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CS3282 Section 6 6.1 BMGC/ 25/08/06

University of Manchester

School of Computer Science

CS3282: Digital Communications 05-06

Section 6: Inter-symbol interference and pulse shaping

Rectangular symbols are not suitable for transmitting data at the highest possible bit-rates over band-limited channels. A rectangular pulse or any other shaped pulse which is time-limited (i.e. is non-zero

for a finite period of time, say T seconds) will require infinite bandwidth if it is not to be distorted,

possibly unrecognizably. A symbol with finite bandwidth must have, in theory, infinite time duration.

Although using a symbol which really does exist from t = - to t = + to send a single 1 or a 0 (or

maybe two or three of them if we are using multi-level signaling) may seem impossible in practice, we

have to keep this in mind as an ideal and produce approximations to this form of signaling. The pulses

we use in practice may not actually go on, and back in time for ever. But they must definitely be non-

zero for considerably more than T seconds when the signaling rate is 1/T symbols per second.

Inevitably this means that one symbol will run into the previous and next symbol, and significantly

affect several more besides. The result could be inter-symbol interference (ISI) where the data

conveyed by one symbol causes the data of other symbols to be misinterpreted. Although we cannot

avoid the overlap of symbols in the time-domain, we must find ways of making sure that the data

carried by the symbols is not affected by this overlap. The solution to this challenge lies in pulse

shaping which means that we must carefully choose the time-domain shape of the symbols and hence

their spectral shape.

A convenient way of generating symbols with the time-domain shape we require is to generate an

impulse of the appropriate strength for each symbol and then to shape this impulse by passing it

through a shaping filter. The impulse-response of the shaping filter is the symbol shape we wish to

launch into the channel. An FIR digital filter followed by a digital to analogue converter will do this

job nicely. The channel will inevitably affect the shape of the symbol and noise will be added.

At the receiver, to optimize the detection process, filtering tasks are required as illustrated below.

A 3A -A

n t

Equal-

iserT T

Samples at

intervals T

ChannelMatched

filterShaping

Filter

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CS3282 Section 6 6.2 BMGC/ 25/08/06

Inter-symbol interference (ISI) can occur due to the ringing of one symbol into the next. However, ISI

can be avoided if the transmitter's pulse shaping filter shapes the symbols so that zero-crossings at the

output of the receiving filters occur T seconds, 2T seconds, and so on after (and before) the centre of

the symbol. So when we sample at t=0, T, 2T, etc, we only see the centre of one symbol, all the other

symbols being zero at those instants. This is nice in theory and possible to a fair degree in practice. Ifwe combine the transmitter's shaping filter, the channel and the receiving filters (i.e. the matched filter

and the equaliser) into a single frequency-response HN(()) say, then our goal is achieved if HN(()) is

a Nyquist frequency-response". To be a classed as a 'Nyquist frequency-response', as well as band-

limiting from -1/T to 1/T Hz, HN((f)) must have a form of odd-symmetry about 1/(2T ) in that:

( ) TffortconsfTHfH NN 1||0tan/1))((* =+

Two purely real frequency-responses satisfying this property are shown below. It may be shown

(Exercise 6.2 below) that this property guarantees that the impulse-response corresponding to HN((f)) ,i.e. its inverse Fourier transform, has zero crossings at t = T, 2T, 3T, .

|HN(())|

0.5

1

-1/T -1/2T 0 1/4T 1/2T 3/4T 1/T

The brick-wall filter: