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ECE 300
Signals and Systems
Exam 1
6 April, 2009
NAME ________________________________________
This exam is closed-book in nature. You are not to use a calculator or computer during the exam. Do not write on the back of any page, use the extra pages at the end of the exam.
Problems 1-4________ / 16 Problem 5 ________ / 29 Problem 6 ________ / 25 Problem 7 ________ / 25
Exam 1 Total Score: _______ / 95
Problems 1-4 are worth 4 points each.
1. Which of the following statements is the best simplification of: (^0) 2
t
x λ t δ λ d λ
−
a) 0 b) x ( t − t 0 ) δ ( t ) c) x ( − t 0 ) ( u t ) d) x ( − t 0 ) δ ( t ) e) none of these
2. The the signal x ( ) t = cos( ) t − j sin( ) t
is
a) and energy signal b) a power signal c) neither energy nor power
3. Given x(t) below, which of the plots labeled (a) − (d) represents ݔሺ2ሺെ ݐെ 2ሻሻ.
x(t)
(b)
(a)
(c) (c)
4. The signal x t ( ) = cos(4 π t + π/ 2) + ej^6^^ π t 1 is
a) not periodic b) periodic with fundamental period 6π seconds c) periodic with fundamental period 1 second d) periodic with fundamental period 3 /2 seconds e) none of the above
6. Impulse Response (25 points ) For each of the following systems, determine the impulse response between the
input
h t ( ) x ( ) t and output y t ( ). Be sure to include any necessary unit step functions.
a) (
2 ( ) )^ ( 1
t
y t e −^ t −^ λ^ x λ ) d λ
−∞
−
b) 2 y t ( ) + y t ( ) = x t ( −1)
c) For the interconnected feedback system below, determine a relationship between the input (^) x ( ) t and the output in terms of and. Your final answer will be of
the form
y t ( ) h t 1 ( )^ h 2^ ( ) t
y t ( ) *[ δ ( ) t + A t ( )] = x t ( ) *[ B t ( )]
You need to determine A ( ) t and B ( ) t.
Hints: e t ( ) = x t ( ) − v t ( )and v t ( ) = y t ( )* h 2 ( ) t
7. System Properties (25 points) a) Fill in the following table with a Y (Yes) or N (No). Only your responses in the table will be graded, not any work. Assume x ( ) t is the system input and is the system
output. Also assume we are looking at all times (positive and negative times).
y t ( )
System Linear?
Time- Invariant? Memoryless? Causal? y t ( ) + t y t^2 ( ) = x t ( +1)
( ) 2
t y t = x ⎛⎜^ − ⎞⎟ ⎝ ⎠ y t ( ) = x t ( ) + 2 y t ( ) =| x t ( ) |
b) For the system described below, determine the value of “c” that will make the system time-invariant. Use a formal technique such as we used in class (and on the homework) and justify your answer. You will be graded more on your method of arriving at an answer than the answer itself!
t t c
y t = e ∫ e −^ λ^ x λ d λ
