Presentation on theme: "AMI 4622 Digital Signal Processing"— Presentation transcript:
1 AMI 4622 Digital Signal Processing Unit 5 The Z Transform
2 Introduction What is the z-transform? enableservice('automationserver',true)The Z Transform is an invaluable tool for:RepresentingAnalysing Discrete Time Signals and SystemsDesigningIt plays a similar role in discrete time systems to that which the Laplace transform plays in continuous time systemsInfer the degree of system stabilityVisualise the system frequency response
3 Discrete Time Signals and Systems A discrete-time system is represented by a sequence of numbers x(n), x(nT), xnA 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 additivityObeys the rules of superposition
4 The z-transform Laplace – analyse the behaviour of electrical circuits z-transform – analyse the behaviour of discrete systemsFourier and z-transforms are related through
5 The z-transformThe z-transform of a sequence x(n) which is valid for all n, is defined as:where z is a complex variableIn 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).
10 The z-transform ROC Region of convergence (everywhere outside of the unit circle)|z| > 1
11 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
12 The z-transformPlot the signal, determine the z-transform, and the region of convergence for the given expressions
14 The inverse z-transform Power SeriesFirst expand the series in either descending powers of z, or ascending powers of z-1Then perform long divisionPartial fractionsExpand into a sum of simple partial fractionsThe inverse z-transform of each partial fraction is then obtained from tables
15 Power series methodExample X(z) = z / (z – 0.5)Self Test Solution…
25 Comparison of the inverse z-transforms Power SeriesDoes not lead to a closed form solutionSimple and lends itself to computer implementationBecause 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.
26 Comparison of the inverse z-transforms Partial FractionsAdvantage is it leads to a closed form solutionDisadvantage there is a need to factorise the denominator polynomialPreferred over residue method for generating the coefficients of parallel structures
27 Comparison of the inverse z-transforms Residue methodAdvantage is it leads to a closed form solutionDisadvantage there is a need to factorise the denominator polynomialWidely used in the analysis of quantisation errors in discrete-time systems
28 Properties of the z-transform LinearityDelays or ShiftsConvolutionDifferentiationRelationship with the Laplace transform
29 Properties of the z-transform Relationship with the Laplace transformLaplace variableletthenand so
30 Properties of the z-transform Show Movie – Laplace z-transform relationship
31 Properties of the z-transform Relationship with the Laplace transformexample for zeros at -0.5+j j j …….exp(-0.5)exp(j0.7071) etc
32 Applications of the z-transform in signal processing Pole-zero description of discrete time systemsFrequency response estimationGeometric evaluation of frequency responseDirect computer evaluation of frequency responseFrequency response estimation via the FFTDetermine the difference equationsObtain the impulse response estimation
33 Pole-zero description of discrete time systems ifthen
34 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.
35 Pole-zero description of discrete time systems Solution:try help zplane and help rootsin Matlab
36 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,j0.7071-0.50.50.25,j
37 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,j0.7071-0.50.50.25,j
38 Frequency response estimation We often need to evaluate the frequency response of a discrete-time systemFor 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…………..
39 Frequency response estimation FIR Filter frequency responsePass BandPass BandStop Band
40 Frequency response estimation The frequency response can be readily obtained from the z-transformTo do this we setThis evaluates the z-transform around the unit circleFrom this we obtain the Fourier transform of the system
41 Frequency response estimation Letzplane_freq_resp.m
42 Geometric evaluation of frequency response This method allows us to estimate the frequency response of a system based on it’s pole-zero diagramTherefore we have to express the z-transform in terms of the poles and zeros of the system and letThe magnitude response is obtained fromThe phase response is obtained fromExample yn = xn + xn ExampleSlide42.m
43 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 yn-1
44 Geometric evaluation of frequency response Solution:zplane_freq_resp_solution.mandExampleSlide43.m
45 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
46 Geometric evaluation of frequency response Do exercise 5.5
47 Direct computer evaluation of frequency response The Geometric method gives a feel of the frequency responseBut can be tedious if the precise response is required at many frequenciesDifficult to find the pole and zero locations for high order filtersThe direct method shown next, makes the substitution directly into the transfer function and then to evaluate the resulting expression.
48 Direct computer evaluation of frequency response Class: Do given example using Matlab
49 Frequency response estimation via the FFT The FFT can be used to evaluate the frequency response of a discrete-time sequenceThis can be done directly for an FIR filterFrequencyResponseEstimationViaFFT.mFor IIR systems it is necessary to first obtain the impulse response of the system for example, using the power series method
50 Frequency units used in discrete time systems Frequency units in the z-plane
51 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 systemThis only applies to systems with an impulse response of infinite durationIf the impulse response is of finite length then it is obvious that the system will always be stableFor an unstable system the impulse response will increase indefinitely with timeStabilityTest.m
52 Difference equationsThe difference equation specifies the operations that must be performed on the input data.The z-transform delay operatorWe can easily move between discrete time and the transfer function....discrete-time transfer function
53 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 finiteIf 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
54 Impulse response estimation We often need to determine the impulse response of a discrete-time systemThe impulse response of a FIR is required to implement the systemThe impulse response of a IIR is required to for stability analysisThe 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 responseIf the z-transform is expressed as a ratio of polynomials (IIR) then the IZT methods shown earlier must be used.
55 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)
56 Impulse response estimation ExampleFind the impulse response of the discrete-time filter characterised by the following transfer function.1) by using the power series method2) by applying a unit impulse to the system