Linear Time Invariant (LTI) Systems Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn
Outline Discrete-time LTI system: The convolution Sum Continuous-time time LTI systems: The convolution integral Property of Linear Time-Invariant Systems Causal LTI Systems Described by Differential and Difference Equations Singularity Functions 2
2.1 Discrete-time LTI system: The convolution Sum Using delta function to represent discretetime signal 3
2.1 Discrete-time LTI system: The convolution Sum 4
2.1 Discrete-time LTI system: The convolution Sum Using delta function to represent discretetime signal An example: 5
2.1 Discrete-time LTI system: The convolution Sum The response of a linear system to x[n] will be the superposition of the scaled responses of fthe system to each of fthese shifted delta functions. The property of time invariance tells us that the responses to the time-shifted delta functions are simply time-shifted version of one another. 6
2.1 Discrete-time LTI system: The convolution Sum Let denote the response of the linear system to With The output of the system 7
2.1 Discrete-time LTI system: The convolution Sum An example: 8
2.1 Discrete-time LTI system: The convolution Sum 9
2.1 Discrete-time LTI system: The convolution Sum 10
2.1 Discrete-time LTI system: The convolution Sum If linear system is time invariant, we have This simplify as This result is referred to as the convolution sum 11
2.1 Discrete-time LTI system: The convolution Sum Example 1: Consider an LTI system with impulse response h[n] and input x[n] as follows. Determine the output t 12
2.1 Discrete-time LTI system: The convolution Sum Solution: 13
2.1 Discrete-time LTI system: The convolution Sum Solution: 14
2.1 Discrete-time LTI system: The convolution Sum Solution: 15
2.1 Discrete-time LTI system: The convolution Sum Solution: 16
2.1 Discrete-time LTI system: The convolution Sum Solution: 17
2.1 Discrete-time LTI system: The convolution Sum Example 2: Consider an LTI system with Determine the output 18
2.1 Discrete-time LTI system: The convolution Sum Solution: 19
2.1 Discrete-time LTI system: The convolution Sum Solution: 20
2.1 Discrete-time LTI system: The convolution Sum Example 3: Consider an LTI system with Determine the output 21
2.1 Discrete-time LTI system: The convolution Sum Solution: Interval 1: n<0, y[n] =0 Interval 2: 0 n 4 22
2.1 Discrete-time LTI system: The convolution Sum Solution: Interval 3: for n>4, but n-6 0, i.e., 4 n 6 23
2.1 Discrete-time LTI system: The convolution Sum Solution: Interval 4: for n>6, but n-6 4, i.e., 6 n 10 24
2.1 Discrete-time LTI system: The convolution Sum Solution: Interval 4: for n-6>4, i.e., n>10 i.e., 6 n 10 25
2.1 Discrete-time LTI system: The convolution Sum Solution: 26
2.2 Continuous-time LTI system: The convolution integral 2.2.12 Representing the continuous-time time signals in terms of impulses 27
2.2 Continuous-time LTI system: The convolution integral 28
2.2 Continuous-time LTI system: The convolution integral If we define a pulse function as Since has unit amplitude, we have In above, for any value of t, only one term in the summation on the right-hand side is nonzero. 29
2.2 Continuous-time LTI system: The convolution integral As 0, the summation approaches an integral. Check the unit step function: 30
2.2 Continuous-time LTI system: The convolution integral 2.2.22 2 the unit impulse response and the convolution integral As x(t) is approximated as the sum of shifted version of the basic versions of the pulse signal The response of the linear system will be 31
2.2 Continuous-time LTI system: The convolution integral 32
2.2 Continuous-time LTI system: The convolution integral 33
2.2 Continuous-time LTI system: The convolution integral 2.2.22 2 the unit impulse response and the convolution integral As, we have 34
2.2 Continuous-time LTI system: The convolution integral 2.2.22 2 the unit impulse response and the convolution integral When the linear system is time invariant, we have Define, we have Referred to the above as the convolution integral. 35
2.2 Continuous-time LTI system: The convolution integral Example 1: Let x(t) be the input to an LTI system with unit impulse response h(t), where determine the output y(t). 36
2.2 Continuous-time LTI system: The convolution integral Solution: 37
2.2 Continuous-time LTI system: The convolution integral When t<0, y(t) = 0 38
2.2 Continuous-time LTI system: The convolution integral When t>0 39
2.2 Continuous-time LTI system: The convolution integral Solution: 40
2.2 Continuous-time LTI system: The convolution integral Example 2: Let x(t) be the input to an LTI system with unit impulse response h(t), where 1 0 t T t 0 t 2 T xt () ht () 0 otherwise 0 otherwise determine the output y(t). 41
2.2 Continuous-time LTI system: The convolution integral Solution: yt () xt () ht () x ( ) ht ( ) d 42
2.2 Continuous-time LTI system: The convolution integral t 1 yt () d t 0 2 2 t 1 y () t d Tt T t T 2 2 43
2.2 Continuous-time LTI system: The convolution integral 2T 1 yt () d 2 T ( tt) tt 2 2 2 yt () 0 44
2.2 Continuous-time LTI system: The convolution integral Example 3: Let x(t) be the input to an LTI system with unit impulse response h(t), where determine the output y(t). 45
2.2 Continuous-time LTI system: The convolution integral Solution: yt () xt () ht () x ( ) ht ( ) d 46
2.2 Continuous-time LTI system: The convolution integral When t-3<=0, i.e., t<=3 When t-3>0, i.e., t>3 47
2.3 Properties of LTI Systems Use convolution sum and convolution integral to obtain the output of discrete- time and continuous-time systems, based on the unit impulse response An LIT system is completely characterized by its impulse response (only for LTI system) 48
2.3 Properties of LTI Systems 2.3.1 The commutative property ( 交换律 ) Proof: 49
2.3 Properties of LTI Systems 2.3.2 The distributive property ( 分配律 ) Definition: 50
2.3 Properties of LTI Systems 2.3.2 The distributive property ( 分配律 ) Definition: 51
2.3 Properties of LTI Systems 2.3.3 The associative property ( 结合律 ) Definition: 52
2.3 Properties of LTI Systems 2.3.3 The associative property ( 结合律 ) Definition: 53
2.3 Properties of LTI Systems 2.3.4 LTI system with and without memory y( 有记忆和无记忆系统 ) Definition: the output only depends on value of the input at the same time. We have ie i.e., 54
2.3 Properties of LTI Systems 2.3.5 Inevitability of LTI system ( 可逆性 ) Identical system 55
2.3 Properties of LTI Systems Example: Delayed or advanced system We have Can we find such that 56
2.3 Properties of LTI Systems Example: accumulator system The inverse system is a first different operation, i.e., 57
2.3 Properties of LTI Systems 2.3.6 Causality for LTI ( 因果性 ) LTI system should satisfy 58
2.3 Properties of LTI Systems 2.3.7 Stability for LTI ( 稳定性 ) LTI system should satisfy: absolutely summable and absolutely integrable How to prove?? 59
Proof: Example: time shift 60
2.3 Properties of LTI Systems 2.3.8 The unit step response of LTI Use unit step response to describe the system behavior 61
2.4 Causal LTI system described by differential & difference equs An important class practical system can be described by using Linear constant-coefficient differential equations Linear constant-coefficient difference equations 62
2.4 Causal LTI system described by differential & difference equs Linear constant-coefficient differential equations a, N a k M k d y () t d x () t b k k k, dt dt k k0 k0 are constant, N is the order k b k The response to an input x(t) generally consists of the sum of particular solution ( 特解 ) to the differential equation & a homogeneous solution ( 其次解 ) 63
2.4 Causal LTI system described by differential & difference equs a homogeneous solution (nature response: 自然相应 ): a solution to the differential equation with input set to zero Different choices of auxiliary conditions leads to different relationships between input and output 64
2.4 Causal LTI system described by differential & difference equs Linear constant-coefficient difference equations a, N M a y( n k) b x( nk) k k0 k0 are constant, N is the order k b k k The response to an input x[n] generally consists of the sum of particular solution ( 特解 ) to the differential equation & a homogeneous solution ( 其次解 ) 65
2.4 Causal LTI system described by differential & difference equs a homogeneous solution (nature response: 自然相应 ): a solution to the differential equation with input set to zero Re-express express difference equation as recursive equation 1 M N yn ( ) b kxn ( k) ayn k ( k ) a 0 k0 k1 66
2.4 Causal LTI system described by differential & difference equs We see that xn ( ) y( 1), y( 2),, y( N) y (0) y( 1), y( 2),, y( N 1) n 0 y(0) y(1) When N = 0, non-recursive equation 67
2.4 Causal LTI system described by differential & difference equs Non-recursive case: M k b k yn ( ) xn ( k) a 0 0 The unit impulse response is Finite impulse response (FIR) system For recursive case: Infinite Impulse Response (IIR) system 68
69
2.4 Causal LTI system described by differential & difference equs Block diagram representation of the first-order systems described by differential and difference equations Addition Multiplication by a coefficient delay 70
2.4 Causal LTI system described by differential & difference equs The following first-order difference equation can be described as 71
2.4 Causal LTI system described by differential & difference equs M N 1 k a 0 k0 k1 y( n) b x( nk) aky( nk) M k 0 wn ( ) bxn ( k ) k xn ( ) wn ( ) D D b 0 b 1 b 2 TypeⅠⅠ D bm 1 b M 72
2.4 Causal LTI system described by differential & difference equs N 1 yn ( ) wn ( ) ayn k ( k) a 0 k 1 wn ( ) yn ( ) 1/a 0 a 1 a 2 an 1 a N D D D 73
2.4 Causal LTI system described by differential & difference equs M N 1 k a 0 k0 k1 y( n) b x( nk) aky( nk) x( n ) y( n) 1/a 0 a 1 a 2 an 1 a N D D D b 0 b 1 b 2 bn 1 b N TypeⅡ 74
2.4 Causal LTI system described by differential & difference equs For differential equation Addition Multiplication by a coefficient differentiation 75
2.4 Causal LTI system described by differential & difference equs The following first-order differential equation can be described as 76
2.4 Causal LTI system described by differential & difference equs N N a k N k d y() t d x() t bk k dt k k k 0 dt k 0 a y () t b x () t k ( N k ) k ( N k ) k0 k0 N 77
2.4 Causal LTI system described by differential & difference equs N N1 1 1 k ( Nk) k ( Nk) N k0 k0 y() t b x () t a y () t a xt () b N wt () wt () a yt () bn 1 bn 2 b 1 b 0 TypeⅠ 1/ N an 1 an 2 aa 1 a 0 78
2.4 Causal LTI system described by differential & difference equs N N1 1 yt () bx k ( N k) () t aky( N k) () t a N k0 k0 xt () 1/aa b yt () 1/ N an 1 a an 2 N bn 1 bn 2 a b a 1 b 1 a 0 b0 TypeⅡⅡ 79
2.5 Singularity functions The delta function is the impulse response of the identity system 80
2.5 Singularity functions Unit doublet function : the derivative of the unit impulse also 81
2.5 Singularity functions We define as the k-th derivative of delta function Property of unit doublet 82
2.5 Singularity functions For t = 0, yields 83
2.5 Singularity functions Unit step function: The unit ramp function 84
2.5 Singularity functions Similarly, we have We have 85