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# TRANSFER - Article type Focus Article Transfer Functions...

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Article type: Focus Article Transfer Functions Article ID Stephen Pollock University of Leicester Keywords Impulse response, Frequency response, Spectral density Abstract In statistical time-series 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 time-invariant (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 time-series 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 pre-sample 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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TRANSFER - Article type Focus Article Transfer Functions...

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