Universiti Malaysia Perlis
2006/2007 Signals & Systems [ EKT230 ]
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Assignment Chapter 2: Time Domain Representation
2.1
2.2
A discrete time LTI system has the impulse response h[n] shown in figure below. Use linearity and
time invariance to determine the system output y[n] if the input is;
(a) x[n]  2 [n]   [n  1]
(b)
x[n]  u[n]  u[n  3]
(c)
x[n]   [n  1]  2 [n]   [n  3]
Evaluate the following discrete-time convolution sums;
(a)
y[n]  u[n] * u[n  3]
(b)
y[n]  2 n u[n  2] * u[n  3]
(c)
1
y[n]    u[n  2] * u[n]
2
(d)
 n 
y[n]  cos u[n] * u[n  1]
 2 
(e)
 n 
y[n]  cos  * 2 n u[n  2]
 2 
n
2.3
Evaluate the following convolution sum.
(a)
a[n]  x[n] * z[n]
(b)
a[n]  x[n] * z[n]
(c)
a[n]  y[n] * z[n]
(d)
a[n]  x[n] * g[n]
(e)
a[n]  y[n] * z[n]
(f)
a[n]  y[n] * g[n]
(g)
a[n]  y[n] * w[n]
1
(h)
2.5
a[n]  y[n] * f [n]
Evaluate the following convolution sum.
(a) y (t )  u (t  1) * u (t  2)
(b) y(t )  e 2t u (t ) * u (t  2)
(c) y (t )  cos(t )(u (t  1)  u (t  3)) * u (t )
(d) y (t )  (u (t  1)  u (t  1)) * u (t  2)
(e) y (t )  {tu(t )  (10  2t )u (t  5)  (10  t )u (t  10)} * u (t )
(f) y(t )  (t  2t 2 )(u(t  1)  u(t  1)) * 2u(t  2)
2.6
Consider the continuous time signals depicted in figure below. Evaluate the following convolution
integral;
(a) m(t )  x(t ) * y (t )
(b) m(t )  x(t ) * z (t )
(c) m(t )  x(t ) * t (t )
2
(d) m(t )  x(t ) * b(t )
(e) m(t )  x(t ) * a(t )
(f) m(t )  y (t ) * z (t )
(g) m(t )  y (t ) * w(t )
(h) m(t )  y (t ) * g (t )
2.16 Write a differential equation description relating the output to the input of the electrical circuit
shown in figure below.
3
2.17 Determine the homogeneous solution for the system described by the following differential
equations;
d
y (t )  10 y (t )  0
dt
(a)
5
(b)
d2
d
y (t )  5 y (t )  6 y (t )  0
2
dt
dt
(c)
d2
d
y (t )  3 y (t )  2 y (t )  0
2
dt
dt
(d)
d2
d
y (t )  2 y (t )  y (t )  0
2
dt
dt
(e)
d2
y (t )  4 y (t )  0
dt 2
(f)
d2
d
y (t )  2 y (t )  2 y (t )  0
2
dt
dt
2.23 Find the difference-equation description for the three systems shown in figure below.
4
2.24 Draw direct form I and direct form II implementations for the following difference equations;
(a)
y[n] 
1
y[n  1]  2 x[n]
2
(b)
y[n] 
1
1
y[n  1]  y[n  2]  x[n]  x[n  1]
4
8
(c)
y[n] 
1
y[n  2]  2 x[n]  x[n  1]
9
(d)
y[n] 
1
y[n  1]  y[n  3]  3x[n  1]  2 x[n  2]
2
2.26 Convert the following differential equations to integral equations, and draw direct form I and direct
form II implementations of the corresponding systems;
(a)
d
y (t )  10 y (t )  2 x(t )
dt
(b)
d2
d
d
y(t )  5 y (t )  4 y (t )  x(t )
2
dt
dt
dt
(c)
d2
d
y (t )  y (t )  3 x(t )
2
dt
dt
5
(d)
2.27
d3
d2
d
d
y
(
t
)

2
y(t )  3 y(t )  x(t )  3 x(t )
3
2
dt
dt
dt
dt
Determine a state variable description for the discrete-time system depicted.
6