tood from both equation 1.9 and Figure 1.7 that if there is no overlap between the main message signal spectrum and its replicas, the original message can be retrieved using a lowpass filter to eliminate the repeated replicas. However, if the sampling frequency used to sample a message signal with a maximum bandwidth of Af is lower than 2 X Af, the overlap will take place between the replicas, SO the message signal cannot be X(jw) P(jw) X(jw) Fig. 1.7 Effect of sampling in the frequency domain. Chapter 1 Voice Transmission X(jw) P(jw) -20s -Ws X(jw) Fig. 1.8 Aliasing distortion due -0m famis to slow sampling. (ws-wm) retrieved by filtering or any other means. The distortion that is caused by slow sampling is known as aliasing distortion. Figure 1.8 shows the aliasing distortion. The fact that efficient sampling needs a sampling frequency at least twice the maximum frequency of the message signal was first proven by Nyquist in 1928 and is known as the Nyquist sampling theory. Obviously, for a message to be presented by sample pulses, the power spectrum should be band limited; otherwise, sampling cannot be used to represent the signal; mathematically speaking, there is no sampling frequency that satisfies the Nyquist criteria. Respectively, an anti-aliasing filter is a filter that is used before a signal is sampled. This is to restrict the bandwidth of the signal to satisfy the Nyquist sampling theorem. Unfortunately, in contrast to digital filters, it is very difficult to implement sharp cut-off analogue anti-aliasing filters. Therefore, a technique called oversampling is used. A signal is called oversampled if it is sampled with a frequency higher than the Nyquist sampling fre- quency limit. The oversampling ratio, B, is defined as (1.10) where again Ws represents the sampling frequency and Wm is the highest frequency of the signal (signal bandwidth). By using the oversampling technique, the transient range of the anti-aliasing filter can be extended. This implies that the filter can be implemented with a reduced