Discrete-time Fourier transform

The Fourier transform of time-discrete signals, as English discrete -time Fourier transform, abbreviated DTFT denotes a linear transformation from the region of the Fourier analysis. It represents a finite time-discrete signal on a continuous, periodic frequency spectrum, which is also referred to as the image area. The DTFT is related to the discrete Fourier transform (DFT), which uses the discrete -time signals, the discrete spectra. The DTFT is different from the DFT is that it forms a continuous spectrum, which, under certain circumstances, can be specified as a partially closed mathematical expression. Like the DFT is the DTFT in the image area a periodically continued frequency spectrum, which is called the image spectrum.

Unlike the DFT, the DTFT has little importance in practical applications, such as digital signal processing, the primary application is in the theoretical signal analysis.

Definition

The spectrum of a sampled (discrete) time signal, represented with as a consequence and the sampling time is:

With the imaginary unit and the angular frequency. The inverse Fourier transform of time-discrete signals, the baseband without periodic spectral components is given as:

In order to avoid the dependence on the sampling time in the expressions, the spectrum is normalized to the sampling frequency and the thus normalized angular frequency

Is the DTFT:

And inverse DTFT:

Property

Some important properties of the Fourier transform of time-discrete signals are presented below.

Offset

The shifted in the time domain sequence corresponds to a phase rotation ( modulation ) in the spectral domain:

Proof:

Similarly, corresponding to a shifted in the frequency domain spectrum of a phase shift in the time domain:

Convolution property

The DTFT of a product of two sequences of values ​​and corresponds to the convolution of the spectra:

Conversely corresponds to the convolution in the time domain, the multiplication in the image area: