The Discrete Fourier Transform: Definition as an inner product
Contents
1 Sampled discrete time signals
Consider an -length finite duration discrete time signal,
, specified at integer sample time indices
.
The signal comprises
samples, and therefore runs from
. In general,
may be complex valued, i.e.,
, though in most audio-related applications it will be purely real.
2 The Discrete Fourier Transform: Definition as an inner product
The DFT, , of the signal
is obtained by evaluating the inner product of
with an orthogonal set of
complex exponentials
(1)
where integer , and
is the imaginary number (
).
Using the definition on the Mathematical tools page, we can evaluate via
(2)
for sample time index , frequency bin index
, and number of time samples included in the transformation
(which is also equal to the number of frequency bins). Note that:
,
, and
are all integers.
- A conjugation operation acts on
, within the summation of equation 2 (as expected).
- There is no explicit definition of `frequency’ when the DFT is defined in this way — i.e., it is agnostic to sample rate (see below for more on this).
- For real valued signals
, only half of the spectrum
is unique — see here[LINK] for more on this.
The values are sometimes referred to as complex amplitudes, since in general they are complex valued for a real valued signal
— i.e., each
value has a magnitude and phase, or equivalently, a real and imaginary part. Analogous with earlier examples, each value of
tells us something about `how much’ of the given complex exponential
is present within
.