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I am trying to understand the periodicity of the DFT. How can this property (both in time and frequency domain) be used and can be helpful while developing on a DSP?
It would be good to see some source code or pseudo code, in MATLAB preferably, where this property is exploited/demonstrated.
If the expression that defines the DFT is evaluated for all integers $k$ instead of just for $k = 0, \dots, N-1$ , then the resulting infinite sequence is a periodic extension of the DFT, periodic with period $N$.
The periodicity can be shown directly from the definition:
$$ X_{k+N} \stackrel{\mathrm{def}}{=} \ \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} (k+N) n} = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} k n} \underbrace{e^{-2 \pi i n}}_{1} = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} k n} = X_k.$$
Similarly, it can be shown that the IDFT formula leads to a periodic extension.
Source: Wikipedia.
While the periodic property of DFT isn't very widely utilized, it often causes aliasing problems.
In the time domain, the DFT is periodic by definition. While DFT stands for Discrete Fourier Transform, the operation is in fact a discrete fourier series. The signal to be analyzed is assumed to be periodic in the lenght of the signal. This periodic signal is decomposed into a series of periodic sequences. The DFT frequency bins are located at f = 1/T and its integer multiples, where T is the duration of the signal to be analyzed.
In the frequency domain, the DFT is periodic because the time domain signal being analyzed is sampled. Recall that any periodic sequence cannot be uniquely represented for frequencies above fs/2 where fs is the sampling frequency of the sequence (also known as the nyquist frequency). Above fs/2 all signal energy is reflected back into the frequency range 0-fs/2. Between fs/2 and fs, the reflection is in reverse order which gives rise to a DFT period equal to fs.
