ECE/355: Signals and Systems

Homework #3 Due day: 02/25/2019

1. Please find the zero-input response

( )

0

y t

of the following systems, if the initial

conditions arey0

(0) = 1, and Dy0

(0) = −1.

1).

( 7 10) ( ) ( )

2 2 D D y t D x t

2).

( 4 4) ( ) ( 1) ( )

2 D D y t D x t

2. Please determine the unit impulse response

h(t) of the following systems

(Given:

h(t) b (t) [P(D)y (t)]u(t)

n n

):

1). ( 6 8) ( ) ( 2) ( )

2 2 D D y t D x t

2). ( 4 4) ( ) ( 5) ( )

2 D D y t D x t

3. Using direct integration to calculate the following convolutions (Don’t use the

convolution table):

1).

e u(t)*e u(t)

at bt

2).

tu(t)*u(t)

4. Please use the convolution table to calculate following convolutions:

1).

( )*(1 ) ( )

2

e u t e u t

t t

2).

[cos(5t) e ]u(t)* (2 t)

t

3).

( 3)*[ ] ( 1)

( 3) 2( 1) ( 1)

e u t e e u t

t t t

5. The unit impulse response of an LTIC system is

h(t) e u(t)

t

. Please use the

convolution table to find the zero-state response y(t) if the input is:

1).

x(t) (t 5)

;

2).

x(t) (t 4)u(t 4)

;

3).

( ) ( 1) +sin(3t)u(t)

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