## 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

*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

^{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].

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Signal Operators

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DFT Problems