whatever the spectrum of the message, thetransmission is directly scaled by P ( f ) . In particular, the support of the spectrum X ( f ) is no larger than the support of the spectrum P ( f ) .

As a concrete example, consider the pulse shape used in Chapter [link] , which is the “blip” function shown in the top plot of [link] . The spectrum of this pulse shapecan readily be calculated using freqz , and this is shown in the bottom plot of [link] . It is a kind of mild lowpass filter.The following code generates a sequence of N 4-PAM symbols, and then carries out the pulse shaping using the filter command.

N=1000; m=pam(N,4,5);                    % 4-level signal of length N M=10; mup=zeros(1,N*M); mup(1:M:N*M)=m;  % oversample by Mps=hamming(M);                           % blip pulse shape of width M x=filter(ps,1,mup);                      % convolve pulse shape with data
pulsespec.m spectrum of a pulse shape (download file)
The Hamming  pulse shape and its magnitude spectrum.
The Hamming pulse shape and its magnitude spectrum.

The program pulsespec.m represents the “continuous-time” or analog signal by oversampling both the data sequenceand the pulse shape by a factor of M . This technique was discussed in [link] , where an “analog” sine wave sine100hzsamp.m was represented digitally at two sampling intervals, a slowsymbol interval T = M T s and a faster rate (shorter interval) T s representing the underlying analog signal. The pulse shape ps is a blip created by the hamming function, and this is also oversampled at the same rate.The convolution of the oversampled pulse shape and the oversampled data sequenceis accomplished by the filter command. Typical output is shown in the top plot of [link] , which shows the “analog” signal over a time interval of about 25 symbols.Observe that the individual pulse shapes are clearly visible, one scaled blip for each symbol.

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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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