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Quantum Field Theory

Feynman's Path integral formulation   |   S-matrix

Theory of relativity

Relativity made simple   |   Special theory of relativity   |   General theory of relativity   |   Tensor calculus  |   Hamiltonian mechanics

Quantum mechanics

Schrodinger's equation   |   Matrix mechanics

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Discrete signals and system

Discrete signal is a signal which takes discrete values as opposed to continuous values. There is always a finite amount of values in certain interval of time , which is processed within any electronics circuits. Time is quantized. Thus we can index the values of such signal with natural number n. Discrete signal is usually denoted by X[n] , which takes definite value for each n. n is set of all natural number as stated : {0,1,2,3,4 ...). Any system which is fed the input X[n] and produces output y[n] is a discrete system.

A discrete signal can be represented as a squence of numbers. Such a sequence of numbers must have a first and a last number so that signal can take only finite number of values . Discrete system is very similar to continuous system but mathematical operations are somewhat different. Same kind of application can be implemented by both continous and discete time signal and systems. In discrete system the method of operation is somewhat different and diferent components are needed in such system. Some examples are digital filters and electronic calculators.

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Nyquest Sampling theorem

In general continuos time signal or continuous signal is sampled to get discrete time signal. Nyquest sampling theorem is applied in any sampling method. The theorem says that the sampling frequency must be at least twice as great as the maximum frequency component of the signal. The condition is expressed by inequality
f(sampling) => 2F(signal).

The Nyquest theorem involves an input signal W(t) and a sinc function ( sinx/x). When input is a signal W(t), the output of the sampler is a(n). a(n) can be called weight of sinc pulse, which is the values of the waveform sampled at the rate f(s) = 1/T. a(n) is the convolution integral of w(t) with sinc function. Such representation is the Nyquest sampling theorem.

nyquest sampling theorem

We can derive the theorem using Discrete fourier transform and concept of convolution integral. By taking DFT of frequency domain signal X(_) which is multiplication of sampled signal with Fourier transform of sinc function h(r), we get the representation of continuous signal in terms of weighted sum of sinc functions. We know that multiplication in frequency domain is equivalent to taking convolution of the same two functions in time domain.

Nyquest criterion is necessary for the reconstruction of the original signal from the sampled signal. In any system if sampling needs to be done, the above condition needs to be satisfied. Otherwise information will be lost and message signal can not be recovered. In other words, sampling frequency should be at least twice the maximum frequency component of message signal.

Digital signal Processing

Digital signal processing is founded upoun the sampling theory and various transform methods. Theoretical aspects are studied under discrete mathematics. Digital system is goverened by boolean logic which is implemented with universal logic gates. One of the most important method is discrete fourier transform which is the discretized version of fourier transform.

Discrete fourier transform treats a signal as periodic with N data point. For k =0, N, 2N the value of discrete fourier transform repeats from one value of k to the next. That is, w takes discrete values of 2(3.14)k/T for each k=0,1,2,... Discrete fourier transform a set of complex number to another set of complex numbers. So DFT can be interpreted as a transformation matrix W which elements are powers of nth root of unity.

The nth root of unity can be computed using euler formula:

nth root of unity
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