MATLAB CODE
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clear all
close all
% Example MATLAB M-file that plots a Fourier series
% Set up some input values
P = 4; %period = 2P
num_terms = 100; %approximate infinite series with finite number of terms
% Miscellaneous setup stuff
format compact; % Gets rid of extra lines in output. Optional.
% Initialize x-axis values from -1.25L to 1.25L. Middle number is step size. Make
% middle number smaller for smoother plots
t = -4:0.001:4;
x=zeros(length(t),1); % reseting original half range expanded function array
x(0<=t & t<2)=1; % forming the original half range expanded function array for the purpose of plotting only
x(2<t & t <4)=0;
x(t<0 & -2<t)=1;
x(t<-2 & t>-4)=0;
figure %plotting original half range expanded function
plot(t,x)
axis([-4.5 4.5 -0.5 1.5])
%Starting to approximate f(t)
% Initialize y-axis values. y = f(t)
f = zeros(size(x'));
% Add a0/2 to series
a0 = 1;
f = f + a0/2;
% Loop num_terms times through Fourier series, accumulating in f.
figure
for n = 1:num_terms
% Formula for an.
an = (2/(n*pi))*sin(n*pi/2);
bn=0;
% Add cosine and sine into f
f = f + an*cos(n*pi*t/P) + bn*sin(n*pi*t/P);
% Plot intermediate f. You can comment these three lines out for faster
% execution speed. The function pause(n) will pause for about n
% seconds. You can raise or lower for faster plots.
plot(t,f);
set(gca,'FontSize',16);
title(['Number of terms = ',num2str(n)]);
grid on;
if n < 5
pause(0.15);
else
pause(0.1);
end
xlabel('even approx');
end;
SOLUTION
In mathematics, a function on the real numbers is called astep function(or staircase function) if it can be written as afinite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constantfunction having only finitely many pieces.
You are given a finite step function xt=-1 0<t<4 1 4<t<8. Hand calculate the FS coefficients of...
MATLAB code starts here --------- clear T0=2; w0=2*pi/T0; f0=1/T0; Tmax=4; Nmax=15; %--- i=1; for t=-Tmax: .01:Tmax T(i)=t; if t>=(T0/2) while (t>T0/2) t=t-T0; end elseif t<=-(T0/2) while (t<=-T0/2) t=t+T0; end end if abs(t)<=(T0/4) y(i)=1; else y(i)=0; end i=i+1; end plot(T,y),grid, xlabel('Time (sec)'); title('y(t) square wave'); shg disp('Hit return..'); pause %--- a0=1/2; F(1)=0; %dc freq C(1)=a0; for n=1:Nmax a(n)=(2/(n*pi))*sin((n*pi)/2); b(n)=0; C(n+1)=sqrt(a(n)^2+b(n)^2); F(n+1)=n*f0; end stem(F,abs,(C)), grid, title(['Line Spectrum: Harmonics = ' num2str(Nmax)]); xlabel('Freq(Hz)'), ylabel('Cn'), shg disp('Hit return...'); pause %--- yest=a0*ones(1,length(T)); for n=1:Nmax yest=yest+a(n)*cos(2*n*pi*T/T0)+b(n)*sin(2*n*pi*T/T0);...
Also : FS (2.8) FS (4) FS (5) The function f(t) is defined by -3t+6, 0<t<4 f(t) = -3, 4 < t < 5. Let f (t) denote the periodic extension of f(t), with period 5. Evaluate f (-2.3), f (O), f (7.5), f (9.2) and state the value to which the Fourier series of f (t), FS(t), converges for each of the following values: t = 0,t = 2.8, t = 4, t = 5. Enter all your answers...
Q#4: (24 points) Given the function y-f(x) shown below: na f(r) -3 (a) Calculate the period of function, T and frequency, (2TT)/T (b) Calculate the Fourier Coefficients Ao. An, and Bn of the Fourier series expansion of function, y-f(x). Here n 1, 2, 3,... (integers) (c) Write the Fourier series approximation of function, yf(x), in terms of numbers, n & x only Q#4: (24 points) Given the function y-f(x) shown below: na f(r) -3 (a) Calculate the period of function,...
Create a file named “toneA.m” with the following MATLAB code: clear all Fs=5000; Ts=1/Fs; t=[0:Ts:0.4]; F_A=440; %Frequency of note A is 440 Hz A=sin(2*pi*F_A*t); sound(A,Fs); Type toneA at the command line and then answer the following: (a) What is the time duration of A? (b) How many elements are there in A? (c) Modify toneA.m by changing “Fs=5000” to “Fs=10000”. Can you hear any difference? (d) Create a file named “tone.m” with the following MATLAB code: function x = tone(frequency,...
Please answer "b" only. %Example code function plotFS(m); %m = user provided number of terms desired in the Fourier series; %this code computes the Fourier series of the function %f(x)=0, for -pi<= x <0, % =cos(x) for 0<= x <pi %generate the interval from -pi to pi with step size h; h = pi/100; xx1=[-pi:h:0]; xx2=[0:h:pi]; xx = [xx1, xx2]; %generate the given function f so that it can be graphed ff = [zeros(size(xx1)), cos(xx2)]; %compute the first partial sum...
Calculate the Fourier Series coefficients of x(t) = cos(2*pi*1*t) + 2*sin(2*pi*4*t). Based on your results which set of FS coefficients corresponding to the positive side of the spectra is correct. a0=0, a1=1/2, a2=1/j, a3 = 0 a1=1, a2=2, a3 = 0 a1=1/2j, a2=1/2, a3 = 0 a1=1/2, a2=2/2j, a3 = 0
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a< x < T, where 0a < T (a) Sketch the odd and even periodic extension of f (x) on the interval -3n < x < 3« for aT/2 (b) Find the half-range Fourier sine series expansion of f(x) for arbitrary a. (e) To what value does the half-range Fourier sine series expansion converge at r a? [8 marks 3. Consider the function defined by...
(4) Consider the function f(0) = 10 € C(T). (a) Show that the Fourier coefficients of f are if n = 0, f(n) (-1)" - 1 if n +0. l n2 (b) Justify why the Fourier series of f converges to f uniformly on T. (c) Taking 0 = 0 in the Fourier series expansion of f, conclude that HINT: First prove that n even
Can you please help me answer Task 2.b? Please show all work. fs=44100; no_pts=8192; t=([0:no_pts-1]')/fs; y1=sin(2*pi*1000*t); figure; plot(t,y1); xlabel('t (second)') ylabel('y(t)') axis([0,.004,-1.2,1.2]) % constrain axis so you can actually see the wave sound(y1,fs); % play sound using windows driver. %% % Check the frequency domain signal. fr is the frequency vector and f1 is the magnitude of F{y1}. fr=([0:no_pts-1]')/no_pts*fs; %in Hz fr=fr(1:no_pts/2); % single-sided spectrum f1=abs(fft(y1)); % compute fft f1=f1(1:no_pts/2)/fs; %% % F is the continuous time Fourier. (See derivation...
Rewrite the following for loop into a whileloop. 1 2 3 4 int s = 0; for (int i = 1; i <= 10; i++) { s = s + i; } Given variables int n and double pi, write a snippet of code that assigns to pi the approximation of π resulting from adding the first nterms in the Gregory-Leibniz series: Given variables int areaBound and int sum, write a snippet of code that assigns to sum the result...