Sampling in Digital Communication

Sampling in digital communication is converting a continuous-time signal into a discrete-time signal. It can also be defined as the process of measuring the discrete instantaneous values of a continuous-time signal.

Digital signals are easier to store and have a higher chance of repressing noise. This makes sampling an important step in converting analog signals to digital signals with its primary purpose as representing analog signals in a discrete format.

Sampling Process in Digital Communication

The sampling process includes the following steps:

1. The continuous signal is taken as an input.
2. Sampling is performed to convert this signal into a digital representation.
3. In addition to sampling, quantization of a signal is performed.
4. After the above step, encoding of the signal is done.
   

Sampling Process

We will going to discuss the sampling process in detail in this article.

Types of Signals in Sampling

There are several types of signals in sampling that are discussed below.

1. Continuous Time Signals
2. Discrete Time Signals
3. Digital Signals

1. Continuous Time Signals

CTS are those signals that are continuous in both time and amplitude. They are represented by functions that stay continuous over a range of time and values of amplitude.

Continuous Time Signal

2. Discrete Time Signals

DTS are those that are continuous in amplitude but discrete in time meaning that they are signals with values at some specific instants of time. It is to be taken care of that these signals are discrete only in time whereas they can be either continuous or discrete in amplitude.

Discrete Time Signal

3. Digital Signals

After sampling and quantization, the resulting signals are in digital format are hence called Digital Signals. These signals have both their time and amplitude in discrete format.

Digital Signal

Important Terminologies of Sampling in Digital Communication

There are few important terminologies of Sampling in Digital Communication discussed below :

• Sampling
• Sample
• Sampling Rate or Sampling Frequency
• Nyquist Rate
• Nyquist Interval
• Quantization

1. Sampling

It is the process by which, we convert CTS (continuous time signal) into DTS (discrete time signal) by taking the signal values at some distinct points in time, meaning that this is used to take samples of analog signals at some points in time (regular or irregular)

2. Sample

It can be defined as the numeric value of an analog signal at a specific time. It is just the signal’s measured amplitude at a particular time and converting it to a digital representation.

3. Sampling Rate or Sampling Frequency

It refers to the number of samples or data points taken per unit of time from an analog signal to convert it into a digital format. It is also known as sampling frequency. It is measured in Hertz (Hz).

The formula for sampling rate or sampling frequency is given by:

[Tex]Sampling Rate = \frac{1}{T_{s}} = f_{s} [/Tex]

where,

Ts = sampling time
fs = sampling

4. Nyquist Rate

It is the minimum sampling rate required to accurately capture an analog signal in digital form without information loss. It is also known as Nyquist Frequency or Nyquist Limit.

It is defined as twice the maximum frequency component present in the analog signal. Mathematically it can be represented as:

[Tex]f_{s} = 2f_{max} [/Tex]

where,

f_s = Sampling Rate or Nyquist Rate (Hz)
f_max = Maximum frequency component (Hz)

Note: The sampling theorem was stated on the basis of Nyquist rate.

5. Nyquist Interval

The Nyquist interval, also known as the Nyquist period, is the time interval between consecutive samples in a digital signal or digital sampling system. It is the reciprocal of the Nyquist rate, which is the smallest sampling rate required to accurately capture an analog signal in digital form without information loss. Mathematically it can be represented as:

[Tex]T = \frac{1}{Nyquist Rate} [/Tex]

Where,

T = Nyquist interval (sec)
Nyquist Rate is the sampling rate (Hz)

6. Quantization

It is the process to represent a continuous-valued signal with a limited set of discrete values. In other words, it involves mapping a continuous signal’s infinite range of potential values to a finite collection of discrete values.

Quantization of a Signal

Nyquist – Shannon Sampling Theorem

The theorem states that for reconstructing a sampled signal accurately from the available samples, the sampling frequency should be at least twice as much as the highest frequency component of the signal.

It can be understood by the following expression:

[Tex]2 \times f_{max} \leqslant f_{s} [/Tex]

Where,

[Tex]f_{max} [/Tex] = maximum frequency component of the original signal

[Tex]f_{s} [/Tex]= sampling frequency

“A signal can be exactly reproduced if it is sampled at the rate fs which is greater than twice the maximum frequency (f_{max} ).”

Sampling Cases

• If [Tex] f_{s} < 2 \times f_{max} [/Tex], then it is the case of undersampling.
• If [Tex]f_{s} = 2 \times f_{max} [/Tex], then it is the case of sampling at Nyquist Rate and it is perfect sampling.
• If [Tex]f_{s} > 2 \times f_{max} [/Tex], then it is the case of oversampling.

Oversampling & Undersampling

1. Undersampling

The spectra of X(ω) are overlapped in this scenario because the sampling rate is less than the Nyquist rate, making it impossible to extract the original signal from the sampled signal. Because the spectra overlap, some frequency components of the original signal will acquire a new frequency; this process is known as frequency aliasing. An application of undersampling can be when the high frequency components are not useful and user needs to reduce the amount of data being processed.

Under Sampling of Signal Causing Aliasing

2. Oversampling

Over-sampling is when more samples are taken that are necessary to capture the signal’s frequency. It can be done to measure more accurately, enhancing SNR, providing more detailed information for further processing. It can be seen in the above figure-‘Sampling cases’.

Aliasing

It is a phenomenon that occurs when a high-frequency signal is represented at lower frequency. Means it occurs when the sampling rate is insufficient and fails to capture the signal properly. When the signals are sampled at lower frequency than the nyquist frequency, high frequency components fold back (gets aliased) in the low frequency range. This may lead to distorted signal representation.

In simple words a high frequency component of a signal taking the identity of low frequency component of a signal when it is undersampled.

Methods to Avoid Aliasing

It can be avoided by the following two methods :

1. Sampling at Nyquist Rate
2. Using Anti-Aliasing Filter (Low Pass Filter): It helps removing the component above the nyquist frequency which may lead to aliasing.

Why Sampling is Required?

Sampling plays an essential role in digital communication systems because it turns continuous analog signals into discrete digital data, allowing them to be processed, transmitted, stored, and manipulated efficiently in the digital world. Noise reduction, error detection and correction, compression, signal processing, and interoperability are all enabled by this conversion, which is crucial for modern communication systems. Digital representation provides for long-distance data transmission with reduced signal deterioration, as well as precise modulation, demodulation, and other signal processing processes, facilitating dependable communication and compatibility among diverse devices and platforms.

What is Anti-Aliasing Filter?

An anti-aliasing filter, often known as a “anti-alias filter” or simply “AAF,” is a filter used in signal processing and digital data collection systems to prevent or eliminate aliasing effects.

Working Anti-Aliasing Filter

1. Filtering High-Frequency Components: The anti-aliasing filter is used to remove high-frequency components in analog signals that exceed the Nyquist frequency, which is half the sampling rate of the ADC. If these high-frequency components are not filtered, they will cause aliasing which results in incorrect information.

2. Preventing Aliasing: The anti-aliasing filter ensures that only the desired frequency are represented in the digital signal by attenuating the high-frequency components.

3. Improved Signal Quality: It improves signal quality and allows for more accurate data gathering. It helps to retain the original signal’s integrity and decreases the possibility of errors in later digital processing.

Methods of Sampling

1. Ideal Sampling

Concept: Ideal sampling, also known as impulse or Dirac sampling, is a theoretical notion in which samples of a continuous signal are taken at specific time intervals, often at the Dirac delta function impulse points.

Sampling Process: Each sample in perfect sampling is an impulse or delta function at the sampling instant. The sampled signal can be represented mathematically as the product of the continuous signal and the Dirac delta function.

Sampling Circuit

Reconstruction: The reconstruction of the original signal from ideal samples, we can use interpolation which uses the functions. Ideal sampling is a simple approach to express and analyze sampling theory, however it is not practical due to the requirement for infinite bandwidth.

Ideal Sampling

2. Natural Sampling

Concept: Natural sampling, also known as zero-order hold sampling, involves taking discrete interval samples of a continuous signal, similar to uniform sampling. The difference, though, is in how the samples are gathered.

Sampling Process: Each sample is taken in natural sampling by retaining the value of the continuous signal constant for the duration of the sampling period.

Sampling Circuit

Reconstruction: The reconstruction of the original signal from natural samples, it usually involves connecting the samples with flat line segments. This method simplifies the reconstruction process compared to ideal sampling.

Natural Sampling

3. Flat-Top Sampling

Concept: Flat-top sampling is a type of natural sampling in which each sample is obtained by maintaining the value of the continuous signal constant for a set period of time, resulting in a flat-top waveform.

Sampling Process: Instead of retaining the value for the whole sample interval, flat-top sampling holds it only for a portion of the interval while allowing it to change at the beginning and end.

Sampling Circuit

Reconstruction: The reconstruction of the original signal from flat-top samples, we can use interpolation techniques. Flat-top sampling is used in applications where it is desirable to minimize the effects of finite bandwidth and aliasing.

Flat Top Sampling

Scope of Fourier Transform

It is well noticed that we seek the assistance of Fourier series and Fourier transforms in analyzing signals and proving theorems. This is because:

• The Fourier Transform is used for a non-periodic signal which is helpful in analysis of the signal.
• The Fourier transform is helps to observe signals in several domains and readily analyze them making it a strong mathematical tool.
• Using this Fourier transform, any signal may be decomposed into the sum of sines and cosines.

Advantages and Disadvantages of Sampling

Advantages of Sampling

• It is an important step in converting analog signal to digital signal which allows efficient digital storage and signal processing
• By taking samples at some specific interval time, it helps compressing the original signal, which in turn helps in efficient transmission
• Digital signals can be easily processed using various algorithms

Disadvantages of Sampling

• When sampling rate is not proper, it may lead to the problem of aliasing, resulting in distorted signals
• In the process of Analog to Digital conversion, the next step after sampling is quantization which may result in: loss of information
• While conversion (Analog to Digital) sampling may introduce errors due to factors such as quantization noise, temperature variations, etc.

Solved Examples on Sampling

Question 1: Given the signal, what will be the sampling frequency for which the signal can be reconstructed

[Tex]x(t) = cos(2\pi10t) [/Tex]

Solution:

Comparing the given signal with [Tex]cos(2\pi ft) [/Tex]

[Tex]f_{max} = 10 Hz [/Tex]

then according to the sampling theorem, the sampling frequency will be [Tex]2 \times f_{max} \leqslant 2 \times 10 \leqslant 20 Hz [/Tex].

Question 2: Sample the given signal at 60Hz

[Tex]x(t) = cos(2 \pi 50 t) [/Tex]

Solution:

We know that x[n] = [Tex]cos(2 \pi \frac{f_{max}}{f_{s}} n ) [/Tex]

Here, x[n] = [Tex](2 \pi \frac{50}{60} n) [/Tex]

Finally, we get the discrete time signal as: [Tex]x[n] = cos(2 \pi \frac{5}{6} n) [/Tex]

FAQs on Sampling in Digital Communication

1. What is the bit depth or resolution in quantization?

The amount of bits utilized to represent each sample is referred to as bit depth, also known as resolution. A higher bit depth enables more precise representation but necessitates more data transmission.

2. What role does sampling rate play in digital communication?

The sampling frequency determines how frequently samples from the continuous signal are obtained. It has a direct impact on the amount of data retained and the bandwidth required for transmission.

3. What effect does the sample method (e.g., uniform, non-uniform) have on digital communication?

The sampling method used influences the accuracy of signal reconstruction, the system’s complexity, and the computer resources required. Non-uniform sampling, for example, is more difficult yet may be required for some signals.