This Jupyter notebook is part of a collection of notebooks in the bachelors module Signals and Systems, Comunications Engineering, Universität Rostock. Please direct questions and suggestions to Sascha.Spors@uni-rostock.de.
The properties, theorems and transforms of the discrete Fourier transform (DFT) as derived in the previous sections are compiled in the following. The corresponding tables serve as a reference for the application of the DFT in the theory of signals and systems. Please refer to the respective sections for details.
The DFT of length $N$ and its inverse are defined as
\begin{align} X[\mu] &= \sum_{k = 0}^{N-1} x[k] \, e^{- j \mu \frac{2 \pi}{N} k} \\ x[k] &= \sum_{k = 0}^{N-1} X[\mu] \, e^{j \mu \frac{2 \pi}{N} k} \end{align}for $k, \mu \in \mathbb{Z}$.
Under the assumption of causal signals of finite length $N$ or peridic signals of period $N$, the properties and theorems of the DTFT are given as
| | $x[k]$ | $X[\mu] = \text{DFT}_N \{ x[k] \}$ |
|---|---|---|
| Linearity | $A \, x_1[k] + B \, x_2[k]$ | $A \, X_1[\mu] + B \, X_2[\mu]$ |
| Real-valued signal | $x^*[k]$ | $X^*[N-\mu]$ |
| Convolution | $x[k] \circledast_N h[k]$ | $X[\mu] \cdot H[\mu]$ |
| Periodic shift | $x[k - \kappa]$ | $e^{-j \mu \frac{2 \pi}{N} \kappa} \cdot X[\mu]$ |
| Multiplication | $x[k] \cdot h[k]$ | $\frac{1}{N} X[\mu] \circledast_{N} H[\mu]$ |
| Modulation | $e^{j M \frac{2 \pi}{N} k} \cdot x[k]$ | $X[\mu - M]$ |
| Parseval's Theorem | $\sum_{k = 0}^{N-1} \lvert x[k] \rvert^2$ | $\frac{1}{N} \sum_{\mu = 0}^{N-1} \lvert X[\mu] \rvert^2$ |
where $A, B \in \mathbb{C}$ and $\kappa, M \in \mathbb{Z}$.
DFTs which are frequently used are given as
| $x[k]$ | $X[\mu] = \text{DFT}_N \{ x[k] \}$ |
|---|---|
| $\delta[k]$ | $1$ |
| $1$ | $N \cdot \delta[\mu]$ |
| $e^{j \Omega_0 k}$ | $e^{-j (\Omega_0 - \mu \frac{2 \pi}{N} ) \frac{N-1}{2} } \cdot \frac{\sin \left( \frac{N (\Omega_0 - \mu \frac{2\pi}{N})}{2} \right)}{\sin \left( \frac{ \Omega_0 - \mu \frac{2\pi}{N}}{2} \right)}$ |
| $\text{rect}_M[k]$ | $e^{-j \pi \mu \frac{M-1}{N} } \cdot \frac{\sin \left( \frac{M \pi \mu}{N} \right)}{\sin \left( \frac{ \pi \mu}{N} \right)}$ |
where $0 \leq \Omega_0 < 2 \pi$ and $0 < M \leq N$. Additional transforms may be found in the literature or online.
Copyright
The notebooks are provided as Open Educational Resource. Feel free to use the notebooks for your own educational purposes. The text is licensed under Creative Commons Attribution 4.0, the code of the IPython examples under the MIT license. Please attribute the work as follows: Lecture Notes on Signals and Systems by Sascha Spors.