## The DFT and its Inverse Restated

Let
, denote an -sample complex sequence,
*i.e.*,
. Then the *spectrum* of is defined by the
*Discrete Fourier Transform (DFT)*:

*inverse DFT*(

*IDFT*) is defined by

*second*.'' (Of course, there's no difference when the sampling rate is really .) Another term we use in connection with the convention is

*normalized frequency*: All normalized radian frequencies lie in the range , and all normalized frequencies in Hz lie in the range .

^{7.1}Note that physical units of seconds and Hertz can be reintroduced by the substitution

### Notation and Terminology

If is the DFT of , we say that and form a *transform
pair* and write

If we need to indicate the length of the DFT explicitly, we will write
and
.
As we've already seen, *time-domain* signals are consistently denoted
using *lowercase* symbols such as ``,'' while *frequency-domain*
signals (spectra), are denoted in *uppercase* (``
'').

### 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,

*periodic extension*,

*i.e.*, for every integer .

Moreover, the DFT also repeats naturally every samples, since

*all*signals in as being single periods from an infinitely long periodic signal with period samples:

**Definition (Periodic Extension): ** For any signal
, we define

As a result of this convention, all indexing of signals and
spectra^{7.2} can be interpreted *modulo* , and we may write
to emphasize this. Formally, ``
'' is defined as
with chosen to give in the range .

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].

**Next Section:**

Signal Operators

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DFT Problems