The Discrete Fourier Transform

The Fourier Transform “The Fourier transform is a mathematical operation with many applications in physics and engineering that expresses a mathematical function of time as a function of frequency, known as its frequency spectrum.” – from

The Fourier Transform “For instance, the transform of a musical chord made up of pure notes (without overtones) expressed as amplitude as a function of time, is a mathematical representation of the amplitudes and phases of the individual notes that make it up.” – from

Amplitude & phase f(x) =  sin(  x +  ) +  –  is the amplitude –  is the frequency –  is the phase –  is the DC offset

More generally f(x) =  1 sin(  1 x +  1 ) +  2 sin(  2 x +  2 ) + 

The Fourier Transform “The function of time is often called the time domain representation, and the frequency spectrum the frequency domain representation.” – from

Applications differential equations geology image and signal processing optics quantum mechanics spectroscopy

REVIEW OF COMPLEX NUMBERS

Complex numbers Complex numbers... extend the 1D number line to the 2D plane are numbers that can be put into the rectangular form, a+bi where i 2 = -1, and a and b are real numbers.

Complex numbers (rectangular form)

Complex numbers Complex numbers... a is the real part; b is the imaginary part If a is 0, then a+bi is purely imaginary; if b is 0, then a+bi is a real number. originally called “fictitious” by Girolamo Cardano in the 16 th century

Complex arithmetic add/subtract – add/subtract the real and imaginary parts separately

Complex arithmetic complex conjugate – often denoted as – negate only the imaginary part

Complex arithmetic inverse where z is a complex number z bar is the length or magnitude of z a is the real part b is the imaginary part

Complex arithmetic multiplication (FOIL)

Complex arithmetic division complex conjugate of denominator

Complex numbers (polar form)

exponential vs. trigonometric Leonhard Euler (phasor form)

DFT (DISCRETE FOURIER TRANSFORM)

DFT Say we have a sequence of N (possibly complex) numbers, x 0 … x N-1. The DFT produces a sequence of N (typically complex) numbers, X 0 … X N-1, via the following:

DFT & IDFT The DFT (Discrete Fourier Transform) produces a sequence of N (typically complex) numbers, X 0 … X N-1, via the following: The IDFT (Inverse DFT) is defined as follows:

Calculating the DFT So how can we actually calculate ?

Calculating the DFT So how can we calculate? Let’s use this relationship: Then So what does this mean?

Interpretation of DFT Back to the polar form: – r/N is the amplitude and  is the phase of a sinusoid with frequency k/N into which x n is decomposed

CALCULATING THE DFT USING EXCEL

Check w/ matlab/octave % see b/ref/fft.html b/ref/fft.html N = 256;% # of samples n = (0:N-1);% subscripts b1 = 0.5;% freq 1 b2 = 2.5;% freq 2 xn = 0.5 * sin( b1*n ) * sin( b2*n ); plot( xn ); Xn = fft( xn ); plot( abs(Xn(1:N/2)) ); X0real= xn.* cos( -2*pi*n*0/N ); X0imag= xn.* sin ( -2*pi*n*0/N ); X1real= xn.* cos( -2*pi*n*1/N ); X1imag= xn.* sin ( -2*pi*n*1/N ); X2real= xn.* cos( -2*pi*n*2/N ); X2imag= xn.* sin ( -2*pi*n*2/N ); X3real= xn.* cos( -2*pi*n*3/N ); X3imag= xn.* sin ( -2*pi*n*3/N );. Note:.* is element-wise (rather than matrix) multiplication in matlab.

Add random noise. % see b/ref/fft.html b/ref/fft.html N = 256;% # of samples n = (0:N-1);% subscripts b1 = 0.5;% freq 1 b2 = 2.5;% freq 2 r = randn( 1, N );% noise xn = 0.5 * sin( b1*n ) * sin( b2*n ) * r; plot( xn ); Xn = fft( xn ); plot( abs(Xn(1:N/2)) ); X0real= xn.* cos( -2*pi*n*0/N ); X0imag= xn.* sin ( -2*pi*n*0/N ); X1real= xn.* cos( -2*pi*n*1/N ); X1imag= xn.* sin ( -2*pi*n*1/N ); X2real= xn.* cos( -2*pi*n*2/N ); X2imag= xn.* sin ( -2*pi*n*2/N ); X3real= xn.* cos( -2*pi*n*3/N ); X3imag= xn.* sin ( -2*pi*n*3/N );.

Signal without and with noise.

Signal with noise. FFT of noisy signal (two major components are still apparent).

Example of differences in phase. xn = 0.5 * sin( b1*n ) * sin( b2*n ) xn = 0.5 * sin( b1*n – 0.5 ) * sin( b2*n )

Computational complexity: DFT vs. FFT The DFT is O(N 2 ) complex multiplications. In 1965, Cooley (IBM) and Tukey (Princeton) described the FFT, a fast way (O(N log 2 N)) to compute the FT using digital computers. – It was later discovered that Gauss described this algorithm in 1805, and others had “discovered” it as well before Cooley and Tukey. – “With N = 106, for example, it is the difference between, roughly, 30 seconds of CPU time and 2 weeks of CPU time on a microsecond cycle time computer.” – from Numerical Recipes in C

Extending the DFT to 2D (and higher) Let f(x,y) be a 2D set of sampled points. Then the DFT of f is the following: (Note that engineers often use i for amps (current) so they use j for  -1 instead.)

Extending the DFT to 2D (and higher) In fact, the 2D DFT is separable so it can be decomposed into a sequence of 1D DFTs. And this can be generalized to higher and higher dimensions as well.

The classical “Gibbs phenomenon” Visit Hear it at uvY. uvY