W. Alexander & C. Williams (NCSU)
FUNDAMENTAL DSP CONCEPTS
ECE 513, Fall 2019
62 / 170
Difference Equation Representation
The system transfer function,
H
(
z
)
, can also be written in terms of
the ratio of the Z Transforms of the output and the input.
H
(
z
) =
Y
(
z
)
X
(
z
)
(67)
It follows that
Y
(
z
)
"
1
.
0
+
L
X
k
=
1
a
(
k
)
z

k
#
=
X
(
z
)
"
L
X
k
=
0
b
(
k
)
z

k
#
(68)
W. Alexander & C. Williams (NCSU)
FUNDAMENTAL DSP CONCEPTS
ECE 513, Fall 2019
63 / 170
Difference Equation Representation
The time shift property of the Z Transform can be used to obtain
the difference equation representation for the discrete time system.
If
x
(
n
)
⇔
X
(
z
)
(69)
then
x
(
n

k
)
⇔
z

k
X
(
z
)
(70)
The region of convergence is the same as that for X
(
z
)
except at z
=
0
for k
>
0
and except for z
=
∞
for k
<
0
.
Thus, if
x
(
n

k
) =
0
∀
n
<
k
, then
z

k
X
(
z
) =
∞
X
n
=
0
x
(
n

k
)
z

n
(71)
W. Alexander & C. Williams (NCSU)
FUNDAMENTAL DSP CONCEPTS
ECE 513, Fall 2019
64 / 170
Difference Equation Representation
Similarly, if
y
(
n

k
) =
0
∀
n
<
k
, then
z

k
Y
(
z
) =
∞
X
n
=
0
y
(
n

k
)
z

n
(72)
This result can be used to modify Equation 68 to obtain
∞
X
n
=
0
[
y
(
n
) +
a
(
1
)
y
(
n

1
) +
· · ·
+
a
(
L
)
y
(
n

L
)]
z

n
(73)
=
∞
X
n
=
0
[
b
(
0
)
x
(
n
) +
b
(
1
)
x
(
n

1
) +
· · ·
+
b
(
L
)
x
(
n

L
)]
z

n
W. Alexander & C. Williams (NCSU)
FUNDAMENTAL DSP CONCEPTS
ECE 513, Fall 2019
65 / 170
Difference Equation Representation
The powers of
z
can be equated to obtain
y
(
n
)
+
a
(
1
)
y
(
n

1
) +
· · ·
+
a
(
L
)
y
(
n

L
)
=
b
(
0
)
x
(
n
) +
b
(
1
)
x
(
n

1
)
+
· · ·
+
b
(
L
)
x
(
n

L
)
(74)
or
y
(
n
)
=
b
(
0
)
x
(
n
) +
b
(
1
)
x
(
n

1
) +
· · ·
+
b
(
L
)
x
(
n

L
)

a
(
1
)
y
(
n

1
)
 · · · 
a
(
L
)
y
(
n

L
)
(75)
W. Alexander & C. Williams (NCSU)
FUNDAMENTAL DSP CONCEPTS
ECE 513, Fall 2019
66 / 170
Difference Equation Representation
It follows that the linear shift invariant discrete time system can be
represented in either of the following standard forms.
H
(
z
)
=
L
X
k
=
0
b
(
k
)
z

k
1
.
0
+
L
X
k
=
1
a
(
k
)
z

k
y
(
n
)
=
L
X
k
=
0
b
(
k
)
x
(
n

k
)

L
X
k
=
1
a
(
k
)
y
(
n

k
)
(76)
Note the relationship between the coefficients
b
(
k
)
and
a
(
k
)
in the
system transfer function and the corresponding coefficients in the
difference equation.
This relationship makes it easy to write the difference equation for
a given system transfer function by inspection or vice versa.
W. Alexander & C. Williams (NCSU)
FUNDAMENTAL DSP CONCEPTS
ECE 513, Fall 2019
67 / 170
Frequency Response
The general form for the transfer function for a causal, discrete
time is
H
(
z
) =
Y
(
z
)
X
(
z
)
=
L
X
k
=
0
b
(
k
)
z

k
1
.
0
+
L
X
k
=
1
a
(
k
)
z

k
(77)
Note that
H
(
z
)
is represented using negative powers of z.
W. Alexander & C. Williams (NCSU)
FUNDAMENTAL DSP CONCEPTS
ECE 513, Fall 2019
68 / 170
Frequency Response
H
(
z
)
can also be written in the form
H
(
z
) =
Y
(
z
)
X
(
z
)
=
L
X
k
=
0
b
(
k
)
z
L

k
z
L
+
L
X
k
=
1
a
(
k
)
z
L

k
(78)
This form can be obtained by multiplying the numerator and
denominator of the previous form by
z
L
.
W. Alexander & C. Williams (NCSU)
FUNDAMENTAL DSP CONCEPTS
ECE 513, Fall 2019
69 / 170
Frequency Response I
The frequency response of a discrete time system can be
obtained by applying a unit magnitude, sinusoidal sequence with
zero phase and arbitrary frequency as an its input.
A unit magnitude sinusoidal sequence with zero phase and
arbitrary frequency can be represented as follows.
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