Himarsha Shreedhara

\[ X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i kn/N}, \]

for \( 0 \le k \le N-1.\)

The \(x_i\) are thought of as the values of a function, or signal, at equally spaced times \(t=0,1,\ldots,N-1.\) The output \(X_k\) is a complex number which encodes the amplitude and phase of a sinusoidal wave with frequency \(\frac kN\) cycles per time unit. \(\big(\)This comes from Euler's formula: \( e^{2\pi i kn/N} = \cos(2\pi kn/N) + i\sin(2\pi kn/N).\big)\) The effect of computing the \(X_k\) is to find the coefficients of an approximation of the signal by a linear combination of such waves. Since each wave has an integer number of cycles per \(N\) time units, the approximation will be periodic with period \(N.\) This approximation is given by the inverse Fourier transform

\[ x_n = \frac1{N} \sum_{k=0}^{N-1} X_k e^{2\pi ikn/N}. \]

The DFT is useful in many applications, including the simple signal spectral analysis outlined above. Knowing how a signal can be expressed as a combination of waves allows for manipulation of that signal and comparisons of different signals:

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