Digital Signal Processing Reference
In-Depth Information
It is standard practice to denote the ratio of polynomials as
Á
b
m
s
m
+ b
m- 1
s
m- 1
+
+ b
1
s + b
0
H(s) =
.
(3.70)
+
Á
+ a
1
s + a
0
a
n
s
n
+ a
n- 1
s
n- 1
We show subsequently that this function is related to the impulse response The
function is called a
transfer function
and is said to be
n
th order. The order of a
transfer function is the same as that of the differential equation upon which the
transfer function is based.
We now summarize this development. Consider an LTI system with the
transfer function
h(t).
H(s)
H(s),
as given in (3.69) and (3.70). If the system excitation is the
Xe
s
1
t
,
complex exponential
the steady-state response is given by, from (3.66) and
(3.69),
x(t) = Xe
s
1
t
: y
ss
(t) = XH(s
1
)e
s
1
t
.
(3.71)
The complex-exponential solution in (3.71) also applies for the special case of sinu-
soidal inputs. Suppose that, in (3.71),
X = ƒXƒ e
jf
and
s
1
= jv
1
,
where
f
and
v
1
are
real. Then
x(t) = Xe
s
1
t
= ƒXƒ e
jf
e
jv
1
t
= ƒXƒ e
j(v
1
t +f)
= ƒXƒ cos (v
1
t + f) + j ƒXƒ sin(v
1
t + f).
(3.72)
H(jv
1
) = ƒH(jv
1
) ƒ e
ju
H
.
Since, in general,
H(jv
1
)
is also complex, we let
The right
side of (3.71) can be expressed as
y
ss
(t) = XH(jv
1
)e
jv
1
t
= ƒXƒ ƒH(jv
1
) ƒ e
j(v
1
t +f+u
H
)
= ƒXƒ ƒH(jv
1
) ƒ [cos [v
1
t + f +
∠
H(jv
1
)] + jsin[v
1
t + f +
∠
H(jv
1
)]],
with From (3.64), since the real part of the input signal produces
the real part of the output signal,
u
H
=
∠
H(jv
1
).
ƒXƒ cos (v
1
t + f) : ƒXƒ ƒH(jv
1
) ƒ cos [v
1
t + f +
∠
H(jv
1
)].
(3.73)
This result is general for an LTI system and is fundamental to the analysis of LTI
systems with periodic inputs; its importance cannot be overemphasized.
Suppose that a system is specified by its transfer function To obtain the
system differential equation, we reverse the steps in (3.67) through (3.70). In fact, in
the numerator coefficients are the coefficients of and the denomi-
nator coefficients are the coefficients of we can consider the transfer
function to be a shorthand notation for a differential equation. Therefore, the system
differential equation can be written directly from the transfer function conse-
quently,
is a complete description of the input-output characteristics of a system,
regardless of the input function
. For this reason, an LTI system can be represented by
the block diagram in Figure 3.21 with the system transfer function given inside the
block. It is common engineering practice to specify an LTI system in this manner.
H(s).
d
i
x(t)/dt
i
,
H(s)
b
i
d
i
y(t)/dt
i
;
a
i
H(s);
H(s)
