Finding DFCS Coefficients of Periodic Signals in Discrete and Continuous Time

1. Problem 1 The signal x(n) is defined as x(-1) = x(1) = 1, x(0) = -2, and x(n) = 0 for all n≠ -1, 0, or 1. x is also periodic, with p = 100. (Source: Quiz 3, Fall 2009) 1. How many DFS coefficients does x have? Find these coefficients. 2. Determine the following quantities: (i) (ii)

2. (1) (1) (1) (1) …zero… 0 …zero… …zero… [ ] 100 1 99 101 98 -1 (-2) (-2) Problem 1 Since x is periodic and discrete, we can use DFS. Necessary equations to use: DFS Synthesis Equation DFS Analysis Equation

3. 1. How many DFS coefficients does x have? Find these coefficients. Here, k will span 100 possibly distinct values in a specific period. The synthesis equation implies that x has 100 Xk coefficients. Use DFS analysis equation to find the coefficients:

4. 1. How many DFS coefficients does x have? Find these coefficients. Any questions?

5. 2i. Determine the quantity of . Note that this looks similar to a certain equation… We have no control over k, p, or w0, but we can set n to be anything…

6. 2ii. Determine the quantity of . Again this looks similar to the synthesis equation… Any questions?

7. Problem 2 The continuous-time signal x(t) is defined as: Find the continuous-time Fourier series coefficients Xk of x. (Source: EE120 Midterm 1, Fall 2011)

8. (1) (1) (1) 3T -T T … … [ ] -2T 0 2T -T/2 3T/2 (-1) (-1) (-1) Problem 2 x(t) is just a series of Dirac deltas at t = nT, where n is an integer. From this, we can draw x(t): x(t) is periodic and continuous, so we need to use the CTFS equations.

9. Problem 2 We need to find Xk, so use the analysis equation:

10. Problem 2 Sifting property of Dirac delta: where I is an interval that contains T

11. Problem 2 when k is even when k is odd Any questions?