SAMPLING THEOREM , proof , definition , application , sampling theorem in digital communication

proof , definition , application , sampling theorem in digital communication pdf :-
SAMPLING THEORY AND PULSE MODULATION
Inside this Chapter

• Introduction
• The Sampling Theorem
• Proof of Sampling Theorem
• Nyquist Rate and Nyquist Interval
• Reconstruction Filter (Low-Pass Filter)
• Signal Reconstruction: The Interpolation Formula
• Effect of Under Sampling: Aliasing
• Sampling of Bandpass Signals
• Sampling Techniques
• Aperture Effect
• Comparison of Various Sampling Techniques
• Analog Pulse Modulation Methods
• Pulse Amplitude Modulation (PAM)
• Pulse Time Modulation
• Comparison of Various Pulse Analog Modulation Methods

3.1 INTRODUCTION
As we know that broadly, there are two types of signals, continuous time signal and discrete-time signals. Due to some recent advance development in digital technology over the past few decades, the inexpensive, light weight, programmable and easily reproducible discrete-time systems are available. Therefore, the processing of discrete-time signals is more flexible and is also preferable to processing of continuous-time signals.
This means that in practice, although we have a large number of continuous-time signals, but we prefer processing of discrete-time signals. For this purpose we should be able to convert a continuous-time signal into discrete-time signal.
This problem is solved by a fundamental mathematical tool known as sampling theorem. The sampling theorem is extremely important and useful in signal processing. With the help of sampling theorem, a continuous-time signal may be completely represented and recovered from the knowledge of samples taken uniformly. This means that sampling theorem provides a mechanism for representing a continuous-time signal by a discrete-time signal. Therefore, sampling theorem may be viewed as a bridge between continuous-time signals and discrete-time signals.
The concept of sampling provides a widely used method for using discrete-time system technology to implement continuous-time systems and process the continuous-time signals. We utilize sampling to convert a continuous-time signal to a discrete-time signal, process the discrete-time signal using a discrete-time system and then convert back to continuous-time signals.
3.2 THE SAMPLING THEOREM    (U.P. Tech-Semester Exam. 2002-2003)(10 marks)

As discussed earlier, sampling of the signals is the fundamental operation in signal-processing. A continuous time signal is first converted to discrete-time signal by sampling process. The sufficient number of samples of the signal must be taken so that the original signal is represented in its samples completely. Also, it should be possible to recover or reconstruct the original signal completely from its
samples. The number of samples to be taken depends on maximum signal frequency present in the signal. Sampling theorem gives the complete idea about the sampling of signals. Different types of samples are also taken like ideal samples, natural samples and flat-top samples.
Let us discuss the sampling theorem first and then we shall discuss different types of sampling processes. The statement of sampling theorem can be given in two parts as:
(i) A bandlimited signal of finite energy, which has no frequency-component higher than f_m Hz, is completely described by its sample values at uniform intervals less than or equal
(ii)    A bandlimited signal of finite energy, which has no frequency components higher than f_m, Hz, may be completely recoverd from the knowledge of its samples taken at the rate of 2 f_m samples per second.
The first part represents the representation of the signal in its samples and minimum sampling rate required to represent a continuous-time signal into its samples.
The second part of the theorem represents reconstruction of the original signal from its samples. It gives sampling rate required for satisfactory reconstruction of signal from its samples.
Combining the two parts, the sampling theorem may be stated as under:
“A continuous-time signal may be completely represented in its samples and recovered back if the sampling frequency is f_s > 2 f_m. Here, f₈. is the sampling frequency and f_m is the maximum frequency present in the signal”.
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EQUATION
Therefore, as long as the sampling frequency f₈ is greater than twice the maximum signal frequency f_m *(signal, bandwidth, f_m), G(w) will consist of non-overlapping repetitions of X(w). this is true, figure 3.1 (f) shows that x(t) can be recovered from its samples g(t) by passing the sampled signal x(t) through an ideal law-pass filter of bandwidth f_m Hz. This proves the sampling theorem.
3.3.1. Important Points about Sampling Theorem
(i)      Figure 3.1 (f) shows the spectrum of sampled signal. According to the figure, as long as, the signal is sampled at rate f_s> 2 f_m, the spectrum G(w) will repeat periodically without overlapping.
(ii)     The spectrum of sampled signal extends upto infinity and the ideal bandwidth of sampled signal is infinite. But here our purpose is to extract our original spectrum X(w) out of the spectrum G(w).
(iii)    The original or desired spectrum X(w) is centred at w = 0 and is having bandwidth or maximum frequency equal to W_m. The desired spectrum may be recovered by passing the sampled signal with spectrum G(w) through a low pass filter with cut-off frequency w_m. This means that since a low-pass filter allows to pass only low frequencies up to cut-off frequency (w_m) and rejects all other higher frequencies, the original spectrum X(w) extended upto wm will be selected and all other successive higher frequency cycles in the sampled-spectrum will be rejected. Therefore, in this way, original spectrum X(w) will be extracted out of spectrum
G(w). This original spectrum X(w) can now be converted into time-domain signal x(t).
(iv)    It may also be observed from figure that for the case f_s> 2f_m, the successive cycles of G(w) are not overlapping each other. Hence in this case, there is no problem in recovering the original spectrum X(w).
(v)     For the case f_s = 2 f_m, although the successive cycles of G(w) are not overlapping each other, but they are touching each other. In this case also, the original spectrum X(w) can be recovered from the sampled spectrum G(w) using a low-pass filter with cut-off frequency w_m.
(vi)    For the case f_s < 2 f_m, the successive cycles, of the sampled spectrum will overlap each other and hence in this case, the original spectrum X(w) cannot be extracted out of the spectrum G(w).
Hence, For reconstruction without distortion, we must have
f_s > 2 f_m