Signals and Systems

Fourier transform for discrete-time periodic function

fourier transform for discrete-time periodic

Here we consider Fourier transform for discrete-time periodic function. Let’s consider the discrete-time Fourier-representation of a function fn=1Nanejn2πnK, where an=1NF(jw)ejn2πNk and F(jw)=xnejn2πNk.

The pair of equations that are called the Fourier pair for discrete-time functions xn=12πF(jw)ejwn and F(jw)=k=xnejn2πNk.

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 f1n=12πF1(jw)ejwndw and f1jw=k=fnejn2πNkf2n=12πF2(jw)ejwn and F2(jw)=k=f2nejn2πNk for the function af1n+bf2n the Fourier transformation will be aF1(jw)+bF2(jw).
  2. Periodicity. For the pair of Fourier transformations fn=12πF(jw)ejwndw and F(jw)=fnejn2πkN. The Fourier transformation of two functions are different within a period are identical F(jw)=F(j(w+2π)).
  3. Shifting property. If we have a shifted discrete-time function f1nn0 so the corresponding Fourier transformation for this function will be ejwn0F1(jw).
  4. Conjugation. For the pair fn and F(jw) there is a conjugation rule that for the function f*n the Fourier transformation is: F(jw).
  5. Differencing property. Let’s consider the discrete-time function fn, so for the sum fnn will correspond the Fourier transformation 11ejwF(jw)+πF(j0)k=δw2πk.
  6. Time reverse. Let’s consider a function fn and it’s Fourier transformation F(jw), so for the function fn the Fourier transformation will be F(jw).
  7. Expansion property. For the discrete-time function f[an] the function of the Fourier transformation will be F(jwa).
  8. Differentiation property. If we have a discrete-time function fn, that is characterised with the Fourier transformation F(jw), so the differential of the Fourier transformation dF(jw)dw will correspond to the function nfnj.
  9. Parseval formula. For the discrete-time function fn with Fourier transformation F(jw) the following Parseval relation is valid fn2=12π2πF(jw)2 . This means that the signal energy can also be integrated by frequency. As for the continuous-time signals F(jw)2 is the energy density spectrum.
  10. Convolution property. Let’s consider two discrete-time functions fn and hn with Fourier transformations F(jw) and H(jw), and discrete-time function gn=fn*hn that is the convolution of two discrete-time functions. So the function g[n] will have the Fourier transformation G(jw)=F(jw)H(jw).
  11. Multiplication property. Let’s consider two discrete-time functions f1n and f2n with the multiplication product hn. Let’s assume they are characterised with Fourier transformations F1(jw) and F2(jw). So the Fourier transformation of the function is hn is H(jw)=12π2πF1(jφ)F2(j(φw))dφ.

Let’s consider the LTI system, which is characterised with the equation bkgk=akfkkk. Functions hk and fk are characterised with the Fourier transformations F(jw) and G(jw)G(jw). So, the LTI system is characterised by the Fourier transformation: H(jw)=F(jw)G(jw).[1]

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

LTI systems and their properties

 

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