The DFT

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). The DFT

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The DFT

For a length $N$ complex sequence $x(n)$, $n=0,1,2,\, thediscrete Fourier transform (DFT) is defined by

\

\


We are now in a position to have a full understanding of the transform kernel:
\

The kernel consists of samples of a complex sinusoid at $N$ discrete frequencies $\ uniformly spaced between $0$ and the sampling rate$\. All that remains is to understand the purpose and function of the summation over $n$ of the pointwise product of $x(n)$times each complex sinusoid. We will learn that this can be interpreted as an inner product operation which computes the coefficient of projection of the signal $x$ onto the complex sinusoid $\. As such, $X(\, the DFT at frequency$\, is a measure of the amplitude and phase of the complex sinusoid at that frequency which is present in the input signal $x$. This is the basic function of all transform summations (in discrete time) and integrals (in continuous time) and their kernels.

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