Discrete-Time s - I Time-Domain Representation CHAPTER 4 These lecture slides are based on "Digital Signal Processing: A Computer-Based Approach, 4th ed." textbook by S.K. Mitra and its instructor materials. U.Sezen Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 1 / 75 Contents Dicrete-Time s Basics Accumulator M-point Moving-Average Linear interpolation Median Filter Linear Shift-Invariant Linear Time-Invariant (LTI) Causal Stable Passive and Lossless s Impulse and Step Responses Tabular Method of Convolution Sum Computation Stability Condition of an LTI Discrete-Time Causality Condition of an LTI Discrete-Time Cascade Connection Parallel Connection Finite-Dimensional LTI Discrete-Time s Finite-Dimensional LTI Discrete-Time s Finite Impulse Response (FIR) Discrete-Time s Innite Impulse Response (IIR) Discrete-Time s Nonrecursive and Recursive s Real and Complex s Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 / 75 Dicrete-Time s Basics Dicrete-Time s Basics Introduction Examples 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 -input, 1-output discrete-time systems: x[n] y[n] x[n] + w[n] w[n] y[n] x[n] Input sequence Discrete-Time H( ) y[n] Output sequence Mathematically, the discrete-time system is characterized by an operator H( ) that transforms the input sequence x[n] into another sequence y[n] at the output The discrete-time system may also have more than one input and/or more than one output Modulator 1-input, 1-output discrete-time systems: x[n] A Multiplier y[n] x[n] z y[n] Unit advance Adder x[n] 1 z y[n] Unit delay Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 3 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 4 / 75 Examples Dicrete-Time s Basics A more complex example of an one-input, one-output discrete-time system is shown below Accumulator: Dicrete-Time s y[n] = = n 1 Accumulator x[l] x[l] + x[n] = y[n 1] + x[n] 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 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 5 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 6 / 75
Dicrete-Time s Accumulator Dicrete-Time s M-point Moving-Average M-point Moving-Average : Input-output relation can also be written in the form y[n] = 1 = y[ 1] + x[l] + x[l] l=0 x[l], n 0 The second form is used for a causal input sequence, in which case y[ 1] is called the initial condition l=0 y[n] = 1 M M 1 x[n k] Used in smoothing random variations in data In most applications, the data x[n] is a bounded sequence, so M-point average y[n] is also a bounded sequence 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 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 7 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 8 / 75 Dicrete-Time s M-point Moving-Average Dicrete-Time s M-point Moving-Average A more ecient implementation is developed next ( y[n] = 1 M ) x[n k] + x[n] x[n M] M = 1 M = 1 M k=1 ( M 1 ( M 1 x[n 1 k] + x[n] x[n M] x[n 1 k] ) ) + 1 (x[n] x[n M]) M Computation of the modied M-point moving average system using the recursive equation now requires 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] Hence y[n] = y[n 1] + 1 (x[n] x[n M]) M Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 9 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 10 / 75 Dicrete-Time s M-point Moving-Average Dicrete-Time s M-point Moving-Average Example: s[n] = [n(0.9) n ] and d[n] is a random signal Exponentially Weighted Running Average Filter: y[n] = αy[n 1] + x[n], 0 < α < 1 Computation of the running average requires only 1 addition, 1 multiplication and storage of the previous running average Does not require storage of past input data samples For 0 < α < 1, the exponentially weighted average lter places more emphasis on current data samples and less emphasis on past data samples as illustrated below y[n] = α(αy[n ] + x[n 1]) + x[n] = α y[n ] + αx[n 1] + x[n] = α (αy[n 3] + x[n ] + αx[n 1]) + x[n] = α 3 y[n 3] + α x[n ] + αx[n 1] + x[n] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 11 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 1 / 75
Dicrete-Time s Linear interpolation Dicrete-Time s Linear interpolation Linear interpolation: Employed to estimate sample values between pairs of adjacent sample values of a discrete-time sequence Factor-of-4 interpolation Factor-of- linear interpolator y[n] = x u [n] + 1 (x u[n 1] + x u [n + 1]) (a) x[n] (b) x u[n] Factor-of-3 linear interpolator y[n] = x u [n] + 1 3 (x u[n 1] + x u [n + ]) + 3 (x u[n ] + x u [n + 1]) (c) y[n] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 13 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 14 / 75 Dicrete-Time s Linear interpolation Dicrete-Time s Median Filter Factor-of- linear -D interpolator Median Filter: The median of a set of (K + 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 Example: Consider the set of numbers Rank-ordered set is given by {, 3, 10, 5, 1} Original (51 51) Downsampled (56 56) Interpolated (51 51) Hence, { 3, 1,, 5, 10} med {, 3, 10, 5, 1} = Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 15 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 16 / 75 Dicrete-Time s Median Filter Dicrete-Time s Median Filter Example: 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 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 (a) Original data (b) Impulse noise corrupted data (c) Median ltered noisy data Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 17 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 18 / 75
Linear Linear Linear Shift-Invariant Causal Stable Passive and Lossless s Denition: If y 1 [n] is the output due to one input x 1 [n] and y [n] is the output due to another input x [n] then for an input the output is given by x[n] = α x 1 [n] + β x [n] y[n] = α y 1 [n] + β y [n] Above property must hold for any arbitrary constants α and β and for all possible inputs x 1 [n] and x [n] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 19 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 0 / 75 Linear Linear Example: Consider two accumulators with y 1 [n] = x 1 [l] and y [n] = x [l] For an input x[n] = α x 1 [n] + β x [n] the output is y[n] = (α x 1 [l] + β x [l) = α x 1 [l] + β x [l] = α y 1 [n] + β y [n] Hence, the above system is linear Example: The median lter described earlier is a nonlinear discrete-time system. To show this, consider a median lter with a window of length 3 Output y 1 [n] of the lter for an input x 1 [n], {x 1 [n]} = {3, 4, 5}, 0 n is {y 1 [n]} = {3, 4, 4}, 0 n Output y [n] of the lter for another input x [n], {x [n]} = {, 1, 1}, 0 n is {y 1 [n]} = {0, 1, 1}, 0 n Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 1 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 / 75 Linear Shift-Invariant Shift-Invariant However, the output y[n] for the input, x[n] = x 1 [n] + x [n], {x[n]} = {5, 3, 4}, 0 n is {y[n]} = {3, 4, 3}, 0 n Note: {y 1 [n] + y [n]} = {3, 3, 3} {y[n]} Hence, the median lter is a nonlinear discrete-time system For a shift-invariant system, if y 1 [n] is the response to an input x 1 [n], then the response to an input is simply x[n] = x 1 [n n 0 ] y[n] = y 1 [n n 0 ] where n 0 is any positive or negative integer The above relation must hold for any arbitrary input and its corresponding output Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 3 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 4 / 75
Shift-Invariant Shift-Invariant Example: Consider the upsampler x[n] L x u [n] 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 specied input, the output is independent of the time the input is being applied with an input-output relation given by { x[n/l], n = 0, ±L, ±L,... x u [n] = 0, otherwise For an input x 1 [n] = x[n n 0 ] the output x 1,u [n] is given by { x 1 [n/l], n = 0, ±L, ±L,... x 1,u [n] = 0, otherwise { x[n/l n 0 ], n = 0, ±L, ±L,... = 0, otherwise Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 5 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 6 / 75 Shift-Invariant Shift-Invariant Linear Time-Invariant (LTI) However from the denition of the up-sampler { x 1 [(n n 0 )/L], n = 0, ±L, ±L,... x u [n n 0 ] = 0, otherwise x 1,u [n] Hence, the upsampler is a time-varying system Linear Time-Invariant (LTI) system is 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 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 7 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 8 / 75 Causal Causal In a causal system, the n 0 -th output sample y[n 0 ] depends only on input samples x[n] for n n 0 and does not depend on input samples for n > n 0 Let y 1 [n] and y [n] be the responses of a causal discrete-time system to the inputs x 1 [n] and x [n], respectively Then implies also that x 1 [n] = x [n] y 1 [n] = y [n] for n < N for n < N Causal Examples: Examples of causal systems: y[n] = α 1 x[n] + α x[n 1] + α 3 x[n ] + α 4 x[n 3] y[n] = b 0 x[n] + b 1 x[n 1] + b x[n ] + a 1 y[n 1] + a y[n ] y[n] = y[n 1] + x[n] Examples of noncausal systems: y[n] = x u [n] + 1 (x u[n 1] + x u [n + 1]) y[n] = x u [n] + 1 3 (x u[n 1] + x u [n + ]) + 3 (x u[n ] + x u [n + 1]) For a causal system, changes in output samples do not precede changes in the input samples Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 9 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 30 / 75
Causal Stable Stable 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- interpolator is given by y[n] = x u [n 1] + 1 (x u[n ] + x u [n]) There are various denitions of stability We consider here the bounded-input, bounded-output (BIBO) stability, i.e., If y[n] is the response to an input x[n] and if x B x for all values of n then y B y where B x < and B y < for all values of n Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 31 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 3 / 75 Stable Example: The M-point moving average lter is BIBO stable y[n] = 1 M For a bounded input we have 1 y[n] = M 1 M M 1 M 1 M 1 1 M (M B x) x[n k] x[n k] x[n k] Passive and Lossless s Passive and Lossless s A discrete-time system is dened to be passive if, for every nite-energy input x[n], the output y[n] has, at most, the same energy, i.e. y[n] x[n] < n= n= For a lossless system, the above inequality is satised with an equal sign for every input B x Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 33 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 34 / 75 Passive and Lossless s Impulse and Step Responses Impulse and Step Responses Example: Consider the discrete-time system dened by y[n] = α x[n N] with N a positive integer Its output energy is given by n= y[n] = α n= x[n] Hence, it is a passive system if α 1 and is a lossless system if α = 1 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]} Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 35 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 36 / 75
Impulse and Step Responses Impulse and Step Responses Example: The impulse response h[n] of the discrete-time accumulator Example: The impulse response h[n] of the factor-of- interpolator is obtained by setting resulting in y[n] = x[l] x[n] = δ[n] h[n] = δ[l] = µ[n] is obtained by setting and is given by y[n] = x u [n] + 1 (x u[n 1] + x u [n + 1]) x u [n] = δ[n] h[n] = δ[n] + 1 (δ[n 1] + δ[n + 1]) The impulse response {h[n]} is thus a nite-length sequence of length 3: {h[n]} = {0.5, 1, 0.5}, 1 n 1 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 37 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 38 / 75 Example: Let h[n] denote the impulse response of a LTI discrete-time system, and compute its output y[n] for the input: A consequence of the linear, time-invariance property is that an LTI discrete-time system is completely characterized by its impulse response Thus, knowing the impulse response one can compute the output of the system for any arbitrary input x[n] = 0.5 δ[n + ] + 1.5 δ[n 1] δ[n ] + 0.75 δ[n 5] 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] Since the system is time-invariant input output δ[n + ] h[n + ] δ[n 1] h[n 1] δ[n ] h[n ] δ[n 5] h[n 5] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 39 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 40 / 75 Now, any arbitrary input sequence x[n] can be expressed as a linear combination of delayed and advanced unit sample sequences in the form Likewise, as the system is linear input output 0.5 δ[n + ] 0.5 h[n + ] 1.5 δ[n 1] 1.5 h[n 1] δ[n ] h[n ] 0.75 δ[n 5] 0.75 h[n 5] Hence, because of the linearity property we get y[n] = 0.5 h[n + ] + 1.5 h[n 1] h[n ] + 0.75 h[n 5] x[n] = x[n] δ[n] = x[k]δ[n k] The response of the LTI system to an input x[k]δ[n k] will be x[k]h[n k] Hence, the response y[n] to the input above is given by y[n] = x[k]h[n k] which can be alternately written as y[n] = x[n k]h[k] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 41 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 4 / 75
The summation y[n] = x[k]h[n k] = x[n k]h[k] is thus the convolution sum of the sequences x[n] and h[n] and represented compactly as y[n] = x[n] h[n] Example: Consider an LTI discrete-time system with an impulse response h[n] generating an output y[n] for a input x[n]: y[n] = x[k]h[n k] = x[n] h[n] Let us determine the output y 1 [n] of an LTI discrete-time system with an impulse response h[n N 0 ] for the same input x[n] Now y 1 [n] = x[k]h[n N 0 k] = x[n] h[n N 0 ] Hence, y 1 [n] = y[n N 0 ] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 43 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 44 / 75 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] In practice, if either the input or the impulse response is of nite length, the convolution sum can be used to compute the output sample as it involves a nite sum of products If both the input sequence and the impulse response sequence are of nite length, the output sequence is also of nite length If both the input sequence and the impulse response sequence are of innite length, convolution sum cannot be used to compute the output For systems characterized by an innite impulse response sequence, an alternate time-domain description involving a nite sum of products will be considered Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 45 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 46 / 75 Tabular Method of Convolution Sum Computation Can be used to convolve two nite-length sequences Consider the convolution of {g[n]}, 0 n 3, with {h[n]}, 0 n, 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 as shown on the next slide. n 0 1 3 4 5 g[n] g[0] g[1] g[] g[3] h[n] h[0] h[1] h[] g[0]h[0] g[1]h[0] g[]h[0] g[3]h[0] + g[0]h[1] g[1]h[1] g[]h[1] g[3]h[1] + g[0]h[] g[1]h[] g[]h[] g[3]h[] y[n] y[0] y[1] y[] y[3] y[4] y[5] The samples y[n] generated by the convolution sum are obtained by adding the entries in the column above each sample 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[] = g[]h[0] + g[1]h[1] + g[0]h[] y[3] = g[3]h[0] + g[]h[1] + g[1]h[] y[4] = g[3]h[1] + g[]h[] y[5] = g[3]h[] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 47 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 48 / 75
The method can also be applied to convolve any two nite-length two-sided sequences as explained below Consider two sequences {x 1 [n]} with N 1 n N and {x [n]} with N a n N b and we are asked to compute the convolution y 1 [n] = x 1 [n] x [n] of these two sequences of size N N 1 + N b N a + 1 1. Create two causal sequences g[n] = x 1 [n + N 1 ] with 0 n N N 1 and h[n] = x [n + N a ] with 0 n N b N a, where both have their rst elements at n = 0. Compute the convolution y[n] = g[n] h[n] using the tabular method explained on the previous slide. Here {y[n]} is dened for 0 n N N 1 + N b N a. 3. Then, obtain the real convolution y 1 [n] as y 1 [n] = y[n N 1 N b ] Convolution Using MATLAB The M-le conv implements the convolution sum of two nite-length sequences If a = [ 0 1 1 3 ] n = [ 1 0 1 ] then conv(a,b) yields [ 4 1 3 1 5 1 ] 3 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 49 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 50 / 75 Stability Condition of an LTI Discrete-Time Stability Condition of an LTI Discrete-Time Example: Stability Condition of an LTI Discrete-Time Consider an LTI discrete-time system with an impulse response BIBO Stability Condition: A discrete-time is BIBO stable if and only if the output sequence {y[n]} remains bounded for all bounded input sequence {x[n]} For this system h[n] = α n µ[n] An LTI discrete-time system is BIBO stable if and only if its impulse response sequence {h[n]} is absolutely summable, i.e. S = n= h[n] < S = n= α n µ[n] = α n 1 = 1 α n=0 Therefore S < if α < 1 for which the system is BIBO stable If α = 1, the system is not BIBO stable Proof can be found in the textbook. Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 51 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 5 / 75 Causality Condition of an LTI Discrete-Time Causality Condition of an LTI Discrete-Time Let x 1 [n] and x [n] be two input sequences with x 1 [n] = x [n] for n n 0 x 1 [n] x [n] for n > n 0 then the system is causal if the corresponding outputs y 1 [n] and y [n] are also given by y 1 [n] = y [n] for n n 0 y 1 [n] y [n] for n > n 0 An LTI discrete-time system is causal if and only if its impulse response {h[n]} is a causal sequence. Proof can be found in the textbook. Examples: The discrete-time system dened by Causality Condition of an LTI Discrete-Time y[n] = α 1 x[n] + α x[n 1] + α 3 x[n ] + α 4 x[n 3] is a causal system as it has a causal impulse response {h[n]} = {[ α 1 α α 3 α 4 ]}, 0 n 3 then the system is causal if the corresponding outputs y 1 [n] and y [n] are also given by The discrete-time accumulator dened by y[n] = x[l] is a causal system as it has a causal impulse response given by h[n] = δ[l] = µ[n] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 53 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 54 / 75
Causality Condition of an LTI Discrete-Time Causality Condition of an LTI Discrete-Time Examples: The factor-of- interpolator dened by y[n] = x u [n] + 1 (x u[n 1] + x u [n + 1]) is noncausal as it has a noncausal impulse response given by {h[n]} = {[ 0.5 1 0.5 ]}, 1 n 1 Note: A noncausal LTI discrete-time system with a nite-length impulse response can often be realized as a causal system by inserting an appropriate amount of delay For example, a causal version of the factor-of- interpolator is obtained by delaying the input by one sample period y[n] = x u [n 1] + 1 (x u[n ] + x u [n]) Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 55 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 56 / 75 Cascade Connection x[n] h 1[n] h [n] y[n] x[n] h [n] h 1[n] y[n] x[n] h 1[n] h [n] y[n] Two simple interconnection schemes are: Cascade Connection Parallel Connection Impulse response h[n] of the cascade of two LTI discrete-time systems with impulse responses h 1 [n] and h [n] is given by h[n] = h 1 [n] h [n] Note: The ordering of the systems in the cascade has no eect 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 also passive (lossless) Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 57 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 58 / 75 An application is in the development of an inverse system as explained below: If the cascade connection satises the relation h 1 [n] h [n] = δ[n] then the LTI system h 1 [n] is said to be the inverse of h [n] and vice-versa An application of the inverse system concept is in the recovery of a signal x[n] from its distorted version ˆx[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 x[n] channel h 1[n] ˆx[n] inverse system h [n] h 1 [n] h [n] = δ[n] x[n] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 59 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 60 / 75
Example: Consider the discrete-time accumulator with an impulse response µ[n] Parallel Connection Its inverse system satisfy the condition µ[n] h [n] = δ[n] It follows from the above that h [n] = 0 for n < 0 and h [0] = 1 h [l] = 0 for n 1 l=0 Thus the impulse response of the inverse system of the discrete-time accumulator is given by h [n] = δ[n] δ[n 1] x[n] h 1[n] h [n] + y[n] x[n] h 1[n] + h [n] y[n] Impulse response h[n] of the parallel connection of two LTI discrete-time systems with impulse responses h 1 [n] and h 1 [n] is given by h[n] = h 1 [n] + h [n] which is called a backward dierence system Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 61 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 6 / 75 Example: Consider the discrete-time system shown in the gure below Simplifying the block-diagram we obtain Overall impulse response h[n] is given by where h[n] = h 1 [n] + h [n] (h 3 [n] + h 4 [n]) = h 1 [n] + h [n] h 3 [n] + h [n] h 4 [n] h 1 [n] = δ[n] + 0.5 δ[n 1] h [n] = 0.5 δ[n] 0.5 δ[n 1] h 3 [n] = δ[n] h 4 [n] = (0.5) n µ[n] Now, ( 1 h [n] h 3 [n] = δ[n] 1 ) δ[n 1] δ[n] 4 = δ[n] 1 δ[n 1] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 63 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 64 / 75 and ( 1 h [n] h 4 [n] = δ[n] 1 ) δ[n 1] ( (0.5) n µ[n]) 4 = (0.5) n µ[n] + 1 (0.5)n 1 µ[n 1] = (0.5) n µ[n] + (0.5) n µ[n 1] = (0.5) n (µ[n] µ[n 1]) = (0.5) n δ[n] = δ[n] Finite-Dimensional LTI Discrete-Time s Finite-Dimensional LTI Discrete-Time s Finite-Dimensional LTI Discrete-Time s An important subclass of LTI discrete-time systems is characterized by a linear constant coecient dierence equation of the form N M d k y[n k] = p k x[n k] where x[n] and y[n] are, respectively, the input and the output of the system, and, {d k } and {p k } are constants characterizing the system Therefore h[n] = δ[n] + 1 δ[n 1] + δ[n] 1 δ[n 1] δ[n] = δ[n] The order of the system is given by max(n, M), which is the order of the dierence equation It is possible to implement an LTI system characterized by a constant coecient dierence equation as here the computation involves two nite sums of products Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 65 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 66 / 75
Finite-Dimensional LTI Discrete-Time s Finite-Dimensional LTI Discrete-Time s If we assume the system to be causal, then the output y[n] can be recursively computed using where d 0 0 N d k y[n] = y[n k] + d 0 k=1 M p k d 0 x[n k] y[n] can be computed for all n n 0, knowing x[n] and the initial conditions y[n 0 ], y[n 1 ],..., y[n N] Finite-Dimensional LTI Discrete-Time s Finite Impulse Response (FIR) Discrete-Time s Finite Impulse Response (FIR) Discrete-Time s Based on Impulse Response Length: If the impulse response h[n] is of nite length, i.e., h[n] = 0 for n < N 1 and n > N, N 1 < N then it is known as a nite impulse response (FIR) discrete-time system The convolution sum description here is y[n] = N k=n 1 h[k] x[n k] Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 67 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 68 / 75 Finite-Dimensional LTI Discrete-Time s Finite Impulse Response (FIR) Discrete-Time s Finite-Dimensional LTI Discrete-Time s Innite Impulse Response (IIR) Discrete-Time s Innite Impulse Response (IIR) Discrete-Time s The output y[n] of an FIR LTI discrete-time system can be computed directly from the convolution sum as it is a nite sum of products Examples of FIR LTI discrete-time systems are the moving-average system and the linear interpolators If the impulse response is of innite length, then it is known as an innite impulse response (IIR) discrete-time system The class of IIR systems we are concerned with in this course are characterized by linear constant coecient dierence equations Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 69 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 70 / 75 Finite-Dimensional LTI Discrete-Time s Innite Impulse Response (IIR) Discrete-Time s Finite-Dimensional LTI Discrete-Time s Innite Impulse Response (IIR) Discrete-Time s Examples: Example: The discrete-time accumulator dened by is seen to be an IIR system y[n] = y[n 1] + x[n] Example: The familiar numerical integration formulas that are used to numerically solve integrals of the form y(t) = t 0 x(τ) dτ can be shown to be characterized by linear constant coecient dierence equations, and hence, are examples of IIR systems If we divide the interval of integration into n equal parts of length T, then the previous integral can be rewritten as nt y(nt ) = y((n 1)T ) + x(τ) dτ (n 1)T where we have set t = nt and used the notation y(nt ) = nt 0 x(τ) dτ Using the trapezoidal method we can write nt (n 1)T x(τ) dτ = T {x((n 1)T ) + x(nt )} Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 71 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 7 / 75
Finite-Dimensional LTI Discrete-Time s Innite Impulse Response (IIR) Discrete-Time s Hence, a numerical representation of the denite integral is given by Finite-Dimensional LTI Discrete-Time s Nonrecursive and Recursive s Nonrecursive and Recursive s y(nt ) = y((n 1)T ) + T {x((n 1)T ) + x(nt )} Let y[n] = y(nt ) and x[n] = x(nt ) Then y(nt ) = y((n 1)T ) + T {x((n 1)T ) + x(nt )} reduces to y[n] = y[n 1] + T {x[n 1] + x[n]} Based on the Output Calculation Process: Nonrecursive : Here the output can be calculated sequentially, knowing only the present and past input samples Recursive : Here the output computation involves past output samples in addition to the present and past input samples which is recognized as the dierence equation representation of a rst-order IIR discrete-time system Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 73 / 75 Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 74 / 75 Finite-Dimensional LTI Discrete-Time s Real and Complex s Real and Complex s Based on the Coecients: Real Discrete-Time : The impulse response samples are real valued Complex Discrete-Time : The impulse response samples are complex valued Umut Sezen (Hacettepe University) EEM401 Digital Signal Processing 06-Nov-01 75 / 75