Digital Signal Processing

Disclaimer

These notes are compiled for personal study from university lectures, faculty materials, and textbooks that I do not own. Sources are credited where possible. Please do not redistribute without permission.

They are shared in the spirit of open, decentralized access to education.

Describe Nyquist sampling theorem.

It states that a periodic signal must be sampled at more than twice the highest frequency component of the signal.
The minimum sampling rate is known as the Nyquist rate.

Describe block diagram of Analog to digital conversion.

Given in class notes

What is quantization error?

It is the difference between the actual analog value and the digital representation after the analog-to-digital conversion.

What is SQNR? Is it better to get high SQNR?

Signal-to-quantization-noise ratio is the measure of the quality of a digitized signal, specifically indicating the ratio of the signal power to the quantization noise power introduced during the ADC process.
A high SQNR means the signal is clean and free from quantization noise from the ADC process.
$SQNR = \frac{P_{Signal}}{P_{Noise}} = \frac{E[x^{2}]]}{E[\tilde{x}^{2}]]}$

What is convolution? What does it represent?

It is a mathematical operation that combines two functions to describe the overlap between them. It takes two functions, and slide one over the other, multiplying the function values at each point where they overlap and adding up the products to create a new function.
Convolution of two signals in their time domain is the multiplication of their representation in frequency domain.
$x\left\lbrack n\right\rbrack\ast h\left\lbrack n\right\rbrack=y\left\lbrack n\right\rbrack$
It represents how input is shaped/filtered by a system/circuit.

What is correlation/ auto-correlation?

Correlation compares two signals to measure their similarity.
Cross-correlation
$\int_{-\infty}^{\infty} x_1(t)x_2(t-\tau)dt$
Auto-correlation is the correlation of a signal with its own time delayed version.
$R_{11}(\tau) = R(\tau) = \int_{-\infty}^{\infty} x(t)x(t-\tau)dt \quad \text{[+ve shift]}$

1
flowchart TD
2

3
Correlation --> Cross-correlation & Autocorrelation

Differences between convolution and correlation.

Feature	Convolution	Correlation
1	Out aim is to calculate the response given by the system.	Our aim is to measure the degree to which two signals are similar and thus to extract information that depends to a large extent of the application.
2	Two signal sequences input signal and the impulse response given by the same system us calculated.	Calculating the how similar two signals are to each other.