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Unformatted text preview: 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 discretetime 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 {xt } to an output sequence {yt } can be represented by the equation
p q aj yt−j =
j =0 bj xt−j , with a0 = 1. (1) j =0 Here, the condition that a0 = 1 serves to identify yt as the current output and the
elements yt−1 , . . . , yt−p as feedback or as lagged dependent variables. The sum of the
input variables on RHS of the equation, weighted by their coefﬁcients, 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 xt+j and
yt+j , which are ahead of time t, are excluded from the equation.
1 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 )
x(z ) = ψ (z )x(z ).
a(z ) (2) Here,
y (z ) = y−p z −p + · · · + y0 + y1 z + · · · + yT −1 z T −1 , x(z ) = x−q z −q + · · · + x0 + x1 z + · · · + xT −1 z T −1 , a(z ) = b(z ) = b0 + b 1 z + · · · + b q z 1 + a1 z + · · · + ap z p and 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. However, 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 indeﬁnitely, as well as forwards in time. Then,
y (z ) and x(z ) become inﬁnite series; and the matter of the initial conditions can be
ignored. THE IMPULSE RESPONSE
The transfer function can be characterised by its effect on certain elementary reference
signals. The simplest of these is the impulse sequence, which is deﬁned by
δt = 1, if t = 0;
0, if t = 0. (4) The corresponding z transform is δ (z ) = 1. The output generated by the impulse is
described as the impulse response function. For an ordinary causal transfer function,
which responds only to present and previous values of the input and output sequences,
the zerovalued inputs and responses at times t < 0 can be ignored.
On substituting x(z ) = δ (z ) = 1 into equation (2), it can be seen that calculating
the impulse response is a matter of ﬁnding coefﬁcients of the series expansion of the
rational function ψ (z ) = b(z )/a(z ).
When a(z ) = 1, the impulse response is just the sequence of the coefﬁcients of b(z ).
Then, there is a ﬁnite impulse response (FIR) transfer function. When a(z ) = 1, the
2 1
0.75
0.5
0.25
0
−0.25
−0.5
−0.75
0 10 20 30 Figure 1: The impulse response of the transfer function b(z )/a(z ) with a(z ) = 1.0 −
0.673z + 0.463z 2 + 0.486z 3 and b(z ) = 1.0 + 0.208z + 0.360z 2 .
impulse response is liable to continue indeﬁnitely, albeit that, in a stable system, it will
converge to zero. Then, there is an inﬁnite impulse response (IIR) transfer function.
When a(z ) = 1, the impulse response can be found via the equation b(z ) = a(z )ψ (z ).
An example is provided by the case where p = 2 and q = 1. Then,
b0 + b1 z = 1 + a1 z + a2 z 2 ψ 0 + ψ1 z + ψ 2 z 2 + · · · . (5) By performing the multiplication on the RHS, and by equating the coefﬁcients of the
same powers of z on the two sides of the equation, it is found that
b0
b1
0 = ψ0 ,
= ψ1 + a1 ψ0 ,
= ψ2 + a1 ψ1 + a2 ψ0 ,
.
.
. 0 = ψn + a1 ψn−1 + a2 ψn−2 , ψ0
ψ1
ψ2
ψn =
=
=
.
.
.
= b0 ,
b1 − a1 ψ0 ,
−a1 ψ1 − a2 ψ0 , (6) −a1 ψn−1 − a2 ψn−2 . The resulting sequence is just the recursive solution of the homogenous difference
equation yt + a1 yt−1 + a2 yt−2 = 0, subject to the initial conditions y0 = b0 and
y1 = b1 − a1 y0 .
The transfer function is fully characterised by its response to an impulse. One is reminded that all of the harmonics of a bell are revealed when it is stuck by a single
blow of the clapper, which constitutes an impulse. Figure 1 provides an example of an
impulse response. STABILITY
The stability of a rational transfer function b(z )/a(z ) can be investigated using its
partialfraction decomposition, which gives rise to a sum of simpler transfer functions
that can be analysed readily.
3 If the degree of the numerator of b(z )/a(z ) exceeds that of the denominator, then long
division can be used to obtain a quotient polynomial and a remainder that is a proper
rational function. The quotient polynomial will correspond to a stable transfer function;
and the remainder will be the subject of the decomposition.
Assume that b(z )/a(z ) is a proper rational function in which the denominator is factorised as
r (1 − z/λj )nj , a(z ) = (7) j =1 where nj is the multiplicity of the root λj , and where j nj = p is the degree of the
polynomial. Then, the socalled Heaviside partialfraction decomposition is
b(z )
=
a(z ) r nj j =1 k=1 cjk
;
(1 − z/λj )k (8) and the task is to ﬁnd the series expansions of the partial fractions. (See Arfken [1] and
Taylor and Mellott [10] for references to the Heaviside expansion.)
Consider, ﬁrst, the case of a partial fraction that contains a distinct (unrepeated) realvalued root λ. The expansion is
c
= c{1 + z/λ + (z/λ)2 + · · · }.
1 − z/λ (9) For this to converge for all z  ≤ 1, it is necessary and sufﬁcient that λ > 1; and this
is necessary and sufﬁcient for the satisfaction of the BIBO condition which is that the
impulse response function must be absolutely summable:
ψj  < ∞. (10) j Here, we are setting ψj = (1/λ)j . For a proof of that (10) is the BIBO condition for
an LTI system, see, for example, Kac [6] or Pollock [7].
Next, consider the case where a(z ) has a distinct pair of conjugate complex roots λ and
λ∗ . These will come from a partial fraction with a quadratic denominator:
cz + d
κ
κ∗
.
=
+
(z − λ)(z − λ∗ )
z − λ z − λ∗ (11) It can be seen that κ = (cλ + d)/(λ − λ∗ ) and κ∗ = (cλ∗ + d)/(λ∗ − λ) are also
conjugate complex numbers.
The expansion of (9) applies to complex roots as well as to real roots:
c
c∗
+
1 − z/λ 1 − z/λ∗ = c 1 + z/λ + (z/λ)2 + · · ·
+c∗ 1 + z/λ∗ + (z/λ∗ )2 + · · ·
4 Im
i −1 1 Re −i Figure 2: The pole–zero diagram for the transfer function b(z −1 )/a(z −1 ) corresponding to the impulse response function of Figure 1. The poles are marked by crosses and
the zeros by circles.
∞ = z t (cλ−t + c∗ λ∗−t ). (12) t=0 The various complex quantities can be represented in terms of exponentials:
λ
c = κ−1 e−iω ,
= ρe−iθ , λ∗
c∗ = κ−1 eiω ,
= ρeiθ . (13) Then, the generic term of the expansion becomes
z t (cλ−t + c∗ λ∗−t ) = z t ρκt ei(ωt−θ) + e−i(ωt−θ)
= z t 2ρκt cos(ωt − θ). (14) The expansion converges for all z  ≤ 1 if and only if κ < 1, which is a condition on
the modulus of the complex number λ. But, κ = λ−1  = λ−1 ; so it is conﬁrmed
that the necessary and sufﬁcient condition for convergence is that λ > 1.
Finally, consider the case of a repeated root with a multiplicity of n. Then, a binomial
expansion is available that gives
1
n(n + 1)
z
=1+n +
n
(1 − z/λ)
λ
2! 2 z
λ + n(n + 1)(n + 2)
3! z
λ 3 + · · · . (15) If λ is real, then λ > 1 is the condition for convergence. If λ is complex, then it can
be combined with the conjugate root in the manner of (13) to create a trigonometric
function; and, again, the condition for convergence is that λ > 1.
5 This result can be understood by regarding the LHS of (15) as a representation of n
transfer functions in series, each of which fulﬁls the BIBO condition.
The general conclusion is that the transfer function is stable if and only if all of the roots
of the denominator polynomial a(z ), which are described as the poles of the transfer
function, lie outside the unit circle in the complex plane.
It is helpful to represent the poles of the transfer function graphically by showing their
locations within the complex plane together with the locations of the roots of the numerator polynomial, which are described as the zeros of the transfer function.
It is more convenient to represent the poles and zeros of b(z −1 )/a(z −1 ), which are the
reciprocals of those of b(z )/a(z ). For a stable and invertible transfer function, these
must lie within the unit circle. This recourse has been adopted for Figure 2, which
shows the pole–zero diagram for the transfer function that gives rise to Figure 1. THE RESPONSE TO A COSINE
One must also consider the response of the transfer function to a simple sinusoidal
signal. Any ﬁnite data sequence can be expressed as a sum of discretely sampled
sine and cosine functions with frequencies that are integer multiples of a fundamental
frequency that produces one cycle in the period spanned by the sequence. The ﬁnite
sequence is to be regarded as a single cycle within a inﬁnite sequence, which is the
periodic extension of the data.
Consider, therefore, the consequences of mapping the signal sequence {xt = cos(ωt)}
through the transfer function with the coefﬁcients {ψ0 , ψ1 , . . . }. The output is
ψj cos ω [t − j ] . y (t) = (16) j By virtue of the trigonometrical identity cos(A − B ) = cos A cos B + sin A sin B , this
becomes
y (t) = ψj cos(ωj ) cos(ωt) +
j ψj sin(ωj ) sin(ωt)
j = α cos(ωt) + β sin(ωt) = ρ cos(ωt − θ), (17) where
α = ψj cos(ωj ),
j ρ2 = α2 + β 2 β= ψj sin(ωj ),
j and θ = tan−1 β
.
α (18) It can be seen that the transfer function has a twofold effect upon the signal. First, there
is a gain effect whereby the amplitude of the sinusoid is increased or diminished by the
factor ρ. Then, there is a phase effect whereby the peak of the sinusoid is displaced by
6 a time delay of θ/ω periods. The frequency of the output is the same as the frequency
of the input, which is a fundamental feature of all linear dynamic systems.
Observe that the response of the transfer function to a sinusoid of a particular frequency
is akin to the response of a bell to a tuning fork. It gives very limited information
regarding the characteristics of the system. To obtain full information, it is necessary
to excite the system over the whole range of frequencies. SPECTRAL DENSITY AND THE FREQUENCY RESPONSE
In a discretetime system, there is a problem of aliasing whereby signal frequencies
(i.e. angular velocities) in excess of π radians per sampling interval are confounded
with frequencies within the interval [0, π ]. To understand this, consider a cosine wave
of unit amplitude and zero phase with a frequency ω in the interval π < ω < 2π that is
sampled at unit intervals. Let ω ∗ = 2π − ω . Then
cos(ωt) = cos (2π − ω ∗ )t
= cos(2π ) cos(ω ∗ t) + sin(2π ) sin(ω ∗ t)
= cos(ω ∗ t); (19) which indicates that ω and ω ∗ are observationally indistinguishable. Here, ω ∗ ∈ [0, π ]
is described as the alias of ω > π .
The maximum frequency in discrete data is π radians per sampling interval and, as
the Shannon–Nyquist sampling theorem indicates, aliasing is avoided only if there are
at least two observations in the time that it takes the signal component of highest frequency to complete a cycle. In that case, the discrete representation will contain all
of the available information on the system. (The classical article of Shannon [9] that
conveys this result is readily available in its reprinted form.)
Any stationary stochastic process deﬁned over the doubly inﬁnite set of positive and
negative integers can be expressed as a weighted combination of the nondenumerable
inﬁnity of sines and cosines that have frequencies in Nyquist interval [0, π ]. Thus, if xt
is an element of such a process, then it can be represented by
π xt = π cos(ωt)dA(ω ) + sin(ωt)dB (ω ) =
0 −π eiωt dZ (ω ). (20) This is commonly described as the spectral representation of the process generating
xt . Here, dA(ω ) and dB (ω ) are the inﬁnitesimal increments of stochastic functions,
deﬁned on the frequency interval, that are everywhere continuous but nowhere differentiable. Moreover, it is assumed that the increments dA(ω ) and dB (ω ) are uncorrelated
with each other and with preceding and succeeding increments. (See Pollock [7] or
Priestley [8] for fuller accounts.)
The concise expression on the RHS of (20) entails the following deﬁnitions:
dZ (ω ) = dA(ω ) − idB (ω )
2 and dZ (−ω ) = dZ ∗ (ω ) =
7 dA(ω ) + idB (ω )
. (21)
2 The expression in terms of sines and cosines can be recovered via the identities
eiωt = cos(ωt) + i sin(ωt) and e−iωt = cos(ω ) − i sin(ωt). (22) 6
4
2
0
−π −π/2 0 π/2 π Figure 3: The gain of the transfer function depicted in Figures 1 and 2.
The spectral density function f (ω ) of the process is deﬁned by
E {dZ (ω )dZ ∗ (ω )} = E {dA2 (ω ) + dB 2 (ω )} = f (ω )dω. (23) The quantity f (ω )dω is the power or the variance of the process that is attributable to
the elements in the frequency interval [ω, ω + dω ]; and the integral of f (ω ) over the
frequency range [−π, π ] is the overall variance of the process. A whitenoise process
2
{εt } of independently and identically distributed elements, with a variance of σε , has
2
a uniform spectral density function fε (ω ) = σε /(2π ).
Let {ψ0 , ψ1 , . . . } be the impulse response of the transfer function. Then the effects
of the transfer function upon the spectral elements of the process deﬁned by (20) are
shown by the equation
yt ψj x(t − j ) = =
j ω j ψj e−iωj dZx (ω ). eiωt = eiω(t−j ) dZx (ω ) ψj ω (24) j These effects are summarised by the complexvalued frequencyresponse function
ψj e−iωj = ψ (ω )e−iθ(ω) , ψ (ω ) = (25) The ﬁnal expression, which is in polar form, entails the following deﬁnitions:
∞ ψ (ω )2 = ∞ 2 ψj cos(ωj )
j =0 +
j =0 8 2 ψj sin(ωj ) θ(ω ) = arctan ψj sin(ωj )
ψj cos(ωj ) (26) . The two components of the frequency response are the amplitude response or the gain
ψ (ω ) and the phase response θ(ω ). Their deﬁnitions subsume those of (18). These
two functions of ω , which are circular or periodic, are plotted in Figures 3 and 4 over
the interval [−π, π ]. π −π
−π −π/2 0 π/2 π Figure 4: The phase plot to accompany Figure 3.
In general, the phase response does not relate in any simple way to the lags of the
transfer function relationship that are perceptible via the impulse response function.
An exception concerns the transfer function ψ (z ) = z that imposes a delay of one
period. Then, the phase response is a line of unit slope passing through the origin,
rising to π when ω = π and falling to −π when ω = −π ; and the absolute time delay
that is imposed on all frequencies is τ = θ(ω )/ω = 1.
The frequency response is just the discretetime Fourier transform of the impulse response function. Therefore, the quantities of (25) and (26) can be expressed in terms
of the z transform of the functions with z = e−iω . They can be obtained in this way
from the following expressions:
ψj z j , ψ (z ) = ψ (z )2 = ψ (z )ψ (z −1 ), θ(z ) = Arg{ψ (z )}. (27) j In that case, there is a minor abuse of notation when we write ψ (ω ) in place of ψ (e−iω ).
As ω progresses from −π to π , or, equally, as z = e−iω travels around the unit circle in
the complex plane, the frequencyresponse function deﬁnes a trajectory that becomes
a closed contour when ω reaches π .
The points on the trajectory are characterised by their polar coordinates. These are the
modulus ψ (ω ), which is the length of the radius vector joining ψ (ω ) to the origin, and 9 Im Re Figure 5: The path described in the complex plane by the frequency response function
corresponding to the gain and phase functions of Figures 3 and 4. The trajectory originates, when ω = 0, in the point on the real axis marked by a dot and it travels in the
direction of the arrow, of which the tip is reached when ω = π/4.
the argument Arg{ψ (ω )} = θ(ω ) which is the (anticlockwise) angle in radians which
the radius makes with the positive real axis. Figure 5 provides an illustration.
The spectral density fy (ω ) of the output process y (t) is given by
fy (ω )dω ∗
= E {dZy (ω )dZy (ω )}
∗
= ψ (ω )ψ ∗ (ω )E {dZx (ω )dZx (ω )} = ψ (ω )2 fx (ω )dω. (28) When the input sequence is a whitenoise process with a uniform spectral density func2
tion fε (ω ) = σε /(2π ), this becomes the spectral density function of an autoregressive
movingaverage (ARMA) process. In the notation of the z transform, the autocovariance generating function of the ARMA process is
2
γ (z ) = σε b(z )b(z −1 )
2
= σε ψ (z )ψ (z −1 ).
a(z )a(z −1 ) (29) Setting z = e−iω and dividing by 2π gives the spectral density function in the form of
fy (ω ) = γ (e−iω )
.
2π (30) In calculating this, we would evaluate the numerator and the denominator of γ (e−iω )
separately. Conclusion
The theory of linear timeinvariant transfer functions is part of the basic grammar of systems analysis. Whereas it ﬁnds numerous applications in electrical
10 and mechanical engineering, it is also used in statistical timeseries analysis in
connection with stationary stochastic processes.
In circumstances where the assumption of stationarity is unsustainable, linear
dynamic models need to be replaced by more sophisticated models. The linear
theory provides a point of departure for such developments. 11 References
[1] Arfken, G. Mathematical Methods for Physicists, Third Edition. Academic Press,
San Diego, California, 1986.
[2] Dhrymes, P.J. Distributed Lags: Problems of Estimation and Formulation. HoldenDay, San Francisco 1971.
[3] Heaviside, O. Electrical Papers. American Mathematical Society, 1970.
[4] Heaviside, O. Electromagnetic Theory. E. Benn, London 1925, reprinted by American Mathematical Society, 1971.
[5] Jury, E.I. Theory and Application of the zTransform Method. John Wiley and Sons,
New York, 1964.
[6] Kac, R. Introduction to Digital Signal Processing. McGrawHill Book Co., New
York, 1982.
[7] Pollock, D.S.G. A Handbook of TimeSeries Analysis, Signal Processing and Dynamics. Academic Press, London 1999.
[8] Priestley, M.B. Spectral Analysis and Time Series, Academic Press, London, 1981.
[9] Shannon, C.E. Communication in the Presence of Noise, Proceedings of the
Institute of Radio Engineers, 37, 10–21, reprinted in Proceedings of the IEEE, 86,
447–457, 1998.
[10] Taylor, F. and Mellott, J. Handson Digital Signal Processing. McGrawHill, New
York, 1998. CrossReferences
Autoregressive process, Discrete Fourier transform, Filtering for time series
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This note was uploaded on 03/02/2012 for the course EC 7087 taught by Professor D.s.g.pollock during the Fall '11 term at Queen Mary, University of London.
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