Chris' Page for HW 4 (Fourier Transforms)

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The Fourier Transform is a process or formula that converts a signal from one domain to another. Often it is used to go between the time domain and the frequency domain.

Developed by Frenchman, Jean Baptiste Joseph Fourier (1768 - 1830), the Fourier Transform stems from the more general Fourier Analysis, which is the representation of a function with sine and cosine terms. Unlike the Fourier Series the Fourier Transform is capable of representing aperiodic signals.

Mathematical Description

The Fourier Transform is detonated by;

The Inverse Fourier Transform is;

Relation to Laplace Transform

Unless otherwise noted, a Laplace Transform is defined by the unilateral or one-sided integral

The Laplace Transform can be applied from to , this is known as the Bilateral Laplace Transform and is denoted by

Setting () gives the equation

which is identical to the Fourier Transform. The same relationship exists between the Inverse Laplace and the Inverse Fourier transforms.

Examples

Below is a very basic example based on a Matlab help file.

The time domain signal is produced by

t = 0:0.001:.1;

y = sin(2*pi*50*t)+sin(2*pi*120*t) +sin(2*pi*150*t) +sin(2*pi*310*t) +sin(2*pi*340*t);

y = 0.2 .* y;

The t matrix represents time vectors and y is the summation of sine functions of various frequencies, then y is redefined as 1/5 the original power.

Plotted with

figure(1)

plot(1000*t(1:100),y(1:100))

title('Time Domain')

xlabel('t (ms)')

yields

Example1time.jpg

Then taking the Fourier Transform via the FFT and plotting

Y = fft(y,512);

Power = Y .* conj(Y) / 512;

f = 1000*(0:256)/512;

figure(2)

plot(f,Power(1:257))

title('Frequancy Domain')

xlabel('f (Hz)')

Example1freq.jpg

And if you want to hear it

wavwrite(y,1000,16,'test2.wav')

then

clear all

I believe that windowing needs to be added. Suggestions?