2.161 Signal Processing: Continuous and Discrete Fall 2008

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1 MIT OpenCourseWare Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit:

2 Massachusetts Institute of Technology Department of Mechanical Engineering Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 13 1 Reading: Proakis & Manolakis, Chapter 3 The z-transform) Oppenheim, Schafer & Buck, Chapter 3 The z-transform) 1 Introduction to Time-Domain Digital Signal Processing Consider a continuous-time filter such as simple first-order RC high-pass filter: + JE K K I B J I O I JA O J D J 0 I L described by a transfer function The ODE describing the system is Hs) RCs RCs +1. τ dy dt + y τ df dt where τ RC is the time constant. Our task is to derive a simple discrete-time equivalent of this prototype filter based on samples of the input ft) taken at intervals ΔT. B, 5 2 C HEJD O 1 copyright c D.Rowell

3 If we use a backwards-difference numerical approximation to the derivatives, that is dx xnδt ) xn 1)ΔT ) dt ΔT and adopt the notation y ynδt ), and let a τ/δt, n ay n y n 1 )+ y n af n f n 1 ) and solving for y n a a a y n y n 1 + f n f n 1 1+ a 1+ a 1+ a which is a first-order difference equation, and is the computational formula for a sampleby-sample implementation of digital high-pass filter derived from the continuous prototype above. Note that The fidelity of the approximation depends on ΔT, and becomes more accurate when ΔT τ. At each step the output is a linear combination of the present and/or past samples of the output and input. This is a recursive system because the computation of the current output depends on prior values of the output. In general, regardless of the design method used, a LTI digital filter implementation will be of a similar form, that is N M y n a i y n i + b i f n i i1 i0 where the a i and b i are constant coefficients. Then as in the simple example above, the current output is a weighted combination of past values of the output, and current and past values of the input. If a i 0 for i 1...N, so that y n M i0 b i f n i The output is simply a weighted sum of the current and prior inputs. Such a filter is a non-recursive filter with a finite-impulse-response FIR), and is known as a moving average MA) filter, or an all-zero filter. If b i 0 for i 1...M, so that N y n a i y n i + b 0 f n i0 only the current input value is used. This filter is a recursive filter with an infiniteimpulse-response IIR), and is known as an auto-regressive AR) filter, or an all-pole filter. 13 2

4 With the full difference equation N M y n a i y n i + b i f n i i1 i0 the filter is a recursive filter with an infinite-impulse response IIR), and is known as an auto-regressive moving-average ARMA) filter. 2 The Discrete-time Convolution Sum For a continuous system + JE K K I B J I O I JA O J D J 0 I the output yt), in response to an input ft), is given by the convolution integral: yt) fτ)ht τ)dτ 0 where ht) is the system impulse response. For a LTI discrete-time system, such as defined by a difference equation, we define the pulse response sequence {hn)} as the response to a unit-pulse input sequence {δ n }, where { 1 n 0 δ n 0 B, 5 2 C HEJD D O If the input sequence {f n } is written as a sum of weighted and shifted pulses, that is f n f k δ n k k then by superposition the output will be a sequence of similarly weighted and shifted pulse responses y n f k h n k k which defines the convolution sum, which is analogous to the convolution integral of the continuous system. 3 The z-transform The z-transform in discrete-time system analysis and design serves the same role as the Laplace transform in continuous systems. We begin here with a parallel development of both the z and Laplace transforms from the Fourier transforms. 13 3

5 The Laplace Transform 1) We begin with causal ft) and find its Fourier transform Note that because ft) is causal, the integral has limits of 0 and ): F jω) 0 ft)e jωt dt 2) We note that for some functions ft) for example the unit step function), the Fourier integral does not converge. 3) We introduce a weighted function and note wt) ft)e σt lim wt) ft) σ 0 The effect of the exponential weighting by e σt is to allow convergence of the integral for a much broader range of functions ft). 4) We take the Fourier transform of wt) The transform 1) We sample ft) at intervals ΔT to produce f t). We take its Fourier transform and use the sifting property of δt)) to produce F jω) f n e jnωδt n0 2) We note that for some sequences f n for example the unit step sequence), the summation does not converge. 3) We introduce a weighted sequence { } {w n } f n r n and note lim {w n } {f n } r 1 The effect of the exponential weighting by r n is to allow convergence of the summation for a much broader range of sequences f n. 4) We take the Fourier transform of w n W jω) F jω σ) ) ft)e σt e jωt dt 0 ft)e σ+jω) dt 0 W jω) F jω r) fn r ) n e jnωδt n0 fn re jωδt ) n n0 and define the complex variable s σ + jω so that we can write F s) F jω σ) 0 ft)e st dt F s) is the one-sided Laplace Transform. Note that the Laplace variable s σ + jω is expressed in Cartesian form. and define the complex variable z re jωδt so that we can write F z) F jω r) f n z n n0 F z) is the one-sided -transform. Note that z re jωδt is expressed in polar form. 13 4

6 The Laplace Transform contd.) 5) For a causal function ft), the region of convergence ROC) includes the s-plane to the right of all poles of F jω). I F A The transform contd.) 5) For a right-sided causal) sequence {f n } the region of convergence ROC) includes the z-plane at a radius greater than all of the poles of F z). F A 4 + : : I : : K E J? E H? A 6) If the ROC includes the imaginary axis, the FT of ft) is F jω): F jω) F s) sjω 7) The convolution theorem states ft) gt) fτ)gt τ)dτ F s)gs) 8) For an LTI system with transfer function Hs), the frequency response is Hs) sjω HjΩ) if the ROC includes the imaginary axis. L 6) If the ROC includes the unit circle, the DFT of {f n }, n 0, 1,...,N 1. is {F m } where F m F z) ze jωm F e jωm ), where ω m 2πm/N for m 0, 1,...,N 1. 7) The convolution theorem states {f n } {g n } f m g n m F z)gz) m 8) For a discrete LSI system with transfer function Hz), the frequency response is Hz) ze jω He jω ) ω π if the ROC includes the unit circle. From the above derivation, the -transform of a sequence {f n } is F z) f n z n n where z r e j ω is a complex variable. For a causal sequence f n 0 for n <0, the transform 13 5

7 can be written F z) f n z n n0 Example: The finite sequence {f 0,...,f 3 } {5, 3, 1, 4} has the z-transform F z) 5z 0 +3z 1 z 2 +4z 3 The Region of Convergence: For a given sequence, the region of the z-plane in which the sum converges is defined as the region of convergence ROC). In general, within the ROC fn r n < n and the ROC is in general an annular region of the z-plane: F A H H 4 +.? L A H C A I B H H H H a) The ROC is a ring or disk in the z-plane. b) The ROC cannot contain any poles of F z). c) For a finite sequence, the ROC is the entire z-plane with the possible exception of z 0 and z. d) For a causal sequence, the ROC extends outward from the outermost pole. e) for a left-sided sequence, the ROC is a disk, with radius defined by the innermost pole. f) For a two sided sequence the ROC is a disk bounded by two poles, but not containing any poles. g) The ROC is a connected region. z-transform Examples: In the following examples {u n } is the unit step sequence, { 0 n< 0 u n 1 n 0 and is used to force a causal sequence. 13 6

8 1) {f n } {δ n } the digital pulse sequence) From the definition of F z): F z) 1z 0 1 for all z. 2) {f n } {a n u n } since n {a } F z) F z) a ) az 1 n n z n n0 n0 1 z 1 az 1 z a n 1 x for x < 1. n0 1 x for z > a. 3) {f n } {u n } the unit step sequence). from 2) with a 1. 1 z F z) z n for z < 1 1 z 1 z 1 n0 4) {f n } { e bn u n }. F z) e bn z n e b z 1) n from 2) with a e b. n 5) {f n } { e b }. n0 n0 { } e bn 1 z F z) for z > e b. 1 e b z 1 z e bn F z) 0 e b z ) n + e b z 1) n 1 n n e b z e b z 1 Note that the item f 0 1 appears in each sum, therefore it is necessary to subtract 1. { } e b n F z) 1 e 2b b for e b < z < e. 1 e b z)1 e b z 1 ) 13 7

9 F A A > A > : : 4 + 6) {f n } { e jω0n u n } {cosω 0 n)u n } j {sinω 0 n)u n }. F z) {cosω 0 n)u n } j {sinω 0 n)u n } From 1) F z) 1 1 e jω 0 z 1 for z > 1 1 cosω 0 )z 1 j sinω 0 ) 1 2 cosω 0 )z 1 + z 2 z 2 cosω 0 )z j sinω 0 )z 2 z 2 2 cosω 0 )z + 1 and therefore {cosω 0 n)u n } z 2 cosω 0 )z z 2 2 cosω 0 )z + 1 for z > 1 {sinω 0 n)u n } sinω 0 )z 2 z 2 2 cosω 0 )z + 1 for z > 1 Properties of the z-transform: Refer to the texts for a full description. We simply summarize some of the more important properties here. a) Linearity: a {f n } + b {g n } af z) + bgz) ROC: Intersection of ROC f and ROC g. b) Time Shift: {f z m n m } F z) ROC: ROC f except for z 0 if k < 0, or z if k > 0. If g n f n m, Gz) f n m z n n k f k z k+m) z m F z). This is an important property in the analysis and design of discrete-time systems. We will often have recourse to a unit-delay block: 13 8

10 B 7 E J, A O O B c) Convolution: {f n } {g n } F z)gz) ROC: Intersection of ROC f and ROC g. where {f n } {g n } f k g n k is the convolution sum. Let Y z) k ) yn z n f k g n k z n n n k ) fk gn k z n k) z k f k z k gm z m k n k m F z)gz) d) Conjugation of a complex sequence: { f n } F z) ROC: ROC f e) Time reversal: {f n } F 1/z) ROC: 1 1 < z < r 1 r 2 where the ROC of F z) lies between r 1 and r 2. e) Scaling in the z-domain: {a n f F a 1 n } z) ROC: a r 1 < z < a r 2 where the ROC of F z) lies between r 1 and r 2. e) Differentiation in the z-domain: df z) {nf n } z dz ROC: r 2 < z < r 1 where the ROC of F z) lies between r 1 and r