We can write
So
Where m is the smallest integer so
that is an
integer
If m = 3, we get
Now to find the fundamental period of the signal.
So the fundamental period of the signal will be
So
The fundamental period is
Any periodic discrete time signal can be represented as
This can be in any one period of the signal.
For this signal
Or
Using
We can write
We have shown that x[n] is periodic
with a period of . So
Comparing, we get
So the DTFS coefficients are
corresponds
to
. The frequency is
So
corresponds
to
. So the
corresponding frequency is
corresponds
to
. So the
corresponding frequency is
corresponds
to
. So the
corresponding frequency is
corresponds
to
. So the
corresponding frequency is
corresponds
to
. So the
corresponding frequency is
Please note that the DTFS coefficients are periodic with the period of the signal. So here the DTFS coefficients will be periodic with a period of 25.
I have found the DTFS coefficients between the period -12 and 12. If you want to get the coefficients in the period 0 to 12, you just have to add 25 to the k value in ck.
For example
MATLAB Code to plot the signal and to plot the DTFS
clc;
clear all;
close all;
n = 0:24;
x = 1 + cos(0.24*pi*n) + 3*sin(0.56*pi*n);
stem(n,x,'linewidth',2);
grid on;
xlabel('Time Index, n');
ylabel('Amplitude');
title('x[n]');
X = 1/25*fft(x);
k = 0:24;
Wk = k*2*pi/25;
figure
stem(Wk,abs(X),'linewidth',2);
grid on;
xlabel('\Omega_k');
ylabel('|c_k|');
title('Plot of |c_k|')
The plots
Determine the DTFS of (B) only. 1. Determine the DTFS of the following signals. Write a...
Objective Conduct DTFT, DTFS on a periodic discrete signal. Task: Consider the system with impulse response Tth sin 8 h(n) S(n) Tn (1) Find the Fourier-series representation for the output y(n) when the input x(n) is the periodic extension of the sequence 3/2, -1,0, -3/2, 1,0 Plot the x(n), h(n), y(n) and Fourier coefficient bk using Matlab or handwriting (Example 7.2.6 irse material) in cour (2) Find the output y(n) of the system with the input 1 Tn Tn x(п)...
I need the solution of c and d only
3. Determine whether the following signals are periodic or not. If periodic, find the fundamental period a. m(t) = (cos(2t - FU/3)] b. x(t) - Even (sin(Art).(t)) c. x(t)= cos(n.1/2) cos(n.rt/4) d. X(t)- cos(np.n/2)
I need help with the following DTFS problems for parts A and B.
Thank you
Prob. 2_Discrete-Time Fourier Series (DTFs) (a) A periodic signal, z[n is shown below. Use the analysis equation to determine the discrete-time Fourier Series (DTFS) coefficients, ak. Please express the ak in terms of cosines. 9-7 -5-335 7 -0.5 Clk (b) Sketch the magnitude spectrum, la vs. k for 100 k 105. Please note each value laJ 100 101 103 104
2. Determine the FS coefficients for each of the following DT periodic signals. (a) x[n] = sin(2 /3) cos(in/2) (b) x[n] periodic with period 4 and x[n] = 1 - sin n for 0 <n<3. (e) a[n) periodic with period 12 and [n] = 1 - sin for 0 <n<11.
Chapter 1: Problem 1. Determine whether or not each of the following signals is periodic. In case a signal is periodic, specify its fundamental period. a.X (n) = 3 cos (5n + 4) b. x(n) = 2 exple-t) C. x(n) = cos () cos (4) d. "(m) = cos (3)- sin (a) + 3 cost 8 cos )
Determine whether the following discrete-time signals are periodic or not? For the periodic ones, find their fundamental period. 2. (5 points each) a. x1 [n] =sin(0.5n +z/4) c. x1n] = cos (tn/3) + sin(nn/5)-2cos(tn/10) d. x4[n]- sin(tn/12) cos(tn/3) (Hint: use trigonometric identities to write the signal as a sum of sinusoids)
During lab 4, we have seen numerical implementation of Fourier Series for periodic signals. As first part of this assignment, you need to write a Matlab function that would take an array representing a single period of a signal (x), corresponding time array (t), and return the Fourier Series coefficients (Ck) in exponential form. The function should also be able to take two (2) optional input arguments: number of Fourier coefficients (Nk) and plot option (p). Use the template ‘fourier_series_exp.m’...
2.For the periodic DT signal shown in Top, a) determine the
Fourier Series Coefficients. b) Use MATLAB to generate a spectral
plot (magnitude plot and a separate phase plot). c) Use MATLAB to
generate and plot the signal as a DTFS expansion of the periodic
signal. Plot over an interval containing several periods. Make sure
to include the MATLAB code
x[ri] -9 63 3 9 12 n 1. For the periodic DT signal shown in Top, a) determine the Fourier...
Answer the following questions based on these DT signals: 2πη x[n] = 1 + cos 6 y[n] = sin 4πη 6 z[n] = x[n]y[n] a) What is the DTFS of x[n]? b) What is the DTFS of y[n]? c) Using the properties you learned in class, find the Fourier series of z[n].
Objective Conduct DTFT, DTFS on a periodic discrete signal. Task: Consider the system with impulse response Tth sin 8 h(n) S(n) Tn (1) Find the Fourier-series representation for the output y(n) when the input x(n) is the periodic extension of the sequence 3/2, -1,0, -3/2, 1,0 Plot the x(n), h(n), y(n) and Fourier coefficient bk using Matlab or handwriting (Example 7.2.6 irse material) in cour (2) Find the output y(n) of the system with the input 1 Tn Tn x(п)...