MATLAB在数字信号处理中的应用3

MATLAB上机系列之
Chapter 2 离散时间傅里叶 变换和z变换
刘尘尘 西昌学院汽车与电子工程学院
May 10th, 2013
Bibliography
Digital Signal Processing Using MATLAB (3rd edition)
Ingle, Proakis
刘树棠 译
subplot(2,2,1); plot(w/pi,magX); grid xlabel('frequency in pi units'); ylabel('Magnitude'); title('Magnitude Part'); subplot(2,2,2); plot(w/pi,angX); grid xlabel('frequency in pi units'); ylabel('Radians'); title('Phase Part'); subplot(2,2,3); plot(w/pi,realX); grid xlabel('frequency in pi units'); ylabel('Real') title('Real Part'); subplot(2,2,4); plot(w/pi,imagX); grid xlabel('frequency in pi units'); ylabel('Imaginary') title('Imaginary Part');
DTFT
3. 时移特性
例5：通过MATLAB检验DTFT的时移特性
n = 0:10;x=rand(size(n)); k = 0:500; w = (pi/500)*k; X = x * (exp(-j*pi/500)).^(n'*k);
% signal shifted by two samples % y(n)=x(n-2) y = x; m = n+2; Y = y * (exp(-j*pi/500)).^(m'*k);
2.1 DTFT of Discrete time
signals
两个重要性质
1. 周期性： 是的周期函数，周期为。
分析时仅需要的一个周期， 或 [, ]
2. 对称性： 对于实序列是共轭对称的。
对于实序列，分析时仅需要的半个周期，
2.1 DTFT of Discrete time
signals
例1：求实指数序列的DTFT并画出频谱图
2.1 DTFT of Discrete time signals
例1：求实指数序列的DTFT并画出频谱图 （幅度，相位，实部，虚部）。
思考：在MATLAB中，如何画出信号的频谱
2.1 DTFT of Discrete time
时域离散信号的DTFT是关于频率的连续函数
MATLAB绘制连续函数： 1. 确定自变量范围 2. 用取样间隔足够小的离散样值代替连续值
高西全 译
Digital Signal Processing—A computer based approach
数字信号处理——基于计算机的方法（第四版）
Contents
2.1 离散时间傅里叶变换(DTFT) 2.1.1 离散时间信号的DTFT和频谱图 2.1.2 验证DTFT的性质 2.1.3 离散时间系统的频率响应 2.2 z 变换(z-Transform) 2.2.1 z 变换 2.2.2 极零点和收敛域 2.2.3 z 反变换 2.2.4 离散时间系统的系统函数
% Linear Combination of X1 & X2 % Difference
19
*2.1.2 verify Properties of
DTFT
4. 频移特性
例6：通过MATLAB检验DTFT的频移特性
% Graphical verification subplot(1,1,1) subplot(2,2,1); plot(w/pi,abs(X)); grid; axis([-1,1,0,60]) xlabel('frequency in pi units'); ylabel('|X|') title('Magnitude of X') subplot(2,2,2); plot(w/pi,angle(X)/pi); grid; axis([-1,1,-1,1]) xlabel('frequency in pi units'); ylabel('radiands/pi') title('Angle of X') subplot(2,2,3); plot(w/pi,abs(Y)); grid; axis([-1,1,0,60]) xlabel('frequency in pi units'); ylabel('|Y|') title('Magnitude of Y') subplot(2,2,4); plot(w/pi,angle(Y)/pi); grid; axis([-1,1,-1,1]) xlabel('frequency in pi units'); ylabel('radians/pi') title('Angle of Y')
DTFT
4. 频移特性
例6：通过MATLAB检验DTFT的频移特性
% x(n)=cos(pi*n/2) n = 0:100; x = cos(pi*n/2); k = -100:100; w = (pi/100)*k; between -pi and +pi X = x * (exp(-j*pi/100)).^(n'*k);
subplot(2,2,1); plot(w/pi,magX); grid xlabel('frequency in pi units'); ylabel('Magnitude'); title('Magnitude Part'); subplot(2,2,2); plot(w/pi,angX/pi); grid xlabel('frequency in pi units'); ylabel(‘Angle'); title('Phase Part'); subplot(2,2,3); plot(w/pi,realX); grid xlabel('frequency in pi units'); ylabel('Real'); title('Real Part'); subplot(2,2,4); plot(w/pi,imagX); grid xlabel('frequency in pi units'); ylabel('Imaginary'); title('Imaginary Part');
magX = abs(X); angX =angle(X);
subplot(2,1,1); plot(w/pi,magX);grid axis([-2,2,0,8]) xlabel('frequency in units of pi'); ylabel('|X|') title('Magnitude Part')
2. 对于有限长信号：
用行向量 表示行向量表示
>>> X = x*exp(-j*n'*w);
具体的推导过程参见实验 指导书或Proakis的《数字 信号处理(MATLAB版)》
2.1 DTFT of Discrete time signals 例2：求有限长序列的DTFT并画出频谱图
n = -1:3; x = 1:5; w = [0:500]*pi/500; X = x * exp(-j*n'*w) ;
X2 = x2 * (exp(-j*pi/500)).^(n'*k); % DTFT of x2
x = alpha*x1 + beta*x2;
% Linear combination of x1 & x2
X = x * (exp(-j*pi/500)).^(n'*k); % DTFT of x
% verification X_check = alpha*X1 + beta*X2; error = max(abs(X-X_check))
% y(n)=exp(j*pi*n/4)x(n) y = exp(j*pi*n/4).*x; multiplied by exp(j*pi*n/4) Y = y * (exp(-j*pi/100)).^(n'*k);% frຫໍສະໝຸດ quency % DTFT of x
% signal % DTFT of y
% Computes Discrete-time Fourier Transform % [X] = dtft(x,n,w) % X = DTFT values computed at w frequency % x = finite duration sequence over n % n = sample position vector % w = frequency location vector
w = [0:1:500]*pi/500; % [0, pi] axis divided into 501 points. X = exp(j*w) ./ (exp(j*w) - 0.5*ones(1,501));
magX = abs(X); angX = angle(X); realX = real(X); imagX = imag(X);