Naturally Sampled Pulse Amplitude Modulated (PAM) Signal , Instantaneous or Idealy Sampled

Naturally Sampled Pulse Amplitude Modulated (PAM) Signal
We have already discussed natural sampling. This natural sampling is basically pulse amplitude modulation (PAM). Therefore, it is called naturally sampled PAM signal.
Thus, time-domain representation of a naturally-sampled PAM signal will be given as
                                                            EQUATION                                         …(3.71)
and the frequency-domain representation, i.e. frequency-spectrum of a naturally-sampled PAM signal will be given as
                                                            EQUATION                                         …(3.72)
3.13.2 Instantaneous or Idealy Sampled Pulse Amplitude Modulated (PAM) Signal
            We have discussed ideal or instantaneous sampling in article 3.9. This instantaneous sampling is basically pulse amplitude modulation (PAM). Therefore, it is called ideally or instantaneously sampled PAM signal.
Thus, time-domain representation of a ideally or instantaneously sampled PAM signal will be given as
                                                            EQUATION                                         …(3.73)
and the frequency-domain representation i.e., frequency-spectrum of a ideally or instantaneously sampled PAM signal will be given as
                                                            EQUATION                                         …(3.74)
3.13.3 Transmission Bandwidth in Pulse Amplitude Modulation (PAM)
In a pulse amplitude modulated (PAM) signal the pulse duration ‘ is considered to be very very small in comparison to time period (i.e., sampling period) T_s between any two samples i.e.,
…(3.75)
Now, if the maximum frequency in the modulating signal x(t) is f_m, then according to sampling theorem, the sampling frequency f_s must be equal to or higher than the Nyquist rate, i.e.,
or                                                                     f_s ≥ 2f_m                                                   …(3.76)
or
or
But according to equation (3.75), we have
therefore                                                                               …(3.77)
                Now, if the ‘ON’ and ‘OFF’ time of the pulse amplitude modulated (PAM) pulse is same as shown in figure 3.26(a) then maximum frequency of the PAM pulse will be
…(3.78)
Therefore, the bandwidth required for the transmission of a PAM signal would be equal the maximum frequency f_max given by the equation (3.78)
Thus, we
Transmission bandwidth
BW ≥ f_max                                            …(3.79)
But
Hence
Again, since
Therefore
or
FIGURE 3.26 (a) Illustration of maximum frequency in PAM signal.
3.13.4 Demodulation of PAM Signals
As discussed earlier, demodulation is the reverse process of modulation in which the
As discussed earlier, demodulation is the reverse process of modulation in which the modulating signal is recoverred back from a modulated signal. For pulse-amplitude modulated (PAM) signals, the demodulation is done using a holding circuit. Figure 3.26(b) shows the block diagram of a PAM demodulator.
DIAGRAM
FIGURE 3.26 (b) A block diagram of PAM demodulator
In this method, the received PAM signal is allowed to pass through a holding circuit and a low pass filter (LPF) as shown in above figure. Now, figure 3.27(a) illustrates a very simple holding circuit. Here the switch `S’ is closed after the arrival of the pulse and it is opened at the end of the pulse. In this way, the capacitor C is charged to the pulse amplitude value and it holds this value during the interval between the two pulses. Hence, the sampled values are held as shown in figure 3.27(c). After this the holding circuit output is smothened in Low Pass filter as shown in figure 3.27(c). It may be observed that some kind of distortion is introduced due to the holding circuit. In fact the circuit of figure 3.27(b) is known as zero-order holding circuit. This zero-order holding circuit considers only the previous sample to decide the value between the two pulses.
NOTE:           It may be noted the first order hold circuit considers the previous two samples whereas a second order holding circuit considers the previous three samples and so on. However, as the order of the holding circuit increases, the distortion decreases at the cost of the circuit complexity. In fact, the amount of permissible distortion decides the order of the holding circuit.
DIAGRAM
FIGURE 3.27 (a) A zero-order holding circuit (b) The output of holding circuit (c) The output of a low pass filter (LPF)
3.13.5. Transmission of PAM Signals
            If the PAM signals are to be transmitted directly i.e., over a pair of wires then no further signal processing is necessary. However, if they are to be transmitted through the space using an antenna, they must first be amplitude or frequency or phase modulated by a high frequency carrier and only then they can be transmitted. Thus, the overall system will be then known as PAM-AM or PAM-FM or PAM-PM respectively. At the receiving end, AM or FM or PM detection is first employed to get the PAM signal and then the message signal is recovered from it.
EXAMPLE 3.8.         For a pulse-amplitude modulated (PAM) transmission of voice signal having maximum frequency equal to f_m= 3kHz, calculate the transmission bandwidth. It is given that the sampling frequency f_s = 8 kHz and the pulse duration
Solution: We know that the sampling period T_s is expressed as
second              …(i)
Also,  is given that
Using (i), we get
x125 = 12.5 seconds                         …(ii)
Now, we know that the transmission bandwidth for PAM signal is expressed as
Using equation (ii), we get
Ans.
3.13.6. Drawbacks of Pulse-Amplitude Modulated (PAM) Signal
            Following are the drawbacks of a PAM signal:
(i)         The bandwidth required for the transmission of a PAM signal is very very large in comparison to the maximum frequency present in the modulating signal.
(ii)        Since the amplitude of the PAM pulses varies in accordance with the modulating signal therefore the interference of noise is maximum in a PAM signal. This noise cannot be removed easily.
(iii)       Since the amplitude of the PAM signal varies, therefore, this also varies the peak power required by the transmitter with modulating signal.