Sunday, April 3, 2011

Fourier Analysis


Commands covered:
          dft
          idft
          fft
          ifft
          contfft
The dft command uses a straightforward method to compute the discrete Fourier transform. Define a vector x and compute the DFT using the command
X = dft(x)
The first element in X corresponds to the value of X(0).
The command idft uses a straightforward method to compute the inverse discrete Fourier transform. Define a vector X and compute the IDFT using the command
x = idft(X)
The first element of the resulting vector x is x[0].
For a more efficient but less obvious program, the discrete Fourier transform can be computed using the command fft which performs a Fast Fourier Transform of a sequence of numbers. To compute the FFT of a sequence x[n] which is stored in the vector x, use the command
X = fft(x)
Used in this way, the command fft is interchangeable with the command dft. For more computational efficiency, the length of the vector x should be equal to an exponent of 2, that is 64, 128, 512, 1024, 2048, etc. The vector x can be padded with zeros to make it have an appropriate length. MATLAB does this automatically by using the following command where N is defined to be an exponent of 2:
X = fft(x,N);
The longer the length of x, the finer the grid will be for the FFT. Due to a wrap around effect, only the first N/2 points of the FFT have any meaning.
The ifft command computes the inverse Fourier transform:
x = ifft(X);
The FFT can be used to approximate the Fourier transform of a continuous-time signal as shown in Section 6.6 of the textbook. A continuous-time signal x(t) is sampled with a period of T seconds, then the DFT is computed for the sampled signal. The resulting amplitude must be scaled and the corresponding frequency determined. An M-file that approximates the Fourier Transform of a sampled continuous-time signal can be downloaded from contfft.m. Let a vector x be defined as the sampled continuous-time signal x(t) and let T be the sampling time.
[X,w] = contfft(x,T);
The outputs are the Fourier transform stored in the vector X and the corresponding frequency vector w.

Source: Georgia Tech
 

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