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lecture_13 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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V + C 1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 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) Introduction to Time-Domain Digital Signal Processing Consider a continuous-time filter C o n t i n u o u s f ( t ) s y s t e m y ( t ) ( h ( t ) , H ( s ) ) such as simple first-order RC high-pass filter: - R v o described by a transfer function H ( s ) = RCs RCs + 1 . The ODE describing the system is τ d y d t + y = τ d f d t 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 f ( t ) taken at intervals Δ T . f n a l g D o S r i t P h m y n 1 copyright c D.Rowell 2008 13–1
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If we use a backwards-difference numerical approximation to the derivatives, that is d x ( x ( n Δ T ) x (( n 1)Δ T ) d t Δ T and adopt the notation y = y ( n Δ T ), and let a = τ/ Δ T , n a ( y n y n 1 ) + y n = a ( f 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 sample- by-sample implementation of digital high-pass filter derived from the continuous prototype above. Note that The “fidelity” of the approximation depends on Δ T , and becomes more accurate when Δ T 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 i =1 i =0 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 M y n = b i f n i i =0 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 i =0 only the current input value is used. This filter is a recursive filter with an infinite- impulse-response (IIR), and is known as an auto-regressive (AR) filter, or an all-pole filter.
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