Signals and Systems

# Fourier transform for discrete-time periodic function Here we consider Fourier transform for discrete-time periodic function. Let’s consider the discrete-time Fourier-representation of a function $f\left[n\right]=\frac{1}{N}\sum _{–\infty }^{\infty }{a}_{n}{e}^{jn\frac{2\mathrm{\pi }}{n}K}$, where ${a}_{n}=\frac{1}{N}\sum _{–\infty }^{\infty }F\left(jw\right){e}^{–jn\frac{2\mathrm{\pi }}{N}k}$ and $F\left(jw\right)=\underset{–\infty }{\overset{\infty }{\sum x\left[n\right]{e}^{–jn\frac{2\mathrm{\pi }}{N}k}}}$.

The pair of equations that are called the Fourier pair for discrete-time functions $x\left[n\right]=\frac{1}{2\mathrm{\pi }}{\int }_{–\infty }^{\infty }F\left(jw\right){e}^{jwn}$ and $F\left(jw\right)=\sum _{k=–\infty }^{\infty }x\left[n\right]{e}^{–jn\frac{2\mathrm{\pi }}{N}k}$.

The Fourier transform for discrete-time periodic function has properties that are similar to the properties of a continuous-time Fourier transformation listed below.

1. Linearity. For the two pairs of Fourier transformation ${f}_{1}\left[n\right]=\frac{1}{2\mathrm{\pi }}{\int }_{–\infty }^{\infty }{F}_{1}\left(jw\right){e}^{jwn}dw$ and ${f}_{1}\left(jw\right)=\sum _{k=–\infty }^{\infty }f\left[n\right]{e}^{–\frac{jn2\mathrm{\pi }}{N}k}$${f}_{2}\left[n\right]=\frac{1}{2\mathrm{\pi }}{\int }_{–\infty }^{\infty }{F}_{2}\left(jw\right){e}^{jwn}$ and ${F}_{2}\left(jw\right)=\sum _{k=–\infty }^{\infty }{f}_{2}\left[n\right]{e}^{–\frac{jn2\mathrm{\pi }}{N}k}$ for the function $a{f}_{1}\left[n\right]+b{f}_{2}\left[n\right]$ the Fourier transformation will be $a{F}_{1}\left(jw\right)+b{F}_{2}\left(jw\right)$.
2. Periodicity. For the pair of Fourier transformations $f\left[n\right]=\frac{1}{2\mathrm{\pi }}{\int }_{–\infty }^{\infty }F\left(jw\right){e}^{jwn}dw$ and $F\left(jw\right)=\underset{–\infty }{\overset{\infty }{\sum f}}\left[n\right]{e}^{–\frac{jn2\mathrm{\pi k}}{N}}$. The Fourier transformation of two functions are different within a period are identical $F\left(jw\right)=F\left(j\left(w+2\mathrm{\pi }\right)\right)$.
3. Shifting property. If we have a shifted discrete-time function ${f}_{1}\left[n–{n}_{0}\right]$ so the corresponding Fourier transformation for this function will be ${e}^{–jw{n}_{0}}{F}_{1}\left(jw\right)$.
4. Conjugation. For the pair $f\left[n\right]$ and $F\left(jw\right)$ there is a conjugation rule that for the function ${f}^{*}\left[n\right]$ the Fourier transformation is: $F\left(–jw\right)$.
5. Differencing property. Let’s consider the discrete-time function $f\left[n\right]$, so for the sum $\underset{n}{\sum f\left[n\right]}$ will correspond the Fourier transformation $\frac{1}{1–{e}^{–jw}}F\left(jw\right)+\mathrm{\pi F}\left(\mathrm{j}0\right)\sum _{\mathrm{k}=–\infty }^{\infty }\mathrm{\delta }\left(\mathrm{w}–2\mathrm{\pi k}\right)$.
6. Time reverse. Let’s consider a function $f\left[n\right]$ and it’s Fourier transformation $F\left(jw\right)$, so for the function $f\left[–n\right]$ the Fourier transformation will be $F\left(–jw\right)$.
7. Expansion property. For the discrete-time function f[an] the function of the Fourier transformation will be $F\left(jwa\right)$.
8. Differentiation property. If we have a discrete-time function $f\left[n\right]$, that is characterised with the Fourier transformation $F\left(jw\right)$, so the differential of the Fourier transformation $\frac{dF\left(jw\right)}{dw}$ will correspond to the function $\frac{nf\left[n\right]}{j}$.
9. Parseval formula. For the discrete-time function $f\left[n\right]$ with Fourier transformation $F\left(jw\right)$ the following Parseval relation is valid $\underset{–\infty }{\overset{\infty }{\sum {\left|f\left[n\right]\right|}^{2}}}=\frac{1}{2\mathrm{\pi }}{\int }_{2\mathrm{\pi }}{\left|F\left(jw\right)\right|}^{2}$ . This means that the signal energy can also be integrated by frequency. As for the continuous-time signals ${\left|F\left(jw\right)\right|}^{2}$ is the energy density spectrum.
10. Convolution property. Let’s consider two discrete-time functions $f\left[n\right]$ and $h\left[n\right]$ with Fourier transformations $F\left(jw\right)$ and $H\left(jw\right)$, and discrete-time function $g\left[n\right]=f\left[n\right]*h\left[n\right]$ that is the convolution of two discrete-time functions. So the function g[n] will have the Fourier transformation $G\left(jw\right)=F\left(jw\right)H\left(jw\right)$.
11. Multiplication property. Let’s consider two discrete-time functions ${f}_{1}\left[n\right]$ and ${f}_{2}\left[n\right]$ with the multiplication product $h\left[n\right]$. Let’s assume they are characterised with Fourier transformations ${F}_{1}\left(jw\right)$ and ${F}_{2}\left(jw\right)$. So the Fourier transformation of the function is $h\left[n\right]$ is $H\left(jw\right)=\frac{1}{2\mathrm{\pi }}{\int }_{2\mathrm{\pi }}{F}_{1}\left(j\phi \right){F}_{2}\left(j\left(\phi –w\right)\right)d\phi$.

Let’s consider the LTI system, which is characterised with the equation $\underset{k}{\sum {b}_{k}g\left[k\right]=\underset{k}{\sum {a}_{k}f\left[k\right]}}$. Functions $h\left[k\right]$ and $f\left[k\right]$ are characterised with the Fourier transformations $F\left(jw\right)$ and $G\left(jw\right)$$G\left(jw\right)$. So, the LTI system is characterised by the Fourier transformation: $H\left(jw\right)=\frac{F\left(jw\right)}{G\left(jw\right)}$.

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 “Signals and systems”, 2nd edition, 1997. Alan V. Oppenheim, Allan S. Willsky.