Determine the Fourier series representation of
Let \(\left.x_{(} t\right)=\left\{\begin{array}{rr}t, & 0 \leq t \leq 1 \\ -t, & -1 \leq t \leq 0\end{array}\right.\), be a periodic signal with fundamental period of \(T=2\) and Fourier series coefficients \(a_{k}\).
a) Sketch the waveform of \(x(t)\) and \(\frac{d x(t)}{d t}\)
b) Calculate \(a_{0}\)
c) Determine the Fourier series representation of \(g(t)=\frac{d x(t)}{d t}
d) Using the results from Part (c) and the property of continuous-time Fourier series to determine the Fourier series coefficients of \(x(t)\)
Homework Answers
Let x(t) = t, 0<t 1 and Fourier series coefficients a , be a periodic signal...
Let x(t) = t, 0<t 1 and Fourier series coefficients a , be a periodic signal with fundamental period of T 2 -t,-1t0 dz(t) a) Sketch the waveform of r(t)d3 marks) b) Calculate ao (3 marks) c) Determine the Fourier series representation of gt)(4 rks) d) Using the results from Part (c) and the property of continuous-time Fourier series to dr(t) determine the Fourier series coefficients of r(t) (4 marks)
Compute the Fourier cosine coefficients for f(x)
Let \(f(x)= \begin{cases}0 & \text { for } 0 \leq x<2 \\ -(4-x) & \text { for } 2 \leq x \leq 4\end{cases}\)- Compute the Fourier cosine coefficients for \(f(x)\).- \(a_{0}=\)- \(a_{n}=\)- What are the values for the Fourier cosine series \(\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos \left(\frac{n \pi}{4} x\right)\) at the given points.- \(x=2:\)- \(x=-3\) :- \(x=5:\)
Consider the heat conduction problem
5. If \(f(x)=\left\{\begin{array}{cc}0 & -2<x<0 \\ x & 0<x<2\end{array} \quad\right.\)is periodio of period 4 , and whose Fourier series is given by \(\frac{a_{0}}{2}+\sum_{n=1}^{2}\left[a_{n} \cos \left(\frac{n \pi}{2} x\right)+b_{n} \sin \left(\frac{n \pi}{2} x\right)\right], \quad\) find \(a_{n}\)A. \(\frac{2}{n^{2} \pi^{2}}\)B. \(\frac{(-1)^{n}-1}{n^{2} \pi^{2}}\)C. \(\frac{4}{n^{2} \pi^{2}}\)D. \(\frac{2}{n \pi}\)\(\mathbf{E}_{1} \frac{2\left((-1)^{n}-1\right)}{n^{2} \pi^{2}}\)F. \(\frac{4}{n \pi}\)6. Let \(f(x)-2 x-l\) on \([0,2]\). The Fourier sine series for \(f(x)\) is \(\sum_{w}^{n} b_{n} \sin \left(\frac{n \pi}{2} x\right)\), What is \(b, ?\)A. \(\frac{4}{3 \pi}\)B. \(\frac{2}{\pi}\)C. \(\frac{4}{\pi}\)D. \(\frac{-4}{3 \pi}\)E. \(\frac{-2}{\pi}\)F. \(\frac{-4}{\pi}\)7. Let \(f(x)\) be periodic...
Please help by writing a MATLAB Code for the this lab Fourier Series Synthesis You will...
Please help by writing a MATLAB Code for the this lab
Fourier Series Synthesis
You will consider five continuous-time signals
1-
2-
for
A
D
4-
We were unable to transcribe this imageWe were unable to transcribe this imager(t) e-t for-1 < t > 1 x(t) 2 2 4 3 3 x(t) -4 2 2 4 2 1, 0sts be a periodic signal with fundamental period T = 2 and Fourier coefficients ak. (a) Determine the value of ao (b)...
(b) Let X(ju) denote the Fourier transform of the signal r(t) shown in the figure x(t) 2 -2 1 2 Using the properties of the Fourier transform (and without explicitly evaluating X(jw)), ii. (5 pts)...
(b) Let X(ju) denote the Fourier transform of the signal r(t) shown in the figure x(t) 2 -2 1 2 Using the properties of the Fourier transform (and without explicitly evaluating X(jw)), ii. (5 pts) Find2X(jw)dw. Hint: Apply the definition of the inverse Fourier transform formula, and you can also recall the time shift property for Fourier Transform. (c) (5 pts) Fourier Series. Consider the periodic signal r(t) below: 1 x(t) 1 -2 ·1/4 Transform r(t) into its Fourier Series...
Find the Fourier series representation of the following periodic signal. The expressions for the coefficients, Dn,...
Find the Fourier series representation of the following periodic
signal. The expressions for the coefficients, Dn, and for the
Fourier series representation of x(t) must not contain complex
expressions (combine complex exponentials into sinusoids).
3 2.5 exp(t/2 1.5 0.5 -4
A periodic signal x(t) is shown below. We want to find the Fourier Series representation for...
A periodic signal x(t) is shown below. We want to find the Fourier Series representation for this signal. x(t) AA -4 -2 1 2 4 6 8 (a) Find the period (T.) and radian frequency (wo) of (t). (b) Find the Trigonometric Series representation of X(t). These include: (a) Fourier coefficients ao, an, and bn ; (b) complete mathematical Fourier series expression for X(t); and (c) first five terms of the series.
3.59. (a) Suppose x[n] is a periodic signal with period N. Show that the Fourier series coefficie...
please
3.59. (a) Suppose x[n] is a periodic signal with period N. Show that the Fourier series coefficients of the periodic signal are periodic with period N. (b) Suppose that x() is a periodic signal with period T and Fourier series coeffi cients a with period N. Show that there must exist a periodic sequence g[n] such that (c) Can a continuous periodic signal have periodic Fourier coefficients?
3.59. (a) Suppose x[n] is a periodic signal with period N. Show...
Problem 2 (20 points) Let (2t +1, Ostsi x() +4 st 3 be a periodic signal...
Problem 2 (20 points) Let (2t +1, Ostsi x() +4 st 3 be a periodic signal with fundamental period T=3 and Fourier coefficients ar. a. Determine the value of an b. Determine ax, k 0, by: 1.first finding the Fourier coefficients of CID II.then using the appropriate property of the continuous-time Fourier series. c. Use the result of part(b) to express the Fourier transform of (t).
Signals and systems Problem 2 (20 points) Let -S2t+1, Osts1 x(t) = -t +4, 1sts 3...
Signals and systems
Problem 2 (20 points) Let -S2t+1, Osts1 x(t) = -t +4, 1sts 3 be a periodic signal with fundamental period T=3 and Fourier coefficients ar a. Determine the value of ao. b. Determine ak, k = 0, by: 1. first finding the Fourier coefficients of ii.then using the appropriate property of the continuous-time Fourier series. c. Use the result of part (b) to express the Fourier transform of x().
Let x(t) = t, 0<t 1 and Fourier series coefficients a , be a periodic signal with fundamental period of T 2 -t,-1t0 dz(t) a) Sketch the waveform of r(t)d3 marks) b) Calculate ao (3 marks) c) Determine the Fourier series representation of gt)(4 rks) d) Using the results from Part (c) and the property of continuous-time Fourier series to dr(t) determine the Fourier series coefficients of r(t) (4 marks)
Please help by writing a MATLAB Code for the this lab
Fourier Series Synthesis
You will consider five continuous-time signals
1-
2-
for
A
D
4-
We were unable to transcribe this imageWe were unable to transcribe this imager(t) e-t for-1 < t > 1 x(t) 2 2 4 3 3 x(t) -4 2 2 4 2 1, 0sts be a periodic signal with fundamental period T = 2 and Fourier coefficients ak. (a) Determine the value of ao (b)...
(b) Let X(ju) denote the Fourier transform of the signal r(t) shown in the figure x(t) 2 -2 1 2 Using the properties of the Fourier transform (and without explicitly evaluating X(jw)), ii. (5 pts) Find2X(jw)dw. Hint: Apply the definition of the inverse Fourier transform formula, and you can also recall the time shift property for Fourier Transform. (c) (5 pts) Fourier Series. Consider the periodic signal r(t) below: 1 x(t) 1 -2 ·1/4 Transform r(t) into its Fourier Series...
Find the Fourier series representation of the following periodic
signal. The expressions for the coefficients, Dn, and for the
Fourier series representation of x(t) must not contain complex
expressions (combine complex exponentials into sinusoids).
3 2.5 exp(t/2 1.5 0.5 -4
A periodic signal x(t) is shown below. We want to find the Fourier Series representation for this signal. x(t) AA -4 -2 1 2 4 6 8 (a) Find the period (T.) and radian frequency (wo) of (t). (b) Find the Trigonometric Series representation of X(t). These include: (a) Fourier coefficients ao, an, and bn ; (b) complete mathematical Fourier series expression for X(t); and (c) first five terms of the series.
please
3.59. (a) Suppose x[n] is a periodic signal with period N. Show that the Fourier series coefficients of the periodic signal are periodic with period N. (b) Suppose that x() is a periodic signal with period T and Fourier series coeffi cients a with period N. Show that there must exist a periodic sequence g[n] such that (c) Can a continuous periodic signal have periodic Fourier coefficients?
3.59. (a) Suppose x[n] is a periodic signal with period N. Show...
Problem 2 (20 points) Let (2t +1, Ostsi x() +4 st 3 be a periodic signal with fundamental period T=3 and Fourier coefficients ar. a. Determine the value of an b. Determine ax, k 0, by: 1.first finding the Fourier coefficients of CID II.then using the appropriate property of the continuous-time Fourier series. c. Use the result of part(b) to express the Fourier transform of (t).
Signals and systems
Problem 2 (20 points) Let -S2t+1, Osts1 x(t) = -t +4, 1sts 3 be a periodic signal with fundamental period T=3 and Fourier coefficients ar a. Determine the value of ao. b. Determine ak, k = 0, by: 1. first finding the Fourier coefficients of ii.then using the appropriate property of the continuous-time Fourier series. c. Use the result of part (b) to express the Fourier transform of x().
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