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(ELEC211)04 fall midterm1-sol.pdf
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Q1: Determine whether or not each of the following signals is periodic. If a signal is periodic, determine its fundamental period.
(a). x(t) = cos(t +)
4

j( t.1)
(b). x(t) = e 2

(c). x[n] = cos 1 n
4

(a).x(t) = cos(t + )
4

x(t + T ) = cos(t ++ T )
4
x(t + T ) = x(t) ifT = 2m, m is an integer
Hence, the fundamental period T (as m = 1), T = 2


j( t.1)
(b).x(t) = e 2


j( t+ T .1) j( t.1+ T )
x(t + T ) = e 22 = e 22


x(t + T ) = x(t) if T = 2m
2
Hence, the fundamental period T (as m = 1), T = 4

1
(c).x[n] = cos n
4
11

x[n + N] = cos( n + N )
44
1

x[n] x[n + N] Q N 2m, for N is an integer.
4
Thus x[n]is aperiodic.

P. 2 of 6
Q2(1): Fill in the table, determine (Yes/No) whether the following impulse response system is (i) memoryless,
(ii) causal, and (iii) stable. Justify your answers below.
Memoryless Causal Stable
( )( ) tch t = [ ][ ] nch n = 0for0( ) th t <= 0for0[ ] nh n <= < . dth t| ( ) | < =. h n n | [ ] |
system (a) ) ( )cos(( ) t u th t = No ( )( ) tch t Q Yes 00( ) <= tforQh t No > dtt || cos 0 Q
system (b) 1)(2( ) ..= u tteh t No ( )( ) tch t Q Yes 00( ) <= tforQh t Yes < .= . 1 2 | ( ) | t dte dth tQ
system (c) 3 ( )( ) th t = Yes ( )( ) tch t =Q Yes 00( ) <= tforQh t Yes < = . 3 | 3 ( ) | dttQ
system (d) ][( 1)[ ] nunh n ..= No [ ][ ] nch n Q No 00[ ] < nforQ h n No > . = . . 0 1 || ( 1) 0 n nQ

(a)
Justify your answers for system (a) here:

(b)
Justify your answers for system (b) here:

(c)
Justify your answers for system (c) here:

(d)
Justify your answers for system (d) here:


P. 3 of 6
Q2(2): Convert the following differential equation to integral equation and draw the block diagram of the 2
dd d
integral equation. y(t) + 5 y(t) + 4 y(t) = x(t)
2 dt dtdt
2
dd d
y(t) + 5 y(t) + 4y(t) = x(t)
2 dt dt
dt tt t
. y(t) + 5 y()d+ 4 y()dd= x()d . .. .
t tt Let u(t) = x()d and y(t) + 5 y()d+ 4 y()dd= u(t) . . .. tt
. u(t) . 5 y()d. 4 y()dd= y(t) . ..


Q3: The discrete-time system below is known as the unit delay element. Determine whether the system is (a) memoryless, (b) causal, (c) linear, (d) time-invariant, or (e) stable. Justify your answers.
x[n]
y[n] = x[n C 1 ]


Unit delay
(a)
Given that y[n] = x[n C 1]. Since the output value at n depends on the input value at n-1, the system is not memoryless.

(b)
Given that y[n] = x[n C 1]. Since the output does not depend on the future input values, the system is causal.

(c)
Let input is ax1[n] then output is ay1[n] = ax1[n-1] Let input is bx2[n] then output is by2[n] = bx2[n-1]


As the input x[n] = ax1[n] + bx2[n] , the output y[n] = x[n-1] = ax1[n-1] + bx2[n