文档介绍：Chapter 3. The Discrete Fourier Transform
The Discrete Fourier Series
Definition: Periodic sequence
N: the fundamental period of the sequences
From FT analysis we know that the periodic functions can be synthesized as a bination plex exponentials whose frequencies are multiples (or harmonics) of the fundamental frequency (2pi/N).
From the frequency-domain periodicity of the DTFT, we conclude that there are a finite number of harmonics; the frequencies are {2pi/N*k,k=0,1,…,N-1}.
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Fourier series of periodic continuous signals
Ω0—period of x(t) in radian;
Let T---sampling period; ω0----smpling period in radian
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So X(k) is also a periodic function with N
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DFS pair
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Properties of DFS
Suppose the following 3 sequences’s period is N
Linearity
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Shifting
Symmetry
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Periodic convolution
Distinction with convolution sum
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The Discrete Fourier Transform
Suppose: x(n)------finite-length sequence, N-----length;
. , x(n)=0 when n<0 or n>N-1
Let x(n) be a period sequence of a periodic sequence
Then we have
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