CS411X - Lecture 6

Lecture Topics:

• Quicktime Media Layer
• Discrete Fourier Transform
• 2-D Fourier Transform
• Sun Au Files
• Sample Quiz

Quick Time Media Layer :

Odd and Even functions :

f(-x) = -f(x) => Odd function

f(-x) = f(x) => Even function

Sin(-x) = -Sin(x) => Odd function.

Cos(-x) = Cos(x) => Even function.

The Discrete Fourier Transform (DFT)

Let x(n) be the sampled waveform :

                k = frequency index
                f = frequency
                fs = sample rate
 
      
        

        ie, for Audio CD's

        fs = 44.4 KHz

Note : DFP summation is performed for every k, so DFT is O(N²)

 
           
               
 
        F(u) = DFT of f(x)
        N = number of input samples
        x = dummy variable used to index the sample
        1/N1/2 = Normalize the spectrum WRT input length

        f(x) is a reconstruction from the integral harmonic of fs/N

Sun Au files

        as read by the Java API
        fs = 8000 Hz
        N = 8000 sample

Thus available Reconstruction harmonic are :

        fs / N = 8000/8000 = 1,2...4KHz
        fs / N = 8000/800 = ± 10,± 20....± 4KHz 

Negative frequencies Occupy the N/2à.N-1 position in the DFT and have no physical meaning.

The maximum allowable f is 1/2 f_s

        fs = 8000 Hz
        fmax = 4000 Hz
        k E [ 0...N/2 ]
        fs/N = 10
        N = 400
        Card { 10 20 30....200 } = 20

The 2D Fourier Transform :

Let f(x,y) it is continuous and integrable

The

u and v are in the frequency domain

The Fourier spectrum

The phase

In 2-D the DFT

u, x = 0,...n-1

x, y = 0,... m-1

Sample Quiz

• Adjacent over Hypotenuse
• Opposite Over Adjacent
• Hypotenuse over Opposite
• Opposite over Hypotenuse
   

Last Update: 04/09/97
Copyright © 1997- Douglas Lyon
Lyon@cse.bridgeport.edu