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| − | [[Category:ECE301Spring2011Boutin]] | + | [[Category:ECE301Spring2011Boutin]] [[Category:Problem_solving]] |
| − | [[Category: | + | = [[:Category:Problem_solving|Practice Question]] on Computing the Fourier Series discrete-time signal = |
| − | = Practice Question on Computing the Fourier Series discrete-time signal= | + | |
| − | + | ||
| − | <math> | + | Obtain the Fourier series the DT signal |
| − | x[n] = \left\{ | + | |
| + | <math>x[n] = \left\{ | ||
\begin{array}{ll} | \begin{array}{ll} | ||
1, & \text{ for } -5\leq n \leq 5,\\ | 1, & \text{ for } -5\leq n \leq 5,\\ | ||
0, & \text{ for } 5< |n| \leq 10. | 0, & \text{ for } 5< |n| \leq 10. | ||
\end{array} | \end{array} | ||
| − | \right. \ </math> | + | \right. \ </math> |
| − | x[n] periodic with period 20. | + | x[n] periodic with period 20. |
---- | ---- | ||
| − | ==Share your answers below== | + | |
| − | You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too! | + | == Share your answers below == |
| + | |||
| + | You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too! | ||
| + | |||
---- | ---- | ||
| − | ===Answer 1=== | + | |
| − | + | === Answer 1 === | |
| − | === | + | |
| − | + | Solution 1: | |
| − | === | + | |
| − | + | 1) <math>w_o = \frac{2\pi}{T} = 2\frac{\pi}{20}=\frac{\pi}{10}</math> | |
| + | |||
| + | 2) a<sub>o</sub> is the DC value of the AC signal and is therefore 1/2 | ||
| + | |||
| + | 3) <math>a_k= 1/20 * \int_{-5}^{5} \! x(t)*e^{-j*k*(\pi/10)*t}\,dx </math> = <math>e^{j*k*\pi/2}-e^{-j*k*\pi/2}/(2*\pi*j*k) = \frac{sin(k\pi/2)}{(k\pi)}</math>([[User:Clarkjv|Clarkjv]] 18:25, 8 February 2011 (UTC)) | ||
| + | |||
| + | Solution 2: | ||
| + | |||
| + | 1)<math>w_o=\frac{\pi}{10}</math> (see solution 1) | ||
| + | |||
| + | From example 3.5 (sec. 3.3 pg 193 Signals and Systems 2nd edition Oppenheim) | ||
| + | where T_1 is half the pulse width, | ||
| + | <math>a_k=\frac{sin(k * w_o * T_1)}{(k*\pi)}</math>, | ||
| + | |||
| + | |||
| + | 2)a<sub>o</sub> is still the DC value of the AC signal and therefore, | ||
| + | a<sub>o</sub> = 1/2 | ||
| + | |||
| + | From <math>a_k=\frac{sin(k * w_o * T_1)}{(k*\pi)}</math> | ||
| + | |||
| + | 3)<math>a_k=\frac{sin(k\pi/2)}{(k\pi)}</math> | ||
| + | ([[User:Clarkjv|Clarkjv]] 18:25, 8 February 2011 (UTC)) | ||
| + | |||
| + | :<span style="color:red">TA's comment: The given signal is a DT signal and not a CT signal.</span> | ||
---- | ---- | ||
[[2011_Spring_ECE_301_Boutin|Back to ECE301 Spring 2011 Prof. Boutin]] | [[2011_Spring_ECE_301_Boutin|Back to ECE301 Spring 2011 Prof. Boutin]] | ||
Latest revision as of 09:25, 11 November 2011
Practice Question on Computing the Fourier Series discrete-time signal
Obtain the Fourier series the DT signal
$ x[n] = \left\{ \begin{array}{ll} 1, & \text{ for } -5\leq n \leq 5,\\ 0, & \text{ for } 5< |n| \leq 10. \end{array} \right. \ $
x[n] periodic with period 20.
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
Solution 1:
1) $ w_o = \frac{2\pi}{T} = 2\frac{\pi}{20}=\frac{\pi}{10} $
2) ao is the DC value of the AC signal and is therefore 1/2
3) $ a_k= 1/20 * \int_{-5}^{5} \! x(t)*e^{-j*k*(\pi/10)*t}\,dx $ = $ e^{j*k*\pi/2}-e^{-j*k*\pi/2}/(2*\pi*j*k) = \frac{sin(k\pi/2)}{(k\pi)} $(Clarkjv 18:25, 8 February 2011 (UTC))
Solution 2:
1)$ w_o=\frac{\pi}{10} $ (see solution 1)
From example 3.5 (sec. 3.3 pg 193 Signals and Systems 2nd edition Oppenheim) where T_1 is half the pulse width, $ a_k=\frac{sin(k * w_o * T_1)}{(k*\pi)} $,
2)ao is still the DC value of the AC signal and therefore,
ao = 1/2
From $ a_k=\frac{sin(k * w_o * T_1)}{(k*\pi)} $
3)$ a_k=\frac{sin(k\pi/2)}{(k\pi)} $ (Clarkjv 18:25, 8 February 2011 (UTC))
- TA's comment: The given signal is a DT signal and not a CT signal.
