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Continuous-time & Discrete-time Systems. • Physical Systems are interconnection of – components, devices, or subsystems. • System can be viewed as a process in which – input signals are transformed by the system or – cause the system to response in some way, – resulting in other signals as outputs. Continuous-time & Discrete-time Systems. x(t) y(t) y(t) x(t) Continuous-time system x[n] Discrete-time System y[n] x[n] y[n] Examples Of Systems i(t) vs(t) vc(t) , R Relationsh ip of current and voltage for a capasitor : dv (t) i(t) C c , and substituti ng this into the above equation : dt We have the differenti al equation describing the relationsh ip From Ohms' s Law : - i(t) between th e input v s (t ) and the output v c (t ) : dvc(t) 1 1 vc(t) vs(t) dt RC RC Example of Mechanical System v (t ) f(t) net force f(t) - v(t) mass * accelerati on of car dv(t ) f(t) - v(t) m * , dt dv(t ) 1 i.e. v(t ) f (t ). dt m m Generally 1st order differenti al equation : dy (t ) ay (t ) bx(t ). dt Example of Discrete-time System Simple Model for Monthly Bank Balance y[n]=present current balance. x[n]=net deposit(deposits-withdrawals). Accrue 1% interest on monthly past balance. y[n]=1.01y[n-1]+x[n]. or y[n]-1.01y[n-1]=x[n]. Digital Simulation of Differential Equation Through Difference Equation. dv(t ) 1 v(t ) f (t ). dt m m dv(t) v[n ] - v[n - 1] By first backward difference : dt The differenti al equation can be expresses as : v[n ] - v[n - 1] 1 v[n] f [n]. m m 1 v[n 1] 1 v[n]( ) f [n]. m m m v[n] v[n 1] f [n], m m Letting, v[n ] v[n] and f[n] f[n ]. m v[n] v[n 1] f [n], m m Interconnections of Systems Series or Cascade Form Input System1 System2 Output Parallel Form Input System 1 + System 2 Output Interconnections of Systems Series - Parallel Form System 1 System 2 + Input System 4 Output System 3 Feedback Form Input System 1 + System 2 Output Interconnections of Systems Simple Electrical Circuit i1 (t ) i2 (t ) V(t) Feedback Block Diagram Form of Circuit Input i (t ) i1 (t ) Capasitor 1 t v(t ) i1 ( )d C + +i2 (t ) Resistor v (t ) i2 (t ) R Output v (t ) Basic System Properties • • • • • • Systems with and without memory. Invertibility and Inverse Systems. Causality. Stability Time Invariance Linearity. Systems without memory. • System is memoryless if its output at any one time depends only on the input at the same time. • E.g. y[n]=(2x[n]-x2[n])2 • A resistor is memoryless because y(t)=Rx(t) • So too an identity system is memoryless because y(t)=x(t), y[n]=x[n]. Systems with memory. • System with memory depicts its output at any one time that is dependent not only on the present input but also past(future) values of input and output. • E.g. accumulator/summer y[n] n x[k ], k Delay, y[n] x[n - 1]. 1 t A capacitor is a memory analog device, y(t) x( )d , C - Invertible System • Systems whereby distinct inputs lead to distinct outputs • As such an inverse system exits that, when cascaded with the original system, yields an output w[n] equal to the input x[n] to the first system. x[n] System y[n] Inverse System w[n]=x[n] Examples of an invertible continuous-time system. 1. y(t)=2x(t) for which the inverse system is w(t)=y(t)/2. 2. y[n] n n 1 k k x[k ], y[n] x[k ] x[n], y[n] y[n 1] x[n], x[n] y[n] y[n 1]. the inverse system is : w[n] x[n] y[n] y[n 1]. Two Examples of Inverse System x(t) y(t)=2x(t) y(t) w(t)=x(t) w(t)=y(t)/2 w[n]=x[n] x[n] y[n] n y[n] w[n] y[n] y[n 1]. x [ k ] k Examples of Noninvertible Systems • y[n]=0. • The output is always zero for any value of input x[n]. • y(t)=x2(t). • The sign for the input x(t) cannot be determined for a certain value of output y(t) • I.e. for both cases the values of the output is not distinct for distinct values of input, Causality • A system is causal because its output depends only on present and past values of the input • Such a system does not anticipate future values of input. • y[n]=x[n]-x[n+1] and y(t)=x(t+1) are noncausal systems. Stability • A stable system is one in which small inputs lead to response that do not diverge. y(t) stable pendulum unstable pendulum Time Invariance • System is time invariant if the behavior and characteristics of the system are fixed over time. • E.g. the RC circuitry where the values of the parameter of the components R and C do not changed with time I.e. constant. • A system is time invariant if a time shift in the input signal results in an identical time shift in the output signal. Linearity • The system is linear if it possesses the superposition property. Let y1 (t ) be the response of a continuous - time system to an input x 1 (t ), and let y 2 (t ) be the output correspond ing to the input x 2 (t ). System is linear if : 1) The response to x1 (t ) x2 (t ) is y1 (t ) y2 (t ). 2)The response to ax1 (t ) is ay1 (t ), where " a" is any complex constant. Combining the two property for linearity. ax1 (t ) bx2 (t ) ay1 (t ) by2 (t ) ax1[n] bx2 [n] ay1[n] by2 [n]. Example 1.12. Causality??? Is the system given by y[n] x[-n] causal or not? If n 0 , e.g n 4, y[4] x[-4]. This says that at n 4, the output y[n] depend on past value of x[n]. However n 0, e.g n -3, y[-3] x[-(-3)], y[-3] x[3], i.e. the output y[n] depends on future value of input x[ 3]. System is not causal.