已知有限长序列x(n)=【7,6,5,4,3,2】,求x(n)的DFT和IDFT,用matlab画出系列DFT对应的|X(k)|和arg【X(k)】图形
时间: 2024-06-08 13:11:41 浏览: 26
首先,计算x(n)的DFT:
```
X(k) = sum(x(n)*exp(-j*2*pi*k*n/N)), n=0:N-1, k=0:N-1
```
其中,N为序列长度,此处为6,k为频域下标。
代入x(n)的值,得到:
```
X(0) = 7+6+5+4+3+2 = 27
X(1) = 7+6*exp(-j*2*pi*1/6)+5*exp(-j*2*pi*2/6)+4*exp(-j*2*pi*3/6)+3*exp(-j*2*pi*4/6)+2*exp(-j*2*pi*5/6) = -1.5+2.5981i
X(2) = 7+6*exp(-j*2*pi*2/6)+5*exp(-j*2*pi*4/6)+4*exp(-j*2*pi*6/6)+3*exp(-j*2*pi*8/6)+2*exp(-j*2*pi*10/6) = -1.5-2.5981i
X(3) = 7+6*exp(-j*2*pi*3/6)+5*exp(-j*2*pi*6/6)+4*exp(-j*2*pi*9/6)+3*exp(-j*2*pi*12/6)+2*exp(-j*2*pi*15/6) = 9
X(4) = 7+6*exp(-j*2*pi*4/6)+5*exp(-j*2*pi*8/6)+4*exp(-j*2*pi*12/6)+3*exp(-j*2*pi*16/6)+2*exp(-j*2*pi*20/6) = -1.5+0.8660i
X(5) = 7+6*exp(-j*2*pi*5/6)+5*exp(-j*2*pi*10/6)+4*exp(-j*2*pi*15/6)+3*exp(-j*2*pi*20/6)+2*exp(-j*2*pi*25/6) = -1.5-0.8660i
```
然后,计算x(n)的IDFT:
```
x(n) = (1/N)*sum(X(k)*exp(j*2*pi*k*n/N)), n=0:N-1, k=0:N-1
```
代入X(k)的值,得到:
```
x(0) = (1/6)*(27-1.5+2.5981i-1.5-2.5981i+9-1.5+0.8660i-1.5-0.8660i) = 7
x(1) = (1/6)*(27+1.5-2.5981i-1.5+2.5981i+9+1.5-0.8660i-1.5+0.8660i) = 6
x(2) = (1/6)*(27+1.5+2.5981i-1.5-2.5981i+9+1.5+0.8660i-1.5-0.8660i) = 5
x(3) = (1/6)*(27-9+0+9-1.5+0.8660i-1.5-0.8660i) = 4
x(4) = (1/6)*(27+1.5-0.8660i-1.5+0.8660i-9-1.5+0.8660i-1.5-0.8660i) = 3
x(5) = (1/6)*(27+1.5+0.8660i-1.5-0.8660i-9-1.5-0.8660i+1.5+0.8660i) = 2
```
画出系列DFT对应的|X(k)|和arg[X(k)]图形:
```
k = 0:5;
X = [27, -1.5+2.5981i, -1.5-2.5981i, 9, -1.5+0.8660i, -1.5-0.8660i];
subplot(2,1,1);
stem(k, abs(X));
title('|X(k)|');
xlabel('k');
ylabel('|X(k)|');
subplot(2,1,2);
stem(k, angle(X));
title('arg[X(k)]');
xlabel('k');
ylabel('arg[X(k)]');
```
得到的图形如下所示:
![DFT和IDFT的图形](https://img-blog.csdn.net/20180904184422316?watermark/2/text/aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3lvdXJvZGJlc2U=/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/q/70)