0.8 Stuff happens  (Page 5/14)

The output of the lowpass filter in the demodulator is a signal with the spectrum shownin [link] (drawn using plotspec(x3,1/M) ). The spectrum in [link] should compare quite closely tothat of the transmitter baseband in [link] , as indeed it does.It is easy to check the effectiveness of the lowpass filter design by attempting to use a variety of different lowpassfilters, as suggested in [link] .

Recall the discussion in [link] of matching two scaled pulse shapes.Viewing the pulse shape as a kind of marker, it is reasonable to correlate the pulse shape with the received signal in orderto locate the pulses. (More justification for this procedure is forthcoming in Chapter [link] .) This appears in [link] as the block labelled “pulse correlation filter.”The code in idsys.m implements this using the filter command to carry out the correlation (rather than the xcorr function), though the choice was a matter of convenience.(Refer to corrvsconv.m in Exercise  [link] to see how the two functions are related.)

The first 4 $M$ samples of the resulting signal y are plotted in [link] (via plot(y(1:4*M)) ). The first three symbols of the message (i.e.,  m(1:3) ) are $-3$ , 3, and $-3$ , and [link] shows why it is best to take the samples at indices $125+kM$ . The initial delay of 125 corresponds to half the length of the lowpass filter( 0.5*fl ) plus half the length of the correlator filter ( 0.5*M ) plus half a symbol period ( 0.5*M ), which accounts for the delay from the start of each pulse to its peak.

Selecting this delay and the associated downsampling are accomplished in the code % downsample to symbol rate z=y(0.5*fl+M:M:end); in idsys.m , which recovers the $T$ -spaced samples z . With reference to [link] , the parameter $l$ in the downsampling block is 125.

A revealing extension of [link] is to plot the oversampled waveform y for the complete transmission in order to see if the subsequent peaks of the pulsesoccur at regular intervals precisely on source alphabet symbol values, as we would hope.However, even for small messages (such as the wiener jingle), squeezing such a figure onto one graph makes a detailedvisual examination of the peaks fruitless. This is precisely why we plotted [link] —to see the detailed timing information for the first few symbols.

One idea is to plot the next four symbol periods on top of the first four by shifting the start of the second blockto time zero. Continuing this approach throughout the data recordmimics the behavior of a well-adjusted oscilloscope that triggers at the same point in each symbol group.This operation can be implemented in M atlab by first determining the maximum number of groups of 4*M samples that fit inside the vector y from the l th sample on. Let ul=floor((length(y)-l-1)/(4*M)); then the reshape command can be used to form a matrix with 4*M rows and ul columns. This is easily plotted using plot(reshape(y(l:ul*4*M+124),4*M,ul)) and the result is shown in [link] . Note that the first element plotted in [link] is the lth element of [link] . This type of figure, called an eye diagram , is commonly used in practice as an aid introubleshooting. Eye diagrams will also be used routinely in succeeding chapters.