0.8 Stuff happens  (Page 10/14)

Just as the receiver oscillator need not be fully synchronized with the transmitter oscillator, the symbol clock at the receiverneed not be properly synchronized with the transmitter symbol period clock. Effectively, the receiver must choose when to samplethe received signal based on its best guess as to the phase and frequency of the symbol clock at the transmitter.In the ideal case, the delay between the receipt of the start of the signal and thefirst sample time was readily calculated using the parameter $l$ . But $l$ cannot be known in a real system because the “first sample” depends, for instance,on when the receiver is turned on. Thus, the phase of the symbol clock is unknown at the receiver.This impairment is simulated in impsys.m using the timing offset parameter toper , which is specified as a percentage of the symbol period.Subsequent samples are taken at positive integer multiples of the presumed sampling interval.If this interval is incorrect, then the subsequent sample times will also be incorrect. The final impairmentis specified by the “symbol period offset,” which sets the symbol period at thetransmitter to so less than that at the receiver.

Using impsys.m , it is now easy to investigate how each impairment degrades the performance of the system.

Additive channel noise

Whenever the channel noise is greater than half the gap between two adjacentsymbols in the source constellation, a symbol error may occur.For the constellation of $\pm 1$ s and $\pm 3$ s, if a noise sample has magnitude larger than 1,then the output of the quantizer may be erroneous.

Suppose that a white, broadband noise is added to the transmitted signal. The spectrumof the received signal, which is plotted in [link] (via plotspec(nv,1/rM) ), shows a nonzero noise floorcompared with the ideal (noise-free) spectrum in [link] . A noise gain factor of cng=0.6 leads to a cluster variance of about $0.02$ and no symbol errors. A noise gain of cng=2 leads to a cluster variance of about $0.2$ and results in approximately 2% symbol errors. When there are 10% symbol errors, the reconstructed textbecomes undecipherable (for the particular coding used in letters2pam.m and pam2letters.m ). Thus, as should be expected, the performance of the systemdegrades as the noise is increased. It is worthwhile taking a closer look to see exactlywhat goes wrong.

The eye diagram for the noisy received signal is shown in [link] , which should be compared with the noise-free eye diagram in [link] . This is plotted using the M atlab commands: ul=floor((length(x3)-124)/(4*rM)); plot(reshape(x3(125:ul*4*rM+124),4*rM,ul)). Hopefully, it is clear from the noisy eye diagram that it would be very difficult to correctly decode the symbolsdirectly from this signal. Fortunately, the correlation filter reducesthe noise significantly, as shown in the eye diagram in [link] . (This is plotted as before, substituting y for x3 .) Comparing Figures [link] and [link] closely, observe that the whole of the latter is shifted over in time by about 50 samples. This is theeffect of the time delay of the correlator filter, which is half the length of the filter.Clearly, it is much easier to correctly decode using y than using x3 , though the pulse shapes of [link] are still blurred when compared with the ideal pulse shapesin [link] .