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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.
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 is turned into the analog pulse train
as it enters the pulse shaping filter. The response of the filter, with impulse response , is the convolution
as suggested by [link] . Since the Fourier transform of a convolution is theproduct of the Fourier transforms (from [link] ), it follows that
Though is unknown, this shows that can have no energy at frequencies where vanishes. Whatever the spectrum of the message, thetransmission is directly scaled by . In particular, the support of the spectrum is no larger than the support of the spectrum .
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 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
and a faster rate (shorter interval)
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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