Mathematics of the Discrete Fourier Transform (DFT)

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Modulo Indexing, Periodic Extension
The DFT sinusoids $$s_k(n) \$ are all periodichaving 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 is periodic extension, i.e., $$x(n+mN)\$ for every integer .
Moreover, the DFT also repeats naturally every samples, since

because $$s_{k+mN}(n) = e^{j2\$ . (The DFT sinusoids behave identically as functions of and.) Accordingly, for purposes of DFT studies, we may define allsignals in $${\$ as being single periods from an infinitely long periodic signal with period samples:

Definition: For any signal $$x\$ , we define

for every integer .
As a result of this convention, all indexing of signals and spectra^8.1 can be interpreted modulo , and we may write $$x(n \$ to emphasize this. Formally, `` $$n \$ '' 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 samefrequency as dc for discrete time signals. In the time domain, we have what is sometimes called the ``periodic extension'' of . This means that the input to the DFT is mathematically treated as samples of a periodic signal with period seconds ( samples). The corresponding assumption in the frequency domain is that the spectrum is zero between frequency samples $$\$ .
It is also possible to adopt the point of view that the time-domain signal consists of samples preceded and followed by zeros. In this case, the spectrum is nonzero between spectral samples $$\$ , and the spectrum between samples can be reconstructed by means of bandlimited interpolation. This ``time-limited'' interpretation of the DFT input is considered in detail in Music 420 and is beyond the scope of Music 320 (except in the discussion of ``zero padding $$\$ interpolation'' below).
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