文档介绍：Outline of chapter 5
The Discrete Fourier Transform
离散傅立叶变换
The discrete Fourier Series(DFS)
Difinition: For periodical Signal
MATLAB Implementation
Example : For N=4, n= k= [0:N-1]
dfs1 function(row dfs)
Function [Xk] = dfs (xn, N)
% Xk and xn are column(not row) vectors
if nargin < 2 N = length(xn)
n = 0: N-1; k = 0: N-1;
WN = exp(-j*2*pi/N);
kn = k’*n;
WNkn = WN.^kn;
Xk = WNkn * xn
DFS of square wave seq.
Note
Envelope like the sinc function;
Zeros occur at N/L (reciprocal of duty cycle);
Relation of N to density of freq. Samples;
The calculating of series summation ΣWNn
( in ; prove of (); and ())
Comments of DFS and DTFT
Relation to z-transform and DTFT
DTFT sampling of example
Show sampling of DTFT equals to
Supplements on periodic sequences
Periodicity: x(n)=x(n+N) N-period
Periodicity of a sinusoid signal(pp7~8 of chinese ref.)
If sin(ω0n)= sin(ω0 ( n+N))
ω0 N=2kπ, N =2kπ/ω0 ---- minimum integer
. ω0 =(4/5) π, N=2k/(4/5)=2k*5/4=5 ( k=2)
. ω0 =(1/4) , N =2kπ*4 not a periodic sig.
Sampling and reconst. in z-domain
The coefficients of DFS are the sampling of DTFT :
The reconstruction of x(n) by is just like an inversion:
Sampling principle in
It is the sum of the r th moving N replica of x(n). If x(n) is actually shorter than N, no overlap occurs. x(n) will keep its shape.
Otherwise reconstruction is impossible.
length(x)=6,N=4,--overlap occurs
show overlap and nonoverlap by plots
Just like the sampling principle in time-domain.