
Modulo Indexing, Periodic Extension
The
DFT sinusoids

are all
periodic
having
periods which divide

. That is,

for any
integer

. Since a length
signal 
can be expressed as a
linear
combination of the
DFT sinusoids in the time domain,

it follows that the ``automatic'' definition of

beyond the
range
![$ [0,N-1]$](http://www.dsprelated.com/josimages_new/mdft/img1128.png)
is
periodic extension,
i.e.,

for every integer

.
Moreover, the DFT also repeats naturally every

samples, since
because

. (The DFT
sinusoids behave identically as functions of

and

.) Accordingly, for purposes of DFT studies, we may define
all
signals in

as being single periods from an infinitely long
periodic
signal with period

samples:
Definition (Periodic Extension): For any signal

, we define
for every integer

.
As a result of this convention, all indexing of signals and
spectra7.2 can be interpreted
modulo 
, and we may write

to emphasize this. Formally, ``

'' is defined as

with

chosen to give

in the range
![$ [0,N-1]$](http://www.dsprelated.com/josimages_new/mdft/img1128.png)
.
As an example, when indexing a
spectrum 
, we have that

which can be interpreted physically as saying that the
sampling rate
is the same frequency as
dc for discrete time signals. Periodic
extension in the time domain implies that the signal input to the DFT
is mathematically treated as being
samples of one period of a
periodic signal, with the period being exactly

seconds (

samples). The corresponding assumption in the
frequency domain is
that the
spectrum is
exactly zero between frequency samples

. It is also possible to adopt the point of view that the
time-domain signal

consists of

samples preceded and
followed by
zeros. In that case, the
spectrum would be
nonzero between spectral samples

, and the spectrum
between samples would be reconstructed by means of
bandlimited
interpolation [
72].
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