数字信号处理之展示 z 变换与 s 变换之间的所有关系 +matlab 源码

题目分析：

要求z变换与s变换的关系，首先考虑z变换与s变换之间运用领域的不同，s域是连续时间表示域，使用连续的时间变量s表示信号的自变量，取值范围为复平面上的所有点。而z域是离散时间表示域，使用离散的时间变量z表示信号的 自变量取值范围虽然也为复平面上的所有点，但对于离散信号而言，主要在单位圆内取值。

另外，从s域到z域的变换关系是通过采样操作实现的，具体关系如下：

1. 采样操作：将连续时间信号进行采样，得到离散时间信号。采样操作可以用冲激函数序列来表示，即将连续时间信号乘以冲激函数序列。
2. 傅里叶变换：对连续时间信号进行傅里叶变换，得到信号在频域上的表示，即s域表示。
3. Z变换：对离散时间信号进行z变换，得到信号在频域上的表示，即z域表示。

设计思路：

首先设置一个列表框，列出输入信号，这里我设置了一个阶跃信号一个冲激信号，之后设置一个按钮与五个轴，在第一个轴中运用heaviside与stem

函数画出离散的阶跃函数，然后在第二个轴中用ztrans函数进行z变换，在第三个轴中用laplace函数进行拉普拉斯变换，随后再定义两个轴用于存放各自变换的定义域，从图中可以看到z变换的收敛域是在单位圆内，而s变换的收敛域是在y轴左边，这是因为对于离散时间系统来说，系统的稳定性要求系统函数的极点位于单位圆内，而对于连续时间系统来说，系统的稳定性要求系统函数的极点位于s平面的左半部分，因此收敛域通常位于轴线的左侧。

而z变换与s变换间数值关系经过推导可知s变换与z变换间关系：z=e^sT

界面设计结果

代码实现

function varargout = untitled1(varargin)
% UNTITLED1 MATLAB code for untitled1.fig
%      UNTITLED1, by itself, creates a new UNTITLED1 or raises the existing
%      singleton*.
%
%      H = UNTITLED1 returns the handle to a new UNTITLED1 or the handle to
%      the existing singleton*.
%
%      UNTITLED1('CALLBACK',hObject,eventData,handles,...) calls the local
%      function named CALLBACK in UNTITLED1.M with the given input arguments.
%
%      UNTITLED1('Property','Value',...) creates a new UNTITLED1 or raises the
%      existing singleton*.  Starting from the left, property value pairs are
%      applied to the GUI before untitled1_OpeningFcn gets called.  An
%      unrecognized property name or invalid value makes property application
%      stop.  All inputs are passed to untitled1_OpeningFcn via varargin.
%
%      *See GUI Options on GUIDE's Tools menu.  Choose "GUI allows only one
%      instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES
 
% Edit the above text to modify the response to help untitled1
 
% Last Modified by GUIDE v2.5 05-Jun-2024 19:33:04
 
% Begin initialization code - DO NOT EDIT
gui_Singleton = 1;
gui_State = struct('gui_Name',       mfilename, ...
                   'gui_Singleton',  gui_Singleton, ...
                   'gui_OpeningFcn', @untitled1_OpeningFcn, ...
                   'gui_OutputFcn',  @untitled1_OutputFcn, ...
                   'gui_LayoutFcn',  [] , ...
                   'gui_Callback',   []);
if nargin && ischar(varargin{1})
    gui_State.gui_Callback = str2func(varargin{1});
end
 
if nargout
    [varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
    gui_mainfcn(gui_State, varargin{:});
end
% End initialization code - DO NOT EDIT
 
 
% --- Executes just before untitled1 is made visible.
function untitled1_OpeningFcn(hObject, eventdata, handles, varargin)
% This function has no output args, see OutputFcn.
% hObject    handle to figure
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
% varargin   command line arguments to untitled1 (see VARARGIN)
 
% Choose default command line output for untitled1
handles.output = hObject;
 
% Update handles structure
guidata(hObject, handles);
 
% UIWAIT makes untitled1 wait for user response (see UIRESUME)
% uiwait(handles.figure1);
 
 
% --- Outputs from this function are returned to the command line.
function varargout = untitled1_OutputFcn(hObject, eventdata, handles) 
% varargout  cell array for returning output args (see VARARGOUT);
% hObject    handle to figure
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
 
% Get default command line output from handles structure
varargout{1} = handles.output;
 
 
% --- Executes on selection change in popupmenu1.
function popupmenu1_Callback(hObject, eventdata, handles)
% hObject    handle to popupmenu1 (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
 
% Hints: contents = cellstr(get(hObject,'String')) returns popupmenu1 contents as cell array
%        contents{get(hObject,'Value')} returns selected item from popupmenu1
 
 
% --- Executes during object creation, after setting all properties.
function popupmenu1_CreateFcn(hObject, eventdata, handles)
% hObject    handle to popupmenu1 (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    empty - handles not created until after all CreateFcns called
 
% Hint: popupmenu controls usually have a white background on Windows.
%       See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
    set(hObject,'BackgroundColor','white');
end
 
 
% --- Executes on button press in pushbutton1.
function pushbutton1_Callback(hObject, eventdata, handles)
% hObject    handle to pushbutton1 (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
a=get(handles.popupmenu1,'Value');
syms z;
syms n;
syms t;
switch a
    case 1
        axes(handles.axes1);
        cla;
        a= -10:10;
        A =heaviside(a);
        stem(a, A);
        xlabel('n');
        ylabel('A');
        title('½×Ô¾º¯Êý');
        ylim([0, 2]);
        yticks(0:1:2); 
        xticks(-10:1:10); 
        
        
        axes(handles.axes2);
        cla
        n = sym('n', 'positive');
        A_z=ztrans(heaviside(n))
        fplot(A_z);
        title('½×Ô¾º¯ÊýµÄz±ä»»');
       ylim([-5, 5]);
       xlim([-5,5]);
        yticks(-5:1:5); 
        xticks(-5:1:5);
        yline(0, 'k--');
        
        
        axes(handles.axes3);
        cla
        f=heaviside(t);
        F = laplace(f)
        fplot(F);
        title('½×Ô¾º¯ÊýµÄs±ä»»');
       ylim([-5, 5]);
       xlim([-5,5]);
        yticks(-5:1:5);
        xticks(-5:1:5);
        yline(0, 'k--');
        
        axes(handles.axes4);
        cla
theta = linspace(0, 2*pi, 1000);
x_circle = cos(theta);
y_circle = sin(theta);
fill(x_circle, y_circle, 'b', 'FaceAlpha', 0.3);
hold on;
plot(x_circle, y_circle, 'b', 'LineWidth', 2);
xlabel('ʵÖá');
ylabel('ÐéÖá');
title('z±ä»»µÄÊÕÁ²Óò');
axis([-3, 3, -3, 3]);
axis equal;
grid on;
hold off;
 
axes(handles.axes5);
cla
x_axis = -10:0.1:10;
y_axis = zeros(size(x_axis));
plot(x_axis, y_axis, 'k', 'LineWidth', 2);
hold on;
fill([-10,-10,0, 0], [-10, 10, 10, -10], 'r', 'FaceAlpha', 0.3);
xlabel('x');
ylabel('y');
title('s±ä»»µÄÊÕÁ²Óò');
axis([-10, 10, -10, 10]);
grid on;
hold off;
case 2
    axes(handles.axes1);
    cla;
x = -10:1:10;
y = zeros(size(x));
y(x == 0) = 1;
 
stem(x, y);
xlabel('x');
ylabel('y');
ylim([-1,2])
title('³å¼¤º¯Êý');
grid on;
    axes(handles.axes2);
    cla;
     x=-5:1:5;
    y=ones(size(x));
    plot(x,y)
    axes(handles.axes3);
    cla;
    x=-5:1:5;
    y=ones(size(x));
    plot(x,y)
    axes(handles.axes4);
    cla;
    
    axes(handles.axes5);
    cla;
    
end
 
 
% --- Executes during object creation, after setting all properties.
function axes3_CreateFcn(hObject, eventdata, handles)
% hObject    handle to axes3 (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    empty - handles not created until after all CreateFcns called
 
% Hint: place code in OpeningFcn to populate axes3