Sampling

There are two types of signals, continuous time signals and discrete time signals. Due to some recent advancement in the digital signal technology over the last few decades the light weight, programmable and easily reproducible discrete time systems are available. Thus in spite of having a large number of continuous time signals we prefer processing of discrete signals. Thus conversion of continuous to discrete time signal is required. Sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave (a continuous signal) to a sequence of samples (a discrete-time signal).

Sampling representation

Sampling is defined as, “The process of measuring the instantaneous values of continuous-time signal in a discrete form.”

Sampling rate

To discretize the signals, the gap between the samples should be fixed. That gap can be termed as a sampling period Ts.

Sampling frequency(fs) can be termed as reciprocal of sampling period (Ts). fs = 1/Ts

Nyquist rate

Nyquist rate is the rate at which sampling of a signal is done so that overlapping of frequency does not take place. When the sampling rate becomes exactly equal to 2fm samples per second, then the specific rate is known as Nyquist rate. It is also known as the minimum sampling rate and given by: fs = 2fm

Sampling Theorem

The sampling rate should be such that the data in the message signal should neither be lost nor it should get over-lapped. The sampling theorem states that, “a signal can be exactly reproduced if it is sampled at the rate fs, which is greater than or equal to twice the maximum frequency of the given signal fm.”

Mathematically it can be stated as: fs ≥ 2fm

There are three cases of sampling according to fs and fm, when fs > 2fm it is called as oversampling; when fs = 2fm it is called as minimum sampling while fs < 2fm is called as under sampling. Lets look into them in brief.

Oversampling

Oversampling is used in most modern analog-to-digital converters to reduce the distortion introduced by practical digital-to-analog converters, such as a zero-order hold instead of idealizations like the Whittaker–Shannon interpolation formula.

Oversampling example when fs > 2fm

The information can be reproduced without any loss. The above figure is a Fourier transform of a signal x(t).

Minimum Sampling

Minimum sampling occurs when the Nyquist rate is observed i.e. fs = 2fm.

Minimum sampling example according to Nyquist rate

The information can be reproduced without any loss like oversampling in Nyquist rate.

Under Sampling

When a bandpass signal is sampled slower than its Nyquist rate, the samples are indistinguishable from samples of a low-frequency alias of the high-frequency signal. That is often done purposefully in such a way that the lowest-frequency alias satisfies the Nyquist criterion, because the bandpass signal is still uniquely represented and recoverable. Such under sampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF to digital conversion.

Under sampling example when fs < 2fm

We can observe from the above pattern that the over-lapping of information is done, which leads to mixing up and loss of information.

The Effect of Aliasing can be reduced by:

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

2) Sampling frequency fs must be selected such that fs > 2fm.

Sampling Techniques

There are basically three types of sampling techniques namely:

1. Ideal sampling
2. Natural top sampling
3. Flat top sampling

We will look into them one by one.

1. Ideal sampling

It is also called as impulse sampling or instantaneous sampling. Train of impulse is used as a carrier signal for ideal sampling. In this sampling technique the sampling function is a train of impulses and the principle used is known as multiplication principle.

Ideal Sampling example

You cannot use this practically because pulse width cannot be zero and the generation of impulse train is not possible practically.

2. Natural sampling

Natural Sampling is a practical method of sampling in which pulse have finite width equal to τ. Sampling is done in accordance with the carrier signal which is digital in nature. With the help of functional diagram of a Natural sampler, a sampled signal g(t) is obtained by multiplication of sampling function c(t) and the input signal x(t).

Natural Sampling

3. Flat Top Sampling

Flat top sampling is like natural sampling i.e.; practical in nature. In comparison to natural sampling flat top sampling can be easily obtained. In this sampling technique, the top of the samples remains constant and is equal to the instantaneous value of the message signal x(t) at the start of the sampling process. Sample and hold circuit are used in this type of sampling.

Flat Top Sampling

Applications of sampling

1. Audio sampling
2. Video sampling
3. 3D sampling
4. Speech sampling

Contributors

1. Shreya Bhadwal
2. Niranjan Bharate
3. Sajal Bhattar
4. Pranav Chandode