Fourier transform for time-discrete signals

The Discrete-Time Fourier Transform , also called English time discrete Fourier transform , abbreviated DTFT called, is a linear transformation in the field of Fourier analysis . It maps an infinite, time-discrete signal onto 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 works with discrete time signals and discrete spectra. The DTFT differs from the DFT in that it forms a continuous spectrum which, under certain circumstances, can be specified as a mathematical expression that is closed in sections . Like the DFT, the DTFT forms a periodically continued frequency spectrum in the image area, which is referred to as the mirror spectrum.

In contrast to the DFT, the DTFT is of little importance in practical applications such as digital signal processing , the primary area of ​​application is in theoretical signal analysis.

definition

The spectrum of a sampled (discrete) time signal, represented as a sequence with and the sampling time , is: ${\displaystyle X(\omega )}$ ${\displaystyle x[n]}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle t_{A}=1/f_{A}}$

${\displaystyle X(\omega )=\sum _{n=-\infty }^{\infty }x[n]\,e^{-\mathrm {j} \omega nt_{A}}=\mathrm {DTFT} \{x[n]\}}$

with the imaginary unit and the angular frequency . The inverse Fourier transform for time-discrete signals over the baseband without periodic spectral components is given as: ${\displaystyle \mathrm {j} }$ ${\displaystyle \omega }$

${\displaystyle x[n]={\frac {1}{\omega _{A}}}\int _{-\omega _{A}/2}^{\omega _{A}/2}X(\omega )\cdot e^{\mathrm {j} \omega nt_{A}}\,\mathrm {d} \omega =t_{A}\int _{-f_{A}/2}^{f_{A}/2}X(2\pi f)\cdot e^{\mathrm {j} 2\pi fnt_{A}}\,\mathrm {d} f=\mathrm {DTFT} ^{-1}\{X(\omega )\}}$

In order to avoid the dependence on the sampling time in the expressions, the spectrum is normalized to the sampling frequency and with the angular frequency normalized in this way ${\displaystyle t_{A}}$ ${\displaystyle f_{A}}$

${\displaystyle \Omega =\omega \cdot t_{A}}$

the DTFT reads:

${\displaystyle X(\Omega )=\sum _{n=-\infty }^{\infty }x[n]\,e^{-\mathrm {j} \Omega n}}$

and the inverse DTFT:

${\displaystyle x[n]={\frac {1}{2\pi }}\int _{-\pi }^{\pi }X(\Omega )\cdot e^{\mathrm {j} \Omega n}\,\mathrm {d} \Omega }$

property

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

Offset

The sequence shifted in the time domain corresponds to a phase shift (modulation) in the spectral domain : ${\displaystyle x[n-n_{0}]}$

${\displaystyle \mathrm {DTFT} \{x[n-n_{0}]\}=e^{-\mathrm {j} \Omega n_{0}}X[\Omega ]\,}$

Proof:

${\displaystyle \mathrm {DTFT} \{x[n-n_{0}]\}=\sum _{n=-\infty }^{\infty }x[n-n_{0}]\,e^{-\mathrm {j} \Omega n}\,,\,mit:\,m=n-n_{0}\leftrightarrow n=m+n_{0}}$

${\displaystyle {\begin{aligned}\rightarrow \sum _{m=-\infty }^{\infty }x[m]\,e^{-\mathrm {j} \Omega [m+n_{0}]}&=\sum _{m=-\infty }^{\infty }x[m]\,e^{-\mathrm {j} \Omega m}\cdot e^{-\mathrm {j} \Omega n_{0}}=e^{-\mathrm {j} \Omega n_{0}}\cdot \sum _{m=-\infty }^{\infty }x[m]\,e^{-\mathrm {j} \Omega m}\\&=e^{-\mathrm {j} \Omega n_{0}}\cdot \mathrm {DTFT} \{x[n]\}=e^{-\mathrm {j} \Omega n_{0}}X[\Omega ]\\\end{aligned}}}$

Similarly, a spectrum shifted in the frequency domain corresponds to a phase shift in the time domain: ${\displaystyle Y[\Omega -\Omega _{0}]}$

${\displaystyle \mathrm {DTFT} \{x[n]e^{\mathrm {j} \Omega _{0}n}\}=X[\Omega -\Omega _{0}]\,}$

Folding property

The DTFT of a product of two series of values and corresponds to the convolution of the spectra: ${\displaystyle x[n]}$ ${\displaystyle y[n]}$

${\displaystyle \mathrm {DTFT} \{x[n]\cdot y[n]\}={\frac {1}{2\pi }}X[\Omega ]*Y[\Omega ]\,}$

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

${\displaystyle \mathrm {DTFT} \{x[n]*y[n]\}=X[\Omega ]\cdot Y[\Omega ]\,}$

literature

• Otto Föllinger : Laplace, Fourier and z transformation . 10th edition. VDE-Verlag, 2011, ISBN 978-3-8007-3257-9 .
• Alan V. Oppenheim, Ronald W. Schafer: Discrete-time signal processing . 3. Edition. Oldenbourg Verlag, 1999, ISBN 3-486-24145-1 .