DFT

Recall, one signal $y(\cdot)$ is projected onto another signal $x(\cdot)$ using an inner product defined by

$\displaystyle \langle y,x \rangle \triangleq \sum_{n=0}^{N-1}y(n)\overline{x(n)}$
If is a sampled, unit amplitude, zero-phase, complex sinusoid,

$\displaystyle x(n) = e^{j\omega_knT}, n=0,1,\hdots, N-1,$
then the inner product computes the Discrete Fourier Transform (DFT).

$\displaystyle \langle y,x \rangle$ $\displaystyle \triangleq$ $\displaystyle \sum_{n=0}^{N-1}y(n)\overline{x(n)}$

$\displaystyle =$ $\displaystyle \sum_{n=0}^{N-1}y(n)e^{-j\omega_knT}$

$\displaystyle \triangleq$ DFT $\displaystyle _k(y) \triangleq Y(\omega_k)$
$Y(\omega_k)$ , the DFT at frequency $\omega_k$ , is a measure of the amplitude and phase of the complex sinusoid which is present in the input signal at that frequency.

``Music 270a: Complex Exponentials and Spectrum Representation'' by Tamara Smyth, Department of Music, University of California, San Diego (UCSD).

$\displaystyle \langle y,x \rangle$	$\displaystyle \triangleq$	$\displaystyle \sum_{n=0}^{N-1}y(n)\overline{x(n)}$
	$\displaystyle =$	$\displaystyle \sum_{n=0}^{N-1}y(n)e^{-j\omega_knT}$
	$\displaystyle \triangleq$	DFT $\displaystyle _k(y) \triangleq Y(\omega_k)$