- Given a discrete-time signal
*x*[*n*], we can determine its frequency response using the Discrete Fourier Transform (DFT): - The complementary inverse DFT
is given by:
- The DFT can be interpreted as the sum of projections of
*x*[*n*] onto a set of*N*sampled complex sinusoids or sinusoidal basis functions at (normalized) radian frequencies given by with . - In this way, the DFT and its inverse provide a “recipe” for reconstructing a given discrete-time signal in terms of sampled complex sinusoids.
- Various computational tools exist that allow for the transformation of a discrete-time signal into its frequency-domain representation.
- The frequency response of a digital filter can be found by taking the DFT of the filter impulse response, assuming the impulse response has sufficiently decayed before being truncated.

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