Chapter 5. The Discrete Fourier Transform. Gao Xinbo School of E.E., Xidian Univ. firstname.lastname@example.org http://see.xidian.edu.cn/teach/matlabdsp/. Review 1. The DTFT provides the frequency-domain ( w ) representation for absolutely summable sequences.
School of E.E., Xidian Univ.
are called the discrete Fourier series coefficients, which are given by
The discrete Fourier series representation of periodic sequences
Let x and X denote column vectors corresponding to the primary periods of sequences x(n) and X(k), respectively.
The DFS X(k) represents N evenly spaced samples of the z-transform X(z) around the unit circle.
The DFS is obtained by evenly sampling the DTFT at w1 intervals.
The interval w1 is the sampling interval in the frequency domain. It is called frequency resolution because it tells us how close are the frequency samples.
DFS & z-transform
RN(n) is called a rectangular window of length N.
THEOREM1: Frequency Sampling
If x(n) is time-limited (finite duration) to [0,N-1], then N samples of X(z) on the unit circle determine X(z) for all z.
Let x(n) be time-limited to [0,N-1]. Then from Theorem 1 we should be able to recover the z-transform X(z) using its samples X~(k).
An interpolation polynomial
This is the DTFT interpolation formula to reconstruct X(ejw) from its samples X~(k)
Since , we have that X(ej2pik/N)=X~(k), which means that the interpolation is exact at sampling points.
The compact relationships between x(n) and x~(n) are
The function rem(n,N) can be used to implement our modulo-N operation.
Note that the DFT X(k) is also an N-point sequence, that is, it is not defined outside of 0<=n<=N-1.
DFT X(k) is the primary interval of X~(k).
1. Linearity: DFT[ax1(n)+bx2(n)]=aDFT[x1(n)]+bDFT[x2(n)]
N3=max(N1,N2): N3-point DFT
2. Circular folding:
4. Symmetry properties for real sequences:
Let x(n) be a real-valued N-point sequence
The real-valued signals
5. Circular shift of a sequence
6. Circular shift in the frequency domain
7. Circular convolution**
9. Parseval’s relation:
In general, the circular convolution is an aliased version of the linear convolution.
If we make both x1(n) and x2(n) N=N1+N2-1 point sequences by padding an appropriate number of zeros, then the circular convolution is identical to the linear convolution.
X3(n) is also causal
Lenx = length(x); M=length(h); M1=M-1; L=N-M1;
H = [h zeros(1,N-M)];
X = [zeros(1,M1), x, zeros(1,N-1);
K = floor((Lenx+M1-1)/L);
For k=0:K, xk = x(k*L+1:k*L+N);
Y(k+1,: ) = circonvt(xk,h,N); end
Number of complex mult. CN=O(N2)
If N=2^10, CN=will reduce to 1/100 times.
Decimation-in-time: DIT-FFT, decimation-in-frequency: DIF-FFT
Three-step procedure: P155
4×4cplx→ 2 ×1+ 1 ×4 cplx 16 →6