AMI 4622 Digital Signal Processing Unit 5 The Z Transform

Introduction What is the z-transform? enableservice('automationserver',true) The Z Transform is an invaluable tool for: Representing Analysing Discrete Time Signals and Systems Designing It plays a similar role in discrete time systems to that which the Laplace transform plays in continuous time systems Infer the degree of system stability Visualise the system frequency response

Discrete Time Signals and Systems A discrete-time system is represented by a sequence of numbers x(n), x(nT), xn A discrete-time system is essentially a mathematical algorithm which takes an input sequence x(n) and produces an output sequence y(n) A discrete-time system may be linear or nonlinear, time invariant or time varying. Linear time invariant (LTI) systems form an important class of systems used in DSP. A discrete-time system is linear if it: Has homogeneity (scaling) Has additivity Obeys the rules of superposition

The z-transform Laplace – analyse the behaviour of electrical circuits z-transform – analyse the behaviour of discrete systems Fourier and z-transforms are related through

The z-transform The z-transform of a sequence x(n) which is valid for all n, is defined as: where z is a complex variable In causal systems x(n) is only non-zero for n >= 0, so the above equation reduces to: ……(power series) The z-transform is a power series with an infinite number of terms and therefore may not converge. The region where the z-transform converges is known as the Region of Convergence (ROC) It is in this region where the value of X(z) is finite (in this case valid).

The z-transform ROC

The z-transform ROC

The z-transform ROC

The z-transform ROC

The z-transform ROC Region of convergence (everywhere outside of the unit circle) |z| > 1

The z-transform Region of Convergence Causal sequences of finite duration the z-transform converges everywhere except at z = 0. Causal infinite duration sequences the z-transform converges everywhere outside a circle bounded by the radius of the pole with the largest radius. For stable causal systems the ROC always encloses the circle of unit radius (important for the systems to have a frequency response) Table of common z-transforms

The z-transform Plot the signal, determine the z-transform, and the region of convergence for the given expressions

The z-transform And one last one!!!

The inverse z-transform Power Series First expand the series in either descending powers of z, or ascending powers of z-1 Then perform long division Partial fractions Expand into a sum of simple partial fractions The inverse z-transform of each partial fraction is then obtained from tables

Power series method Example 1 X(z) = z / (z – 0.5) Self Test Solution…

Power series method Try this yourself

Power series method Solution: (error 0.5 -> 0.5z) (try impz() in Matlab)

Partial fractions method Example 1 – X(z) contains simple first order poles Solution……..

Partial fractions method Example 1 – X(z) contains simple first order poles

Partial fractions method Example 2 – X(z) contains first order conjugate poles Lets try this Solution……….

Partial fractions method Example 2 – X(z) contains first order conjugate poles

Residue method The IZT is obtained by evaluating the contour integral.

Residue method Example Solution…..

Residue method Example

Comparison of the inverse z-transforms Power Series Does not lead to a closed form solution Simple and lends itself to computer implementation Because of its recursive nature care is required to minimise possible build up of numerical error when the number of data points in the IZT is large.

Comparison of the inverse z-transforms Partial Fractions Advantage is it leads to a closed form solution Disadvantage there is a need to factorise the denominator polynomial Preferred over residue method for generating the coefficients of parallel structures

Comparison of the inverse z-transforms Residue method Advantage is it leads to a closed form solution Disadvantage there is a need to factorise the denominator polynomial Widely used in the analysis of quantisation errors in discrete-time systems

Properties of the z-transform Linearity Delays or Shifts Convolution Differentiation Relationship with the Laplace transform

Properties of the z-transform Relationship with the Laplace transform Laplace variable let then and so

Properties of the z-transform Show Movie – Laplace z-transform relationship

Properties of the z-transform Relationship with the Laplace transform example for zeros at -0.5+j0.7071 0+j0.7071 +5+j0.7071 …….exp(-0.5)exp(j0.7071) etc

Applications of the z-transform in signal processing Pole-zero description of discrete time systems Frequency response estimation Geometric evaluation of frequency response Direct computer evaluation of frequency response Frequency response estimation via the FFT Determine the difference equations Obtain the impulse response estimation

Pole-zero description of discrete time systems if then

Pole-zero description of discrete time systems Express the following transfer function in terms of its poles and zeros and sketch the pole zero diagram.

Pole-zero description of discrete time systems Solution: try help zplane and help roots in Matlab

Pole-zero description of discrete time systems Determine the transfer function H(z) of a discrete time filter with the pole-zero diagram shown: 0.25,j0.7071 -0.5 0.5 0.25,j-0.7071

Pole-zero description of discrete time systems Determine the transfer function H(z) of a discrete time filter with the pole-zero diagram shown: Solution: 0.25,j0.7071 -0.5 0.5 0.25,j-0.7071

Frequency response estimation We often need to evaluate the frequency response of a discrete-time system For example, in the design of a discrete filter we need to examine the spectrum of the filter to ensure that the filter specification is satisfied…………..

Frequency response estimation FIR Filter frequency response Pass Band Pass Band Stop Band

Frequency response estimation The frequency response can be readily obtained from the z-transform To do this we set This evaluates the z-transform around the unit circle From this we obtain the Fourier transform of the system

Frequency response estimation Let zplane_freq_resp.m

Geometric evaluation of frequency response This method allows us to estimate the frequency response of a system based on it’s pole-zero diagram Therefore we have to express the z-transform in terms of the poles and zeros of the system and let The magnitude response is obtained from The phase response is obtained from Example yn = xn + xn-1 ExampleSlide42.m

Geometric evaluation of frequency response Example Determine using the geometric method, the frequency response at dc, the sampling frequency, of the causal discrete-time system with the following difference equation. yn = xn + xn-1 + 0.7071 yn-1

Geometric evaluation of frequency response Solution: zplane_freq_resp_solution.m and ExampleSlide43.m

Geometric evaluation of frequency response Determine using the geometric method, the frequency response at dc, Fs/4 and Fs/2, of the causal discrete-time system with the following z-transform

Geometric evaluation of frequency response Do exercise 5.5

Direct computer evaluation of frequency response The Geometric method gives a feel of the frequency response But can be tedious if the precise response is required at many frequencies Difficult to find the pole and zero locations for high order filters The direct method shown next, makes the substitution directly into the transfer function and then to evaluate the resulting expression.

Direct computer evaluation of frequency response Class: Do given example using Matlab

Frequency response estimation via the FFT The FFT can be used to evaluate the frequency response of a discrete-time sequence This can be done directly for an FIR filter FrequencyResponseEstimationViaFFT.m For IIR systems it is necessary to first obtain the impulse response of the system for example, using the power series method

Frequency units used in discrete time systems Frequency units in the z-plane

Stability considerations One stability criterion for a LTI system is that all bounded inputs produce bounded outputs - bounded input bounded output (BIBO) A LTI system said to be BIBO satisfies: where h(k) is the impulse response of the system This only applies to systems with an impulse response of infinite duration If the impulse response is of finite length then it is obvious that the system will always be stable For an unstable system the impulse response will increase indefinitely with time StabilityTest.m

Difference equations The difference equation specifies the operations that must be performed on the input data. The z-transform delay operator We can easily move between discrete time and the transfer function.... discrete-time transfer function

Difference equations The transfer function H(z) can then be obtained If the denominator coefficients are zero then this is…… referred to an finite impulse response (FIR) since the length of the expression is finite If at least one of the denominator coefficients are non-zero (at least one of the poles will be non-zero) then this is…… referred to as an infinite impulse response (IIR) system

Impulse response estimation We often need to determine the impulse response of a discrete-time system The impulse response of a FIR is required to implement the system The impulse response of a IIR is required to for stability analysis The impulse response of a discrete time system is defined as the inverse z-transform of the systems transfer function, H(z) If the z-transform is available as a power series……… then the coefficients of the z-transform give directly the impulse response If the z-transform is expressed as a ratio of polynomials (IIR) then the IZT methods shown earlier must be used.

Impulse response estimation The impulse response may also be viewed as the response of a discrete-time system to a unit impulse. ….this provides a simple method of computing h(n) (as well as another method of obtaining the IZT)

Impulse response estimation Example Find the impulse response of the discrete-time filter characterised by the following transfer function. 1) by using the power series method 2) by applying a unit impulse to the system