Understanding Sampling, Convolution, and Fourier Transforms
Sampling Theorem
The sampling theorem, also known as the Nyquist-Shannon theorem, states that to accurately reconstruct a continuous signal from its samples, the sampling frequency must be at least twice the highest frequency present in the signal.
Nyquist Frequency
The highest frequency present in a signal is known as the Nyquist frequency. It represents half of the sampling frequency.
Aliasing
If the sampling frequency is too low relative to the signal frequency, a phenomenon called aliasing occurs. This results in distortion of the reconstructed signal, as higher frequencies fold back into lower frequencies.
Nyquist Rate
The minimum sampling rate required to avoid aliasing is called the Nyquist rate. It’s defined as twice the highest frequency component of the signal.
Applications
The sampling theorem is fundamental in digital signal processing, telecommunications, and various fields where analog signals need to be digitized and then reconstructed accurately. It ensures fidelity in digital representation of analog signals.
Convolution Theorem
F(ω) represents the Fourier Transform of the function f(t) with respect to frequency ω.
f(t) is the function being transformed.
ω is the angular frequency variable.
e−iωt is the complex exponential function.
Inverse Transform
Definition: The operation that reverses the effect of a transform. For the Fourier Transform, the Inverse Fourier Transform converts a function from the frequency domain back to the time (or space) domain.
Application: Essential in reconstructing a signal or image after it has been manipulated in the frequency domain using Fourier Transform. It brings the processed data back to its original domain.
f(t) is the original function.
F(ω) is its Fourier Transform.
ω represents frequency.
eiωt is the inverse complex exponential function.
Aliasing
Aliasing is an undesirable effect that is seen in sampled systems. When the input frequency is greater than half the sample frequency, the sampled points do not adequately represent the input signal. Inputs at these higher frequencies are observed at a lower, aliased frequency.
Anti-aliasing
Anti-aliasing is a technique used in computer graphics to remove the aliasing effect. The aliasing effect is the appearance of jagged edges or “jaggies” in a rasterized image. The problem of jagged edges technically occurs due to distortion of the image when scan conversion is done with sampling at a low frequency, which is also known as Undersampling. Aliasing occurs when real-world objects which comprise smooth, continuous curves are rasterized using pixels.