NYQUIST RATE AND NYQUIST INTERVAL , defined for correct sampling is , EFFECT OF UNDER SAMPLING

defined for correct sampling is , EFFECT OF UNDER SAMPLING , NYQUIST RATE AND NYQUIST INTERVAL
(i) Nyquist Rate
          When the sampling rate becomes exactly equal to 2 f_m samples per second, then it is called Nyquist rate. Nyquist rate is also called the minimum sampling rate. It is given by
f_s = 2 f_m                                                                           … (3.8)
 
(ii) Nyquist Interval
          Similarly, maximum sampling interval is called Nyquist interval. It is given by
Nyquist interval T_s =  seconds                        …(3.9)
Equation (3.17) is known as the interpolation formula, which provides values of x(t) between samples as a weighted sum of all the sample values.
In the proof of sampling theorem, it is assumed that the signal x(t) is strictly bandlimited. But, in general, an information signal may contain a wide range of frequencies and cannot be strictly bandlimited. This means that the maximum frequency f_m in the signal x(t) cannot he predictable. Therefore, it is not possible to select suitable sampling frequency f_s.
3.7 EFFECT OF UNDER SAMPLING : ALIASING
          When a continuous-time band limited signal is sampled at a rate lower than Nyquist rate f_s < 2f_m, then the successive cycles of the spectrum G(w) of the sampled signal g(t) overlap with each other as shown in figure 3.7.
DIAGRAM
FIGURE 3.7 Spectruym of the sampled signal for the case f_s < 2f_m.
Hence, the signal is under-slampled in this case (f_s < 2 f_m) and some amount of aliasing is produced in this under-sampling process. In fact, aliasing is the phenomenon in which a high frequency component in the frequency-spectrum of the signal takes identity of a lower-frequency component in the spectrum of the sampled signal.
From figure 3.7, it is obvious that because of the overlap due to aliasing phenomenon, it is not possible to recover original signal x(t) from sampled signal g(t) by low-pass filtering since the spectral components in the overlap regions add and hence the signal is distorted.

Since any information signal contains a large number of frequencies, so, to decide a sampling frequency is always a problem. Therefore, a signal is first passed through a low-pass filter. This low-pass filter blocks all the frequencies which are above f_m Hz. This process is known as band-limiting of the original signal x(t). This low-pass filter is called prelias filter because it is used to prevent aliasing effect. After bandlimiting, it becomes easy to decide sampling frequency since the maximum frequency is fixed at f_m Hz.
In short, to avoid aliasing:
SAMPLING OF BANDPASS SIGNALS
In previous sections, we discussed sampling theorem for low-pass signals. However, when the given signal is a bandpass signal, then a different criteria must be used to sample the signal. Therefore, the sampling theorem for bandpass signals may be expressed as under:
Hence, this is the interpolation formula to reconstruct x(t) from its samples x(n T_s). Therefore, from all above, it is clear that the signal may be completely represented into and recovered from its samples if the spacing between the successive samples is  seconds i.e., f_s= 2f_m samples per second.
Sampling Frequency for Bandpass Signal
          Since the spectral range of the bandpass signal is 20 kHz to 82 kHz
Therefore
Bandwidth = 2f_m = 82 kHz – 20 kHz = 62 kHz
Hence, Minimum Sampling rate = 2 x bandwidth = 2 x 62 = 124 kHz
Generally, the range of minimum sampling frequencies is specified for bandpass signals. It lies between 4f_m to 8f_m samples per second.
Therefore,
Range of minimum sampling frequencies
= (2 x bandwidth) to (4 x bandwidth)
= 2 x 62 kHz to 4 x 62 kHz = 124 kHz to 248 kHz Ans.
3.9     SAMPLING TECHNIQUES
In the last article, we discussed how sampling of a continuous-time signal is done. This sampling of a signal is done in several ways. Therefore, in this section, we shall discuss different types of sampling i.e., sampling techniques.
Basically, there are three types of sampling techniques as under:
(i) Instantaneous sampling
(ii) Natural sampling
(iii) Flat-top sampling.
Out of these three, instantaneous sampling is called ideal sampling whereas natural sampling and flat-top sampling are called practical sampling methods. Now, let us discuss three different types of sampling techniques in detail.
3.9.1. Ideal Sampling or Instantaneous Sampling or Impulse Sampling
          In the proof of sampling theorem, we used ideal or impulse sampling. In this type of sampling, the sampling function is a train of impulses. Figure 3.11(b) shows this sampling function.
x(t) is the input signal (i.e., signal to be sampled) as shown in figure 3.11(a).
Figure 3.11(c) shows a circuit to produce instantaneous or ideal sampling. This circuit is known as the switching sampler.
The working principle of this circuit is quite easy. The circuit simply consists of a switch. Now if we assume that the closing time ‘t’ of the switch approaches zero, then the output g(t) of this circuit will contain only
instantaneous value of the input signal x(t). Since the width of the pulse approaches zero, the instantaneous sampling gives a train of impulses of height equal to instantaneous value of the input signal x(t) at the sampling instant.
We know that the train of impulses may be represented as