Part B:Discrete-Time Systems Chapter 2B Part B ·Linear System ●●● ●●● Shift(Time)-Invariant System ●●●南 Linear Time-Invariant System Discrete-Time Signals and Discrete-Time Systems 。Causal System Systems in the Time-Domain Stable System Passive and Lossless Systems 1.Linear System 1.Linear System 2.Shift Invariant System Definition In other words,the system considered satisfies Ify (n)is the output due to an inputx (n)and the principle of Superposition. Definition y2(n)is the output due to an inputx(n)then for Recall that the inputx(n)can be written as For a shift-invariant system,if y(n)is the an input m=立k)6a- response to an input x(n),then the response to x(n)=ax (n)+Bx2(n) an input x(n)=x(n-mo)is simply y(n)=y(n- the output is given by Hence,the output n)can be computed as no)where no is any positive or negative integer y(n)=ay (n)+By2(n) follows according to the principle of superposition y(m)=(n) The above relation must hold for any arbitrary Above property must hold for any arbitrary input and its corresponding output constants a andand for all possible inputs where h(n,k)is the output due to the input x(n)and x2(m) 4 6(n-k3
Chapter 2B Discrete-Time Signals and Systems in the Time-Domain Part B Discrete-Time Systems 3 Part B: Discrete-Time Systems Linear System Shift (Time) Shift (Time)-Invariant System Linear Time Linear Time-Invariant System Causal System Stable System Passive and Lossless Systems Passive and Lossless Systems 4 1. Linear System 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)= ¢x1(n) +£x2(n) the output is given by y(n)= ¢y1(n) +£y2(n) Above property must hold for any arbitrary constants ¢and £and for all possible inputs x1(n) and x2(n) 5 1. Linear System In other words, the system considered satisfies the principle of Superposition. Recall that the input x(n) can be written as Hence, the output y(n) can be computed as follows according to the principle of superposition where h(n,k) is the output due to the input ¥(nˉk) () ()( ) n xn xk n k () ()(, ) n yn xkhnk 6 2. Shift Invariant System Definition 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(nˉn0) is simply y(n)= y1(nˉ n0) where n0 is any positive or negative integer The above relation must hold for any arbitrary input and its corresponding output
2.Shift Invariant System 3.Linear Time-Invariant Systems 3.Linear Time-Invariant Systems For a linear system,the output y(n)due to the Time-invariance property ensures that for a .LTI is the abbreviation of"Linear Time- inputx(n)can be written as specified input,the output is independent of Invariant" (m)=∑x(k)h(n,k) the time the input is being applied .LTI systems are mathematically easy to Question .If this system is also time-invariant,the time- analyze and characterize,and consequently, varying function h(n,k)becomes a time- Consider the linearity and time-invariance easy to design invariant function /(n-k),and the output is properties of the following systems Highly useful signal processing algorithms now given by (1)y(n)=x(n-n)(2)y(n)=ax(n)+b have been developed utilizing this class of m=Σ- systems over the last several decades The above operation is called a linear convolution sum. 4.Causal System 4.Causal System 5.Stable System Definition ●Then x(n=x(,fornno Question If wn)is the response to an input x(n)and if x()is bounded.i.e. Let y (n)and y,(n)be the responses of a causal .Consider the Causality of the following discrete-time system to the inputs x(n)and (n)B,for all values of n systems x2(n),respectively then yn)is bounded.i.e (1)y(n)=ax(n)+b (2)y(n=nx(n-no) l(n)B,for all values of n 11
7 2. Shift Invariant System Time-invariance property ensures that for a specified input, the output is independent of the time the input is being applied Question Consider the linearity and time-invariance properties of the following systems (1) y(n)=x(nˉn0) (2) y(n)=ax(n)+b 8 3. Linear Time-Invariant Systems LTI is the abbreviation of “Linear Time Linear TimeInvariant Invariant ” LTI systems 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 9 3. Linear Time-Invariant Systems For a linear system, the output y(n) due to the input x(n) can be written as If this system is also time-invariant, the timevarying function h(n,k) becomes a timeinvariant function h(nˉk) , and the output is now given by The above operation is called a linear convolution sum. () ()(, ) n yn xkhnk () ()( ) n yn xkhn k 10 4. Causal System Definition In a causal system, the n0-th output sample y(n0) depends only on input samples x(n) for nİn0 and does not depend on input samples for n>n0 Let y1(n) and y2(n) be the responses of a causal discrete-time system to the inputs x1(n) and x2(n), respectively 11 4. Causal System Then 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 Question Consider the Causality of the following systems (1) y(n)= ax(n)+b (2) y(n)= nx(nˉn0) 12 5. Stable System Definition There are various definitions of stability. We consider here the bounded-input, boundedoutput (BIBO) stability ) If y(n) is the response to an input x(n) and if x(n) is bounded, i.e. |x(n)|<Bx for all values of n then y(n) is bounded, i.e |y(n)|<By for all values of n
6.Passive and Lossless Systems Definition .A discrete-time system is defined to be Chapter 2C Part C passive,if for every finite-energy input x(n), ●●● ●●● the output n)has,at most,the same energy, ●●● i.e. 立ofe2of< Discrete-Time Signals and Time-Domain Characterization Systems in the Time-Domain of LTI Discrete-Time Systems For a lossless system,the above inequality is satisfied with an equal sign for every input 13 Time-Domain Characterization of LTI Discrete-Time Systems 1.Input-Output Relationship 1.Input-Output Relationship Input-Output Relationship Unit impulse response h(n)is defined by the A consequence of the linear,time invariance response of a digital filter to a unit sample property is that a LTI discrete time system is Simple interconnection schemes sequence {6(n) completely characterized by its impulse Stability condition in terms of the Impulse .Unit sample response (s(n))is defined by the response response response of a discrete system to a unit step ·◆Knowing the impulse response one can Causality condition in terms of the Impulse sequence fu(n) compute the output of the system for any arbitrary input response 常 18
13 6. Passive and Lossless Systems Definition A discrete-time system is defined to be passive passive, if for every finite-energy input x(n), the output y(n) has, at most, the same energy, i.e. For a lossless lossless system, the above inequality is satisfied with an equal sign for every input 2 2 () () n n yn xn Chapter 2C Discrete-Time Signals and Systems in the Time-Domain Part C Time-Domain Characterization of LTI Discrete-Time Systems 16 Time-Domain Characterization of LTI Discrete-Time Systems Input-Output Relationship Simple interconnection schemes Simple interconnection schemes Stability condition in terms of the Impulse Stability condition in terms of the Impulse response response Causality condition in terms of the Impulse Causality condition in terms of the Impulse response response 17 1. Input-Output Relationship Unit impulse response Unit impulse response {h(n)}is defined by the response of a digital filter to a unit sample sequence {¥(n)} Unit sample response Unit sample response {s(n)} is defined by the response of a discrete system to a unit step sequence {u(n)} 18 1. Input-Output Relationship A consequence of the linear, time invariance property is that a LTI discrete time system is completely characterized by its impulse response Knowing the impulse response one can compute the output of the system for any arbitrary input
1.Input-Output Relationship 1.Input-Output Relationship 1.Input-Output Relationship Let h(n)denote the impulse response of an .Since h(n)is the response of input 6(n)and Taking advantage of the property of linear, LTI discrete-time system the system is time invariant,we have we have x(n) h(n) m 6(n-)→n-k) Likewise,as the system is linear -)十2a- Recall that x(n)can be written as x(6(n-)→xk)hn-) Eventually,the I-O relationship of an LTI system can be written as follows x(m)=x(k)6(n-k) Note that.x(A)is considered as a constant in this case ym)=∑xkhn-k)=∑)xn-k) 二 20 2 1.Input-Output Relationship 1.Input-Output Relationship 1.Input-Output Relationship Solution This operation is called a linear convolution Schematic Representation sum and can be represented compactly as Time-reverse h(k)to form h) h(n-k) y(n)=x(n)*h(n) Shift /n-k)to the right by n sampling periods "⑧ y(n) if n>0 or shifth(k)to the left by n sampling The linear convolution sum has periods if n<0 to form h(n) n Commutative property .Form the product vx()h(n-k) If the lengths of x(n)and h(n)are N and N Associative property Sum all samples of v)to develop the n-th respectively,the length of the convolution Distributive property sample ofy(n)of the convolution sum sum y(n)will be N +N2-1 22
19 1. Input-Output Relationship Let h(n) denote the impulse response of an LTI discrete-time system Recall that x(n) can be written as h(n) x(n) y(n) () ()( ) n x n xk n k 20 1. Input-Output Relationship Since h(n) is the response of input ¥(n) and the system is time invariant, we have ¥(nˉk) ėh(nˉk) Likewise, as the system is linear x(k)¥(nˉk) ė x(k)h(nˉk) Note that, x(k) is considered as a constant in this case 21 1. Input-Output Relationship Taking advantage of the property of linear, we have Eventually, the I-O relationship of an LTI system can be written as follows () ()( ) ()( ) k k y n xkhn k hkxn k ()( ) ()( ) k k x k n k xkhn k 22 1. Input-Output Relationship This operation is called a linear convolution linear convolution sum and can be represented compactly as The linear convolution sum has Commutative property Associative property Distributive property yn xn hn () () () 23 1. Input-Output Relationship Solution Time-reverse reverse h(k) to form h(ˉk) Shift h(ˉk) to the right by n sampling periods if n > 0 or shift h(ˉk) to the left by n sampling periods if n <0 to form h(nˉk) Form the product product v(k)=x(k)h(nˉk) Sum all samples of v(k) to develop the n-th sample of y(n) of the convolution sum 24 1. Input-Output Relationship Schematic Representation If the lengths of x(n) and h(n) are N1 and N2 respectively, the length of the convolution sum y(n) will be N1+N2ˉ1 z h(ˉk) n h h(nˉk) k y(n) x(n)
1.Input-Output Relationship 1.Input-Output Relationship 1.Input-Output Relationship Example Correlation of Signals Ifx(n)=v(n),we obtain the definition of the Calculate the convolution sum:y(n)=R(n)*R(n) Definition autocorrelation of x(n) Method 1:analytical method Method 1:graphically method A measure of similarity between a pair of r()()(n-D) energy signals,x(n)and y(n),is given by the Application cross-correlation sequence r()defined by 。Note,,0-立()is the energy of the Using Convolution to calculate the ,0=2mm-小1=0L2 signal x(m）- Correlations of sequence 1.Input-Output Relationship 2.Simple Interconnection 2.Simple Interconnection Properties Cascade Connection An application of the inverse system concept ·0=2 oa-0=立0m+nm=5- h(n)=h(n)*h(n)=h(n)*h(m) is in the recovery of a signal x(n)from its An application is in the development of an It follows that r(=r(-1)implying that r() distorted version(m)appearing at the output inverse system.if the cascade connection of a transmission channel is an even function for real x(n) satisfies the relation 。,0=2xm(-1-m)=x0-0 If the impulse response of the channel is (n)*九，(n)=δn) known,then x(n)can be recovered by x (-m) .ro(n) Then the LTI system h(n)is said to be the designing an inverse system of the channel inverse of h(n)and vice-versa
25 1. Input-Output Relationship Example Example Calculate the convolution sum: Method 1: analytical method Method 1: graphically method Application Using Convolution to calculate the Correlations of sequence 3 4 y() () () n Rn Rn 26 1. Input-Output Relationship Correlation of Signals Correlation of Signals Definition A measure of similarity between a pair of energy signals, x(n) and y(n), is given by the cross-correlation sequence rxy(l) defined by ( ) ( ) ( ), 0, 1, 2, xy n r l xnyn l l 27 1. Input-Output Relationship If x(n)=y(n), we obtain the definition of the autocorrelation of x(n) Note, is the energy of the signal x(n) () ( ) ( ) xx n r l xnxn l 2 (0) ( ) xx n r xn 28 1. Input-Output Relationship Properties Properties sd It follows that rxx(l)=rxx(ˉl) implying that rxx(l) is an even funct even fu ion for real x(n) sd () ( ) ( ) ( ) ( ) ( ) yx xy n m r l y nxn l y m lxm r l xy () ( ) ( ) () ( ) n r l xny l n xl y l x(n) y(ˉn) rxy(n) 29 2. Simple Interconnection Cascade Connection An application is in the development of an inverse system, if the cascade connection satisfies the relation Then the LTI system h1(n) is said to be the inverse of h2(n) and vice-versa 12 21 hn h n h n h n h n () () () () () 1 2 hn hn n () () () 30 2. Simple Interconnection An application of the inverse system concept is in the recovery of a signal x(n) from its distorted version 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 ˆ( ) n
2.Simple Interconnection 3.Stability Condition 4.Causality Condition BIBO Stability Condition- Parallel Connection Causality Condition- h(n)=h (n)+h(n) .A discrete-time system is BIBO stable if the An LTI discrete-time system is causal if and output sequence n)remains bounded for The parallel connection of two stable systems only if its impulse response(n)!is a causal all bounded input sequence (x(n)) sequence is stable An LTI discrete-time system is BIBO stable if However,the parallel connection of two .A non-causal LTI discrete-time system with a and only if its impulse response sequence finite-length impulse response can often be passive systems may or may or may not be (h(n)is absolutely summable.i.e. realized as a causal system by inserting an passive appropriate amount of delay 31 33 5.Differential Equations An LTI discrete-Time system can also be described by a linear constant coefficient differential equation of the form a4n-k)=∑hxn-m) Ifav≠0，then the difference equation is of order N If N=0,we call this system an FIR filter If N0,we call this system an IIR filter
31 2. Simple Interconnection Parallel Connection The parallel connection of two stable systems is stable However, the parallel connection of two passive passive systems may or may or may not be may or may not be passive passive 1 2 hn h n h n () () () 32 3. Stability Condition BIBO Stability Condition BIBO Stability Condition -- A discrete-time system is BIBO stable BIBO stable if the output sequence {y(n)} remains bounded for all bounded input sequence {x(n)} An LTI discrete-time system is BIBO stable if and only if its impulse response sequence {h(n)} is absolutely summable summable, i.e. ( ) n S hn 33 4. Causality Condition Causality Condition Causality Condition -- An LTI discrete-time system is causal if and only if its impulse response {h(n)} is a causal sequence sequence A non-causal LTI discrete-time system with a finite-length impulse response can often be realized as a causal system by inserting an appropriate amount of delay 34 5. Differential Equations An LTI discrete-Time system can also be described by a linear constant coefficient linear constant coefficient differential equation of the form If aNĮ0, then the difference equation is of order N If N=0, we call this system an FIR filter FIR filter If NĮ0, we call this system an IIR filter IIR filter 0 0 () ( ) N M k m k m ayn k bxn m