Discrete-Time LTI Systems And Analysis
Discrete-Time LTI SystemsDiscrete-time SystemsInput-Output Description of Dst-Time Systemsinput/excitationDiscrete-Time LTI Systems and Discrete-timesignalDiscrete-timesignalDr. Deepa KundurUniversity of TorontoIInput-output description (exact structure of system is unknownor ignored):y (n) T [x(n)]I“black box” representation:Tx(n) y (n)Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and AnalysisDiscrete-Time LTI Systems1 / 61Dr. Deepa Kundur (University of Toronto)Discrete-time SystemsDiscrete-Time LTI Systems and AnalysisDiscrete-Time LTI SystemsSystem Properties2 / 61Discrete-time SystemsTerminology: ImplicationIf “A” then “B”Shorthand: A BWhy is this so important?IImathematical techniques developed to analyze systems are oftencontingent upon the general characteristics of the systems beingconsideredfor a system to possess a given property, the property must holdfor every possible input to the systemIIExample 1:it is snowing it is at or below freezing temperatureExample 2:α 5.2 α is positiveNote: For both examples above, B 6 Ato disprove a property, need a single counter-exampleto prove a property, need to prove for the general caseDr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis3 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis4 / 61
Discrete-Time LTI SystemsDiscrete-time SystemsDiscrete-Time LTI SystemsTerminology: EquivalenceDiscrete-time SystemsCommon PropertiesIf “A” then “B”Shorthand: A BandIf “B” then “A”ITime-invariant system: input-output characteristics do notchange with timeIa system is time-invariant iffShorthand: B ATTx(n) y (n) x(n n0 ) y (n n0 )can be rewritten asfor every input x(n) and every time shift n0 .Shorthand: A B“A” if and only if “B”We can also say:IA is EQUIVALENT to BIA BDr. Deepa Kundur (University of Toronto) Discrete-Time LTI Systems and AnalysisDiscrete-Time LTI Systems5 / 61Discrete-time SystemsDiscrete-Time LTI Systems and AnalysisDiscrete-Time LTI Systems6 / 61Discrete-time SystemsAdditivity:Common PropertiesIDr. Deepa Kundur (University of Toronto)Linear system: obeys superposition principleIa system is linear iffT [a1 x1 (n) a2 x2 (n)] a1 T [x1 (n)] a2 T [x2 (n)]for any arbitrary input sequences x1 (n) and x2 (n), and anyarbitrary constants a1 and a2 .Homogeneity:Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis7 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis8 / 61
Discrete-Time LTI SystemsDiscrete-time SystemsDiscrete-Time LTI SystemsCommon PropertiesIThe Convolution SumThe Convolution SumCausal system: output of system at any time n depends only onpresent and past inputsIa system is causal iffRecall:y (n) F [x(n), x(n 1), x(n 2), . . .] Xx(n) for all n.x(k)δ(n k)k IBounded Input-Bounded output (BIBO) Stable: every boundedinput produces a bounded outputIa system is BIBO stable iff x(n) Mx y (n) My for all n and for all possible bounded inputs.Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and AnalysisDiscrete-Time LTI Systems9 / 61Dr. Deepa Kundur (University of Toronto)The Convolution SumDiscrete-Time LTI Systems and AnalysisDiscrete-Time LTI SystemsThe Convolution Sum10 / 61The Convolution SumThe Convolution SumLet the response of a linear time-invariant (LTI) system denoted T tothe unit sample input δ(n) be h(n).Therefore,Tδ(n) h(n)Tδ(n k) h(n k)y (n) Tα δ(n k) α h(n k) Xx(k)h(n k) x(n) h(n)k Tx(k) δ(n k) x(k) h(n k) XXTx(k)δ(n k) x(k)h(n k)k for any LTI system.k Tx(n) y (n)Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis11 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis12 / 61
Discrete-Time LTI SystemsThe Convolution SumDiscrete-Time LTI SystemsCausality and ConvolutionStability and ConvolutionFor a causal system, y (n) only depends on present and past inputsvalues. Therefore, for a causal system, we have: Xy (n) It can also be shown that Xh(k)x(n k)k 1X Xh(k)x(n k) k X h(n) LTI system is BIBO stablen h(k)x(n k)Note:I means that the two statements are equivalentI BIBO bounded-input bounded-outputk 0h(k)x(n k) The Convolution Sumk 0where h(n) 0 for n 0 to ensure causality.Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and AnalysisDiscrete-Time LTI Systems13 / 61Dr. Deepa Kundur (University of Toronto)The Convolution SumDiscrete-Time LTI Systems and AnalysisDiscrete-Time LTI SystemsPROOFFor a stable system, y (n) is bounded if x(n) is bounded. What are the implications on h(n)?We have: X y (n) h(k)x(n k) h(k)x(n k) k XPTo prove the reverse implication (i.e., necessity), assuming n h(n) we must find abounded input x(n) that will always result in an unbounded y (n). Recall, X h(k) Mx MxTherefore,k y (0) X Consider x(n) sgn(h( n)); note: x(n) 1.y (0) X h(k)x( k)k h(k) X h(k)sgn(h( ( k))) k and we can write: X h(n) h(k)x( k)k h(k) k X Xh(k)x(0 k) k k Xh(k)x(n k)k h(k) is a sufficient condition to guarantee:y (n) Mx h(k) · x(n k) {z }k x(n) Mx k P y (n) X XThe Convolution SumPROOFk X14 / 61LTI system is stable Xh(k)sgn(h(k))k h(n) n n Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis15 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis16 / 61
Discrete-Time LTI SystemsThe Convolution SumDiscrete-Time LTI SystemsThe z-Transform and System FunctionThe Direct z-TransformPROOFTherefore, XI h(n) Direct z-Transform:n guarantees that there exists a bounded input that will result in an unbounded output, so it isalso a necessary condition and we can write:X (z) Xx(n)z nn X h(n) LTI system is stableIn Notation:Putting sufficiency and necessity together we obtain: X h(n) X (z)LTI system is stableZ{x(n)}Zx(n) X (z)n Note: means that the two statements are equivalent.Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and AnalysisDiscrete-Time LTI Systems17 / 61Dr. Deepa Kundur (University of Toronto)The z-Transform and System FunctionDiscrete-Time LTI SystemsRegion of ConvergenceI18 / 61The z-Transform and System Functionz-Transform PropertiesPropertyNotation:IDiscrete-Time LTI Systems and Analysisthe region of convergence (ROC) of X (z) is the set of all valuesof z for which X (z) attains a finite valueThe z-Transform is, therefore, uniquely characterized by:1. expression for X (z)2. ROC of X (z)Linearity:Time shifting:Time Domainx(n)x1 (n)x2 (n)a1 x1 (n) a2 x2 (n)x(n k)z-DomainX (z)X1 (z)X1 (z)a1 X1 (z) a2 X2 (z)z k X (z)z-Scaling:Time an x(n)x( n)x (n)n x(n)x1 (n) x2 (n)X (a 1 z)X (z 1 )X (z )dX (z) z dzX1 (z)X2 (z)ROCROC: r2 z r1ROC1ROC2At least ROC1 ROC2ROC, exceptz 0 (if k 0)and z (if k 0) a r2 z a r11 z r1r12ROCr2 z r1At least ROC1 ROC2among others . . .Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis19 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis20 / 61
Discrete-Time LTI SystemsThe z-Transform and System FunctionDiscrete-Time LTI SystemsCommon Transform Pairs123456Signal, x(n)δ(n)u(n)an u(n)nan u(n) an u( n 1) nan u( n 1)7(cos(ω0 n))u(n)8(sin(ω0 n))u(n)9(an10(an sin(ω0 n)u(n)cos(ω0 n)u(n)Dr. Deepa Kundur (University of Toronto)The System Functionz-Transform, X (z)111 z 111 az 1az 1(1 az 1 )211 az 1az 1(1 az 1 )21 z 1 cos ω01 2z 1 cos ω0 z 2z 1 sin ω01 2z 1 cos ω0 z 21 az 1 cos ω01 2az 1 cos ω0 a2 z 21 az 1 sin ω01 2az 1 cos ω0 a2 z 2ROCAll z z 1 z a z a z a z a Zh(n) H(z)Ztime-domain z-domainZimpulse response system functionZy (n) x(n) h(n) Y (z) X (z) · H(z) z 1 z 1Therefore, z a H(z) z a Discrete-Time LTI Systems and AnalysisDiscrete-Time Fourier AnalysisThe z-Transform and System Function21 / 61Dr. Deepa Kundur (University of Toronto)DTFTDiscrete-Time LTI Systems and AnalysisDiscrete-Time Fourier AnalysisDiscrete-Time Fourier Transform (DTFT)Y (z)X (z)22 / 61DTFTPeriodicity of the DTFTConsiderIDTFT pair:X (ω 2π) Z1x(n) X (ω)e jωn dω2π 2π XX (ω) x(n)e jωn n I X (ω) is the decomposition of x(n) into its frequencycomponents. Xn Xn Xn x(n)e j(ω 2π)nx(n)e jωn · e j2πnx(n)e jωn·1 Xx(n)e jωn X (ω)n Therefore, X (ω) is periodic with a period of 2π.Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis23 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis24 / 61
Discrete-Time Fourier AnalysisDTFTDiscrete-Time Fourier AnalysisPeriodicity of the DTFTIIIIPeriodicity of the DTFTISince X (ω) X (ω 2π), when dealing with discretefrequencies, only a continuous frequency range of length 2π(representing one period) needs to be considered.IDTFTContinuous-Time Sinusoids: Frequency and Rate of Oscillation:x(t) A cos(ωt φ)Minimum frequency for ω 2kπ, k ZMaximum frequency for ω (2k 1)π, k ZConvention is to use ω [0, 2π) or ω ( π, π]T 2π1 ωfFrequency range of a discrete-time signal is considered to beω ( π, π]Rate of oscillation increases as ω increases (or T decreases).Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and AnalysisDiscrete-Time Fourier Analysis25 / 61DTFTDiscrete-Time LTI Systems and AnalysisDiscrete-Time Fourier Analysisω smallerDr. Deepa Kundur (University of Toronto)Dr. Deepa Kundur (University of Toronto)26 / 61DTFTω larger, rate of oscillation higherDiscrete-Time LTI Systems and Analysis27 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis28 / 61
Discrete-Time Fourier AnalysisDTFTDiscrete-Time Fourier AnalysisDTFTPeriodicity of the DTFTMINIMUM OSCILLATIONIDiscrete-Time Sinusoids: Frequency and Rate of Oscillation:x[n] A cos(Ωn φ)MAXIMUM OSCILLATIONRate of oscillation increases as Ω increases UP TO A POINT thendecreases again and then increases again and then decreases again.MINIMUM OSCILLATIONDr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and AnalysisDiscrete-Time Fourier Analysis-3-3-2-2Dr. Deepa Kundur (University of Toronto)29 / 61Dr. Deepa Kundur (University of Toronto)DTFT-101-10122Discrete-Time LTI Systems and AnalysisDiscrete-Time Fourier Analysis345345Discrete-Time LTI Systems and Analysis6-36-331 / 61-2-2Dr. Deepa Kundur (University of Toronto)30 / 61DTFT-101-10122345345Discrete-Time LTI Systems and Analysis6632 / 61
Discrete-Time Fourier AnalysisDTFTDiscrete-Time LTI FilteringDTFT Theorems and PropertiesLTI Filteringy (n) PropertyNotation:Linearity:Time shifting:Time reversalConvolution:Correlation:Time Domainx(n)x1 (n)x2 (n)a1 x1 (n) a2 x2 (n)x(n k)x( n)x1 (n) x2 (n)rx1 x2 (l) x1 (l) x2 ( l)Wiener-Khintchine:rxx (l) x(l) x( l) XFrequency DomainX (ω)X1 (ω)X1 (ω)a1 X1 (ω) a2 X2 (ω)e jωk X (ω)X ( ω)X1 (ω)X2 (ω)Sx1 x2 (ω) X1 (ω)X2 ( ω) X1 (ω)X2 (ω) [if x2 (n) real]Sxx (ω) X (ω) 2Y (ω) H(ω)X (ω)whereFx(n) Fh(n) Fy (n) H(ω)among others . . . H(ω) H(ω) Θ(ω) Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysisx(k)h(n k)k 33 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI FilteringX (ω)H(ω)Y (ω) H(ω) e jΘ(ω)system gain for freq ωphase shift for freq ωDiscrete-Time LTI Systems and Analysis34 / 61Discrete-Time LTI FilteringComplex Nature of X (jω)Complex Nature of X (jω)Recall, Fourier Transform:ZX (jω) x(t)e jωt dt IRectangular coordinates: rarely used in signal processingC X (jω) XR (jω) j XI (jω)and Inverse Fourier Transform:Z 1x(t) X (jω)e jωt dω2π Z 0Z 11jωt X (jω)e dω X (jω)e jωt dω2π 2π 0where XR (jω), XI (jω) R.IX (jω) X (jω) e j X (jω)Note: If x(t) is real, then the imaginary part of the negative frequency sinusoids(i.e., e jωt for ω 0) cancel out the imaginary part of the positive frequencysinusoids (i.e., e jωt for ω 0)Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and AnalysisPolar coordinates: more intuitive way to represent frequency content35 / 61where X (jω) , X (jω) R.Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis36 / 61
Discrete-Time LTI FilteringDiscrete-Time LTI FilteringMagnitude and Phase of X (jω)Magnitude and Phase of X (jω)x(t)I X (jω) : determines the relative presence of a sinusoid e jωt inx(t) I X (jω): determines how the sinusoids line up relative to oneanother to form x(t)Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis37 / 61Z 1X (jω)e jωt dω2π Z 1 X (jω) e j X (jω) e jωt dω2π Z 1 X (jω) e j(ωt X (jω)) dω2π IRecall, e j(ωt X (jω)) cos(ωt X (jω)) j sin(ωt X (jω)).IThe larger X (jω) is, the more prominent e jωt is in forming x(t).I X (jω) determines the relative phases of the sinusoids (i.e. how they lineup with respect to one another).Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI FilteringDiscrete-Time LTI Systems and Analysis38 / 61Discrete-Time LTI FilteringLTI FilteringLTI Systems as Frequency-Selective Filtersy (n) X x(k)h(n k)k Y (ω) H(ω)X (ω)whereFx(n) Fh(n) Fy (n) X (ω)H(ω)IFilter: device that discriminates, according to some attribute ofthe input, what passes through itIFor LTI systems, given Y (ω) H(ω)X (ω)IY (ω)IH(ω) Y (ω) Y (ω)Dr. Deepa Kundur (University of Toronto)H(ω) acts as a weighting or spectral shaping function of thedifferent frequency components of the signalLTI system is known as a frequency shaping filterLTI system filter H(ω) e jΘ(ω) H(ω) X (ω) Θ(ω) X (ω)Discrete-Time LTI Systems and Analysis39 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis40 / 61
Discrete-Time LTI FilteringDiscrete-Time LTI FilteringCausal FIR FiltersCausal IIR FiltersDefinition: a discrete-time finite impulse response (FIR) filter is onein which the associated impulse response has finite duration.y (n) XDefinition: a discrete-time infinite impulse response (IIR) filter is onein which the associated impulse response has infinite duration.h(k)x(n k)k M 1X Xy (n) k h(k)x(n k) X k 0IIh(k)x(n k)k 0lower limit of k 0 is from causality requirementupper limit of 0 M 1 is from the finite durationrequirement; in this case the support is M consecutive pointsstarting at time 0 and ending at M 1Dr. Deepa Kundur (University of Toronto)h(k)x(n k)Discrete-Time LTI Systems and AnalysisII41 / 61lower limit of k 0 is from causality requirementnecessary upper limit of is from the infinite durationrequirementDr. Deepa Kundur (University of Toronto)Discrete-Time LTI FilteringDiscrete-Time LTI Systems and Analysis42 / 61Discrete-Time LTI FilteringLCCDEsLCCDEsLinear constant coefficient difference equations (LCCDEs) are animportant class of filters that we consider in this course:Q: Why does an LCCDE have a rational system function?y (n) NXak y (n k) k 1y (n) NXk 1ak y (n k) MXbk x(n k)a0 y (n) k 0They have a rational system function:PM kpolynomial in zk 0 bk zH(z) PNanother polynomial in z1 k 1 ak z kak y (n k) k 0Z{ak y (n k) NXak y (n k)}NX43 / 61ak Z{y (n k)}k 0Dr. Deepa Kundur (University of Toronto)MXMXbk x(n k)a0 1k 0bk x(n k)k 0 k 0Depending on the values of N, M, ak and bk they can correspond toeither FIR or IIR filters.Discrete-Time LTI Systems and AnalysisNXbk x(n k)k 0k 1NXDr. Deepa Kundur (University of Toronto) MXZ{MXbk x(n k)}k 0 MXbk Z{x(n k)}k 0Discrete-Time LTI Systems and Analysis44 / 61
Discrete-Time LTI FilteringDiscrete-Time LTI Filteringz-Transform PropertiesPropertyNotation:LCCDEsLinearity:Time shifting:Time Domainx(n)x1 (n)x2 (n)a1 x1 (n) a2 x2 (n)x(n k)z-DomainX (z)X1 (z)X1 (z)a1 X1 (z) a2 X2 (z)z k X (z)z-Scaling:Time an x(n)x( n)x (n)n x(n)x1 (n) x2 (n)X (a 1 z)X (z 1 )X (z )dX (z) z dzX1 (z)X2 (z)NXROCROC: r2 z r1ROC1ROC2At least ROC1 ROC2ROC, exceptz 0 (if k 0)and z (if k 0) a r2 z a r11 z r1r12ROCr2 z r1At least ROC1 ROC2k 0ak Z{y (n k)} {z} k 0z k Y (z)NXak z k Y (z) k 0Y (z)MXMXbk z k X (z)ak z k X (z)Y (z)X (z)PMk 0H(z) 45 / 61MXbk z kk 0 H(z) Discrete-Time LTI Systems and Analysisz k X (z)k 0NXk 0Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Filteringbk z ka0 1 kk 0 ak zPM kk 0 bk zPN· z 0 k 1 ak z kPNamong others . . .Dr. Deepa Kundur (University of Toronto)bk Z{x(n k)} {z}1Discrete-Time LTI Systems and AnalysisPM k 01 PNbk z kk 1 ak z k46 / 61Discrete-Time LTI FilteringFIR LCCDEsBlock Diagram Represenationy (n) M 1Xbk x(n k) H(z) Adder:h(k)x(n k) k k 0M 1X XUnit delay:bk z kk 0Constant multiplier:Please note: upper limit is M 1 opposed to M (which is used for the generalLCCDE case) to meet common FIR convention of an M-length filter.Unit advance: h(n) Dr. Deepa Kundur (University of Toronto)bn 0 n M 10 otherwiseDiscrete-Time LTI Systems and AnalysisSignal multiplier:47 / 61Dr. Deepa Kundur (University of Toronto) By inspection:Discrete-Time LTI Systems and Analysis48 / 61
Discrete-Time LTI FilteringDiscrete-Time LTI FilteringFIR Filter Implementationy (n) M 1XIIR LCCDEsbk x(n k)k 0y (n) . . ak y (n k) k 1PMH(z) NX 1 k 0PNbk z kk 1ak z k MXk 0MXbk x(n k)1PN1 k 1 ak z k{z}} bk z k · k 0 {zH1 (z)H2 (z) H1 (z) · H2 (z)Requires:I M multiplicationsI M 1 additionsI M 1 memory elementsDr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis49 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI FilteringAdder:Discrete-Time LTI Systems and Analysis50 / 61Discrete-Time LTI Filtering Direct Form I IIR Filter ImplementationDirect Form II IIR Filter ImplementationUnit delay: Unit LTI All-zero system.advance: LTI All-pole systemDiscrete-Time LTI Systems and Analysis LTI All-pole systemRequires: M N 1 multiplications, M N additions, M N memory locationsDr. Deepa Kundur (University of Toronto). . . .Signal multiplier: Constant multiplier: 51 / 61LTI All-zero systemRequires: M N 1 multiplications, M N additions, M N memory locationsDr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis52 / 61
Discrete-Time LTI FilteringDiscrete-Time LTI FilteringDirect Form II IIR Filter Implementation Stability of Rational System Function Filtersy (n) NXak y (n k) k 1.H(z) 1 k 0PN bk x(n k)k 0PM . MXbk zk 1 kak z k.Recall, for BIBO stability of a causal system the system poles mustbe strictly inside the unit circle.For N MWhy?Requires: M N 1 multiplications, M N additions, max(M, N) memorylocationsDr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis53 / 61Discrete-Time LTI FilteringAdder:Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis54 / 61Discrete-Time LTI Filtering Stability of Rational System Function FiltersStability of Rational System Function FiltersUnit delay:Constant multiplier:H(z) Unit advance:Recall, H(z) n Xh(n)z n h(n)z n X h(n) z n XSignal multiplier: X h(n) LTI system is stablen n n When evaluated for z 1 (i.e., on the unit circle), H(z) X h(n) n Therefore, BIBO stability ROC includes unit circleROC includes unit circle BIBO stability is also true.Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis55 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis56 / 61
Discrete-Time LTI FilteringDiscrete-Time LTI FilteringStability of Rational System Function FiltersStability of Rational System Function Filters. . . becauseTherefore, X h(n) n H(z) ROCLTI system includesis stableunit circleFor a causal rational system function, the ROC includes the unitcircle if all the poles are inside the unit circle.Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis57 / 61Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI FilteringDiscrete-Time LTI Systems and AnalysisDiscrete-Time LTI FilteringARMA, MA and AR FiltersARMA, MA and AR FiltersOther commonly used terminology for the filters described include:I Autoregressive moving average (ARMA) filter:Other commonly used terminology for the filters described include:I Moving average (MA) filter:y (n) H(z) II1NXak y (n k) k 1PM kk 0 bk zP Nk 1 ak z kMXbk x(n k)y (n) k 0H(z) Discrete-Time LTI Systems and AnalysisMXk 0MXbk x(n k)bk z kk 0has both poles and zerosIIRDr. Deepa Kundur (University of Toronto)58 / 61II59 / 61has zeros only; no poles; is BIBO stableFIRDr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis60 / 61
Discrete-Time LTI FilteringARMA, MA and AR FiltersOther commonly used terminology for the filters described include:I Autoregressive (AR) filter:y (n) NXak y (n k)k 1H(z) II1 1PNk 1ak z khas poles only; no zerosIIR Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis61 / 61
IThe z-Transform is, therefore, uniquely characterized by: 1.expression for X(z) 2.ROC of X(z) Dr. Deepa Kundur (University of Toronto)Discrete-Time LTI Systems and Analysis19 / 61 Discrete-Time LTI SystemsThe z-Transform and System Function z-Trans
Two important basic Properties of systems: Linearity. Time-invariance (superposition property). Plays an important role in signals and systems analysis. Many of physical processes possess these properties. They are modeled as LTI systems Any system possess these two properties is called linear time- invariant (LTI) system. 3/17
2.1 Sampling and discrete time systems 10 Discrete time systems are systems whose inputs and outputs are discrete time signals. Due to this interplay of continuous and discrete components, we can observe two discrete time systems in Figure 2, i.e., systems whose input and output are both discrete time signals.
Safari Montage has implemented a way to solve this problem using IMS Learning Tools Interoperability (LTI) . Although Simple LTI Search is not a formal part of the Learning Tools Interoperability Standard, this application note provides information on how this functionality can be achieved using LTI
y[n] X k h[n k]x[k] (discrete-time) where h is called the impulse response of the system. Another important property of LTI systems is their action on complex exponentials: if S is LTI, then S n ejωt o c(ω)ejωt for some complex number c(ω). So LTI systems can attenu
Time-domain analysis of discrete-time LTI systems Discrete-time signals Di erence equation single-input, single-output systems in discrete time The zero-input response (ZIR): characteristic values and modes The zero (initial) state response (ZSR): the unit-pulse response, convolution System stability The eigenresponse .
2.1 Discrete-time Signals: Sequences Continuous-time signal - Defined along a continuum of times: x(t) Continuous-time system - Operates on and produces continuous-time signals. Discrete-time signal - Defined at discrete times: x[n] Discrete-time system - Operates on and produces discrete-time signals. x(t) y(t) H (s) D/A Digital filter .
Definition and descriptions: discrete-time and discrete-valued signals (i.e. discrete -time signals taking on values from a finite set of possible values), Note: sampling, quatizing and coding process i.e. process of analogue-to-digital conversion. Discrete-time signals: Definition and descriptions: defined only at discrete
Apprendre à accorder la guitare par vous même. Laguitaretousniveaux 11 Se familiariser avec le manche Ce que je vous propose ici, c'est de travailler la gamme chromatique, pour vous entraîner à faire sonner les notes. C'est un exercice qui est excellent pour cela, ainsi que pour s'échauffer avant de jouer. Le principe est très simple, il s'agit de placer consécutivement chaque doigt sur .
