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Z-transform

Z-transform - Chapter 4 ZTransform 4.1 Introduction In...

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Chapter 4 Z–Transform 4.1 Introduction In signal processing, the Z-transform converts a discrete time domain signal (a sequence of real numbers), into a complex frequency domain representa- tion. The Z-transform is to discrete time domain signals what the Laplace transform is to continuous time domain signals. 4.2 Definition If we discretise a time series with a constant sampling interval T , we can write the time series in the following manner: [ x k ] = x 0 , x 1 , x 2 , . . . , x k , . . . (4.1) where each x k represents the value (number) of the variable x ( t ) at time t = kT . We can represent this data in the following form: x 0 + x 1 z - 1 + x 2 z - 2 + . . . + x k z - k . . . (4.2) this form is a (symbolic) power series in the variable z - 1 . We can also write it in the following more compact form: Z [ x k ] = k =0 x k z - k . (4.3) This representation is called the Z–transform of the sequence x k . We write the Z–transform of x k as Z [ x k ] or X ( z ). We assume here the time-series begins at the time t = 0. Note that z only plays the role of a timestamp. Additionally, z is not a complex variable nor does it have to fulfil any criteria 1

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for convergence of the power series. The coefficients of the power series are simply the amplitudes of the time series at a given time. Mathematically, the Z–transform is the discrete counterpart to the contin- uum Laplace transform. It allows us to implement many complex operations a in simple way and to represent the characteristics of digital filters in com- pact form. If z is regarded as a complex variable, additional useful results from the complex analysis are available. Reference: In various texts the Z–transform is defined with a positive sign in the exponent, i.e. as the power series z k instead of z - k . (eg. Gub- bins 1 , Buttkus 2 ). Although both ways of writing are equivalent, the relationships with the Fourier and Laplace transforms, as well as the convergence criteria, will be defined differently. 4.2.1 Characteristics of the Z–transform In the following, x k and y k are discrete timeseries and X ( z ) and Y ( z ) as their respective Z–transforms. Linearity The Z–transform is a linear operation: αx k + βy k ←→ αX ( z ) + βY ( z ) . (4.4) Time shift What happens if we shift the values of the time series by a time nT ? Z [ x k - n ] = k x k - n z - k (4.5) or Z [ x k - n ] = z - n k x k - n z - ( k - n ) = z - n m x m z - m (4.6) thus Z [ x k - n ] = z - n Z [ x k ] . (4.7) We gave here a first interpretation of z - 1 ( z - 1 is a shift operator). The multiplication of the Z – transform of a time sequence x k by z - 1 corresponds 1 D. Gubbins, Time Series Analysis and Inverse Theory for Geophysicists, Cambridge (2004) 2 B. Buttkus, Spektralanalyse und Filtertheorie in der angwandten Geophysik, Springer Verlag, Berlin (1991) 2
to a delay in the time series around the time interval T . z - 1 is known as the unit delay operator .

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Z-transform - Chapter 4 ZTransform 4.1 Introduction In...

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