1. 1 ELEC 361/W: Midterm exam Solution: Fall 2005 Professor: A. Amer TA: M. Ghazal Q1: 1. True: According to the “Shifting property” of the FT 2. False: Causality is not determined based on the input signal x(t) 3. True: Using shifting and linearity properties of the FS 4. True: If (x(t) is bounded and since |cos(1/t)| is bounded by 1 5. False: The fundamental period of this signal is 24 which is the least common multiple of 3 (the fundamental period of the first term) and 8 (the fundamental period of the second term)

2. 2 Q2: Fourier Transform

3. We have Taking the inverse transform, we get 3 Q2: Solution

4. We have Taking the inverse transform, we get 4 Q2: Solution

5. 5 Q3: Convolution

6. 6 Q3 Solution: Step 1 (a) Write down the x(t) and h(t) functionally and graphically Note that h(t) is a scaled version of x(t)

7. 7 Q3 Solution: Step 2 Sketch h(-τ) and h(t-τ) h(-τ) Rreflection around y-axis Chage t to τ h(t-τ) = h(-τ+t) Add t to all axis points Move the graph away to the left

8. 8 Q3 Solution: Step 3 Slide h(t-τ) to the right and collect the overlap As you go, find Limits for y(t) Limits for integration

9. 9 Q3 Solution: Step 3

10. 10 Q3 Solution: Step 3

11. 11 Q3 Solution: Step 3

12. 12 Q3 Solution: Step 4 Put the limits together to make y(t)

13. 13 Q3 Solution: Step 5 (b) find the first derivative of y with respect to t both functionally and graphically This function has 4 discontinuities, only when  = 1, it has 3 discontinuities (two discontinuities become one) Note that we know 0<  1

14. Q4 Fourier Series

15. Q4 Solution (1) Graph both x[n] and x[n-1] to get g[n] From the graph, we can write g[n] as Note that g[n] is periodic with N = 10 (2) From the expression for g[n], the FS coefficients are

16. Q4 Solution (3) Since g[n] = x[n] – x[n-1], the FS coefficients a k and b k are related by Therefore,