**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.

DO YOU KNOW? |

An ideal reconstruction filter would include the signal bandwidth with no distortion and exclude all the aliases. |

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