Ch 2: Linear Time-Invariant System A system is said to be Linear Time-Invariant (LTI) if it possesses the basic system properties of linearity and time-invariance. Consider a system with an output signal y(t) corresponding to an input signal x t. The system will be called a timeinvariant system, if for an arbitrary time shift T 0 in the input signal, i.e., x(t + T 0 ), the output signal is time-shifted by the same amount T 0, i.e., y(t + T 0 ). If x k [n], k = 1, 2, 3,, are a set of inputs to a discrete-time linear system with corresponding outputs y k [n], k = 1, 2, 3,, then we get INPUT: x n = a k x k [n] = a 1 x 1 n + a 2 x 2 n + a 3 x 3 n + k OUTPUT: y n = a k y k [n] = a 1 y 1 n + a 2 y 2 n + a 3 y 3 n + k CEN340: Signals and Systems - Dr. Ghulam Muhammad 1
2. Discrete-Time LTI Systems: the Convolution Sum Any discrete-time signal x[n] can be represented as a function of shifted unit impulses [n-k], where the weights in this linear combination are x[k]. x[ n] x[ k] [ n k] k Original Signal + + + + +. CEN340: Signals and Systems - Dr. Ghulam Muhammad 2
The Convolution Sum x[ n] x[ k] [ n k] k Scaled impulses For a particular case of unit step function: 1 for n 0 un [ ] 0 for n 0 We can write: u[ n] u[ k ] [ n k ] k 0 Shifting property of the discrete-time unit impulse CEN340: Signals and Systems - Dr. Ghulam Muhammad 3
The Convolution Sum contd. [n] Impulse input with zero initial conditions Linear Time-Invariant System, Impulse response (?) h[n] Impulse Response x[n] Input Linear Time-Invariant System, Impulse response, h[n] y[n] Output y[n] = x[n]h[n] y [ n] x [ k ] h[ n k ] k Convolution CEN340: Signals and Systems - Dr. Ghulam Muhammad 4
Example: Convolution (1) Impulse response of an LTI system There are only two non-zero values for the input. y[n] = x[0]h[n-0] + x[1]h[n-1] = 0.5h[n] + 2h[n-1] Input Find output. Output CEN340: Signals and Systems - Dr. Ghulam Muhammad 5
Example: Convolution (2) Solution: Length = 3 Length = 3 Convolution length = 3 +3 1 = 5 CEN340: Signals and Systems - Dr. Ghulam Muhammad 6
Example: Convolution (3) Solution: Length = 3 Length = 2 Convolution length = 3 + 2 1 = 4 CEN340: Signals and Systems - Dr. Ghulam Muhammad 7
Example: Convolution (4) Find the output of an LTI system having unit impulse response, h[n], for the input, x[n], as given below. n x [ n] u[ n], 0 1 h[ n] u[ n] CEN340: Signals and Systems - Dr. Ghulam Muhammad 8
Example: Convolution (4) contd. k, 0k n x [ k ] h[ n k ] 0, otherwise Thus, for n 0, n n 1 k 1 y [ n], for n 0 1 k 0 Thus, for all n, n 1 1 y [ n] u[ n]. 1 CEN340: Signals and Systems - Dr. Ghulam Muhammad 9
Representation of Continuous-Time Signals in Terms of Impulses A continuous-time signal, x(t), is approximated in terms of pulses or a staircase. Defining: 1/, 0 t () t 0, otherwise 1, 0 t () t 0, otherwise (t) Pulse or staircase approximation of x(t) at t = 0: 1/ x(0), 0 t xˆ(0) x (0) ( t ) 0, otherwise 0 CEN340: Signals and Systems - Dr. Ghulam Muhammad 10 t
Continuous-Time Signals in Terms of Impulses contd. Going one step further, shifted can be written as: (t - ) 1/, t 2 ( t ) 0, otherwise 1/ 0 2 t Pulse or staircase approximation of x(t) at t = : x( ), t 2 xˆ( ) x ( ) ( t ) 0, otherwise CEN340: Signals and Systems - Dr. Ghulam Muhammad 11
Continuous-Time Signals in Terms of Impulses contd. In general, for an arbitrary k, we write 1/, k t ( k 1) ( t k) 0, otherwise Pulse or staircase approximation of x(t) at t = k: x ( k ), k t ( k 1) xˆ( k ) x ( k ) ( t k ) 0, otherwise Combining all individual approximations, we get the complete pulse/staircase approximation of x(t) as: xˆ ( t )... xˆ ( ) xˆ (0) xˆ ( )... xˆ ( k ) x ( k ) ( t k ) k k CEN340: Signals and Systems - Dr. Ghulam Muhammad 12
Continuous-Time Signals in Terms of Impulses contd. If we keep on reducing the value of, the approximation becomes closer and closer to the original value. x ( t ) lim xˆ ( t ) lim x ( k ) ( t k ) 0 0 k In the limiting case: 0; ( t) ( t) Consequently, k...... d x ( t ) x ( ) ( t ) d CEN340: Signals and Systems - Dr. Ghulam Muhammad 13
The Convolution Integral y ( t ) x ( ) h( t ) d x ( t )* h( t ) Let, the input x(t) to an LTI system with unit impulse response h(t) be given as x(t) = e -at u(t) for a > 0 and h(t) = u(t). Find the output y(t) of the system. CEN340: Signals and Systems - Dr. Ghulam Muhammad 14
The Convolution Integral contd. a y ( t ) x ( ) h( t ) d e h( t ) d, for t 0 t 1 1 = e d e. 1e a a 0 a a t at 0 Thus, for all t, we can write 1 at y ( t ) 1 e u ( t ) a 0 CEN340: Signals and Systems - Dr. Ghulam Muhammad 15
Example: The Convolution Integral Find y(t) = x(t)*h(t), where x t e u t 2t ( ) ( ) h( t ) u ( t 3) We observe that there is nonzero overlap. t-3 0: t 3 1 For t 3, y ( t ) e d e 2 t-3 > 0: 0 2 1 For 0, y ( t ) e d 2 2 2( t 3) CEN340: Signals and Systems - Dr. Ghulam Muhammad 16
Example: The Convolution Integral y ( t ) x ( ) h( t ) d x ( t )* h( t ) CEN340: Signals and Systems - Dr. Ghulam Muhammad 17
Example: The Convolution Integral contd. x() x() t < 2 CEN340: Signals and Systems - Dr. Ghulam Muhammad 18
Example: The Convolution Integral contd. x() x() CEN340: Signals and Systems - Dr. Ghulam Muhammad 19
Properties of LTI Systems The characteristics of an LTI system are completely determined by its impulse response. This property holds in general for LTI systems only. The unit impulse response of a nonlinear system does not completely characterize the behavior of the system. Consider a discrete-time system with unit impulse response: 1, n 0,1 hn [ ] 0, otherwise If the system is LTI, we get (by convolution): y [ n] x [ n] x [ n 1] There is only one such LTI system for the given h[n]. However, there are many nonlinear systems with the same response, h[n]. y [ n] x [ n] x [ n 1] y [ n] max x [ n], x [ n 1] 2 CEN340: Signals and Systems - Dr. Ghulam Muhammad 20
Proof: (discrete domain) x [ n]* h[ n] x [ k ] h[ n k ] k Commutative Property Put r = n k k = n - r x ( t )* h( t ) h( t )* x ( t ) x [ n]* h[ n] h[ n]* x [ n] x [ n]* h[ n] x [ n r ] h[ r ] h[ r ] x [ n r ] h[ n]* x [ n] r r Similarly, we can prove it for continuous domain. CEN340: Signals and Systems - Dr. Ghulam Muhammad 21
Distributive Property x[ n]*( h [ n] h2[ n]) x[ n]* h1[ n] x[ n]* h2[ 1 n ] CEN340: Signals and Systems - Dr. Ghulam Muhammad 22
y[n] = x[n] * h[n] Example: Distributive Property n 1 n x [ n] u[ n] 2 u[ n] 2 h[ n] u[ n] x[n] in nonzero for entire n, so direct convolution is difficult. Therefore, we will use commutative property. y [ n] x [ n]* h[ n] x [ n] x [ n] * h[ n] x [ n]* h[ n] x [ n]* h[ n] y [ n] y [ n] 1 2 1 2 1 2 1 y 1[ n] x 1[ n]* h[ n] x 1[ k ] h[ n k ] u[ k ] u[ n k ] k k 2 n 1 1 (1/ 2) n 1 u[ n] 21 (1/ 2) u[ n] 1 (1/ 2) k y [ n] x [ n]* h[ n] 2 u[ k ] u[ n k ] 2 2 2 k 1 2 n1 y [ n] y [ n] y [ n] 2 1 (1/ 2) u[ n] 2 CEN340: Signals and Systems - Dr. Ghulam Muhammad 23 k n1 n1
Associative Property x ( t )* h ( t )* h ( t ) x ( t )* h ( t ) * h ( t ) 1 2 1 2 x [ n]* h [ n]* h [ n] x [ n]* h [ n] * h [ n] 1 2 1 2 (A) x[n] h 1 [n] w[n] h 1 [n] y[n] (B) x[n] h[n] = h 1 [n]*h 2 [n] y[n] Proof: y [ n] w [ n]* h [ n] x [ n]* h [ n] * h [ n] From (A), 2 1 2 From (B), y [ n] x [ n]* h[ n] x [ n]*( h1[ n]* h2[ n]) CEN340: Signals and Systems - Dr. Ghulam Muhammad 24
LTI Systems With and Without Memory A discrete-time LTI system can be memoryless if only: h[ n] 0, for n 0 Thus, the impulse response have the form: h[ n] K [ n], K is a constant y [ n] Kx [ n] If K = 1, then the system is called identity system. Similarly for continuous LTI systems. Invertibility of LTI Systems: The system with impulse response h 1 [n] is inverse of the system with impulse response h(t), if h( t )* h ( t ) ( t ) 1 CEN340: Signals and Systems - Dr. Ghulam Muhammad 25
Example: LTI Systems Properties Consider, the following LTI system with pure time-shift. y ( t ) x ( t t ) 0 Such a system is a delay if t 0 > 0, and an advance if t 0 < 0. If 0 t = 0, the system is an identity system and is memoryless. For t 0 0, the system has memory. The impulse response of the system is h(t) = (t t 0 ), therefore, x ( t t ) x ( t )* ( t t ) 0 0 The convolution of a signal with a shifted impulse simply shifts the signal. To recover the input (i.e. to invert the system), we simply need to shift the output back. The impulse response of the inverted system: h ( t ) ( t t ) 1 0 h( t )* h ( t ) ( t t )* ( t t ) ( t ) 1 0 0 CEN340: Signals and Systems - Dr. Ghulam Muhammad 26
Example: LTI Systems Properties Consider, an LTI system with impulse response: h[ n] u[ n] y [ n] x [ k ] u[ n k ] x [ k ] k The system is a summer or accumulator. n k The system is invertible and its inverse is given by: y[n] = x[n] x[n-1], which is a first difference equation. By putting, x[n] = [n], we find the impulse response of the inverse system: h r [n] = [n] [n-1] To check that h[n] and h r [n] are impulse responses of the systems that are inverse of each other, we do the following calculation: h[ n]* h [ n] u[ n]* [ n] [ n 1] 1 u[ n]* [ n] u[ n]* [ n 1] u[ n] u[ n 1] [ n] Therefore, the two systems are inverses of each other. CEN340: Signals and Systems - Dr. Ghulam Muhammad 27
y [ n] x [ k ] h[ n k ] k Causality of LTI Systems y[n] must not depend on x[k] for k > n, to be causal. Therefore, for a discrete-time LTI system to be causal: h[n] = 0, for n < 0. n y [ n] x [ k ] h[ n k ] h[ k ] x [ n k ] k k 0 Similarly, for a continuous-time LTI system to be causal: y ( t ) h( ) x ( t ) d 0 Both the accumulator ( h[n] = u[n]) and its inverse (h[n] = [n] - [n-1]) are causal. CEN340: Signals and Systems - Dr. Ghulam Muhammad 28
Stability of LTI Systems Consider, an input x[n] to an LTI system that is bounded in magnitude: x [ n] B, for all n Suppose that we apply this to the LTI system with impulse response h[n]. x[n k] < B, for all n and k y [ n] h[ k ] x [ n k ] k k h[ k ] x [ n k ] B h[ k ] for all n k Therefore, if h[ k ], then y [ n] k If the impulse response is absolutely summable, then y[n] is bounded in magnitude, and hence the system is stable. Similar case in continuous-time LTI system. CEN340: Signals and Systems - Dr. Ghulam Muhammad 29
Example: Stability of LTI Systems An LTI system with pure time shift is stable. n h[ n] [ n n ] 1 n 0 An accumulator (DT domain) system is unstable. h[ n] u[ n] u[ n] n n n 0 Similarly, an integrator (CT domain) system is unstable. CEN340: Signals and Systems - Dr. Ghulam Muhammad 30
Unit Step Response of An LTI System Discretetime domain s[ n] u[ n]* h[ n] h[ n]* u[ n] s[ n] h[ k ] n k h[ n] s[ n] s[ n 1] Running Sum First Difference Continuoustime domain t s ( t ) h( ) d ds () t h( t ) s ( t ) ds Running Integral First Derivative CEN340: Signals and Systems - Dr. Ghulam Muhammad 31
Linear Constant-Coefficient Differential Equation Consider an LTI system described by the following differential equation: dy () t 2 y ( t ) x ( t ) dt 3 where the input to the system is: x ( t ) Ke t u ( t ) Solve for y(t). y () t Y e p Determine 3t y ( t ) y ( t ) y ( t ) p h Particular solution Homogeneous solution y () t Ae h st From differential equation: 3 3 3 3 t 2 t t 3 2 K Y e Y e Ke Y Y K Y K y t e t 5 3t p ( ), for 0 5 st st st Ase 2Ae 0 A ( s 2) e 0 s 2 y () t Ae CEN340: Signals and Systems - Dr. Ghulam Muhammad 32 h Complete solution: 2t K y () t e Ae 5 3t 2t
Solution contd. Still, the value of A is unknown. We can find it by using the auxiliary condition. Different auxiliary conditions lead to different solutions of y(t). Suppose that the auxiliary condition is y(0) = 0, i.e., at t = 0, y(t) = 0. Using this condition into the complete solution, we get: K K 0 A A 5 5 K y t e e t 5 K e e u t 5 3t 2t ( ), 0 3t 2t = ( ) A general N-th order linear constant-coefficient differential equation is given by: N k M k d y ( t ) d x ( t ) ak k bk k k 0 dt k 0 dt M k 1 d x ( t ) y () t bk k A particular case when N = 0: a0 k 0 dt CEN340: Signals and Systems - Dr. Ghulam Muhammad 33
Convolution Integral: at x ( t ) e u ( t ) at h( t ) e u ( t ) y ( t ) x ( t )* h( t ) x ( ) h( t ) d Workout (1) x() 1 h() h(-) t < 0 h(t - ) 1 Reflect 1 Shift 1 t h(t - ) 1 t > 0 CEN340: Signals and Systems - Dr. Ghulam Muhammad 34 t
Workout (1) contd. For t < 0, the overlap between x() and h(t - ) is between the range = - and = t. t t 2a at a a( t ) at 2a at e t e y () t e e d e e d e 2a 2a For t > 0, the overlap between x() and h(t - ) is between the range = - and = 0. By combining, y () t 0 t 2a at a a( t ) at 2a at e 0 e y () t e e d e e d e 2a 2a 1 e 2a at y(t) 1/2a t CEN340: Signals and Systems - Dr. Ghulam Muhammad 35
Acknowledgement The slides are prepared based on the following textbook: Alan V. Oppenheim, Alan S. Willsky, with S. Hamid Nawab, Signals & Systems, 2 nd Edition, Prentice-Hall, Inc., 1997. Special thanks to Prof. Anwar M. Mirza, former faculty member, College of Computer and Information Sciences, King Saud University Dr. Abdul Wadood Abdul Waheed, faculty member, College of Computer and Information Sciences, King Saud University CEN340: Signals and Systems - Dr. Ghulam Muhammad 36