n Fig. 1.6 Example of sampling process: (a) continuous analogue signal; (b) discrete analogue PAM signal. Pulse Amplitude Modulation (PAM) is a direct result of sampling a signal. It is a pulse modulation technique in which the amplitude of the sample pulses is in direct proportion to the message signal amplitude. Therefore, the result of the sampling process sometimes is called a PAM-sampled signal. Figure 1.6 shows the sampling process. From a mathematical point of view, the sampling process can be considered to be the multiplication of the message signal by a train of Dirac impulses. A Dirac impulse is math- ematically defined as a very narrow, very high pulse, represented by S: (1.2) otherwise If the input speech signal, x(t), is sampled by a sampling function, p(t), which is a train of pulses with period of T, the sampling result, xp(t), would be xp(t)=x(t).p(t) (1.3) On the other hand, because S(t - to) is 0 for all values of t except to, we have (1.4) The preceding equation shows the sampling property of a Dirac impulse. Using this, equation 1.3 can be written as (1.5) It is known that multiplication of two waveforms in the time domain is equivalent to the convolution of their frequency domain transforms: (1.6) Pulse Code Modulation Using the Fourier transform tables we have: P(ja)) (1.7) where Ws is the sampling frequency (cos=27). It is also known that convolving any waveform with a shifted Dirac impulse will shift the waveform. This means that X(jw)*d(w-wo)=X(j(w-w0)) (1.8) Now equation 1.6 can be written as: X(Ja)= (1.9) This equation is important because it clearly shows the effect of sampling in the frequency domain. According to this, the sampling causes a periodic repetition of the message spec- trum in the frequency domain. The period of this repetition is the same as the sampling period. Figure 1.7 shows the effect of sampling on the power spectrum of a message signal sampled by a train of Dirac impulses with sampling frequency Ws. Nyquist Sampling Theorem and Aliasing Distortion It can be unders