Introduction
In the realm of Signal Processing and Fourier analysis, sampling theory plays a pivotal role in understanding how continuous signals can be represented in a discrete form.
Objective :-
To provide a brief exploration of sampling theory within the context of Fourier analysis.
We will delve into :-
fundamental concepts
very basic mathematical underpinnings
Practical implications of sampling with a particular focus on Nyquist frequency and its significance in signal processing
Visualization and Intuition in both frequency and time domain
Understanding Sampling Theory
Sampling theory revolves around the process of converting continuous signals into discrete representations by selecting a finite number of samples from the continuous signal.
The key idea is to capture enough information from the continuous signal to accurately reconstruct it back from the discrete collection to the continuous form.
Ideal Sampling
Suppose a signal x(t) has a frequency representation X(f)
X(f) is bounded, i.e there exist a f_max beyond which there exists no more frequency contributes to x(t)
For understanding purpose it's shown as a simple triangle in the figure.
Now suppose we collect all the values of x(t) that occurs after some fixed interval T, i.e we after every T we record what is the value of x(t).
We do this by multiplying x(t) with an impulse train signal, g(t).
$$g(t) = \sum_{n=-\infty}^{\infty} \delta(t - nT)$$
So the sampled signal x_s(t)
$$x_s(t) = \sum_{n=-\infty}^{\infty} x(t) \delta(t - nT)$$
Important Understanding
How does the sampled signal's frequency profile look ?
Let's take the Fourier Transform on both the side
$$\mathcal{F}\biggl\{x_s(t)\biggl\} {}{}{}{}{}{}{}{} ={}{}{}{}{}{}{}{}{}{}{} \mathcal{F} \Biggl\{ \sum_{n=-\infty}^{\infty} x(t) \delta(t - nT) \Biggl\}$$
$$X_{sampled}(t) = f_s \cdot \sum_{n=-\infty}^{\infty} X(f - nf_s)$$
How to understand this ?
We get the frequency profile of x(t) , i.e X(f) ,
CENTRED AT EVERY n . f_s
So we have a the graph at n = 0 ( f = 0 ) { triangle at the centre }
At n = ± 1 ( f = ± f_s ) { the triangle on either side of the triangle at 0 }
So we have the frequency profile of original signal replicating itself throughout the frequency domain
We can see this in the bottom image of frequency domain representation.
The same result can be arrived at if we convolve X(f) with G(f) which is what was mentioned in the bottom image.
So to recap :-
We sampled a continuous signal to get a vector of discrete values of signal ( after every fixed time instant )
The frequency profile of original signal bounded between [ -f_max , f_max ]
Frequency profile of sampled signal is a periodically occuring X(f) centered around n . f_s
By using a low pass filter of cut-off frequency greater than f_max, we can reconstruct the original.
Nyquist-Shannon Sampling Theorem
At the heart of sampling theory lies the Nyquist-Shannon sampling theorem, which states that a continuous signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the maximum frequency component present in the signal, known as the Nyquist frequency. Mathematically, the Nyquist criterion can be expressed as:
$$f_s \geq 2 \cdot f_{\text{max}}$$
Where:
f_s is the sampling frequency.
*f_*max is the maximum frequency component in the signal.
Proof of Nyquist Criterion
To prove the Nyquist criterion, we can use the concept of aliasing, which occurs when high-frequency components in the signal fold back onto lower frequencies during under sampling. By ensuring that the sampling frequency is greater than twice the maximum frequency, we prevent aliasing and enable accurate reconstruction of the original signal.
Consider a continuous signal x(t) with a maximum frequency component *f_*max. According to the Nyquist-Shannon sampling theorem, this signal can be perfectly reconstructed if it is sampled at a rate f_s that is at least twice the maximum frequency *f_*max.
Let's assume that the sampling frequency f_s is less than twice the maximum frequency component, i.e : -
$$f_s < 2 \cdot f_{\text{max}}$$
Now, when we sample the signal at f_s, any frequency component above f_max will undergo aliasing. This means that the high-frequency components will fold back onto lower frequencies, creating ambiguity in the sampled signal.
Mathematically, aliasing occurs due to the periodic nature of the Fourier transform. When the sampling frequency is less than 2⋅*f_*max, the higher frequency components are not adequately captured by the samples, leading to aliasing.
To prevent aliasing and accurately reconstruct the original signal, we need to ensure that the sampling frequency f_s satisfies the Nyquist criterion:
$$f_s \geq 2 \cdot f_{\text{max}}$$
This inequality guarantees that the samples contain sufficient information to represent the original signal without distortion or ambiguity caused by aliasing.
Therefore, by satisfying the Nyquist criterion, we ensure that the sampling process captures all frequency components of the original signal, allowing for faithful reconstruction and preserving the integrity of the signal in the discrete domain.
Aliasing and Frequency Folding
Aliasing occurs when under-sampling leads to frequency components folding back onto lower frequencies, causing distortion and loss of information. By satisfying the Nyquist criterion, we can prevent aliasing and maintain the fidelity of the reconstructed signal.
Practical Sampling
- Instead of impulses, we use very narrow Rect signals
Practical Implications
In practical signal processing applications, adhering to the Nyquist criterion is crucial for avoiding distortion and preserving the integrity of the signal. For example, in audio processing, the Nyquist frequency determines the minimum sampling rate required to faithfully capture the audible frequency range, ensuring that high-frequency components are not lost or misrepresented.
Conclusion
Sampling theory, particularly the Nyquist-Shannon sampling theorem, forms the cornerstone of Fourier analysis and signal processing. By understanding the Nyquist criterion and its implications, engineers and researchers can design robust sampling systems and ensure accurate reconstruction of continuous signals in a discrete domain. From audio processing and digital communications to data analysis and AI/ML, sampling theory underpins a wide range of applications, making it an indispensable tool in modern technology.
Reference :-
This guy did a really good job on the diagrams
https://www.12000.org/my_notes/sampling_theory/index.htm
https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem
P.S.
it's kinda frustrating that Hashnode does not support inline latex rendering. Hashnode also does not have support for many languages - verilog, VHDL, Matlab.. Sigh, hope it gets better;