LECTURE 3c-ECC3107 200110

LECTURE 3c-ECC3107 200110 - Discrete-Time Systems Examples...

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Unformatted text preview: Discrete-Time Systems: Examples 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: Discrete − time System x [n ] 1 Input sequence • 2-input, 1-output discrete-time systems Modulator, adder • 1-input, 1-output discrete-time systems Multiplier, unit delay, unit advance y [n ] Output sequence 2 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Discrete-Time Systems: Examples Discrete- Discrete-Time Systems:Examples Discrete- n • Accumulator - y[n] = ∑ x[l] • Accumulator - Input-output relation can also be written in the form l = −∞ n −1 = ∑ x[l] + x[ n] = y[n − 1] + x[n] −1 3 • 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 y[n − 1] at time instant n − 1, which is the sum of all previous input sample values from − ∞ to n − 1 • The system cumulatively adds, i.e., it accumulates all input sample values l = −∞ l =0 n = y[−1] + ∑ x[l], n ≥ 0 l=0 • The second form is used for a causal input sequence, in which case y[−1] is called the initial condition 4 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Discrete-Time Systems:Examples Discrete- Discrete-Time Systems:Examples Discrete- • M-point moving-average system - • If there is no bias in the measurements, an improved estimate of the noisy data is obtained by simply increasing M • A direct implementation of the M-point moving average system requires M − 1 additions, 1 division, and storage of M − 1 past input data samples • A more efficient implementation is developed next M −1 1 y[ n] = M ∑ x[ n − k ] k =0 • Used in smoothing random variations in data • In most applications, the data x[n] is a bounded sequence • M-point average y[n] is also a bounded sequence 5 n y[n] = ∑ x[l] + ∑ x[l] l = −∞ 6 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 1 Discrete-Time Systems:Examples Discrete- Discrete-Time Systems:Examples Discrete- 1 ⎛ M −1 ⎜ ∑ x[ n − l] + M⎝ l =0 1⎛M ⎞ x[ n − M ] − x[ n − M ] ⎟ ⎠ ⎞ = ⎜ ∑ x[ n − l] + x[ n] − x[ n − M ] ⎟ M⎝ ⎠ l =1 M −1 ⎞ 1⎛ = ⎜ ∑ x[ n − 1 − l] + x[ n] − x[ n − M ] ⎟ M⎝ ⎠ l =0 Hence y[ n] = y[ n] = y[ n − 1] + • Computation of the modified M-point moving average system using the recursive equation now requires 2 additions and 1 division • An application: Consider x[n] = s[n] + d[n], where s[n] is the signal corrupted by a noise d[n] 1 ( x[ n] − x[ n − M ]) M 7 8 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Discrete-Time Systems:Examples Discrete- Discrete-Time Systems:Examples Discrete- s[ n] = 2[n(0.9) ], d[n] - random signal n • Exponentially Weighted Running Average Filter y[ n] = αy[ n − 1] + x[ n], 0 < α < 1 8 d[n] s[n] x[n] Amplitude 6 4 2 • Computation of the running average requires only 2 additions, 1 multiplication and storage of the previous running average • Does not require storage of past input data samples 0 -2 0 10 20 30 Time index n 40 50 7 s[n] y[n] 6 Amplitude 5 4 3 2 1 9 0 0 10 20 30 Time index n 40 50 10 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Discrete-Time Systems:Examples Discrete- Discrete-Time Systems:Examples Discrete• Linear interpolation - Employed to estimate sample values between pairs of adjacent sample values of a discrete-time sequence • Factor-of-4 interpolation • For 0 < α < 1, the exponentially weighted average filter places more emphasis on current data samples and less emphasis on past data samples as illustrated below y[ n] = α(αy[ n − 2] + x[ n − 1]) + x[ n] = α 2 y[ n − 2] + αx[ n − 1] + x[ n] = α 2 (αy[ n − 3] + x[ n − 2]) + αx[ n − 1] + x[ n] y[n] = α3 y[ n − 3] + α 2 x[ n − 2] + αx[ n − 1] + x[ n] 3 11 12 Copyright © 2005, S. K. Mitra 0 1 2 4 5 6 7 8 9 10 11 12 n Copyright © 2005, S. K. Mitra 2 Discrete-Time Systems: Examples Discrete-Time Systems: Examples • Factor-of-2 interpolator - • Factor-of-2 interpolator y[n] = xu [ n] + 1 ( xu [n − 1] + xu [n + 1]) 2 • Factor-of-3 interpolator y[n] = xu [n] + 1 ( xu [n − 1] + xu [ n + 2]) 3 + 2 ( xu [n − 2] + xu [n + 1]) 3 13 Original ( 512×512 ) Down −sampled ( 256×256 ) Interpolated ( 512 × 512 ) 14 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Discrete-Time Systems: Examples Discrete-Time Systems: Examples Median Filter – • The median of a set of (2K+1) numbers is the number such that K numbers from the set have values greater than this number and the other K numbers have values smaller • Median can be determined by rank-ordering the numbers in the set by their values and choosing the number at the middle 15 Median Filter – • Example: Consider the set of numbers {2, − 3, 10, 5, − 1} • Rank-order set is given by {− 3, − 1, 2, 5, 10} • Hence, med{2, − 3, 10, 5, − 1} = 2 16 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Discrete-Time Systems: Examples Discrete-Time Systems: Examples Median Filter – • Implemented by sliding a window of odd length over the input sequence {x[n]} one sample at a time • Output y[n] at instant n is the median value of the samples inside the window centered at n 17 Median Filter – • Finds applications in removing additive random noise, which shows up as sudden large errors in the corrupted signal • Usually used for the smoothing of signals corrupted by impulse noise 18 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 3 Discrete-Time Systems: Examples Discrete-Time Systems: Classification Median Filtering Example – • • • • • 19 Linear System Shift-Invariant System Causal System Stable System Passive and Lossless Systems 20 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Linear Discrete-Time Systems Linear Discrete-Time Systems n • Definition - If y1[ n] is the output due to an input x1[n] and y2 [n] is the output due to an input x2 [n] then for an input x[n] = α x1[n] + β x2 [n] the output is given by y[n] = α y1[n] + β y2 [n] 21 • Above property must hold for any arbitrary constants α and β , and for all possible inputs x1[n] and x2 [n] n • Accumulator - y1[n] = ∑ x1[l], y2 [n] = ∑ x2 [l] l = −∞ l = −∞ For an input x[n] = α x1[n] + β x2 [n] the output is n y[n] = ∑ (α x1[l] + β x2 [l]) l = −∞ n n l = −∞ l = −∞ = α ∑ x1[l] + β ∑ x2 [l] = α y1[n] + β y2 [n] • Hence, the above system is linear 22 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Linear Discrete-Time Systems Linear Discrete-Time Systems • The outputs y1[n] and y2 [n] for inputs x1[n] and x2 [n] are given by n y1[n] = y1[−1] + ∑ x1[l] y2 [n] = y2 [−1] + • Now α y1[n] + β y2 [n] = α ( y1[−1] + l =0 n ∑ x2 [l] y[n] = y[−1] + ∑ (α x1[l] + β x 2 [l]) Copyright © 2005, S. K. Mitra l =0 n n l =0 ∑ x2 [l]) l =0 • Thus y[n] = α y1[n] + β y2 [n] if n l =0 l =0 = (α y1[−1] + β y2 [−1]) + (α ∑ x1[l] + β l =0 • The output y[n] for an input α x1[n] + β x 2 [n] is given by 23 n n ∑ x1[l]) + β ( y2 [−1] + ∑ x2 [l]) y[−1] = α y1[−1] + β y2 [−1] 24 Copyright © 2005, S. K. Mitra 4 Nonlinear Discrete-Time System Linear Discrete-Time System 25 • For the causal accumulator to be linear the condition y[−1] = α y1[−1] + β y2 [−1] must hold for all initial conditions y[−1], y1[−1] , y2 [−1] , and all constants α and β • This condition cannot be satisfied unless the accumulator is initially at rest with zero initial condition • For nonzero initial condition, the system is nonlinear 26 • The median filter described earlier is a nonlinear discrete-time system • To show this, consider a median filter with a window of length 3 • Output of the filter for an input {x1[ n]} = {3, 4, 5}, 0 ≤ n ≤ 2 is {y1[ n]} = {3, 4, 4}, 0 ≤ n ≤ 2 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Nonlinear Discrete-Time System Nonlinear Discrete-Time System • Output for an input {x2 [ n]} = {2, − 1, − 1}, 0 ≤ n ≤ 2 is • Note {y1[ n] + y2 [ n]} = {3, 3, 3} ≠ {y[ n]} {y2 [ n]} = {0, − 1, − 1}, 0 ≤ n ≤ 2 • Hence, the median filter is a nonlinear discrete-time system • However, the output for an input {x[ n]} = {x1[ n] + x2 [ n]} is {y[ n]} = {3, 4, 3} 27 28 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Shift-Invariant System Shift-Invariant System • For a shift-invariant system, if y1[n] is the response to an input x1[n], then the response to an input x[n] = x1[n − no ] is simply y[n] = y1[n − no ] 29 where no is any positive or negative integer • The above relation must hold for any arbitrary input and its corresponding output Copyright © 2005, S. K. Mitra • In the case of sequences and systems with indices n related to discrete instants of time, the above property is called time-invariance property • Time-invariance property ensures that for a specified input, the output is independent of the time the input is being applied 30 Copyright © 2005, S. K. Mitra 5 Shift-Invariant System Shift-Invariant System • Example - Consider the up-sampler with an input-output relation given by x[n / L], n = 0, ± L, ± 2 L, ..... xu [n] = ⎧ ⎨ 0, otherwise ⎩ 31 • For an input x1[n] = x[n − no ] the output x1,u [n] is given by x [n / L], n = 0, ± L, ± 2 L, ..... x1,u [n] = ⎧ 1 ⎨ 0, otherwise ⎩ x[(n − Lno ) / L], n = 0, ± L, ± 2 L, ..... =⎧ ⎨ 0, otherwise ⎩ Copyright © 2005, S. K. Mitra • However from the definition of the up-sampler xu [n − no ] x[(n − no ) / L], n = no , no ± L, no ± 2 L, ..... =⎧ ⎨ 0, otherwise ⎩ ≠ x1,u [n] • Hence, the up-sampler is a time-varying system 32 Copyright © 2005, S. K. Mitra Linear Time-Invariant System Causal System • Linear Time-Invariant (LTI) System A system satisfying both the linearity and the time-invariance property • LTI systems are mathematically easy to analyze and characterize, and consequently, easy to design • Highly useful signal processing algorithms have been developed utilizing this class of systems over the last several decades • In a causal system, the no -th output sample y[no ] depends only on input samples x[n] for n ≤ no and does not depend on input samples for n > no • Let y1[n] and y2 [n] be the responses of a causal discrete-time system to the inputs x1[n] and x2 [n] , respectively 33 34 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Causal System Causal System • Examples of causal systems: y[n] = α1x[n] + α 2 x[n − 1] + α 3 x[n − 2] + α 4 x[n − 3] • Then y[n] = b0 x[n] + b1x[n − 1] + b2 x[n − 2] + a1 y[n − 1] + a2 y[n − 2] y[n] = y[n − 1] + x[n] x1[n] = x2[n] for n < N implies also that y1[n] = y2[n] for n < N • For a causal system, changes in output samples do not precede changes in the input samples 35 • Examples of noncausal systems: 1 y[n] = xu [n] + ( xu [n − 1] + xu [n + 1]) 2 1 36 Copyright © 2005, S. K. Mitra y[n] = xu [n] + ( xu [n − 1] + xu [n + 2]) 3 2 + ( xu [n − 2] + xu [n + 1]) 3 Copyright © 2005, S. K. Mitra 6 Stable System Causal System • There are various definitions of stability • We consider here the bounded-input, bounded-output (BIBO) stability • If y[n] is the response to an input x[n] and if x[n] ≤ Bx for all values of n then y[n] ≤ B y for all values of n • A noncausal system can be implemented as a causal system by delaying the output by an appropriate number of samples • For example a causal implementation of the factor-of-2 interpolator is given by 1 2 y[n] = xu [n − 1] + ( xu [n − 2] + xu [n]) 37 38 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Stable System Passive and Lossless Systems • Example - The M-point moving average filter is BIBO stable: y[n] = 1 M • A discrete-time system is defined to be passive if, for every finite-energy input x[n], the output y[n] has, at most, the same energy, i.e. M −1 ∑ x[n − k ] k =0 • For a bounded input x[n] ≤ Bx we have y[n] = ≤ 1 M 1 M M −1 ∑ x[n − k ] ≤ k =0 1 M ∞ n = −∞ ∑ x[n − k ] k =0 40 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Impulse and Step Responses Passive and Lossless Systems • The response of a discrete-time system to a unit sample sequence {δ[n]} is called the unit sample response or simply, the impulse response, and is denoted by {h[n]} • The response of a discrete-time system to a unit step sequence {µ[n]} is called the unit step response or simply, the step response, and is denoted by {s[n]} • Example - Consider the discrete-time system defined by y[n] = α x[n − N ] with N a positive integer • Its output energy is given by ∞ 2 ∑ y[n] = α n = −∞ 41 n = −∞ • For a lossless system, the above inequality is satisfied with an equal sign for every input ( MBx ) ≤ Bx 39 ∞ 2 2 ∑ y[n] ≤ ∑ x[n] < ∞ M −1 2∞ ∑ x[n] 2 n = −∞ • Hence, it is a passive system if α < 1 and is a lossless system if α = 1 Copyright © 2005, S. K. Mitra 42 Copyright © 2005, S. K. Mitra 7 Impulse Response Impulse Response • Example - The impulse response of the system y[n] = α1x[n] + α 2 x[n − 1] + α 3 x[n − 2] + α 4 x[n − 3] 43 is obtained by setting x[n] = δ[n] resulting in h[n] = α1δ [n] + α 2δ [n − 1] + α 3δ [n − 2] + α 4δ [n − 3] • The impulse response is thus a finite-length sequence of length 4 given by {h[n]} = {α1, α 2 , α 3 , α 4} ↑ • Example - The impulse response of the discrete-time accumulator y[n] = l = −∞ is obtained by setting x[n] = δ[n] resulting in n h[n] = ∑ δ [l] = µ [n] l = −∞ 44 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Time-Domain Characterization of LTI Discrete-Time System Impulse Response • Example - The impulse response {h[n]} of the factor-of-2 interpolator 1 y[n] = xu [n] + ( xu [n − 1] + xu [n + 1]) 2 • is obtained by setting xu [n] = δ [n] and is given by 1 h[n] = δ [n] + (δ [n − 1] + δ [n + 1]) • Input-Output Relationship A consequence of the linear, timeinvariance property is that an LTI discretetime system is completely characterized by its impulse response • Knowing the impulse response one can compute the output of the system for any arbitrary input 2 45 n ∑ x[l] • The impulse response is thus a finite-length sequence of length 3: {h[n]} = {0.5, 1 0.5} ↑ 46 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Time-Domain Characterization of LTI Discrete-Time System Time-Domain Characterization of LTI Discrete-Time System • Let h[n] denote the impulse response of a LTI discrete-time system • We compute its output y[n] for the input: • Since the system is time-invariant input output δ[n + 2] → h[n + 2] x[ n] = 0.5δ[ n + 2] + 1.5δ[n − 1] − δ[ n − 2] + 0.75δ[ n − 5] δ[n − 1] → h[n − 1] • As the system is linear, we can compute its outputs for each member of the input separately and add the individual outputs to determine y[n] δ[n − 2] → h[n − 2] 47 δ[n − 5] → h[n − 5] 48 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 8 Time-Domain Characterization of LTI Discrete-Time System Time-Domain Characterization of LTI Discrete-Time System • Likewise, as the system is linear input 49 • Now, any arbitrary input sequence x[n] can be expressed as a linear combination of delayed and advanced unit sample sequences in the form output 0.5δ[n + 2] → 0.5h[n + 2] 1.5δ[n − 1] → 1.5h[n − 1] − δ[n − 2] → − h[n − 2] 0.75δ[n − 5] → 0.75h[n − 5] • Hence because of the linearity property we get y[n] = 0.5h[n + 2] + 1.5h[n − 1] − h[n − 2] + 0.75h[n − 5] ∞ x[n] = ∑ x[k ] δ[n − k ] k = −∞ • The response of the LTI system to an input x[ k ] δ[n − k ] will be x[k ] h[n − k ] 50 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Time-Domain Characterization of LTI Discrete-Time System Convolution Sum • Hence, the response y[n] to an input • The summation ∞ y[n] = k = −∞ will be ∞ k = −∞ which can be alternately written as 51 ∞ k = −∞ y[n] = x[n] * h[n] ∑ x[n − k ] h[k ] k = −∞ ∞ ∑ x[k ] h[n − k ] = ∑ x[n − k ] h[n] is called the convolution sum of the sequences x[n] and h[n] and represented compactly as y[n] = ∑ x[k ] h[ n − k ] y[n] = ∞ k = −∞ x[ n] = ∑ x[k ] δ[ n − k ] 52 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum Convolution Sum • Properties • Commutative property: x[n] * h[n] = h[n] * x[n] • Associative property : (x[n] * h[n]) * y[n] = x[n] * (h[n] * y[n]) • Distributive property : x[n] * (h[n] + y[n]) = x[n] * h[n] + x[n] * y[n] 53 54 Copyright © 2005, S. K. Mitra • Interpretation • 1) Time-reverse h[k] to form h[− k ] • 2) Shift h[− k ] to the right by n sampling periods if n > 0 or shift to the left by n sampling periods if n < 0 to form h[n − k ] • 3) Form the product v[k ] = x[k ]h[n − k ] • 4) Sum all samples of v[k] to develop the n-th sample of y[n] of the convolution sum Copyright © 2005, S. K. Mitra 9 Convolution Sum Convolution Sum • Schematic Representation h[− k ] zn h[n − k ] v[k ] ∑ × y[n] k x[k ] 55 • The computation of an output sample using the convolution sum is simply a sum of products • Involves fairly simple operations such as additions, multiplications, and delays 56 • We illustrate the convolution operation for the following two sequences: ⎧1, 0 ≤ n ≤ 5 x[ n] = ⎨ ⎩0, otherwise ⎧1.8 − 0.3n, 0 ≤ n ≤ 5 h[ n] = ⎨ 0, otherwise ⎩ • Figures on the next several slides the steps involved in the computation of y[n] = x[n] * h[n] Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum Convolution Sum Plot of x[-4- k] and h[k] h[k]x[-4- k] 2 3 Amplitude Amplitude 1.5 1 0.5 0 -0.5 -10 2 1 0 0 10 k→ -10 y[n] 6 6 Amplitude Amplitude 8 4 2 0 -10 57 10 k → 0 y[-4] 8 0 4 2 0 -10 10 0 n 58 10 n Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum 3 Amplitude 1.5 Amplitude Plot of x[0- k] and h[k] 1 0.5 0 10 1 k→ -10 3 1.5 2 1 0.5 0 0 0 y[-1] h[k]x[0- k] 2 Amplitude 2 -0.5 -10 Convolution Sum h[k]x[-1- k] Amplitude Plot of x[-1- k] and h[k] 0 10 y[n] -0.5 -10 k→ 2 1 0 0 10 y[0] -10 k→ 10 6 6 6 4 2 0 10 n 4 2 0 -10 0 60 Copyright © 2005, S. K. Mitra 4 2 0 -10 10 n Amplitude 6 k→ 8 Amplitude 8 Amplitude Amplitude 8 0 -10 59 0 y[n] 8 0 10 n 4 2 0 -10 0 10 n Copyright © 2005, S. K. Mitra 10 Convolution Sum h[k]x[1- k] Amplitude Amplitude 1 0.5 0 -0.5 -10 Plot of x[3- k] and h[k] 3 2 1 10 y[1] -10 k→ 3 1 0.5 0 0 0 h[k]x[3- k] 2 1.5 Amplitude 2 1.5 Amplitude Plot of x[1- k] and h[k] Convolution Sum 0 10 y[n] -0.5 -10 k→ 2 1 0 0 10 y[3] -10 k→ 10 6 6 6 6 4 2 0 -10 0 4 2 0 -10 10 0 n 61 4 2 0 -10 10 n 0 4 2 0 -10 10 0 n 62 10 n Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum h[k]x[5- k] 1 0 2 1 10 y[5] -10 k→ 3 1 0.5 0 0 0 h[k]x[7- k] 2 1.5 Amplitude Amplitude Amplitude Plot of x[7- k] and h[k] 3 0.5 -0.5 -10 Convolution Sum Amplitude Plot of x[5- k] and h[k] 2 1.5 0 10 y[n] -0.5 -10 k→ 2 1 0 0 10 k→ -10 10 6 6 6 6 4 2 0 4 2 0 -10 10 0 n 63 4 2 0 -10 10 n Amplitude 8 Amplitude 8 0 -10 0 4 2 0 -10 10 0 n 64 10 n Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum 3 Amplitude 1.5 Amplitude Plot of x[10- k] and h[k] 1 0.5 0 10 1 y[9] -10 k→ 3 1.5 2 1 0.5 0 0 0 h[k]x[10- k] 2 Amplitude 2 -0.5 -10 Convolution Sum h[k]x[9- k] Amplitude Plot of x[9- k] and h[k] 0 10 y[n] -0.5 -10 k→ 2 1 0 0 y[10] 10 -10 k→ 0 10 y[n] 6 6 6 k→ 8 6 4 2 0 -10 0 10 n 4 2 0 -10 0 66 Copyright © 2005, S. K. Mitra 4 2 0 -10 10 n Amplitude 8 Amplitude 8 Amplitude 8 Amplitude k→ y[n] 8 Amplitude Amplitude 0 y[7] 8 65 k→ 8 Amplitude 8 Amplitude 8 Amplitude Amplitude 0 y[n] 8 0 10 n 4 2 0 -10 0 10 n Copyright © 2005, S. K. Mitra 11 Convolution Sum Convolution Sum h[k]x[12- k] 1 0.5 0 2 1 0 y[12] 10 -10 k→ 3 1 0.5 0 0 -0.5 -10 h[k]x[13- k] 2 1.5 Amplitude Amplitude Amplitude Plot of x[13- k] and h[k] 3 Amplitude Plot of x[12- k] and h[k] 2 1.5 0 10 -0.5 -10 k→ y[n] 2 1 0 0 y[13] 10 k→ -10 10 k→ 8 6 6 6 6 4 2 0 -10 0 10 4 2 0 -10 n 67 0 4 2 0 -10 10 n 68 Amplitude 8 Amplitude 8 Amplitude Amplitude 0 y[n] 8 0 4 2 0 -10 10 0 n 10 n Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Time-Domain Characterization of LTI Discrete-Time System Time-Domain Characterization of LTI Discrete-Time System • In practice, if either the input or the impulse response is of finite length, the convolution sum can be used to compute the output sample as it involves a finite sum of products • If both the input sequence and the impulse response sequence are of finite length, the output sequence is also of finite length • If both the input sequence and the impulse response sequence are of infinite length, convolution sum cannot be used to compute the output • For systems characterized by an infinite impulse response sequence, an alternate time-domain description involving a finite sum of products will be considered 69 70 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Time-Domain Characterization of LTI Discrete-Time System Time-Domain Characterization of LTI Discrete-Time System • Example - Develop the sequence y[n] generated by the convolution of the sequences x[n] and h[n] shown below x[n] • As can be seen from the shifted timereversed version {h[n − k ]} for n < 0, shown below for n = −3 , for any value of the sample index k, the k-th sample of either {x[k]} or {h[n − k ]} is zero h[n] 3 2 3 1 h[−3 − k ] 1 1 0 2 –1 4 n 3 0 1 n 2 2 1 –1 –6 –2 –5 –4 –3 –2 –1 0 k –1 71 72 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 12 Time-Domain Characterization of LTI Discrete-Time System Time-Domain Characterization of LTI Discrete-Time System • As a result, for n < 0, the product of the k-th samples of {x[k]} and {h[n − k ]} is always zero, and hence y[n] = 0 for n < 0 • Consider now the computation of y[0] • The sequence h[ − k ] 2 {h[− k ]} is shown 1 k on the right –1 • The product sequence {x[k ]h[−k ]} is plotted below which has a single nonzero sample x[0]h[0] for k = 0 x[k ]h[ −k ] 0 –5 –4 –3 –2 –1 73 –2 –1 0 1 2 2 k 3 –2 • Thus y[0] = x[0]h[0] = −2 –3 –6 –5 –4 1 3 74 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Time-Domain Characterization of LTI Discrete-Time System Time-Domain Characterization of LTI Discrete-Time System • For the computation of y[1], we shift {h[−k ]} to the right by one sample period to form {h[1 − k ]} as shown below on the left • The product sequence {x[k ]h[1 − k ]} is shown below on the right • To calculate y[2], we form {h[2 − k ]} as shown below on the left • The product sequence {x[k ]h[2 − k ]} is plotted below on the right –5 –4 –3 –2 –1 1 –2 –1 0 1 2 3 1 2 x[ k ]h[ 2 − k ] 2 0 2 –5 –4 –3 h[ 2 − k ] x[k ]h[1 − k ] h[1 − k ] 3 k 1 1 –1 –4 –3 –2 k 1 0 2 3 4 5 k –3 –2 –1 0 1 2 3 4 5 6 k –1 –1 –4 75 • Hence, y[1] = x[0]h[1] + x[1]h[0] = −4 + 0 = −4 76 y[2] = x[0]h[2] + x[1]h[1] + x[2]h[0] = 0 + 0 + 1 = 1 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Time-Domain Characterization of LTI Discrete-Time System Time-Domain Characterization of LTI Discrete-Time System • Continuing the process we get y[3] = x[0]h[3] + x[1]h[2] + x[2]h[1] + x[3]h[0] = 2 + 0 + 0 +1 = 3 • From the plot of {h[n − k ]} for n > 7 and the plot of {x[k]} as shown below, it can be seen that there is no overlap between these two sequences • As a result y[n] = 0 for n > 7 y[4] = x[1]h[3] + x[2]h[2] + x[3]h[1] + x[4]h[0] = 0 + 0 − 2 + 3 =1 y[5] = x[2]h[3] + x[3]h[2] + x[4]h[1] = −1 + 0 + 6 = 5 y[6] = x[3]h[3] + x[4]h[2] = 1 + 0 = 1 77 y[7] = x[4]h[3] = −3 x[k] 2 1 0 3 1 2 –1 78 Copyright © 2005, S. K. Mitra h[8 − k ] 3 –2 4 k 1 5 2 3 4 –1 6 7 8 9 10 11 k Copyright © 2005, S. K. Mitra 13 Time-Domain Characterization of LTI Discrete-Time System Time-Domain Characterization of LTI Discrete-Time System • Note: The sum of indices of each sample product inside the convolution sum is equal to the index of the sample being generated by the convolution operation • For example, the computation of y[3] in the previous example involves the products x[0]h[3], x[1]h[2], x[2]h[1], and x[3]h[0] • The sum of indices in each of these products is equal to 3 • The sequence {y[n]} generated by the convolution sum is shown below y[n] 5 3 0 1 1 1 1 7 2 –2 –1 3 4 5 6 8 9 n –2 –3 –4 79 80 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Tabular Method of Convolution Sum Computation Time-Domain Characterization of LTI Discrete-Time System • Can be used to convolve two finite-length sequences • Consider the convolution of {g[n]}, 0 ≤ n ≤ 3 , with {h[n]}, 0 ≤ n ≤ 2 , generating the sequence y[n] = g[n] * h[n] • Samples of {g[n]} and {h[n]} are then multiplied using the conventional multiplication method without any carry operation • In the example considered the convolution of a sequence {x[n]} of length 5 with a sequence {h[n]} of length 4 resulted in a sequence {y[n]} of length 8 • In general, if the lengths of the two sequences being convolved are M and N, then the sequence generated by the convolution is of length M + N − 1 81 82 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Tabular Method of Convolution Sum Computation n: g[ n ]: h[ n ]: y[ n ]: Tabular Method of Convolution Sum Computation 0 1 2 3 4 5 g[ 0 ] g[1] g[ 2 ] g[ 3 ] h[ 0 ] h[1] h[ 2 ] g[ 0 ]h[ 0 ] g[1]h[ 0 ] g[ 2 ]h[ 0 ] g[ 3]h[ 0 ] g[ 0 ]h[1] g[1]h[1] g[ 2 ]h[1] g[ 3 ]h[1] g[ 0 ]h[ 2 ] g[1]h[ 2 ] g[ 2 ]h[ 2 ] g[ 3]h[ 2 ] y[ 0 ] y[1] y[ 2 ] y[ 3 ] y[ 4 ] y[ 5 ] • The samples of {y[n]} are given by y[0] = g[0]h[ 0] y[1] = g[1]h[ 0] + g[ 0]h[1] y[2] = g[2]h[0] + g[1]h[1] + g[0]h[2] y[3] = g[3]h[0] + g[2]h[1] + g[1]h[2] y[ 4] = g[3]h[1] + g[2]h[2] y[5] = g[3]h[2] • The samples y[n] generated by the convolution sum are obtained by adding the entries in the column above each sample 83 84 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 14 Tabular Method of Convolution Sum Computation Tabular Method of Convolution Sum Computation • The method can also be applied to convolve two finite-length two-sided sequences • In this case, a decimal point is first placed to the right of the sample with the time index n = 0 for each sequence • Next, convolution is computed ignoring the location of the decimal point • Finally, the decimal point is inserted according to the rules of conventional multiplication • The sample immediately to the left of the decimal point is then located at the time index n = 0 85 86 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Simple Interconnection Schemes Convolution Using MATLAB • The M-file conv implements the convolution sum of two finite-length sequences a =[− 2 0 1 − 1 3 • If • Two simple interconnection schemes are: • Cascade Connection • Parallel Connection b =[1 2 0 -1] then conv(a,b) yields [− 2 − 4 1 3 1 5 1 − 3 87 88 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Cascade Connection Cascade Connection ≡ • Note: The ordering of the systems in the cascade has no effect on the overall impulse response because of the commutative property of convolution • A cascade connection of two stable systems is stable • A cascade connection of two passive (lossless) systems is passive (lossless) h1[n] h2[n] ≡ h2[n] h1[n] h1[n] * h[n]=h2[n] h[n] 1 • Impulse response h[n] of the cascade of two LTI discrete-time systems with impulse responses h1[n] and h2[n] is given by h[n] = h1[n] * h2[n] 89 90 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 15 Cascade Connection Cascade Connection • An application of the inverse system concept is in the recovery of a signal x[n] ˆ from its distorted version x[n] appearing at the output of a transmission channel • If the impulse response of the channel is known, then x[n] can be recovered by designing an inverse system of the channel • An application is in the development of an inverse system • If the cascade connection satisfies the relation h1[n] * h2[n] = δ[ n] then the LTI system h1[n] is said to be the inverse of h2[n] and vice-versa 91 x[ n ] 92 inverse system channel ^ x[ n ] h1[n] h2[n] h1[n] * h 2[n] = δ[ n] Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Cascade Connection Cascade Connection • Example - Consider the discrete-time accumulator with an impulse response µ[n] • Its inverse system satisfy the condition µ[n] * h 2[n] = δ[n] • It follows from the above that h2[n] = 0 for n < 0 and h2[0] = 1 n ∑ h2[l] = 0 93 • Thus the impulse response of the inverse system of the discrete-time accumulator is given by h2[ n] = δ[n] − δ[n − 1] which is called a backward difference system for n ≥ 1 l =0 94 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Simple Interconnection Schemes Parallel Connection h1[n] h2[n] 95 x[ n ] + ≡ • Consider the discrete-time system where h1[ n] = δ[n] + 0.5δ[n − 1], h1[n] + h[n]=h2[n] h[n] 1 • Impulse response h[n] of the parallel connection of two LTI discrete-time systems with impulse responses h1[n] and h2[n] is given by h[n] = h1[n] + h2[n] Copyright © 2005, S. K. Mitra h2[ n] = 0.5δ[n] − 0.25δ[n − 1], h3[n] = 2δ[n], h1[n] 96 + h3[n] h4[ n] = −2(0.5) µ[ n] n + h2[n] h4[n] Copyright © 2005, S. K. Mitra 16 Simple Interconnection Schemes Simple Interconnection Schemes • Simplifying the block-diagram we obtain • Overall impulse response h[n] is given by h[n] = h1[n] + h2[n] * (h3[n] + h4[n]) = h1[n] + h 2[n] * h3[n] + h 2[n] * h 4[n] • Now, + h1[n] ≡ h2[n] h1[n] + h 2[ n ] * ( h3[ n ]+ h 4[ n ]) h 3[ n ] + h 4[ n ] h2[n] * h3[n] = ( 1 δ[ n] − 1 δ[n − 1]) * 2δ[n] 2 4 = δ[n] − 1 δ[n − 1] 2 97 98 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Simple Interconnection Schemes ( ) h2[n] * h4[n] = ( 1 δ[n] − 1 δ[n − 1]) * − 2( 1 ) n µ[n] 2 2 • Therefore 4 = − ( 1 ) n µ[ n] + 1 ( 1 ) n −1µ[n − 1] 2 22 1 ) n µ[ n] + ( 1 ) n µ[ n − 1] = − (2 2 = − ( 1 ) n δ[n] = − δ[n] 2 h[ n] = δ[n] + 1 δ[n − 1] + δ[n] − 1 δ[n − 1] − δ[n] = δ[n] 2 2 99 Copyright © 2005, S. K. Mitra 17 ...
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This note was uploaded on 01/11/2010 for the course ECC ECC3107 taught by Professor Drmakhfudzah during the Spring '09 term at Punjab Engineering College.

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