Nyquist Sampling Theorem

Definition

The Nyquist sampling theorem states that a continuous-time signal can be completely and uniquely recovered from its samples if it is sampled at a rate greater than or equal to twice the highest frequency present in the signal.

If the highest frequency component of a signal is $f_{max}$ , then the minimum sampling frequency required is:

$f_s \geq 2f_{max}$

The value $2f_{max}$ is called the Nyquist rate, and half of the sampling frequency $\frac{f_s}{2}$ is called the Nyquist frequency.

Main Content

1. Sampling and Its Meaning

Sampling is the process of measuring the value of a continuous-time signal at uniform time intervals.
Instead of keeping every point of the analog waveform, only selected points are recorded, and these values represent the signal in digital form.

A continuous waveform $x(t)$ becomes a sequence $x(nT_s)$ , where:

$T_s$ = sampling interval
$n$ = sample index

If a signal is sampled at equal intervals, the digital representation becomes easier to store, transmit, and process. For example, a voice signal in analog form may vary smoothly with time, but after sampling, it is represented as a list of numerical values.

Example:

Suppose a signal has frequency components up to 3 kHz.
Then the minimum sampling frequency should be: $f_s \geq 2 \times 3\,\text{kHz} = 6\,\text{kHz}$
In practice, a slightly higher rate is often used to provide a safety margin.

A simple view of sampling:

Analog signal:   ~~~~~~\~~~~~~/~~~~~\~~~~~~
Samples:           |   |   |   |   |   |
Values:           x1  x2  x3  x4  x5  x6

2. Nyquist Rate and Nyquist Frequency

The Nyquist rate is the minimum sampling rate required to avoid information loss for a bandlimited signal.
It is equal to twice the maximum frequency present in the signal.

If the signal contains frequencies only up to $f_{max}$ , then:

Nyquist rate = $2f_{max}$
Nyquist frequency = $\frac{f_s}{2}$

This concept is central to digital communication because it tells us the boundary between safe sampling and inadequate sampling.

Example:

For a signal bandlimited to 5 kHz:
Nyquist rate = 10 kHz
Sampling at 8 kHz is not sufficient
Sampling at 10 kHz is the minimum acceptable
Sampling at 12 kHz or 16 kHz gives extra safety and is commonly preferred

Why it matters:

If sampling is below the Nyquist rate, the high-frequency components can distort the sampled representation.
If sampling is at or above the Nyquist rate, the original waveform can, in theory, be reconstructed exactly.

3. Aliasing and Signal Reconstruction

Aliasing is the distortion that occurs when a signal is sampled below the Nyquist rate.
In aliasing, different frequency components become indistinguishable after sampling, making the original signal impossible to recover accurately.

When the sampling rate is too low:

High-frequency components appear as lower frequencies in the sampled signal.
The sampled data misleadingly represents the original signal.

A conceptual example:

A 7 kHz signal sampled at 10 kHz violates the Nyquist rule because the maximum recoverable frequency is only 5 kHz.
The 7 kHz component folds into an incorrect lower-frequency component in the sampled data.

This is why anti-aliasing filters are used before sampling:

They remove frequency components above $\frac{f_s}{2}$
They help ensure the signal is bandlimited
They protect the system from irreversible information loss

Relation to reconstruction:

If a signal is bandlimited and sampled properly, reconstruction is possible using interpolation methods such as sinc interpolation.
In practical communication systems, reconstruction involves digital-to-analog conversion followed by smoothing filters.

A simple aliasing illustration:

Original high-frequency wave:   /\/\/\/\/\/\/\/\/\
Too-slow samples:               |    |    |    |
Recovered incorrectly:          /\/\_/\/\_/\/\_/

Working / Process

1. Identify the highest frequency component of the signal

Determine the maximum frequency $f_{max}$ present in the analog signal.
If the signal is not already bandlimited, first apply a low-pass filter to remove frequencies above the desired limit.

2. Choose a sampling frequency at least twice the highest frequency

Use the Nyquist formula: $f_s \geq 2f_{max}$
In real systems, choose a sampling rate greater than the minimum to allow practical filter design and reduce the chance of aliasing.

3. Sample, digitize, and reconstruct

Take samples at uniform time intervals $T_s = \frac{1}{f_s}$ .
Convert each sample into digital form using quantization and encoding.
At the receiver or output stage, reconstruct the signal using interpolation and filtering.

Advantages / Applications

Enables accurate conversion of analog signals into digital form for storage, processing, and transmission.
Prevents aliasing when the sampling frequency is properly selected.
Forms the foundation of modern systems such as PCM telephony, audio CD recording, digital video, medical imaging, and sensor networks.
Helps engineers design practical anti-aliasing filters and ADC systems.
Supports reliable digital communication by ensuring the transmitted sample sequence preserves the original information content.

Summary

The Nyquist sampling theorem tells us the minimum sampling rate needed to represent a signal correctly.
It requires sampling at least twice the highest signal frequency.
If sampling is too slow, aliasing occurs and the original signal cannot be recovered properly.
Important terms to remember: Nyquist rate, Nyquist frequency, sampling, aliasing, anti-aliasing filter, reconstruction