Question

I wanted to understand about Fourier Series and Fourier Transform.I need some one to explain from basics.

Like what is Signal,What is sine wave,What is cos wave .Why we need fourier series and fourier transform.I wanted to understand everything visuallyI tried watching you tube videos .As i am not sure the basics,I am not able to understand any of the videos.

Also i wanted to know the sinusoida wave equation.Not able to understand this formula.I wanted to know "Why" behind this formula.y(x,t)=Asin(kx−ωt+ϕ) y ( x , t ) = A sin ( kx − ω t + ϕ ) , where A is the amplitude of the wave, ω is the wave's angular frequency, k is the wavenumber, and ϕ is the phase of the sine wave given in radians.e meaning of sinusoidal wave equation .

Here i am not sure what is A,K,pi,omega. I am pretty confused about this. Could someone explain me from basics.

Answer 1

A Signal is a function that conveys information about a phenomenon. In electronics and telecommunications, it refers to any time varying voltage, current or electromagnetic wave that carries information. A signal may also be defined as an observable change in a quality such as quantity.

A Sine wave is a curve representing a periodic Oscillations of a Constant Amplitude as given by Sine function.

Similarly, A Cosine wave is a curve representing a periodic Oscillations of a Constant Amplitude as given by Cosine function.

Basic Purpose of Fourier transform is to change the domain of a Signal or waves or even images like from frequency Or time to Spatial domain. The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components. The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent.

The main advantage of Fourier analysis is that very little information is lost from the signal during the transformation. The Fourier transform maintains information on amplitude, harmonics, and phase and uses all parts of the waveform to translate the signal into the frequency domain.

Fourier Series:

The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials.

A Fourier series is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions. ... It may be the best application of Fourier analysis. Approximation Theory. We use Fourier series to write a function as a trigonometric polynomial.

Sinusoidal wave equation is given by

Y(x,t)= A Sin( kx-wt+ Φ).

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Here A, is called the amplitude which is maximum displacement of particle from mean or equilibrium position.

K Is called the wave number or propagation vector

(k= 2π/ wavelength)

w(omega) is called the angular frequency (w=2π* frequency)  

Actually this (Kx -wt +Φ) is called a phase of wave. This phase determines the status of the particle in A harmonic or periodic Motion . For Example if phase is zero at certain instant, this means that the particle is crossing the mean position and is going towards the positive direction. If phase is (π/2) it means that the particle is at the positive extreme position.

whereas the Φ is called the phase constant which gives us the choice of the instant t=0.

To describe the motion quantitatively, a particular instant should be called t = 0 and measurement of time should be made from this instant. This instant may be chosen according to the convenience of the problem. Suppose we choose t = 0 at an instant when the particle is passing through its mean position and is going towards the positive direction. The phase should then be zero. As t = 0 this means Φ will be zero. The equation for displacement can then be written as x = A Sin (wt).

If we choose t = 0 at an instant when the particle is at its positive extreme position, the phase is π/2 at this instant. Thus, = π/2 and hence Φ= π/2 The equation for the displacement is x = A sin(wt + π/2) or, x = A Cos(wt). Any instant can be chosen as t = 0 and hence the phase constant can be chosen arbitrarily.

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