Chapter z Transforms 7 In the study of discrete time signal and systems we have thus far considered the time domain and the frequency domain The zdomain gives us a third representation All three domains are related to each other A special feature of the z transform is that for the signals and system of interest to us all of the analysis will be in terms of ratios of polynomials Working with these polynomials is relatively straight forward Definition of the z Transform Given a finite length signal x n the z transform is defined as X z N k 0 x k z k N 1 k x k z 7 1 k 0 where the sequence support interval is 0 N and z is any complex number This transformation produces a new representation of x n denoted X z Returning to the original sequence inverse z transform x n requires finding the coefficient associated with the nth power 1 of z ECE 2610 Signal and Systems 7 1 Definition of the z Transform Formally transforming from the time sequence n domain to the z domain is represented as z n Domain z Domain x n N z x k n k X z k 0 N x k z k k 0 A sequence and its z transform are said to form a z transform pair and are denoted z x n X z 7 2 In the sequence or n domain the independent variable is n In the z domain the independent variable is z Example x n n n 0 Using the definition X z N x k z k 0 k N k n 0 z k z n0 k 0 Thus z n n0 z ECE 2610 Signals and Systems n0 7 2 The z Transform and Linear Systems Example x n 2 n 3 n 1 5 n 2 2 n 3 By inspection we find that X z 2 3z Example X z 4 5z 2 z 1 3 5z 2z 2 2z 3 4 By inspection we find that x n 4 n 5 n 2 n 3 2 n 4 What can we do with the z transform that is useful The z Transform and Linear Systems The z transform is particularly useful in the analysis and design of LTI systems The z Transform of an FIR Filter We know that for any LTI system with input x n and impulse response h n the output is y n x n h n 7 3 We are interested in the z transform of h n where for an FIR filter h n M bk n k 7 4 k 0 ECE 2610 Signals and Systems 7 3 The z Transform and Linear Systems To motivate this consider the input n x n z n 7 5 The output y n is y n M M bk x n k k 0 M bk z n k k 0 n k bk z z k 0 M k 0 k n bk z z 7 6 The term in parenthesis is the z transform of h n also known as the system function of the FIR filter j Like H e was defined in Chapter 6 we define the system function as M H z bk z k M k 0 h k z k 7 7 k 0 The z transform pair we have just established is z H z h n M z bk n k k 0 M bk z k k 0 Another result similar to the frequency response result is n y n h n z H z z ECE 2610 Signals and Systems n 7 8 7 4 The z Transform and Linear Systems j Note if z e we in fact have the frequency response result of Chapter 6 The system function is an Mth degree polynomial in complex variable z As with any polynomial it will have M roots or zeros that is there are M values z 0 such that H z 0 0 These M zeros completely define the polynomial to within a gain constant scale factor i e H z b0 b1 z 1 1 bM z M 1 1 1 z1 z 1 z2 z 1 zM z z z1 z z2 z zM M z where z k k 1 M denote the zeros Example Find the Zeros of 1 1 h n n n 1 n 2 6 6 The z transform is 1 1 1 2 H z 1 z z 6 6 1 1 1 1 1 z 1 z 3 2 2 1 1 z z z 2 3 ECE 2610 Signals and Systems 7 5 Properties of the z Transform The zeros of H z are 1 2 and 1 3 The difference equation y n 6x n x n 1 x n 2 has the same zeros but a different scale factor proof Properties of the z Transform The z transform has a few very useful properties and its definition extends to infinite signals impulse responses The Superposition Linearity Property z ax 1 n bx 2 n aX 1 z bX 2 z 7 9 proof X z ax 1 n bx 2 n z n 0 N a x 1 n z 1 b n 0 N 1 x 2 n z 1 n 0 aX 1 z bX 2 z ECE 2610 Signals and Systems 7 6 Properties of the z Transform The Time Delay Property z 1 z n0 X z 1 N z x n 1 z X z 7 10 and x n n0 z 7 11 proof Consider X z 0 1 z N then x n N k n k k 0 0 n 1 n 1 N n N Let 1 Y z z X z 0 z 1 1 z 2 N z N 1 so y n 0 n 1 1 n 2 N n N 1 x n 1 Similarly n Y z z 0X z y n x n n0 ECE 2610 Signals and Systems 7 7 The z Transform as an Operator A General z Transform Formula We have seen that for a sequence x n having support interval 0 n N the z transform is X z N x n z n 7 12 n 0 This definition extends for doubly infinite sequences having support interval n to X z x n z n 7 13 n There will be discussion of this case in Chapter 8 when we deal with infinite impulse response IIR filters The z Transform as an Operator The z transform can be considered as an operator Unit Delay Operator x n x n ECE 2610 Signals and Systems Unit Delay z y n x n 1 1 7 8 The z Transform as an Operator In the case of the unit delay we observe that 1 y n z x n x n 1 7 14 unit delay operator 1 which is motivated by the fact that Y z z X z Similarly the filter y n x n x n 1 can …