Discrete_time_systems

Discrete_time_systems - Discrete-Time Systems Linear...

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Discrete-Time Systems Linear Systems and Signals – Lectures 17-20 Dr. J. K. Aggarwal The University of Texas at Austin
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Discrete-Time Signals Signals defined over a continuous range of t are continuous-time signals Denoted by Signals defined only at discrete instants of time are discrete-time signals Denoted by where n is an integer Usually, , for all n are further simplified to () , () f t yt xt , , n nn f t yt xt 0 12 ,, n t tt t  1 t tT + −= , , f nT y nT x nT [] , [] fn yn xn
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Discrete-Time Systems . ..contd Discrete-Time signals arise naturally in situations like national income models, stock market, etc. The also arise as a result of sampling continuous-time signals in digital filtering 0 1 2 3 4 5 n [] () n fn or f nT or f t
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Signal Processing Processing a continuous-time signal by a discrete-time system Bank deposit/Interest example deposit at n th instant account balance interest per unit per period Discrete-time System C/D D/C x ( t ) y ( t ) Continuous to Discrete Discrete to Continuous [] xn yn = = r = [ ] [1 ] ][ ] ( 1 )[ 1 ] [] ryn r yn = −+ = +
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Signal Processing ...contd Figure shows a realization of the equation - Delaying it by T, we generate and a=1+r a Delay [] xn yn [ 1] [ 1]
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Signal Processing ...contd Moving Average example Input is x [ n ] , Output is y [ n ] You would like the average of the past 4 values of f Delay Delay Delay c 0 c 1 c 2 c 3 [] xn [ 1] [ 2] [ 3] yn
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Signal Processing ...contd If we choose we are emphasizing the most recent value 01 2 3 3 0 [] [ 1 ] [ 2 ] [ 3 ] [ ] N N yn c xn cxn c xn yn c xn N = = + −+ + = 0 1 23 11 1 24 8 c c cc = = =
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Realization of Discrete-Time Systems The examples show the basic elements required for the realization of discrete-time systems: Time delays Scalar multipliers Adders Discrete-time systems or digital filters are used to implement such systems Advantages: Precision and stability Flexibility and variety Other advantages listed on page 268, Lathi
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Difference Equations Two examples - Bank Balance and Moving Average - give rise to difference equations Difference equations may be written in two forms: Using delay terms, such as Using advance terms, such as Both forms are useful [ 1], [ 2], [ 2], yn xn −− [ 2], [ 2], ++
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Difference Equations ...contd For causality, In this way, present depends upon past inputs and outputs. So, the general case is 11 [ ] [ 1] [ [ ] [ ] [ [ NN NM N N yn N ayn N a yn b xn M b b xn −+ + + + + ++ = + + + + MN [] yn N + 01 1 [ ] [ [ [ ] [ [ bxn N bxn N + + + + = + + + + Advance Operator Form
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Difference Equations ...contd Replacing n by n-N gives rise to You can go from the Advance operator form to Delay operator form or vice versa 11 01 1 [ ] [ 1] [ [ ] [ [ [ ] NN yn ayn a yn N bxn bxn b xn N + −+ + ++ = + + Delay Operator Form
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Example 1 Example 3.9, Lathi, page 273 Substituting n = -2 Substituting n = -1 Substituting n = 0 … and so on [ 2] [ 1] 0.24 [ ] [ 2] 2 [ 1] yn xn +− + + = + [1 ] 2 [2 ] 1 [ 0 ]0 [ 1 [ ] yy x x n −= = = [0] 2 0.24(1) 0 0 1.76 y = +−= [1] 1.76 0.24(2) 1 0 2.28 y = = [2] 2.28 0.24(1.76) 2 2(1) 1.8576 y = =
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Example 1 ...contd
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This note was uploaded on 02/08/2011 for the course EE 313 taught by Professor Cardwell during the Spring '07 term at University of Texas.

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Discrete_time_systems - Discrete-Time Systems Linear...

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