Fourier Transform

What is Fourier transform? Answer Fourier transform is use when there is aperiodic signal. This is pretty much an extension of fourier series to aperiodic signals.

Fourier Transform show below: The basic relationships are x(t)=0, absolute t is T/2. Useful Fourier transform to learn : Fourier transform pair OR PDF form

Required condition to be Fourier transform:

Need to satisfies satisfies the Dirichlet condition the one from Fourier series

1. x(t) is absolutely integrate; that is have to be smaller than infinity.

2. x(t) has a finite number of maxima and minima within any finite interval

x(t) has a finite number of discontinuities within any finite interval. Further more, each of these discontinuities must be finite.

Properties of Fourier transform
1. linearity

we learn it in early lecture. If we do a Fourier transform to a function x(t) and y(t) to x(w) and y(w). Then ax(t)+by(t) Fourier transform can be ax(w) and by(w) 2. Time shift

The equation is listed below. Proof for time shift 3. Scaling

For a non-zero real number a, if h(x) = f(ax). The case a = −1 leads to the time-reversal property, which states: if h(x) = f(−x). Shown above



4. Differentiation This properties replaces the operation of differentiation in the time domain with that of

multiplication by jw in the frequency domain.

5. Integration 6. Duality + symetry

Below show the two forms of symetric form

7. Pars val relation

This is special case-Parseval's equation The Parseval's equation indicates that the  energy  or  information  contained in the signal is reserved, i.e., the signal is represented equivalently in either the time or frequency domain with no energy gained or lost.

8. convolution

The convolution theorem states that convolution in time domain corresponds to multiplication in frequency domain and vice versa: 9. multiplication

Youtube video: part 1, part 2

Practice Problems
Practice problem Solution

MIT practice problem and solution

MIT Fourier transform properties practice problem and solution

Extra Resources
MIT course and video

Very useful website for Fourier transform : http://www.thefouriertransform.com/

http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/

http://mathworld.wolfram.com/FourierTransform.html