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