离散序列频域抽样的实验说明.docx

DFT实质是对离散时间序列的频域抽样而来，抽样后得到的是DFS,DFT只是DFS的时域和频域的主值序列。DTFT的表达式(假设x(n)为N点有限长)：对频域抽样其中，k=0,1,M-1当M=N，即频域的抽样点数等于时域的点数时，此频域的抽样表达式即是DFT.编写程序验证不同的抽样值对于时域信号的影响：1、当频域采样点数比时域多1：x=1:16; %测试的序列N=length(x);n=0:(N-1);k=0:N;M=length(k);X=x*(exp(-j*2*pi/M).(n*k) %采用matlab向量化来计算Y=fft(x)x_=ifft(X)结果：X = 1.0e+02 * Columns 1 through 4 1.3600 -0.2435 + 0.3933i -0.2106 + 0.1049i -0.1608 - 0.0149i Columns 5 through 8 -0.1007 - 0.0760i -0.0385 - 0.0993i 0.0174 - 0.0933i 0.0595 - 0.0653i Columns 9 through 12 0.0821 - 0.0234i 0.0821 + 0.0234i 0.0595 + 0.0653i 0.0174 + 0.0933i Columns 13 through 16 -0.0385 + 0.0993i -0.1007 + 0.0760i -0.1608 + 0.0149i -0.2106 - 0.1049i Column 17 -0.2435 - 0.3933iY = 1.0e+02 * Columns 1 through 4 1.3600 -0.0800 + 0.4022i -0.0800 + 0.1931i -0.0800 + 0.1197i Columns 5 through 8 -0.0800 + 0.0800i -0.0800 + 0.0535i -0.0800 + 0.0331i -0.0800 + 0.0159i Columns 9 through 12 -0.0800 -0.0800 - 0.0159i -0.0800 - 0.0331i -0.0800 - 0.0535i Columns 13 through 16 -0.0800 - 0.0800i -0.0800 - 0.1197i -0.0800 - 0.1931i -0.0800 - 0.4022ix_ = Columns 1 through 4 1.0000 + 0.0000i 2.0000 + 0.0000i 3.0000 + 0.0000i 4.0000 + 0.0000i Columns 5 through 8 5.0000 + 0.0000i 6.0000 + 0.0000i 7.0000 - 0.0000i 8.0000 - 0.0000i Columns 9 through 12 9.0000 - 0.0000i 10.0000 - 0.0000i 11.0000 - 0.0000i 12.0000 - 0.0000i Columns 13 through 16 13.0000 - 0.0000i 14.0000 - 0.0000i 15.0000 - 0.0000i 16.0000 - 0.0000i Column 17 -0.0000 + 0.0000i频域采样M个点，时域以M为周期延拓，当M时域点数N时，可以看到延拓没有混叠，时域信号可以恢复，一个周期内的最后一个值是0.2、当频域采样点数等于时域点数：x=1:16; %测试的序列N=length(x);n=0:(N-1);k=0:N-1;M=length(k);X=x*(exp(-j*2*pi/M).(n*k) %采用matlab向量化来计算Y=fft(x)x_=ifft(X)结果：X = 1.0e+02 * Columns 1 through 5 1.3600 -0.0800 + 0.4022i -0.0800 + 0.1931i -0.0800 + 0.1197i -0.0800 + 0.0800i Columns 6 through 10 -0.0800 + 0.0535i -0.0800 + 0.0331i -0.0800 + 0.0159i -0.0800 + 0.0000i -0.0800 - 0.0159i Columns 11 through 15 -0.0800 - 0.0331i -0.0800 - 0.0535i -0.0800 - 0.0800i -0.0800 - 0.1197i -0.0800 - 0.1931i Column 16 -0.0800 - 0.4022iY = 1.0e+02 * Columns 1 through 5 1.3600 -0.0800 + 0.4022i -0.0800 + 0.1931i -0.0800 + 0.1197i -0.0800 + 0.0800i Columns 6 through 10 -0.0800 + 0.0535i -0.0800 + 0.0331i -0.0800 + 0.0159i -0.0800 -0.0800 - 0.0159i Columns 11 through 15 -0.0800 - 0.0331i -0.0800 - 0.0535i -0.0800 - 0.0800i -0.0800 - 0.1197i -0.0800 - 0.1931i Column 16 -0.0800 - 0.4022ix_ = Columns 1 through 5 1.0000 + 0.0000i 2.0000 + 0.0000i 3.0000 + 0.0000i 4.0000 + 0.0000i 5.0000 + 0.0000i Columns 6 through 10 6.0000 + 0.0000i 7.0000 + 0.0000i 8.0000 + 0.0000i 9.0000 + 0.0000i 10.0000 - 0.0000i Columns 11 through 15 11.0000 - 0.0000i 12.0000 - 0.0000i 13.0000 - 0.0000i 14.0000 - 0.0000i 15.0000 - 0.0000i Column 16 16.0000 - 0.0000i频域采样M个点，时域以M为周期延拓。当M=N时可以看到延拓恰好没有混叠， 此时的抽样即是DFT(程序中采用FFT来计算，FFT只是DFT的快速算法)。3、当频域抽样点数小于时域点数：x=1:16; %测试的序列N=length(x);n=0:(N-1);k=0:N-2;M=length(k);X=x*(exp(-j*2*pi/M).(n*k) %采用matlab向量化来计算Y=fft(x)x_=ifft(X)实验结果：X = 1.0e+02 * Columns 1 through 5 1.3600 0.2242 + 0.2373i 0.1798 + 0.0203i 0.1156 - 0.0682i 0.0444 - 0.1002i Columns 6 through 10 -0.0198 - 0.0914i -0.0642 - 0.0534i -0.0800 - 0.0000i -0.0642 + 0.0534i -0.0198 + 0.0914i Columns 11 through 14 0.0444 + 0.1002i 0.1156 + 0.0682i 0.1798 - 0.0203i 0.2242 - 0.2373iY = 1.0e+02 * Columns 1 through 5 1.3600 -0.0800 + 0.4022i -0.0800 + 0.1931i -0.0800 + 0.1197i -0.0800 + 0.0800i Columns 6 through 10 -0.0800 + 0.0535i -0.0800 + 0.0331i -0.0800 + 0.0159i -0.0800 -0.0800 - 0.0159i Columns 11 through 15 -0.0800 - 0.0331i -0.0800 - 0.0535i -0.0800 - 0.0800i -0.0800 - 0.1197i -0.0800 - 0.1931i Column 16 -0.0800 - 0.4022ix_ = Columns 1 through 5 16.0000 + 0.0000i 18.0000 + 0.0000i 3.0000 - 0.0000i 4.0000 - 0.0000i 5.0000 - 0.0000i Columns 6 through 10 6.0000 - 0.0000i 7.0000 - 0.0000i 8.0000 - 0.0000i 9.0000 + 0.0000i 10.0000 + 0.0000i Columns 11 through 14 11.0000 + 0.0