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

Recall the overall block diagram of the system in [link] , where it was assumed that the portion of the system from upconversion (to passband)to final downconversion (back to baseband) is done perfectly and that the channel is just the identity.Thus, the central portion of the systemis effectively transparent (except for the intrusion of noise). This simplifies the systemto the baseband model in [link] .

The task is to design an appropriate pair of filters: a pulse shape for the transmitter, anda receive filter that is matched to the pulse shape and the presumed noise description.It is not crucial that the transmitted signal itself have no intersymbol interference.Rather, the signal after the receive filter should have no ISI. Thus, it is not the pulse shape that should satisfy the Nyquist pulsecondition, but the combination of the pulse shape and the receive filter.

The receive filter should simultaneously

1. (i) allow no intersymbol interference at the receiver, and
2. (ii) maximize the signal-to-noise ratio.

Hence, it is the convolution of the pulse shape and the receive filter that should be a Nyquist pulse, and the receive filter should bematched to the pulse shape. Considering candidate pulse shapes that are both symmetric and evenabout some time $t$ , the associated matched filter (modulo the associated delay) is the same as the candidate pulse shape.What symmetric pulse shapes, when convolved with themselves, form a Nyquist pulse?Previous sections examined several Nyquist pulse shapes, the rectangle, the sinc, and the raised cosine.When convolved with themselves, do any of these shapes remain Nyquist?

For a rectangle pulse shape and its rectangular matched filter, the convolution isa triangle that is twice as wide as the original pulse shape. With precise timing, (so that the sample occursat the peak in the middle), this triangular pulse shape is also a Nyquist pulse. This exact situation will beconsidered in detail in [link] .

The convolution of a sinc function with itself is more easily viewed in the frequency domainas the point-by-point square of the transform. Since the transform of the sinc is a rectangle,its square is a rectangle as well. The inverse transform is consequently still a sinc,and is therefore a Nyquist pulse.

The raised cosine pulse fails. Its square in the frequency domain does not retain theodd symmetry around the band edges, and the convolution of the raised cosine with itselfdoes not retain its original zero crossings. But the raised cosine was the preferred Nyquist pulsebecause it conserves bandwidth effectively and because its impulse response dies away quickly.One possibility is to define a new pulse shape that is the square root of the raised cosine(the square root is taken in the frequency domain, not the time domain).This is called the square-root raised cosine filter (SRRC). By definition, the square in frequency of the SRRC (which is the raised cosine) is a Nyquist pulse.

The time domain description of the SRRC pulse is found by taking the inverse Fourier transform of thesquare root of the spectrum of the raised cosine pulse. The answer is a bit complicated:

Though the SRRC is not itself a Nyquist pulse, the convolution in time of two SRRCs is a Nyquist pulse. The square root raised cosineis the most commonly used pulse in bandwidth-constrained communication systems.