**NATURAL SAMPLING:**

- Natural sampling is performed by multiplying w(t) by a train of pulses:

$$ W_s(t) = w(t) s(t) $$

where

$$ s(t) = \sum_{k=-\infty}^\infty P \left( \frac{t-kT_s}{\tau} \right) $$

- Natural sampling takes a slice of the waveform and the top of the slice preserves the shape of the waveform.
- PAM using natural sampling is as shown below:

**FLAT TOP SAMPLING:**

- In flat top sampling, the top of the samples remains constant and equal to the instantaneous value of the modulating signal at the start of the sampling.
- Thus the amplitude of the pulse after sampling is kept constant and the top of the sampled pulse do not follow the contour of the modulating signal unlike Natural sampling.
- The duration of each sample is τ and the sampling rate is fs=1/Ts
- Therefore, Ts=1/fs
- Sample and hold circuit is used for the generation of the sampled signal to attain flat top sampling.

**Shannon's Sampling theorem:**

- Sampling theorem states that in any pulse modulation system if the sampling rate of the samples exceeds twice the maximum signal frequency, then this ensures the reconstruction of the original signal in the receiver with minimum distortion.
- Sampling theorem can be expressed as given below: fs≥2fm Where, fs is the sampling frequency and fm is the maximum modulating signal frequency
- Sampling is a process of translating continuous analog signal into discrete analog signal, where the sampled signal is the discrete time representation of the original analog signal.

**Aliasing:**

Aliasing is an effect that causes different signals to become indistinguishable from each other during sampling.

Signal loss may occur due to aliasing effect.

Now, we may state the sampling theorem for strictly band limited signals of finite energy into two equivalent parts:

A band limited signal of finite energy, which only has frequency components less than

**W**Hertz, is completely described by specifying the values of the signal at instants of time separated by**1/2W**secondsA band limited signal of finite energy which only has frequency components less than

**W**Hertz , may be completely recovered from a knowledge of its sample taken at the rate of**2W**samples per second.

The sampling rate of **2W** samples per second for a signal bandwidth of **W** Hertz is called the Nyquist rate, its reciprocal **1/2W** measured in seconds is called the Nyquist interval.

Fig: a) spectrum of a strictly band limited signal g(t)

b) spectrum of a sampled version of g(t) for $T_s$ = $\frac{1}{2W}$

NOTE : The concept of under sampling and over sampling is explained below.

- When sampling frequency $f_r =2W$ then this type of sampling is called correct sampling and here there is no aliasing effect seen in this mechanism i.e when $f_s = 2W$.

- When $f_s \lt 2W$ then it is under sampling and there will be aliasing effect induced here.

- When $f_s \gt 2W$ then it is over sampling and there will no aliasing effect.

**The effect of aliasing can be reduced by :**

Pre-alias filter must be used to limit band of frequency of the required signal fm Hz.

Sampling frequency fs must be selected such that sampling frequency is greater than twice the maximum modulating signal frequency.