4.5. Исследование эффекта подмены частот

4.5.1 Примеры введения различных частот дискретизации:

В данном случае нарушаются условия теоремы Котельникова. Слишком мала частота дискретизации.

В данном случае выполняются условия теоремы Котельникова.

4.5.2. Текст программы:

function varargout = fig_4(varargin)

% FIG_4 M-file for fig_4.fig

% FIG_4, by itself, creates a new FIG_4 or raises the existing

% singleton*.

%

% H = FIG_4 returns the handle to a new FIG_4 or the handle to

% the existing singleton*.

%

% FIG_4('CALLBACK',hObject,eventData,handles,...) calls the local

% function named CALLBACK in FIG_4.M with the given input arguments.

%

% FIG_4('Property','Value',...) creates a new FIG_4 or raises the

% existing singleton*. Starting from the left, property value pairs are

% applied to the GUI before fig_4_OpeningFunction gets called. An

% unrecognized property name or invalid value makes property application

% stop. All inputs are passed to fig_4_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

% Copyright 2002-2003 The MathWorks, Inc.

% Edit the above text to modify the response to help fig_4

% Last Modified by GUIDE v2.5 24-Apr-2006 14:19:31

% Begin initialization code - DO NOT EDIT

gui_Singleton = 1;

gui_State = struct('gui_Name', mfilename, ...

'gui_Singleton', gui_Singleton, ...

'gui_OpeningFcn', @fig_4_OpeningFcn, ...

'gui_OutputFcn', @fig_4_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 fig_4 is made visible.

function fig_4_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 fig_4 (see VARARGIN)

% Choose default command line output for fig_4

handles.output = hObject;

% Update handles structure

guidata(hObject, handles);

% This sets up the initial plot - only do when we are invisible

% so window can get raised using fig_4.

if strcmp(get(hObject,'Visible'),'off')

genSIN_1(handles);

end

% UIWAIT makes fig_4 wait for user response (see UIRESUME)

% uiwait(handles.figure1);

% --- Outputs from this function are returned to the command line.

function varargout = fig_4_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 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)

genSIN_1(handles);

%popup_sel_index = 1;%get(handles.popupmenu1, 'Value');

%switch popup_sel_index

% case 1

% plot(rand(5));

% case 2

% plot(sin(1:0.01:25.99));

% case 3

% bar(1:.5:10);

% case 4

% plot(membrane);

% case 5

% surf(peaks);

%end

% --------------------------------------------------------------------

function FileMenu_Callback(hObject, eventdata, handles)

% hObject handle to FileMenu (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles structure with handles and user data (see GUIDATA)

% --------------------------------------------------------------------

function OpenMenuItem_Callback(hObject, eventdata, handles)

% hObject handle to OpenMenuItem (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles structure with handles and user data (see GUIDATA)

file = uigetfile('*.fig');

if ~isequal(file, 0)

open(file);

end

% --------------------------------------------------------------------

function PrintMenuItem_Callback(hObject, eventdata, handles)

% hObject handle to PrintMenuItem (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles structure with handles and user data (see GUIDATA)

printdlg(handles.figure1)

% --------------------------------------------------------------------

function CloseMenuItem_Callback(hObject, eventdata, handles)

% hObject handle to CloseMenuItem (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles structure with handles and user data (see GUIDATA)

selection = questdlg(['Close ' get(handles.figure1,'Name') '?'],...

['Close ' get(handles.figure1,'Name') '...'],...

'Yes','No','Yes');

if strcmp(selection,'No')

return;

end

delete(handles.figure1)

% --- 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 = 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

set(hObject,'BackgroundColor','white');

else

set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor'));

end

%set(hObject, 'String', {'plot(rand(5))', 'plot(sin(1:0.01:25))', 'bar(1:.5:10)', 'plot(membrane)', 'surf(peaks)'});

function edit2_Callback(hObject, eventdata, handles)

% hObject handle to edit2 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles structure with handles and user data (see GUIDATA)

% Hints: get(hObject,'String') returns contents of edit2 as text

% str2double(get(hObject,'String')) returns contents of edit2 as a double

% --- Executes during object creation, after setting all properties.

function edit2_CreateFcn(hObject, eventdata, handles)

% hObject handle to edit2 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles empty - handles not created until after all CreateFcns called

% Hint: edit 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 pushbutton4.

function pushbutton4_Callback(hObject, eventdata, handles)

% hObject handle to pushbutton4 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles structure with handles and user data (see GUIDATA)

function edit3_Callback(hObject, eventdata, handles)

% hObject handle to edit3 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles structure with handles and user data (see GUIDATA)

% Hints: get(hObject,'String') returns contents of edit3 as text

% str2double(get(hObject,'String')) returns contents of edit3 as a double

% --- Executes during object creation, after setting all properties.

function edit3_CreateFcn(hObject, eventdata, handles)

% hObject handle to edit3 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles empty - handles not created until after all CreateFcns called

% Hint: edit 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

function edit4_Callback(hObject, eventdata, handles)

% hObject handle to edit4 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles structure with handles and user data (see GUIDATA)

% Hints: get(hObject,'String') returns contents of edit4 as text

% str2double(get(hObject,'String')) returns contents of edit4 as a double

% --- Executes during object creation, after setting all properties.

function edit4_CreateFcn(hObject, eventdata, handles)

% hObject handle to edit4 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles empty - handles not created until after all CreateFcns called

% Hint: edit 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

% unsere

function genSIN_1(handles)

FREQstr = get(handles.edit2, 'String');

FREQnum = str2num(FREQstr);

FREQSstr = get(handles.edit3, 'String');

FREQSnum = str2num(FREQSstr);

TMAXstr = get(handles.edit4, 'String');

TMAXnum = str2num(TMAXstr);

STEP = TMAXnum/FREQSnum;

T = 0:STEP:TMAXnum;

SIGSIN = sin(2*pi*FREQnum*T);

axes(handles.axes1);

cla;

plot(T, SIGSIN);

NFFTstr = get(handles.edit5, 'String');

NFFTnum = str2num(NFFTstr);

POWERS = fft(SIGSIN, NFFTnum);

POWERSPECTR = POWERS.*conj(POWERS);

axes(handles.axes3);

cla;

%plot((1:20), POWERSPECTR(1:20));

plot(POWERSPECTR);

function edit5_Callback(hObject, eventdata, handles)

% hObject handle to edit5 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles structure with handles and user data (see GUIDATA)

% Hints: get(hObject,'String') returns contents of edit5 as text

% str2double(get(hObject,'String')) returns contents of edit5 as a double

% --- Executes during object creation, after setting all properties.

function edit5_CreateFcn(hObject, eventdata, handles)

% hObject handle to edit5 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB

% handles empty - handles not created until after all CreateFcns called

% Hint: edit 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

4.5.3. Файлы, относящиеся к програме:

- fig_4.fig– файл графического интерфейса

- fig_4.m– файл с функциями, обрабатывающими графическую форму

4.5.4. Запуск программы:

>> fig_4

4.5.5. Вывод:

Эффект подмены частот проявляется при выборе слишком низкой частоты дискретизации сигнала, т. е. ниже удвоенной максимальной частоты фурье-спектра сигнала.

В этом случае нарушаются условия теоремы Котельникова и высокочастотные гармоники сигнала или гармоники, частота которых выше половины выбранной частоты дискретизации, воспринимаются в полученной выборке как низкочастотные гармоники, т. е. гармоники, частота которых ниже половины частоты дискретизации.

Теорема Котельникова:Если сигналx(t) имеет преобразование ФурьеX(jw) такое, чтоX(jw) = 0 приw>= 2πF_N(F_N– частота Найкриста), тоx(t) может быть восстановлен единственным образом по последовательности равноотстоящих отсчетовx(nT), гдеn= -... (то есть должно быть так:F_Д2F_N).

Частота дискретизации непрерывного сигнала не должна быть меньше удвоенной ширины спектра: иначе произойдет наложение спектров (рис. б) и будет невозможно с помощью фильтра нижних частот выделить спектр исходного непрерывного сигнала.