Digital Signal Processing Reference
In-Depth Information
Be sure to review how to convert between
b
and
a
coefficients suitable for a call to the function
filter
and the coefficients in a difference equation. Note particularly Eqs. (2.8) and (2.9). For situations
where the system is an FIR, you can also use the script
LV
_
LT I of X
.
a) The system defined by the difference equation
y
[
n
]=
x
[
n
]+
x
[
n
−
2
]
b) The system defined by the difference equation
y
[
n
]=
x
[
n
]−
0
.
95
y
[
n
−
2
]
c) The system defined by the coefficients
a
= [1] and
b
=[
0
.
1667
,
0
.
5
,
0
.
5
,
0
.
1667
]
d) The system defined by the coefficients
b
= [1] and
a
=[
1
,
0
,
0
.
3333
]
e) The system defined by the coefficients
b
=[
0
.
1667
,
0
.
5
,
0
.
5
,
0
.
1667
]
a
=[
1
,
0
,
0
.
3333
]
24. Use paper and pencil and the graphical method to compute the first five values of the convolution
sequence of the following sequence pairs, then check your answers by using the MathScript function
conv
.
(a)
[(-1).ˆ(0:1:7)], [0.5*ones(1,10)]
(b)
[0.1,0.7,1,0.7,0.1], [(-1).ˆ(0:1:9)]
(c)
[1,1], [(-0.9*j).ˆ(0:1:7)]
(d)
[(exp(j*pi)).ˆ(0:1:9)], [ones(1,3)]
25. Verify that the commutative property of convolution holds true for the argument pairs given
below, using the script
LV
_
LT I of X
, and then, for each argument pair, repeat the exercise using the
MathScript function
conv
. Plot results for comparison.
(a) chirp([0:1/99:1],0,1,50) and [1,1];
(b) chirp([0:1/99:1],0,1,50) and [1,0,1];
(c) [1,0,-1] and chirp([0:1/999:1],0,1,500)
(d) [1,0,-1] and chirp([0:1/999:1],-500,1,500)
26. Does a linear system of the form
c
where
k
and
c
are constants, and
x
is an independent variable obey the law of superposition, i.e., is it true
that
y
=
kx
+
y(ax
1
+
bx
2
)
=
ay(x
1
)
+
by(x
2
)
where
a
and
b
are constants? Prove your answer.
