 complexity (i.e., more cheaply). For example, considering a 4-kHz speech signal, without oversampling, the anti-aliasing filter should sharply cut off all frequencies above 4 kHz. Figure 1.9(a) shows an eighth-order Chebyshev II lowpass filter. It can be seen that the filter roll-off is very sharp, and the corresponding phase response is nonlinear toward the higher frequencies. These variations in phase can cause undesirable effects in audio signal. By contrast, if the oversampling technique is employed, with B=64, a lowpass filter with a cutoff frequency of 256 kHz can be used. Figure 1.9(b) shows a very simple first-order Chebyshev II lowpass Pulse Code Modulation Bode Diagram Bode Diagram Frequency (Hz) Frequency (Hz) Fig. 1.9 Analogue filter at (a) Nyquist limit (no oversampling) and (b) 64x Nyquist limit. filter. As it is shown, the roll-off is much gentler but the phase response is remarkably better. Needless to mention, the implementation of a first-order filter is much easier compared to an eighth-order filter. Another advantage of oversampling is the noise reduction. In fact it is shown by Watkinson (1994) that by taking N samples from the same quantity with a random noise signal, the noise is reduced by factor 1/N when the samples are averaged. Over- sampling can also be used to reduce the quantization noise. A detailed description is given in the next section. After all, the signal will be again down-sampled to the target rate. In fact, oversampling enables digital processing of the signal. Mention should also be made of the impact of the sampling pulses. In fact, in contrast with impulses, the practical sample pulses have limited amplitude and finite duration. If x(t) represents the message signal, the sampled waveform, xs(t), can be written as (1.11) where p(t) represents the sampling pulse. Comparing this with equation 1.3, it can be seen that the sampling waveform has appeared as a new term in equation 1.11. The convolution of this term in the time domain corresponds to a multiplication of its spe