ANALOG PULSE MODULATION METHODS , Pulse Amplitude Modulation (PAM) , Time Modulation (PTM)

Pulse Amplitude Modulation (PAM) , Time Modulation (PTM) ? :-
ANALOG PULSE MODULATION METHODS

          We know that in analog modulation systems, some parameter of a sinusoidal carrier is varied according to the instantaneous value of the modulating signal. In pulse modulation methods, the carrier is no longer a continuous signal but consists of a pulse train. Some parameter of which is varied according to the instantaneous value of the modulating signal. There are two types of pulse modulation systems as under:
(i) Pulse Amplitude Modulation (PAM)
(ii) Pulse Time Modulation (PTM)
In pulse amplitude modulation (PAM), the amplitude of the pulses of the carrier pulse train is varied in accordance with the modulating signal whereas in pulse time modulation (PTM), the timing of the pulses of the carrier pulse train is varied.
They are two types of PTM as under:
(i) Pulse width modulation (PWM)*
(ii) Pulse position modulation (PPM)
In Pulse width modulation, the width of the pulses of the carrier pulse train is varied in accordance with the modulating signal whereas in Pulse position modulation (PPM), the position of pulses of the carrier pulse train is varied. Figure 3.21 shows three types of pulse analog modulation methods.
According to the sampling theorem, if a modulating signal is bandlimited to f_m Hz,** the sampling frequency must be at least 2f_m Hz and, hence the frequency of the carrier pulse train must also be least 2f_m Hz.
*        Pulse width modulation is also known as Pulse duration modulation (PDM).
**      If a signal is said to be bandlimited to f_m Hz, then it means that the maximum frequency component in this signal is f_m Hz.
At this point, it may he noted that all the above pulse modulation methods (i.e., PAM, PWM and PPM) are called analog Pulse modulation methods because the modulating signal is analog in nature in PAM, PWM and PPM.
DIAGRAM
FIGURE 3.21 Different types of pulse analog modulation methods.
3.13   PULSE AMPLITUDE MODULATION (PAM)
Pulse amplitude modulation may be defined as that type of modulation in which the amplitudes of regularly spaced rectangular pulses vary according to instantaneous value of the modulating or message signal. In fact, the pulses in a PAM signal may be of flat top type or natural type or ideal type. Actually all the
sampling methods which have been discussed in last sections are basically pulse amplitude modulation methods.
Out of these three pulse amplitude modulation methods, the Flat top PAM is most popular and is widely used. The reason for using flat top PAM is that during the transmission, the noise interferes with the top of the transmitted pulses and this noise can be easily removed if the PAM pulse has flat top.
DIAGRAM
FIGURE 3.22 (a) Sample and hold circuit generating flat top sampled PAM (b) Waveforms of flat top sampled PAM
However, in case of natural samples PAM signal, the pulse has varying top in accordance with the signal variation. Now, when such type of flat top sampled of pulse is received at the receiver, it is always contaminated by noise. Then it becomes quite difficult to determine the shape of the top of the, pulse and thus amplitude detection of the pulse is not exact. Due to this, errors are introduced in the received signal.
Therefore, flat top sampled PAM is widely used.
Figure 3.22 shows the sample and hold circuit waveform for flat top sampled PAM.
Working Principle
A sample and hold circuit shown in figure 3.22 is used to produce flat top sampled PAM. The working principle of this circuit is quite easy. The sample and hold (S/H) circuit consists of two field effect transistors (FET) switches and a capacitor. The sampling switch is closed for a short duration by a short pulse applied to the gate G₁ of the transistor. During this period, the capacitor ‘C’ is quickly charged upto a voltage equal to the instantaneous sample value of the incoming signal x(t). Now, the sampling switch is opened and the capacitor ‘C’ holds the charge. The discharge switch is then closed by a pulse applied to gate G₂ of the other transistor. Due to this, the capacitor ‘C’ is discharged to zero volts. The discharges switch is then opened and thus capacitor has no voltage.
Hence, the output of the sample and hold circuit consists of a sequence of Flat top samples as shown in figure 3.23.
DIAGRAM
          FIGURE 3.23      (a) Baseband signal x(t)
(b) Instantaneously sample single s(t)
(c) Constant pulse width function h(t)
(d) Flat top sampled PAM signal g(t) obtained through                                            convolution of h(t) and s(t).
Mathematical Analysis
          In a Flat top PAM, the top of the samples remains constant and is equal to the instantaneous value of the baseband signal x(t) at the start of sampling. The duration or width of each sample is t and sampling rate is equal to .  From figure 3.22 (b), it may be noted that only starting edge of the pulse represents instantaneous value of the baseband signal x(t). Also, the flat top
pulse of g(t) is mathematically equivalent to the convolution of instantaneous sample and a pulse h(t) as depicted in figure 3.24.
This means that the width of the pulse in g(t) is determined by the width of h(t) and the sampling instant is determined by the delta function.
DIAGRAM
FIGURE 3.24 Convolution of any function with delta function is equal to that function.
In figure 3.22 (b), the starting edge of the pulse represents the point where baseband singal is sampled and width is determined by function h(t).
Therefore, g(t) will be expressed as
g(t) = s(t) Ä h(t)                                               …..(3.58)
This equation has been explained in figure 3.23.
Now, from the property of delta function, we know that for any functionf(t)
f(t) Ä  s(t) = f(t)                                                …..(3.59)
This property is used to obtain flat top samples. It may be noted that to obtain flat top sampling, we are not applying the equation (3.59) directly here i.e., we are applying a modified from of equation (3.59). This modified equation is equation (3.58).
Thus, in this modified equation, we are taking s(t) in place of delta functions . Observe that  is a constant amplitude delta function wheres s(t) is a verying amplitude train of impulses. This mean that we are taking s(t) which is an instantaneously sampled signal and this is convolved with function h(t) as in equation (3.58)
Therefore, on convolution of s(t) and h(t), we get a pulse whose duration is equal to h(t) only but amplitude is defined by s(t).
Now, we know that the train of impulses may be represented mathematically as
The signal s(t) is obtained by multiplication of baseband signal x(t) and .
Thus,                                                                                   …(3.61)
Or,                                                                   Equation
Now, sampled signal g(t) is given as equation (3.58)
                                                                        g(t) = s(t) Ä h(t)                                     …(3.63)
or                                                         EQUATION                         
or                                                         EQUATION
According to shifting property of delta function we know that
Using equation (3.65) and (3.66), we get
This equation represents value of g(t) in terms of sampled value x(nT_s) and function h(t-nT_s) for flat top sampled signal.
Now, again from equation (3.58), we have
g(t) = s(t)Ä h (t)
Taking Fourier transfrom of both sides of above equation, we get
G(f) = S(f) H(f)                                               …(3.67)
We know that S(f) is given as
EQUATION
Therefore, equation (3.67) will become
EQUATION
Thus, spectrum of flat top PAM signal:
EQUATION
Here, H(f) is the Fourier transform of the rectangular pulse. The spectrum of this rectangular pulse is shown in figure 3.18(b). Let the spectrum of s(t) be the rectangular pulse train as shown in figure 3.25(a) and the spectrum of h(t) i.e., H(f) is shown in figure 3.25(b).
By equation (3.67), we know that
G(ff) = S(f). H(f)
Thus, according to above equation, we can plot the spectrum G(f) as shown in figure 3.25 (b).
NOTE: It may be observed in figure 3.25(b) that higher frequencies in S(f) are attenuated due to roll-off characteristics of the ‘sinc’ pulse. This effect is popularly known as aperture effect.
An equalizer is needed to overcame this effect.
DIAGRAM
FIGURE 3.25 (a) Spectrum of some arbitrary signal. The signal is sampled at f_s and maximum frequency in the signal is f_m, (b) Spectrum of flat top signal. The dotted curve is H(f) =