Discrete-Time Systems a discrete-time system processes a given input sequence xn] to generates an output sequence yln] with more desirable roperties In most applications, the discrete-time system is a single-input, single-output system Discrete-timel System Input sequence Output sequence Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 1 Discrete-Time Systems • A discrete-time system processes a given input sequence x[n] to generates an output sequence y[n] with more desirable properties • In most applications, the discrete-time system is a single-input, single-output system: System Discrete− time x[n] y[n] Input sequence Output sequence
Discrete-Time Systems Examples 2-input, 1-output discrete-time systems Modulator. adder input, I-output discrete-time systems Multiplier, unit delay, unit advance tn-1] rn -1n vIn Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 2 Discrete-Time Systems: Examples • 2-input, 1-output discrete-time systems - Modulator, adder • 1-input, 1-output discrete-time systems - Multiplier, unit delay, unit advance
Discrete-Time Systems: Examples Accumulator-yn]=∑x(] ∑x]+x[n]=yn-1x[n The output yn at time instant n is the sum of the input sample x[n] at time instant n and the previous output yIn -l at time instant n-1. which is the sum of all previous input sample values from -oo to n-1 The system cumulatively adds,i.e,it accumulates all input sample values Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 3 Discrete-Time Systems: Examples • Accumulator - • The output y[n] at time instant n is the sum of the input sample x[n] at time instant n and the previous output at time instant which is the sum of all previous input sample values from to • The system cumulatively adds, i.e., it accumulates all input sample values = =− n y n x [ ] [] [ ] [ ] [ 1] [ ] 1 x x n y n x n n = + = − + − =− y[n −1] n −1, − n −1
Discrete-Time Systems. Examples Accumulator- Input-output relation can also be written in the form yn=∑x[(+∑x[( C=0 =y-1]+∑x[(ln≥0 C=0 The second form is used for a causal input sequence, in which case y-l is called the initial condition Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 4 Discrete-Time Systems:Examples • Accumulator- Input-output relation can also be written in the form • The second form is used for a causal input sequence, in which case is called the initial condition = + = − =− n y n x x 0 1 [ ] [ ] [ ] [ 1] [ ], 0 = − + = n y x y[−1] n 0
Discrete-Time Systems. Examples M-point moving-average system vn- 1 M M ∑x[n-k k=0 Used in smoothing random variations in data An application in denoising: Consider x团n]=s[n]+an], where sin is the signal corrupted by a noise dn Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 5 Discrete-Time Systems:Examples • M-point moving-average system - • Used in smoothing random variations in data • An application in denoising: Consider x[n] = s[n] + d[n], where s[n] is the signal corrupted by a noise d[n] = − − = 1 0 [ ] 1 [ ] M k x n k M y n
Discrete-Time Systems: Examples sEnf=2(n 0.9 dnf-fandom signal xn] 10 50 Time index n yIn Time index n Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 6 [ ] 2[ (0.9) ], d[n] - random signal n s n = n Discrete-Time Systems:Examples 0 10 20 30 40 50 0 1 2 3 4 5 6 7 Time index n Amplitude s[n] y[n]
Discrete-Time Systems: Examples Linear interpolation -employed to estimate sample values between pairs of adjacent sample values of a discrete-time sequence Factor-of-4 interpolation xnI x团n j 6789101112 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 7 Discrete-Time Systems:Examples • Linear interpolation - Employed to estimate sample values between pairs of adjacent sample values of a discrete-time sequence • Factor-of-4 interpolation
Discrete-Time Systems Examples Factor-of-2 interpolator yn=x小]+5(xn-1]+xn+) Factor-of-3 interpolator =x]+( 3xln-1]+x[n+2]) +4(xnn-2]+xn+1) 8 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 8 Discrete-Time Systems: Examples • Factor-of-2 interpolator - • Factor-of-3 interpolator - ( [ 1] [ 1]) 2 1 y[n] = xu [n]+ xu n − + xu n + ( [ 1] [ 2]) 3 1 y[n] = xu [n]+ xu n − + xu n + ( [ 2] [ 1]) 3 2 + xu n − + xu n +
Discrete-Time Systems Classification Linear Systems Shift-Invariant Systems Causal systems Stable systems Passive and Lossless systems Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 9 Discrete-Time Systems: Classification • Linear Systems • Shift-Invariant Systems • Causal Systems • Stable Systems • Passive and Lossless Systems
Linear Discrete-Time Systems Definition-If yin]is the output due to an input xin and y2n is the output due to an input x,n then for an input x[n]=axn+Bx2n the output is given b yIn]=ayIn]+By2ln Above property must hold for any arbitrary constants a and B, and for all possible inputs x[n]and x2[n Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 10 Linear Discrete-Time Systems • Definition - If is the output due to an input and is the output due to an input then for an input the output is given by • Above property must hold for any arbitrary constants and and for all possible inputs and [ ] y1 n [ ] x1 n [ ] x2 n [ ] 2 y n [ ] [ ] [ ] x n = x1 n + x2 n [ ] [ ] [ ] y n = y1 n + y2 n , [ ] x1 n [ ] x2 n