Discrete-Time Signals & Digital Filtering
Steady State and Transient Response
The Discrete Fourier Transform (DFT)
Given a discrete-time signal
, we can determine its frequency response using the
Discrete Fourier Transform (DFT)
is given by:
The DFT can be interpreted as the sum of projections of
onto a set of
sampled complex sinusoids or sinusoidal basis functions at (normalized) radian frequencies given by
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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Gary P. Scavone