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0.10 Pulse shaping and receive filtering  (Page 7/13)

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In pulseshape2.m , examine the effect of using different oversampling rates M . Try M= 1, 5, 100.

Change pulseshape2.m so that the data sequence is drawn from the alphabet ± 1 , ± 3 , ± 5 . Can you visually identify the correct values in thepulse shaped signal?

In pulseshape2.m , examine the effect of using sinc approximations of different lengths L . Try L= 1, 5, 100, 1000.

In pulseshape2.m , examine the effect of adding noise to the received signal x . Try M atlab commands randn and rand . How large can the noise be and still allow the data to be recognizable?

The goal is to design a frequency-division multiplexed (FDM) system with a square root raised cosine as thetransmitter pulse shape. The symbol period is T = 137 msec. The design uses T / 4 sampling, pulse lengths of 8 T , and a rolloff factor of 0.9, but it does not work, since only three modulated carrier signalsfit into the alloted bandwidth without multiuser interference. Five are needed. What parameters in the design would you change and why?

Using the code from [link] , examine the effects of adding noise in pulseshape2.m . Does the same amount of noise in the 6-level datahave more or less effect than in the 2-level data?

Modify pulseshape2.m to include the effect of a nonunity channel. Try both a highpass channel and abandpass channel. Which appears worse? What are reasonable criteria for “better” and “worse” in this context?

A M atlab question: In pulseshape2.m , examine the effect of using the filtfilt command for the convolution instead of the filter command. Can you figure out why the results are different?

Another M atlab question: In pulseshape2.m , examine the effect of usingthe conv command for the convolution instead of the filter command. Can you figure out how to make this work?

Nyquist pulses

Consider a multilevel signal drawn from a finite alphabet with values w ( k T ) , where T is the sampling interval. Let p ( t ) be the impulse response of the linear filter representing the pulse shape. The signal just afterpulse shaping is

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

where w a ( t ) is the pulse train signal [link] .

The corresponding output of the received filter is

y ( t ) = w a ( t ) * p ( t ) * h c ( t ) * h R ( t ) ,

as depicted in [link] , where h c ( t ) is the impulse response of the channel and h R ( t ) is the impulse response of the receive filter. Let h e q u i v ( t ) = p ( t ) * h c ( t ) * h R ( t ) be the overall equivalent impulse response. Thenthe equivalent overall frequency response (i.e.,  F { h equiv ( t ) } ) is

H equiv ( f ) = P ( f ) H c ( f ) H R ( f ) .

One approach would be to attempt to choose H R ( f ) so that H e q u i v ( f ) attained a desired value (such as a pure delay) forall f . This would be a specification of the impulseresponse h e q u i v ( t ) at all t , since the Fourier transform is invertible.But such a distortionless response is unnecessary, since it does not reallymatter what happens between samples, but only what happens at the sample instants.In other words, as long as the eye is open, the transmitted symbols are recoverableby sampling at the correct times. In general, if the pulse shape is zeroat all integer multiples of k T but one, then it can have any shape in between without causing intersymbol interference.

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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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