Article type:
Focus Article
Transfer Functions
Article ID
Stephen Pollock
University of Leicester
Keywords
Impulse response, Frequency response, Spectral density
Abstract
In statistical timeseries analysis, signal processing and control engineering, a
transfer function is a mathematical relationship between a numerical input to
a dynamic system and the resulting output. The theory of transfer functions
describes how the input/output relationship is affected by the structure of the
transfer function.
The theory of the transfer functions of linear timeinvariant (LTI) systems has
been available for many years. It was developed originally in connection with
electrical and mechanical systems described in continuous time.
The basic
theory can be attributed largely to Oliver Heaviside (1850–1925) [3] [4].
With the advent of digital signal processing, the emphasis has shifted to discrete
time representations.
These are also appropriate to problems in statistical
timeseries analysis, where the data are in the form of sequences of stochastic
values sampled at regular intervals.
REPRESENTATIONS OF THE TRANSFER FUNCTION
In the discrete case, a univariate and causal transfer function mapping from an input
sequence
{
x
t
}
to an output sequence
{
y
t
}
can be represented by the equation
p
j
=0
a
j
y
t
−
j
=
q
j
=0
b
j
x
t
−
j
,
with
a
0
= 1
.
(1)
Here, the condition that
a
0
= 1
serves to identify
y
t
as the current output and the
elements
y
t
−
1
, . . . , y
t
−
p
as feedback or as lagged dependent variables. The sum of the
input variables on RHS of the equation, weighted by their coef
fi
cients, is described as
a distributed lag scheme. (See Dhrymes [2] for a treatment of distributed lags in the
context of econometric estimation.) The condition of causality implies that
x
t
+
j
and
y
t
+
j
, which are ahead of time
t
, are excluded from the equation.
1
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Consider
T
realisations of the equation (1), with the successive outputs indexed by
t
= 0
, . . . , T
−
1
. By associating each equation with the corresponding power
z
t
of an
indeterminate algebraic quantity
z
and by adding them together, a polynomial equation
is derived that can be denoted by
a
(
z
)
y
(
z
) =
b
(
z
)
x
(
z
)
or, equivalently,
y
(
z
) =
b
(
z
)
a
(
z
)
x
(
z
) =
ψ
(
z
)
x
(
z
)
.
(2)
Here,
y
(
z
)
=
y
−
p
z
−
p
+
· · ·
+
y
0
+
y
1
z
+
· · ·
+
y
T
−
1
z
T
−
1
,
x
(
z
)
=
x
−
q
z
−
q
+
· · ·
+
x
0
+
x
1
z
+
· · ·
+
x
T
−
1
z
T
−
1
,
a
(
z
)
=
1 +
a
1
z
+
· · ·
+
a
p
z
p
and
b
(
z
)
=
b
0
+
b
1
z
+
· · ·
+
b
q
z
q
(3)
are described as the
z
transforms of the corresponding sequences. This representation
allows the algebra of polynomials to be deployed in analysing the dynamic system. An
extensive account of the
z
transform has been provided by Jury [5].
Equation (2) comprises the presample elements
y
−
p
, . . . , y
−
1
and
x
−
q
, . . . , y
−
1
of
the input and output sequences, which provide initial conditions for the system. How
ever, if the transfer function is stable in the sense that a bounded input will lead to a
bounded output—described as the BIBO condition—then it is permissible to extend
the two sequences backwards in time inde
fi
nitely, as well as forwards in time. Then,
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 Fall '11
 D.S.G.Pollock
 Digital Signal Processing, Signal Processing, LTI system theory, stephen pollock

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