DSP用Matlab学习的笔记

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Matlab 笔记

学习DSP时记录的笔记

Matlab中的常用信号

  1. sin, cos, sinc, exp

  2. heaviside(unit step function), diract(delta function)

  3. dsolve(eqn, conditions): 解微分方程, simpify(func)：化简函数表达式func

  4. lsim(sys, f,t), tf(b,a)

    lsim: t is the time axis. f is the input signal. sys is the model of the LTI system

    For the differential equation:

    \[a_3y'''(t)+a_2y''(t)+a_1y'(t)+a_0=b_3f'''(5)+b_2f''(t)+b_1f'(t)+b_0f(t)\] the value of a and b is: \[ \begin{matrix}a=[a_3,a_2,a_1,a_0]\\b=[b_3,b_2,b_1,b_0]\end{matrix} \] $$

  5. impluse(sys, t) , step(sys, t) ：脉冲响应， 阶跃响应

  6. tripuls(t, width, center): 三角波

  7. sawtooth(t, center): 锯齿波

  8. rectangle(t, center)：单个矩形波

  9. square(t, range)：连续的方波，range表示高电平占周期的比例 ## 微分&积分

  10. 微分diff

    • symbolic: 
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      syms x
      y1 = heaviside(x);
      y2 = diff(y1,x);
    • numerical, 记得除以dt 
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      t = -5:dt:5;
      f1 = heaviside(t);
      f2 = diff(f1)/dt; % diff 用于计算离散序列的差分，默认元素间的间隔为 1

  11. 不定积分int, cumtrapz。定积分int加上区间、trapz

    • symbolic 
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      syms x
      c = 0;
      y1 = heaviside(x);
      y2 = int(y1,x)+c;
      % 定积分
      syms t1
      int(heaviside(t1),-1,2)
    • numerical 
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      dt = 0.01;
      t = -5:dt:5;
      f1 = heaviside(t);
      f2 = cumtrapz(t, f1)+c;
      % 定积分
      t2 = -1:0.01:2;
      trapz(t2,heaviside(t2))

Chapter 2: 时序的信号分析

\[ y(t)=y_{zi}(t)+y_{zs}(t) \]

其中\(y_{zi}(t)\)是零输入响应，输入为0；\(y_{zs}(t)\)是零状态响应，初始条件为0

零输入响应

𝑦𝑦′′(𝑡𝑡) + 4𝑦𝑦(𝑡𝑡) = 0, 𝑦𝑦(0-) = 1, 𝑦𝑦′(0-) = 1, find the zero_input response

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clear;
close all;
clf;
syms y(t)
D2y = diff(y,t,2);
Dy = diff(y,t);
eqn = D2y+4*y==0; % zero input response 右边的输入是0
conds = [y(0)==1, Dy(0)==1];
ysol = dsolve(eqn,conds);
yzi = simplify(ysol);

零状态响应

dsolve求零状态响应

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clear;
close all;
clf;

syms y(t)
D2y=diff(y,t,2);
Dy=diff(y,t);
eqn=D2y-5*Dy+6*y==exp(-2*t)*heaviside(t); % 零状态响应右边输入不是零
conds=[y(0)==0, Dy(0)==0];                % 零状态响应的状态是0
ysol=dsolve(eqn,conds);
yzs=simplify(ysol)

lsim和tf求零状态响应

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clear;
close all;
clf;

sys = tf([1],[1,-5,6]);
t = 0:0.01:5;
f = exp(-2*t).*heaviside(t);
y = lsim(sys,f,t);
plot(t,y);
xlabel('t');ylabel('y(t)');title('Zero-State Response'),grid on;

使用卷积求零状态响应

step response和impulse response可以反映系统的特性，