Signals and Systems Midterm 10:20a.m. ~ 12:20p.m., May 1, Fri., 2009 Closed book, but open 1 sheet (both sides, 2 pages) of personal notes of A4 size Total score: 120 Total 4 pages in one B4 sheet

1.

[12] Suppose x and y denote input and output, respectively, of each of the three systems: System A: y (t ) = x(t + 2) sin(ω t + 2), ω ≠ 0 System B: y[n] = ( − 1 ) ( x[n] + 1) 2 System C: y[n] = ∑ x 2 [k + 1] − x[k ] k =1 n n

Answer the following questions for each system and justify your answer. (a) Is the system linear? (b) Is the system time invariant? (c) Is the system causal? (d) Is the system stable?

2.

We want to develop an edge detector that is robust against additive noise. Consider a discrete-time (DT) linear time-invariant (LTI) system H 2 with h2 [ n] = h[ n] ∗ h[ n + 1] as its impulse response shown below, where h[ n] = δ [ n] − δ [ n − 1] .

(a) [4] Assume there is no noise, i.e., d [n] = 0 and x[n] = p[n] . Sketch the output y[n] of the system assuming the input p[n] to the system is the following signal:

(b) [4] Assume the noise is d [ n] = −δ [ n + 1] and the input p[n] remains the same. Sketch the output y[n] of the system. (c) [4] In order to use system H 2 as a part of an edge detector, we would like to add an LTI system H s whose unit impulse response hs [n] is shown below. System H s smoothes out effect of noise on x[n] . The overall system can be represented as below:

Sketch the output ys [n] of the system with d [n] and p[n] specified in Part (b). 3.

[15] You are given the following 5 facts about a discrete time sequence x[n] :
i. ii. iii. x[n] is real and odd. x[n] is periodic with period N = 6 .

iv. v.

∑ n =< N >

(−1) n / 3 x[n] = 6 j .

1 N

∑ n =< N >

x[n] = 10 .

2

x[1] > 0 .

Find an expression of x[n] in the form of sines and cosines.

4.

[6] Suppose x(t ) =

(

sin(4π t ) πt

)(

sin(2π t ) πt

(−1)t . Determine the frequency range of x(t ) .

)

5.

Let x(t ) be a periodic signal with fundamental period T and Fourier series coefficients ak . Derive the Fourier series coefficients of each of the following signals in terms of ak : (a) [3] The odd part of x ( t − T ) 2 (b) [3] x ( T − t ) 4

6.

[10] Suppose x(t ) = below: sin(4π t ) πt

, p(t ) = cos(2π t ) , and q(t ) =

sin(2π t ) πt

. Consider the system depicted

where the frequency response of H ( jω ) is given by

Determine the output c(t ) and its Fourier transform C ( jω ) . Sketch and clearly label C ( jω ) .

NOTE: There are problems in the back.

7.

Assume that a linear time-invariant (LTI) system has an input x(t ) and output y (t ) relationship given by y (t ) = ∫ e −α x(t − α )dα .
0 ∞

(a) [4] Find the system impulse response h(t ) . (b) [4] Is the system causal? Why? (c) [4] Determine y (t ) if the input x(t ) is set to u (t + 1) . (d) [4] Following (a), consider an interconnection of LTI systems given as following graph: System A is a series interconnection of System 1 with impulse response − h(t ) and System 2 with impulse response δ (t − 1) . Then the overall system is a parallel interconnection of System A and System B with impulse response h(t ) . Find the impulse response of the overall system. (Note: Remember to substitute your result in (a) for h(t ) .)

8.

Consider a first-order LTI system with the linear constant coefficient differential equation (LCCDE) given by

dy (t ) + 2 y (t ) = x(t ) for t ≥ 0 and y (0) = 4 . dt

(a) [4] Find the natural response of the LTI system. (b) [4] Find the forced response of the LTI system when the input x(t ) = 2 . (c) [4] Find the forced response of the LTI system when the input x(t ) = e3t u (t ) .

9.

Suppose that an LTI system with unit impulse response h[n] has an input signal x[n] . (a) [5] Find the output y[n] of the system at n = 0 in terms of X (e jω ) and H (e jω ) by using the convolution property of Fourier transform, where X (e jω ) and H (e jω ) are the Fourier transforms of x[n] and h[n] , respectively. (b) [5] Find the output y[n] of the system at n = 0 in terms of x[n] and h[n] by using the convolution sum. (c) [5] Based on the above results, derive the Parseval’s relation between the DT signal x[n] and its Fourier transform X (e jω ) .

10. Here, we consider a scheme to generate signals for communication applications. Assume that two continuous-time (CT) signals f1 (t ) and f 2 (t ) are given as follows: f1 (t ) = 2 cos(200π t ) and f 2 (t ) = 5cos(1000π t ) , respectively. Let the signal f 3 (t ) be the product of the two CT signals f1 (t ) and f 2 (t ) . (a) [3] Find the frequency spectrum of f 3 (t ) . (b) [3] Describe the properties of the Fourier transform you use to derive the result in part (a). 11. Consider the following DT signals x1[n] and x2 [n] : π sin n2 nπ  sin 5 nπ 9  9sin nπ  9 x2 [n] =  5  9 

x1[n] =

if n is not a multiple of 9 with period N = 9 if n is a multiple of 9

(a) [5] Find the discrete-time Fourier transform of x1[n] by using the duality between the discrete-time Fourier transform and continuous-time Fourier series expansion. (b) [5] Find the Fourier series expansion of x2 [n] by using the duality for the discrete-time Fourier series expansion.