PETER PAZMANY CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

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1 PETER PAZMANY CATHOLIC UNIVERSITY SEMMELWEIS UNIVERSITY Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework** Consortium leader PETER PAZMANY CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund *** **Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben ***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg. 0/5/20. TÁMOP /2/A/KMR

2 Peter Pazmany Catholic University Faculty of Information Technology Digital- and Neural Based Signal Processing & Kiloprocessor Arrays Digitális- neurális-, és kiloprocesszoros architektúrákon alapuló jelfeldolgozás Description digital signals and systems in time domain Digitális jelek és rendszerek időbeli leírása és analízise András Olás, Gergely Treplán, Dávid Tisza 0/5/20. TÁMOP /2/A/KMR

3 Introduction Contents Review of sampling and quantizing Most important categories of discrete time signals Basic definitions and operations on discrete time signals Elementary discrete time signals Elementary operations on discrete time signals Convolution Even-odd decomposition Discrete Time system Linearity Time invariance 0/5/20. TÁMOP /2/A/KMR

4 Causality Stability, BIBO stability LTI systems Contents Definitions of an LTI system Impulse response of an LTI system Properties of the convolution relation to the LTI systems FIR, IIR systems Block diagrams, signal flow diagrams of LTI systems Basic flow graph types of a system Direct form, 2 Transposed forms Serial, parallel forms Description of the LTI systems by difference equation 0/5/20. TÁMOP /2/A/KMR

5 Contents General method for solving LTI system type difference equations Examples Stability Time invariance Causality LTI system analysis in time domain 0/5/20. TÁMOP /2/A/KMR

6 Introduction review of signals A signal is a physical quantity in space, time and other dimensions in the physical reality (e.g. voltage, current) In mathematical sense a signal is a model of the physical quantity, a function of one or more independent variables (usually time or frequency) e.g.: f t = a si n t exp t, a = 2, t A discrete time (DT) signal is a function of time where the domain of time consist of a discrete set. It is a sequence in a mathematical sense e.g.: = exp, f k a sin k k a = 2, k {0,, 2,, 30} + 0/5/20. TÁMOP /2/A/KMR

7 Introduction review of signals A DT signal has values only where the domain has elements, it is undefined at other places. f ( k) = a sin( k) exp( k), a = 2, k {0,, 2,, 30} 0/5/20. TÁMOP /2/A/KMR

8 Introduction review of sampling, quantization x(t) Sampling Quantization Coding T ΔT x(nt) x(n) ˆx ( n ) Optimal representation Compressing c n Signal Time Value Analog signal x(t) Continuous Continuous Sampled signal x(n) or x(nt) Discrete Continuous Quantized signal x ˆk Discrete Discrete Coded signal c n Discrete Binary 0/5/20. TÁMOP /2/A/KMR

9 Introduction review of sampling, quantization We assume that we have a sufficiently precise quantizer so we neglect the effect of quantization when we are analyzing discrete time systems and signals (we are working only with the sampled signal) The consequence of the sampling theorem is that the signals are to be analyzed are reconstructable without loss only from the sampled versions. 0/5/20. TÁMOP /2/A/KMR

10 Categories of discrete time signals The most important properties of the DT signals are Time behavior, amplitude behavior, periodicity: support (finite, infinite, entrant), energy, power, even-oddness 0/5/20. TÁMOP /2/A/KMR

11 Categories of discrete time signals 0/5/20. TÁMOP /2/A/KMR

12 Categories of discrete time signals 0/5/20. TÁMOP /2/A/KMR

13 Categories of discrete time signals If the energy of a signal is finite, the average power is null. There exist several signal of which it s energy is infinite but the average power is finite. E.g. x n = sin( n) N 2 x n n= N lim = N 2N + 2 0/5/20. TÁMOP /2/A/KMR

14 Elementary DT signals Kronecker delta or unit sample, if n = 0 x( n) = δ ( n) = 0, otherwise Unit step function, if n 0 x( n) = u( n ) = 0, otherwise Unit ramp function n, if n 0 x( n) = ur ( n) = 0, otherwise 0/5/20. TÁMOP /2/A/KMR

15 Exponential function Digital Signal Processing: Time domain description Elementary DT signals n, if ( jϕ = a = ) x n a a, x n, if, x n re = r e n n jϕn 0/5/20. TÁMOP /2/A/KMR

16 Operations on DT signals basic operations Addition: Multiplication with constant: Multiplication: y n = g( n) h n Time shift: Accumulation: Digital Signal Processing: Time domain description = + y n g n h n k Discrete time convolution: = a g( n) y n { } y n = S g n = g n k y n n = g( k) k = 0 = = y n g( n) h n : g( k) h( n k) h( k) g( n k) k= k= 0/5/20. TÁMOP /2/A/KMR

17 Operations on DT signals basic operations Addition: = + = sin ( n) = u( n) y n g( n) h( n) g n h n Digital Signal Processing: Time domain description Multiplication with constant: = a g( n) = sin ( n) y n g n a = 2 0/5/20. TÁMOP /2/A/KMR

18 Operations on DT signals basic operations Multiplication: = = sin ( n) = y n g n h n u n Digital Signal Processing: Time domain description g( n) h( n) r Time shift: k = u( n) k = 3 { } y n = S g n = g n k g n 0/5/20. TÁMOP /2/A/KMR

19 Operations on DT signals basic operations Accumulation: Digital Signal Processing: Time domain description n = = y n g( k), g n u n 0.9 k = 0 n 0/5/20. TÁMOP /2/A/KMR

20 Operations on DT signals basic operations Discrete Time convolution: g ( n) Upper figure: Blue dots: Purple dots: Red bars: Lower figure: Digital Signal Processing: Time domain description ( n) Blue dots: gk hn ( k) k = Red dot (sum of red bars): value of convolution at time instant n y n = g( n) h n : = g( k) h( n k) sin 0. 2π 5 n 5 0.2n 5 n 5 = h( n) = 0 otherwise 0 otherwise g( k) h( n k) g( k) h( n k) k = 0/5/20. TÁMOP /2/A/KMR

21 Operations on DT signals basic operations Discrete Time convolution: g ( n) ( n) sin 0. 2π 5 n 5 0.2n 5 n 5 = h( n) = 0 otherwise 0 otherwise y n = g( n) h n : = g( k) h( n k) k = 0/5/20. TÁMOP /2/A/KMR

22 Operations on DT signals basic operations Discrete Time convolution: g ( n) ( n) sin 0. 2π 5 n 5 0.2n 5 n 5 = h( n) = 0 otherwise 0 otherwise y n = g( n) h n : = g( k) h( n k) k = 0/5/20. TÁMOP /2/A/KMR

23 Operations on DT signals algebraic properties of the discrete time convolution: Linear operator Commutativity Associativity Distributivity Associativity with scalar multiplication ( f g) ( f ) g f ( g) o Multiplicative identity Complex conjugation Digital Signal Processing: Time domain description f g = g f f ( g h) = ( f g) h f ( g + h) = ( f g) + ( f h) λ = λ = λ, λ r λ δ, f δ = f f g = f g 0/5/20. TÁMOP /2/A/KMR

24 Operations on DT signals algebraic properties of the discrete time convolution: Integration: Differentiation: Digital Signal Processing: Time domain description f g x dx= f x dx g x dx d d d R R R = ( f ( n) g( n) ) f ( n) g( n) n= n= n= d df dg ( f g) = g = f dx dx dx for discrete case, if the operator Df n : f n f n ( ) = = D f g Df g f Dg = ( + ) Time invariance: ( ) = = k k k S f g S f g f S g 0/5/20. TÁMOP /2/A/KMR

25 Operations on DT signals Signal representation by convolution (multiplicative identity): Similarly as the definition of the Dirac delta we can use the same structure to define an arbitrary DT signal with convolution. Dirac delta (continuous time): + + t = 0 δ() t =, and δ() t has the property of δ() t = 0 t 0 Kronecker delta alternate definition (the property comes from definition): n = 0 δ( n) =, and δ( n) has the property of δ( n) = 0 n 0 n= Signal representation by convolution: every signal can be represented as a series of weighted Kronecker deltas yn y n = δ( n) y n = δ( k) k = y( k) δ( n k) k= k= 0/5/20. TÁMOP /2/A/KMR

26 Operations on DT signals even-odd decomposition An important property of a DT signal for it s analysis is that it can be decomposed to even and odd parts: x( n) = sin( 2n+ 2) xeven ( n) = ( x( n) + x( n) ) 2 xodd ( n) = ( x( n) x( n) ) 2 = + x n x n x n even odd 0/5/20. TÁMOP /2/A/KMR

27 Discrete Time system A DT system is an object which operates in a discrete time fashion. We can describe a DT system with the input-output model: A DT system has an input (inputs) A DT system has an output (outputs) Single input single output SISO Single input multiple output SIMO Multiple input single output MISO Multiple input multiple output MIMO In this course we are dealing with SISO systems only 0/5/20. TÁMOP /2/A/KMR

28 Discrete Time system A DT system s input-output model is described by a mapping, a rule between the input and the output. A DT system is a function which operates on the input. x( n ) y ( n) Input, stimulus Discrete Time system =Ψ x( n) y n Output, system response We can describe the simplest DT systems with the basic signal operations mentioned earlier. 0/5/20. TÁMOP /2/A/KMR

29 Operations of DT systems basic operations Addition: Multiplication with constant: Multiplication: Digital Signal Processing: Time domain description y( n) = g( n) + h( n) y( n) = g( n) h( n) k Time shift: Shift register: n Accumulation: y( n) = g k k = 0 Discrete time convolution: = a g( n) y n { } y n = S g n = g n k yn = gn hn: = gk hn ( k) k = g( n) h( n) g n h( n) y( n) g( n ) g( n 4) T T T T g( n ) y( n) h( n) y ( n) g( n ) y ( n) a g( n ) g( n ) T g( n ) y ( n) y( n ) T 0/5/20. TÁMOP /2/A/KMR

30 Discrete Time system - Linearity A DT system is linear if: y n =Ψ α x n + β x n = α Ψ x n + β Ψ x n A DT system is non linear if that does not hold. Example of a linear DT system: ( 2 ) ( ) ( 2) ( u( n) ) u( n) 2u( n ) Ψ = + ( ) LHS: Ψ α x n + β x2 n = α x n + β x2 n + 2 α x n + β x2 n ( ) 2un RHS: α Ψ ( x ) ( ) n + β Ψ x2 n = α x( n) + 2x n + β x2( n) + 2x2 n LHS=RHS linear system un un 2 ( ) un un 2un ( ) 0/5/20. TÁMOP /2/A/KMR

31 Discrete Time system - Linearity Example of a non linear DT system: 2 ( u( n) ) u ( n) u( n ) ( ) Ψ = + ( ) ( n) 2 2 LHS: Ψ α x n + β x2 n = α x n + β x2 n + α x n + β x2 n ( ) ( 2 ) 2 RHS: α Ψ x n + β Ψ x n = α x ( n) + x n + β x ( n) + x n LHS RHS non linear system u 2 2 u ( n) un ( ) u ( n) un ( ) un 0/5/20. TÁMOP /2/A/KMR

32 Discrete Time system Time invariance A DT system is time invariant if the system operator is invariant to time shifts (the system does the same on Monday, Tuesday, and on every holidays as well) { } k k{ } y n =Ψ S u n = S Ψ u n A DT system is time variant if the system is dependent of the time when is it evaluated. { } k k{ } y n =Ψ S u n S Ψ u n = y n 2 0/5/20. TÁMOP /2/A/KMR

33 Discrete Time system Time invariance Example of a time invariant DT system: ( u( n) ) 2 u ( n) u( n ) k { } k ( ) k Ψ = + 2 LHS: Ψ S u n =Ψ u n k = u n k + u n k { } { 2 } 2 Ψ = + = ( ) + ( ) RHS: S u n S u n u n u n k u n k LHS=RHS time invariant system 0/5/20. TÁMOP /2/A/KMR

34 Discrete Time system Time invariance Example of a time variant DT system: 2 ( u( n) ) u( n ) u( 2n ) Ψ = + { } ( k ) ( ) { } k ( 2 ) LHS: Ψ S u n =Ψ u n k = u n k + u 2 n k { 2 } 2 k RHS: S Ψ u n = S u n + u 2n = u n k + u 2n k LHS RHS time variant system 0/5/20. TÁMOP /2/A/KMR

35 Discrete Time system - Causality A DT system is causal if the system s next state can be fully determined by the combination of the input at that time instant and the input s previous values. In other words the system is fully determined by the past and the present input. ( n) = Ψ x( n), x( n ), ( n ) y x k A typical causal system could be a real time audio processing codec or physical phenomenon 0/5/20. TÁMOP /2/A/KMR

36 Discrete Time system - Causality A DT system is anticausal if it only depends on the future input values y n = Ψ x n+ k,, x n+ 2, x n+ A DT system is acausal or non-causal if the system s response needs both future and past input values y n = Ψ x n+ k,, x n+ 2, x n+, x n, x n,, x n m A typical acausal system could be an offline compression algorithm (e.g. zip), where we can seek for future samples to determine the best value at the present 0/5/20. TÁMOP /2/A/KMR

37 Dynamical system stability A dynamical system is said to be stabile if after a time in an abstract wide sense it stays somewhere and does not move elsewhere. There are different types of stability defined for a dynamical system. E.g.: Lyapunov stability if all solutions of a DS start near an equilibrium point and stays close to it, the system is said to be Lyapunov stabile Marginal stability, asymptotic stability Orbital stability, structural stability, BIBO stability, etc. 0/5/20. TÁMOP /2/A/KMR

38 Discrete Time system BIBO stability A DT system is said to be Bounded Input Bounded Output (BIBO) stabile if for every finite amplitude excitation it produces a finite amplitude response. x ( n) system Ψ. is BIBO stable iff M < ( n) the system response: y n = Ψ x M <, n i i i y x i i 0/5/20. TÁMOP /2/A/KMR

39 LTI systems A discrete time system is an LTI (linear time invariant system) if the linearity and the time invariance property holds. x( n ) y( n) Input, stimulus LTI system ( n) x( n) y =Ψ LTI Output, system response An LTI system is fully characterized by its h(n) impulse response function. ΨLTI u( n) = h n u n = h( n) x( n) y n 0/5/20. TÁMOP /2/A/KMR

40 LTI systems Because an LTI system can be viewed as a convolution by it s impulse response function (h(n)) every property which the convolution holds is true for an LTI system as well. x LTI system ( n ) y( n) Input, stimulus h n Output, system response 0/5/20. TÁMOP /2/A/KMR

41 LTI systems impulse response (constructive definition) The impulse response of a system can be defined with the multiplicative identity property of the convolution: The impulse response of a system is the system s response to the Kronecker delta. =, = δ = y n h n x n x n n y n h n δ LTI system ( n) h( n) Input, stimulus h n Output, system response 0/5/20. TÁMOP /2/A/KMR

42 LTI systems consequences of the convolution properties Associativity serial combination of LTI systems = y n h n x n ( 2 ) hn y n = h n h n x n x( n) y n h n h n x n = 2 g n LTI system LTI system h n g ( n) h n h n h n = LTI system 2 h n 2 2 h( n) y( n) 0/5/20. TÁMOP /2/A/KMR

43 LTI systems consequences of the convolution properties Commutativity switch ability of LTI systems y( n) = ( h( n) h2( n) ) x( n) h( n) = h( n) h2( n) = h2( n) h( n) y( n) h ( n) h ( n) x n x( n) ( 2 ) = LTI system LTI system h n g ( n) LTI system 2 h n 2 h( n) y( n) x( n) LTI system LTI system 2 h n 2 f ( n) LTI system h n h( n) y ( n) 0/5/20. TÁMOP /2/A/KMR

44 LTI systems consequences of the convolution properties Commutativity switch ability of impulse response and excitation y ( n) = h( n) x( n) y ( n) = x( n) h( n) x LTI system ( n ) y( n) h n h LTI system 2 ( n ) y ( n) x n 0/5/20. TÁMOP /2/A/KMR

45 LTI systems consequences of the convolution properties Distributivity parallel combination of LTI systems y( n) = h( n) x( n) y( n) = h( n) x( n) + h2( n) x( n) h( n) = h( n) + h2( n) y( n) ( h( n) h2( n) ) x( n) = + LTI system h( n) g( n LTI system ) x( n ) h ( n) y ( n) LTI system 2 h n 2 h( n) 0/5/20. TÁMOP /2/A/KMR

46 LTI systems causality An LTI system is causal iff it s impulse response is an entrant DT signal. y n = hn x n = hk xn ( k) = k = = hk xn ( k) + hk xn ( k) k = is an entrant signal k= 0 future x s past and present x s "future" term must be 0 due to causality h n h n 0, n=, 2, 0/5/20. TÁMOP /2/A/KMR

47 LTI systems BIBO stability An LTI system is BIBO stable iff the system s impulse response is absolute summable. = If a y n h( n) x n = h( k) xn ( k) system is BIBO stabile, M x y( n) n, x n M < and M <, n must hold. y n = h( k) xn ( k) h( k) xn ( k) y n ( n) x h( k) for y M <, n to hold, S = h( k) < must hold y k = k= k= < k = h y k = 0/5/20. TÁMOP /2/A/KMR

48 LTI systems BIBO stability Consequently if an LTI system has an impulse response with finite number elements (limited support), the system is always BIBO stabile. h n h {, } h n \ if A n B = 0 otherwise B ( n) = S = h h n k= k= A 0/5/20. TÁMOP /2/A/KMR

49 LTI systems FIR, IIR systems An LTI system (or equivalently an LTI filter) is said to be of FIR type (finite impulse response) if it s impulse response has limited support (finite length). An system/filter is said to be of IIR type (infinite impulse response) if it s impulse response has infinite support. h h FR I IIR ( n) = {,0,0,,.2,0.5,-0.4,0.3,-0.2,0.,0,0, } ( n) n 0.9 n 0 = 0 otherwise 0/5/20. TÁMOP /2/A/KMR

50 LTI systems flow graph representation Every LTI system can be derived by the combination of the following basic operations: addition, multiplication, time shift Every FIR system is an IIR system There exist IIR systems where the impulse response function can be represented by a closed form formula so they are implementable by a finite number of basic operations. They are called recursive IIR systems. We deal with FIR and Recursive IIR. IIR type systems Recursive IIR systems FIR type systems 0/5/20. TÁMOP /2/A/KMR

51 LTI systems flow graph representation A FIR type LTI system can be viewed as a purely feed forward M type flow graph: y( n) = h( n) x( n) = h( k) x( n k) FIR LTI k = 0 Input, stimulus Output, system response 0/5/20. TÁMOP /2/A/KMR

52 LTI systems flow graph representation Most practically useful IIR filter can be represented by a finite element feedback and feed forward type flow graph: x( n) y n = h( n) = h( k) x( n k), if the system can be represented by: N ( ) = ( ) k k= 0 k= 0 k = 0 a y n k b x n k M without the loss of generality we usually assume a = + ( ) + + N ( ) = 0 + ( ) + + M ( ) = ( n) + b ( n ) + + b x( n M) a y( n ) a y( n N) 0 k y n a y n a y n N b x n b x n b x n M y n b x x M 0 N 0/5/20. TÁMOP /2/A/KMR

53 LTI systems flow graph representation y n = b x n + b x n + + b x n M a y n a y n N 0 M N un~ feed forward term gn~ feedback term IIR LTI 0/5/20. TÁMOP /2/A/KMR

54 LTI systems system equation Implementable systems can be represented by finite element addition, multiplication and time shift operations. We are dealing with such systems. These LTI systems can be described by a linear difference equation (system equation): N M M N a y n k b x n k, a y n b x n k a y n k ( ) = ( ) = = ( ) ( ) k k 0 k k k= 0 k= 0 k= 0 k= The order of this system/filter is the maximum number of time shift used either in the feed forward or the feedback path for the system: filter order= max ( NM, ) 0/5/20. TÁMOP /2/A/KMR

55 Basic flow graph types Direct form I Direct form I implementation of a filter is the direct readout implementation of the system equation: = + ( ) + + ( ) ( ) ( ) y n b x n b x n b x n M a y n a y n N 0 M IIR LTI N 0/5/20. TÁMOP /2/A/KMR

56 Basic flow graph types Direct form II Direct form II implementation of a filter is the direct readout of the modified system equation: = a un ( ) an un ( ) = bun + + b u( n M) un xn N yn 0 M DF-II is a canonical representation respect to time delays. 0/5/20. TÁMOP /2/A/KMR

57 Basic flow graph types Transposed forms Transposed SISO filters can be constructed by exchanging the signal flow directions. q ( n) = S bu( n) + q n M q = { }, + i = M + i i i IIR LTI DF-I Transposed, : 0 0/5/20. TÁMOP /2/A/KMR

58 Other flow graph types There exist several other type of block diagrams which we don t deal with within this course. E.g. Lattice ladder type construction SOS (second order sections) Can be connected in serial or parallel fashion Special type serial and parallel constructed filters Every structure type can implement the same LTI system, but for the actual implementation (let it be by hardware or software) all have consequences operating properties. E.g. internal numerical stability, scalability, numerical output precision, number of multiplication, adder, time shift elements used. 0/5/20. TÁMOP /2/A/KMR

59 Canonical representations A canonical representation respect to an element type (e.g. time shift) is the implementation which has the least number of element from that type. Direct form II type is the canonical implementation respect to the time shift operation. DF-I DF-II # of time shifts=n + M # of time shifts=max ( NM, ) 0/5/20. TÁMOP /2/A/KMR

60 System complexity SISO, MIMO SISO single input single output SIMO single input multiple output MISO multiple input single output MIMO multiple input multiple output It is often useful or necessary to break more complex systems apart to SISO systems and join them together via known operations to get the original behavior and be analyzable. 0/5/20. TÁMOP /2/A/KMR

61 LTI system description by difference equation Every LTI system can be described in the time domain by it s system equation which is a discrete time difference equation. N M N M i j ( ) = ( ) = a y n k b x n k a S y n b S x n k k i j k= 0 k= 0 i= 0 j= 0 without the loss of generality we usually assume a = + ( ) + + N ( ) = 0 + ( ) + + M ( ) = ( ) ( ) + + ( ) + + x( n M) N 0 This equation can be analyzed by well known mathematical analytical methods. D y n a y n a y n N b x n b x n b x n M y n a y n a y n N b x n b x n b N 0 D M M 0/5/20. TÁMOP /2/A/KMR

62 LTI system description by difference equation A difference equation for a specific time step can be computed recursively if we know the initial conditions: y( n) = a y( n ) an y( n N) + b0 x( n) + + bm x( n M ) initial conditions: y( N), y( N + ),, y( ) given, x( i) entrant e.g. = y u ( 0) 0 () = y + u () = + ( 2) = y () u ( 2) 2 y n = 0. 5y n + x n, x n u n 0.5 y( i) = 0, i < 0 (relaxed system) y y y y 0 = = = = = 0.75 ( k) = y( k ) + u( k ) k 0/5/20. TÁMOP /2/A/KMR n

63 Difference equation analysis in time domain review A solution to a difference equation can be composed to a homogenous and a particular solution: = ( n) + y n y y n H P The homogeneous solution is when the system has no excitation: N H 0 D y n = The particular solution is the composition: N H = 0 N M + = D y n D yp n D x n 0/5/20. TÁMOP /2/A/KMR

64 Homogenous solution For solving the homogenous part we need the characteristic equation of the difference equation: N ( N) i D yh( n) = 0 as i y( n) = 0 i= 0 Sφ( n) = λ φ( n) φ( n ) = λ φ( n) n n n- - n i n n-i -i n φ λ λ λ λ λ λ λ λ λ ( n) = S = = S = = applying: N N N i n i n n i as i λ = aiλ λ = aiλ = 0 i= 0 i= 0 i= 0 the characteristic equation (note that a0 =) n i aiλ = 0 λ, λ, λn 2 i 0/5/20. TÁMOP /2/A/KMR

65 Homogenous solution By solving the characteristic polynomial we get the homogenous solution with unknown coefficients: N If the characteristic polynomial has complex conjugate root pairs then the corresponding coefficients will be also complex conjugate: j i j i j i j i i re ϕ i, i re ϕ i Ci Nie ρ λ = λ, Ci Nie ρ + = = + = If the characteristic polynomial has multiplicity in it s roots, then we speak of internal resonance, and the solution will have the form of: if 2 = r i j r N i n n H = i λr + jλ j i= j= r+ y n Cn C y ( n) = H C l l= λ = λ = λ, and λ λ, i, j > r, i j λ n l 0/5/20. TÁMOP /2/A/KMR

66 Particular solution The particular part is determined by the excitation of the system: D y ( n) = D x( n) ( N) ( M) P There exist a couple of methods for solving the particular solution, we use the method of undetermined coefficients. With this method, we expect the response to be of a specific type and we want to guess the coefficients of the specified function family. For the most common excitations we give a table with the guessed response type. s n y n G 0/5/20. TÁMOP /2/A/KMR

67 The guessed function families: Particular solution exponential polynomial poly,exp response type excitation s n yg n constant C A Ca p 2 p Cn A0 + An + A2n + + Apn 2 ( Ap ) ( ϕn) + ( ϕn) ( ϕn) + ( ϕn) n ( ϕn) + ( ϕn) ϕn + ϕn triganometric Csin Dcos Asin Bcos ( ) trig,exp Csin Dcos a Asin Bcos a n exite resonance Cλ, λ simple root of char.poly. i i n Aa Cn a A A n A n n a p n p n n Ana n n 0/5/20. TÁMOP /2/A/KMR

68 Particular solution For finding the particular solution with the guessed functions, we need to solve the linear eq. system to find the values of the undetermined coefficients by back substituting the guessed function and the excitation to the system equation having all terms (n>max(n,m)). The terms having the time inside should cancel out each other. This part of the particular solution is valid from time n M+ To find the part of the particular solution for times n< M+ we need to correct the values with adding weighted Kronecker deltas to the solution after that we solved the general part (see later). 0/5/20. TÁMOP /2/A/KMR

69 ( N) ( M ) P Digital Signal Processing: Time domain description Particular solution i G i= 0 j= 0 P,after D y ( n) = D x( n) a y ( n i) = b x( n j), K = max M, N E.g. n yn + 4 yn ( ) 3 yn ( 2) = xn xn = 2 u n y n = A 2 back substituting to system equation having all terms: A 2 + 4A 2 3A 2 = 2 n n n n A+ 4A 3A= A= 2 yp,after( n) is valid for n M + = N y M j ( n) = u n 2 so later on we may need to correct the time instant n < M + = alas n=0 with adding a proper weighted Kronecker delta to the general solution G n 0/5/20. TÁMOP /2/A/KMR

70 General solution The general solution is the sum of the homogenous and particular solutions: = y ( n) + y ( n) y n y N H n ( n) C λ + y ( n) = l= l l P But we don t know the full particular solution yet, but we can correct that later. For the general solution we need to determine the coefficients of the homogenous part from the initial conditions of the system: y( N), y( N + ),, y( ) given P () i for relaxed systems, y 0, i< 0 0/5/20. TÁMOP /2/A/KMR

71 General solution We use the recursively computed values for times M-N+ n M. WehaveN independent coefficients from the homogenous solution, so we need to use N equations: N M N+ ( + ) = lλl + P,after( + ) l= from recursion ym N C y M N ( N M N ) = lλl + P,after( + 2) l= ym N C y M N N ym = Cλ + y ( M) l= M l l P,after 0/5/20. TÁMOP /2/A/KMR

72 General solution After solving the coefficients for the homogenous part we need to correct the analytical solution to match the recursive solution for times smaller than M+ with weighted Kronecker deltas: we need to compute the weights for the Kronecker deltas: N k wδ, k = yrecursive ( k) Ciλi + yp,after ( k), 0 k < M + i= we need to correct the particular solution for times n < M + P = ( n) + w δ ( n i) y n y P,after i= 0 δ, i with this we get the general solution M = y ( n) + y ( n) y n H P = ( n) Note that if N > M, y n y, there is no need to add weighted Kron. deltas P P,after 0/5/20. TÁMOP /2/A/KMR

73 Getting the impulse response Applying a special type of excitation (Kronecker delta function) to the system, the impulse response can be computed for a system by solving the difference equation with particular solution 0 after time n M+). N M system equation: y( n) + a y( n k) = b x( n k) recursi ve ( n) = δ ( n) ( n) = P,after k k= k= 0 initial conditions: (relaxed system) yn 0, n< 0 excitation: x particular soluti on: y we need to solve back substituting the recursive values y N M N 0 n k ( n) = Cλ + y ( k) Cλ δ( n k) i i recursive i i i= k= 0 i= k 0/5/20. TÁMOP /2/A/KMR

74 Example An LTI system is given by it s system equation: y( n) + y( n 2) = x( n) 2x( n ) 0.5x( n 4) a) Give the impulse response of this system, by solving it in the time domain and verify it by comparing with the recursive solution for 7 time instants. b) Give the response of the system for the x(n)=2u(n)(0.5) n input and verify it by comparing it to the recursive solution for 7 time instants. c) Draw the flow graph representation of this system in DF-I and DF-II form. 0/5/20. TÁMOP /2/A/KMR

75 Example solution a + ( 2) = 2 ( ) 0.5 ( 4) y n y n x n x n x n system parameters: N = 2, M = 4 a) A system s impulse response can be derived by applying the Kronecker delta function as the excitation of the system. x n = δ n i. Getting the characteristic polynomial + ( 2) = 2 ( ) 0.5 ( 4) y n y n x n x n x n 2 0 λ + 0λ + λ = 0 Solving the characteristic polynomial 2 2 λ + = 0 λ = j, λ2 = j, where j is the imaginary num ber j = 0/5/20. TÁMOP /2/A/KMR

76 ii. iii. iv. Digital Signal Processing: Time domain description Example solution a The parametric homogenous solution is: N n n n n H = lλl = λ + 2λ2 = l= The particular solution is null for times n M+ Thus we need to solve: but for this we need the recursive solutions for the first M time instant for the relaxed system: y ( n) C C C C j C j n recursion recursion ( ) y M N + = Cλ ( 2) recursion N l= y M N + = Cλ N l= y M = Cλ N l= l l l M N+ l M N+ 2 l M l 0/5/20. TÁMOP /2/A/KMR

77 Example solution a ( M ) = ( 2) + δ 2δ ( ) 0.5δ ( 4) ( 0) = = () = = 2 ( 2) = ( 3) = ( 2 ) ( 4) = ( ) ( 5) = = = 2 = 2 = 2 recursive solutions for solution times needed: n =4 y n y n n n n y y y y y y y y ( ) 6 = = 7 = = 2 ( ) 2 0/5/20. TÁMOP /2/A/KMR

78 Example solution a The particular solution is 0 for times n>=m+ so we need to solve for the homogenous part s coefficients: 3 3 yrecursion (3) = 2 = C () j + C2 (-j) 4 4 C = j, C2 = 0.25 j yrecursion (4) = = C () j + C2 (-j 2 ) But we need to correct the particular solution with the additional terms for the time n<m+ n = λ + = recursi l= ve ( n) δ, n N g n Cl l yp,afte r n wδ, n yrecursive n g n y n g n w 0 for n M /5/20. TÁMOP /2/A/KMR

79 Example solution a So the general solution, the impulse response is we need to correct the particular solution for times n< M + P P = + δ ( n i) y n y n w y ( n) = 0.5δ ( n) 0.5δ ( n 2) ( n) P,after 0 i= 0 δ, i with this we get the general solution y = y H M ( n) + y ( n) P n n = = ( j) j + ( 0.25 j)( j) δ 0.5δ ( 2) y n h n n n 0/5/20. TÁMOP /2/A/KMR

80 Example solution b + ( 2) = 2 ( ) 0.5 ( 4) y n y n x n x n x n system parameters: N = 2, M = 4 b) Give the response of the system for the given excitation and verify it by comparing it to the recursive solution for 0 time instants. x( n) = 2u( n)( 0.5) n i. Getting the characteristic polynomial (same as for the impulse response) y n + y n 2 = x n 2x n 0.5x n λ + 0λ + λ = 0 Solving the characteristic polynomial 2 2 λ + = 0 λ = j, λ2 = j, where j is the imaginary num ber j = 0/5/20. TÁMOP /2/A/KMR

81 ii. iii. Example solution b The parametric homogenous solution is: (same as at the impulse response computation) N n n n n H = lλl = λ + 2λ2 = l= The particular solution for times n>=m+ using the method of undetermined coefficients: y ( n) C C C C j C j n n = 2 ( 0.5) = ( 0.5) x n u n y n K u n G we need to back substitute the excitation and the guessed function back to the system equation using all terms. n 0/5/20. TÁMOP /2/A/KMR

82 Example solution b + ( 2) = 2 ( ) 0.5 ( 4 ), y n y n x n x n x n x n y n K 2 ( 0.5) + K( 0.5) = 2( 0.5) 2 2( 0.5) 0.5 2( 0.5) solving this K n n n n n 4 ( 0.5) + K( 0.5) = 2( 0.5) 2 2( 0.5) 0.5 2( 0.5) K = 4.4 y P,after ( n) = 4.4 u( n)( 0.5) but this part is valid after n M + n G 0/5/20. TÁMOP /2/A/KMR

83 iv. Digital Signal Processing: Time domain description We need to solve: Example solution b recursion recursion M N+ ( + ) = λ + ( + ) l= M N+ 2 ( + 2) = λ + ( + 2) recursion N y M N C y M N N y M N C y M N l= N M = λ + y M C y M l= l l l l l l P P P but for this we need the recursive solutions for the first M time instant for the relaxed system: 0/5/20. TÁMOP /2/A/KMR

84 Example solution b ( M ) x( n) = u( n) = ( 2) + 2 ( ) 0.5 ( 4) ( 0) = = 2 () = 0 + 2/ = 3 ( 2) = ( 2 ) ( 3) = ( 3 ) ( 4) = ( 7/2 ) + 2 / / 8 + 2/6 2 2/2 2 2/4 2 2 / = 7 / 2 = 9 / 4 = 7 / 8 ( 5) = ( 9/4 ) + 2 / / /2 = 47 /6 recursive solutions for solution times needed: n = y n y n x n x n x n y y y y y y y y 6 = / / / 4 = 79 / = / / /8 = 77 / 64 6 n 0/5/20. TÁMOP /2/A/KMR

85 Example solution b we need to solve for the homogenous part s coefficients: () yrecursion (3) = 9 = C j + C2 -j C = j, C2 =.2.4 j yrecursion (4) = 7 = C () j + C2 (-j) 4.4 ( ) But we need to correct the particular solution with the additional terms for the time n<m+ n = λ + = recurs l= ive ( n) δ, n N g n C y n w y n g n y n g n w l l P,after δ, n recursive /2 9/ 4 7 / 8 47/6 79/ / /2 9/ 4 7 / 8 47/6 79 / / /5/20. TÁMOP /2/A/KMR

86 Example solution b The corrected the particular solution for times n< M + M δ ( n i) y n = w + y ( n) y P P i= 0 δ, i P,after ( n) = 4δ ( n) + 2δ ( n ) 4.4 u( n)( 0.5) So the general solution for the given exitation is: = y ( n) + y ( n) y n H P n n = ( j) j + (.2.4 j)( j) + 4δ ( n) y n n + 2δ n 4.4 u n 0.5 n 0/5/20. TÁMOP /2/A/KMR

87 Example solution c + ( 2) = 2 ( ) 0.5 ( 4) y n y n x n x n x n DF-I 0/5/20. TÁMOP /2/A/KMR

88 Example solution c + ( 2) = 2 ( ) 0.5 ( 4) y n y n x n x n x n DF-II 0/5/20. TÁMOP /2/A/KMR

89 Categories of DT signal types Summary Elementary operations on DT signals Discrete time convolution and properties DT systems Additional properties of DT systems (linearity, causality, time invariance, BIBO stability) LTI systems, system description as DT convolution Properties of LTI systems FIR, IIR type LTI systems Flow graph representations of LTI systems (DF-I, DF-II, transposed forms) LTI system equation linear constant coefficient difference equation A method to analyze LCCDE in the time domain. 0/5/20. TÁMOP /2/A/KMR