0.10 Pulse shaping and receive filtering  (Page 8/13)

The condition that one pulse does not interfere with other pulses at subsequent $T$ -spaced sample instants is formalized by saying that $h_{NYQ}\left(t\right)$ is a Nyquist pulse if there is a $\tau$ such that

for all integers $k$ , where $c$ is some nonzero constant. The timing offset $\tau$ in [link] will need to be found by the receiver.

A rectangular pulse with time-width less than $T$ certainly satisfies [link] , as does any pulse shape that is less than $T$ wide. But the bandwidth of the rectangular pulse(and other narrow pulse shapes such as the Hamming pulse shape) may be too wide. Narrow pulse shapesdo not utilize the spectrum efficiently. But if just any wide shape is used(such as the multiple- $T$ -wide Hamming pulses), then the eye may close. What is needed is a signalthat is wide in time (and narrow in frequency) that also fulfills the Nyquist condition [link] .

One possibility is the sinc pulse

with $f_0=1/T$ . This has the narrowest possible spectrum, since it forms a rectangle in frequency(i.e., the frequency response of a lowpass filter). Assuming that the clocks at the transmitter and receiverare synchronized so that $\tau =0$ , the sinc pulse is Nyquist because $h_{sinc}\left(0\right)=1$ and

for all integers $k\ne 0$ . But there are several problems with the sinc pulse:

• It has infinite duration. In any real implementation, the pulse must be truncated.
• It is noncausal. In any real implementation, the truncated pulse must be delayed.
• The steep band edges of the rectangular frequency function $H_{sinc}\left(f\right)$ are difficult to approximate.
• The sinc function $\sin \left(t\right)/t$ decays slowly, at a rate proportional to $1/t$ .

The slow decay (recall the plot of the sinc function in [link] ) means that samples that are far apart in timecan interact with each other when there are even modest clock synchronization errors.

Fortunately, it is not necessary to choose between a pulse shape that is constrained to lie within a singlesymbol period $T$ and the slowly decaying sinc. While the sinc has the smallest dispersionin frequency, there are other pulse shapes that are narrower in time and yet are only a little wider in frequency.Trading off time and frequency behaviors can be tricky. Desirable pulse shapes

1. (i) have appropriate zero crossings (i.e., they are Nyquist pulses),
2. (ii) have sloped band edges in the frequency domain, and
3. (iii) decay more rapidly in the time domain (compared with the sinc), while maintaining a narrow profile in thefrequency domain.

One popular option is called the raised cosine-rolloff (or raised cosine) filter. It is defined by its Fourier transform

where

• $B$ is the absolute bandwidth,
• $f_0$ is the 6 dB bandwidth, equal to $\frac{1}{2T}$ , one half the symbol rate,
• $f_{\Delta }=B-f_0$ , and
• $f_1=f_0-f_{\Delta }$ .

The corresponding time domain function is

Define the rolloff factor $\beta =f_{\Delta }/f_0$ . [link] shows the magnitude spectrum $H_{RC}\left(f\right)$ of the raised cosine filter in the bottom and the associated time response $h_{RC}\left(t\right)$ on the top, for a variety of rolloff factors. With $T=\frac{1}{2f_0}$ , $h_{RC}\left(kT\right)$ has a factor $\sin \left(\pi k\right)/\pi k$ which is zero for all integer $k\ne 0$ . Hence the raised cosine is a Nyquist pulse. In fact, as $\beta \to 0$ , $h_{RC}\left(t\right)$ becomes a sinc.