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. Samples at A 3A -A intervals T Shaping Matched Equal- Channel T T Filter filter iser n(t) 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. If we 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((ƒ)) say, then our goal is achieved if HN((ƒ)) 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: * H N (( f )) + H N ()1/T − f = cons tant for 0 ≤ | f | ≤ 1 T 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((ƒ))| 1 0.5 ƒ -1/T -1/2T 0 1/4T 1/2T 3/4T 1/T 1 : | f |< 1/(2T) The “brick-wall” filter: H B (( f )) = 0 : | f |≥ 1/(2T) has the property of having a 'Nyquist frequency-response' and we already know that its impulse response is a sinc function with zero crossings at t = ±T, ±2T, etc. The 'brick-wall' frequency- response achieves the required zero-crossing property with the minimum possible bandwidth, but is difficult to deal with because the side-lobes of its "sinc" impulse response do not die away very quickly. To design a pulse-shaping filter with a reasonable approximation to this 'brick-wall' frequency-response, we would need a very long impulse-response and hence a high order FIR digital filter. Also, assuming we could implement a good enough approximation, if there is the slightest error in the timing of our sampling point, significant ISI will occur. CS3282 Section 6 6.3 BMGC/ 25/08/06 Raised cosine frequency response A commonly used family of Nyquist frequency-responses is a range of “raised cosine” frequency- responses parameterized by r (sometimes called α) and defined by the following formula: 1 : | f | ≤ (1− r) /(2T) H rc (( f )) = 0.5[1+ cos(πT{| f | −(1− r) /(2T)}/ r)] : (1− r) /(2T) ≤ | f | ≤ (1+ r) /(2T) 0 : (1+ r) /(2T) <| f | The impulse-response of a filter with such a frequency-response may be shown to be: sin c(t /T )cos(πrt /T) sin(πx) h(t) = where sinc(x) = lim T − 4r 2t 2 /T πx When r=0, this becomes the “brick-wall” filter mentioned earlier with the narrowest bandwidth of all the family: -1/(2T) to 1/(2T) Hz. When r = 0.5, Hrc(f) is as shown below, assuming T=0.001 seconds so that the symbol rate 1/T = 1 kHz.. From the general formula above, its spectrum is: 1 : | f | ≤ 250 Hz H rc (( f )) = 0.5[1+ cos(2π{| f | −250}/1000)] : 250 ≤ | f | ≤ 750 0 : 750 < | f | 1.2 1 50%RC spectrum 0.8 0.6 Modulus 0.4 0.2 0 Frequency -1000 -750 -500 -250 0 250 500 750 1000 In the graph above, drawn using "Microsoft Excel" , note the "odd-symmetry" about f=500 Hz for f >0 and about f = -500 Hz = - 1/(2T) for f <0. It may be shown that Hrc((f)) + H*rc(( 1000 - f ) ) = 1. CS3282 Section 6 6.4 BMGC/ 25/08/06 When r=1, Hrc((ƒ)) becomes a pure raised cosine shaped frequency-response with no “flat-top” and the widest bandwidth of the family: -1/T to 1/T Hz. With 1/T=1 kHz, the bandwidth would be -1000 to 1000 Hz. The impulse-responses hrc(t) corresponding to Hrc((ƒ)) with r=0 and r=1 are shown below taking T to be 10 seconds. Such graphs are easily drawn using a spreadsheet such as Microsoft Excel or MATLAB. Notice the reduction of side-lobe ripples when r=1, but this is at the expense of a doubling of the bandwidth. A raised cosine spectrally shaped pulse with a given value of r is called a "100r per- cent RC symbol" and has band-width from -(1+r)/T Hz to (1+r)/T. When r=0.5, we have a 50% RC spectrally shaped symbol (see spectrum above) with bandwidth from -750 Hz to 750 Hz if 1/T = 1 kHz.. This is 50% more than the absolute minimum band-width needed to avoid ISI with the techniques discussed up to now. The minimum bandwidth would be achieved with a 0% RC symbol which has a "brick-wall" spectrum from -1/(2T) to +1/(2T) Hz. Apart from the practical difficulties of generating a pulse with such a spectrum, the disadvantage of the brick-wall spectrum , as mentioned above, is that its time-domain shape is a "sinc" pulse which does not die away quickly enough for our liking. The 100% RC spectrum (r=1) produces a time-domain pulse which dies away much faster. So by increasing r from 0 to 1 we improve the rate of dying away at the expense of extra bandwidth. 0.1 0.08 0.06 h(t) for T=10, r=0 & 1 0.04 0.02 0 -40 -30 -20 -10 0 10 20 30 40 -0.02 t A good way to implement a shaping filter is to use an FIR digital filter with multiplier coefficients equal to samples of h(t). A succession of shaped pulses generated by a the transmission of a continuous bit-stream are often viewed on an oscilloscope as “eye-diagrams”. The oscilloscope is CS3282 Section 6 6.5 BMGC/ 25/08/06 triggered approximately at the beginning of each pulse, and an open eye is hopefully seen as each new shaped pulse is superimposed on the previous pulses. If noise is added, the eye is seen to close, making threshold comparisons for detection difficult. Remember that HN((ƒ)) is the required overall response of transmitting filter, channel and receiving filters. Within the receiver an equalising filter is introduced to cancel out the filtering effect of the channel. We discuss this later. If the overall symbol pulse shape as seen at the output of the receiving filters is to have a raised cosine spectrum and the receiving filters include a matched filter, it would not be correct for the transmitter to send symbols with raised cosine spectra. If we did send such a symbol its matched filter would have gain-response equal to the magnitude spectrum of the pulse, and the resulting output would have this spectrum squared.