The discrete Fourier transform (DFT) is a basic tool for digital signal processing. From a mathematical point of view the DFT transforms a digital signal from the original time domain into another discrete series, in the transformed frequency domain. This enables the analysis of the discrete-time signal in both the original and (especially) the transformed domains.

The frequency analyses of a digital filter and of a discrete-time signal are very similar problems. It will be seen in the next chapter that, for a digital filter, it consists of evaluating the transfer function *H*(*z*) on the unit circle (*z* = *e*^{j2πvT}). This is the same for the digital signal analysis using the z-transform, which is closely related to the DFT.

As the calculations are performed on a digital computer there are 3 types of errors related to the transition from the Fourier transform of the continuous-time signal to the DFT of the discrete-time signal:

– errors due to time sampling (transition from *x*(*t*) to *x _{s}* (

– errors due to time truncation (transition from *x*_{s}(*t*) to *x*_{s}(*t*)Π_{T}(*t*)),

– errors due to frequency sampling.

The DFT of finite time 1D digital signals, denoted by DFT_{ID}, is defined by:

where: *W*_{N} = *e*^{−j2π/N} and *n*, *k* = 0..*N* –1.

In the above equations, the index of the vectors *x*[*n*] and *X*[*k*] should begin with 1 instead of 0, ...

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