Sigature
2022S - Midterm
ID:
.
Name:
.
(1) [30 points] answer the following question with O or X. If an answer is correct, 5 points. If an
answer is incorrect, -3 point. If there is no answer, 0 point.
2
(1-1) The fundamental period of 𝑒 𝑗7𝜋𝑡 is .
(
)
(1-2) The fundament period of 𝑥(𝑡) = 2 cos(10𝑡 + 1) − sin⁡(4𝑡 − 1) is 𝜋.
(
)
7
(1-3) A continuous time system with input 𝑥(𝑡) and 𝑦(𝑡) output related by 𝑦(𝑡) = 𝑥(sin⁡(𝑡)) is
causal.
(
)
(1-4) A continuous time system with input 𝑥(𝑡) and 𝑦(𝑡) output related by 𝑦(𝑡) = 𝑥(sin⁡(𝑡)) is
linear.
𝜋
2
(1-5) A continuous time signal 𝑥(𝑡) = [cos (2𝑡 − )] has a period of
3
𝜋
2
.
(
)
(
)
4 𝑛
(1-6) The impulse response of a discrete time LTI system defined by ℎ[𝑛] = ( ) 𝑢[𝑛 + 2] is causal.
5
(
)
1 𝑛
(1-7) The impulse response of a discrete time LTI system defined by ℎ[𝑛] = ( ) 𝑢[−𝑛] is stable.
2
(
)
(1-8) The impulse response of a continuous time LTI system defined by ℎ(𝑡) = 𝑒 −4𝑡 𝑢(𝑡 − 2) is causal.
(
)
(1-9) The impulse response of a continuous time LTI system defined by ℎ(𝑡) = 𝑒 −4𝑡 𝑢(𝑡 − 2) is stable.
(
)
(1-10) The impulse response of a continuous time LTI system defined by ℎ(𝑡) = 𝑒 2𝑡 𝑢(−1 − 𝑡) is
stable.
(
Number
Correct
Incorrect
No answer
Total score of No. (1)
Score
)
Sigature
2022S - Midterm
ID:
.
Name:
.
(2) [20 points] Answer the following questions when all the systems are initially at rest.
1
(2-1) [5 points] If the system is defined by 𝑦[𝑛] − 𝑦[𝑛 − 1] = 𝑥[𝑛], find the closed-form of the
2
impulse response ℎ[𝑛].
(2-2) [5 points] If the system is defined by 𝑦[𝑛] = 𝑥[𝑛] + 2𝑥[𝑛 − 1], find the closed-form expression
of the impulse response ℎ[𝑛].
(2-3) [5 points] Find the difference equation with 𝑥[𝑛] and 𝑦[𝑛] of the overall system described
below.
(2-4) [5 points] Find the closed-form expression of the impulse response of (2-3).
Sigature
2022S - Midterm
ID:
.
Name:
.
(3) [20 points] Consider the following three continuous-time signals with a fundamental period of
1
𝑇= :
2
𝑥(𝑡) = cos(4𝜋𝑡) ,
𝑦(𝑡) = sin(4𝜋𝑡) ,
𝑧(𝑡) = 𝑥(𝑡)𝑦(𝑡).
(3-1) [5 points] Determine the Fourier series coefficients of 𝑥(𝑡)?
(3-2) [5 points] Determine the Fourier series coefficients of 𝑦(𝑡)?
(3-3) [5 points] Use the results of parts (a) and (b), along with the multiplication property of the
continuous-time Fourier series, to determine the Fourier series coefficients of 𝑦(𝑡).
(3-4) [5 points] Determine the Fourier series coefficients of z(𝑡) through direct expansion of 𝑧(𝑡) in
trigonometric form.
Sigature
2022S - Midterm
ID:
.
Name:
.
(4) [30 points] Suppose we are given the following information about a signal 𝑥(𝑡):
1. 𝑥(𝑡) is a real signal.
2. 𝑥(𝑡) is periodic with period 𝑇 = 6 and has Fourier coefficients 𝑎𝑘 .
3. 𝑎𝑘 = 0 for 𝑘 = 0 and 𝑘 > 2.
4. 𝑥(𝑡) = −𝑥(𝑡 − 3)
5.
3
1
∫ |𝑥(𝑡)|2 𝑑𝑡 = 2.
6 −3
1
6. 𝑎1 is a positive real number.
(4-1) [5 points] Determine the fundamental frequency 𝜔0 .
(4-2) [10 points] Determine all 𝑎𝑘 s.
(4-3) [15 points] Determine the values of the constants A, B, and C when 𝑥(𝑡) = 𝐴𝑐𝑜𝑠(𝐵𝑡 + 𝐶).