Classification of Systems: Part 1

Classification of Systems of the I unit in Signals and Systems

1. Systems and its Classification: Part-1 Dr.K.G.SHANTHI Professor/ECE [email protected]

2. System s : Systems process input signals to produce output signals A system is combination of elements that manipulates one or more signals to accomplish a function and produces some output 2 Or Excitation Or Response System Input Signal Output Signal

3. System - An entity that responds to a signal 3 3 Examples Circuit system input output

4. Systems- Examples The Camera The Speech Recognition System 4 Identified Image Light

5. Automobile System : System is the automobile Pressure on accelerator pedal is the input signal The automobile speed is the response or output signal. 5 Systems- Examples

6. Systems- Examples 6 Music Music System

7. Classification of Systems Continuous time and Discrete time system Linear and Non-Linear system Static and Dynamic system Time invariant and Time variant system Causal and Non-Causal system Stable and Unstable system Invertible & Inverse Systems 7

8. Continuous time and Discrete time system Continuous time system Operates on a continuous time signal T denotes transformation Response 𝑦(t)=𝑇{𝑥(t)} 8 T 𝑥(𝑡) y(𝑡) Discrete time system Operates on a discrete time signal T denotes transformation Response 𝑦(n)=𝑇{𝑥(n)} T 𝑥(n) y(n)

9. Static and Dynamic system Static system (Memory-less): If the response of the system is due to present input alone. Output at any instant of time depends only on present inputs Eg : 𝑦(𝑡)= 2𝑥(t) t=0, 𝑦(0)= 2𝑥(0) t= 1, 𝑦(1)= 2𝑥(1) t= -1, 𝑦(-1)= 2𝑥(-1) A resistor is a memory less system ; with the input x(t) taken as the current and with the voltage taken as the output y(t), the input-output relationship of a resistor is 𝑦(𝑡)= R 𝑥(t) 9 𝑦(n)= 𝑥 2 (n)+ 𝑥(n) n=0, 𝑦(0)= 𝑥 2 (0)+ 𝑥(0) n=1, 𝑦(1)= 𝑥 2 (1)+ 𝑥(1) n=-1, 𝑦(-1)= 𝑥 2 (-1)+ 𝑥(-1)

10. Static and Dynamic system( contd ) Dynamic system (Memory): If the response of the system depends on factors other than present input also. Output at any instant of time depends only on past and future inputs Eg : 𝑦(𝑡)= 2𝑥(t)+𝑥(-t), t=0, 𝑦(0)= 2𝑥(0)+𝑥(-0), t= 1, 𝑦(1)=2𝑥(1)+𝑥(-1), t= -1, 𝑦(-1)=2𝑥(-1)+𝑥(1), A capacitor is an example of a continuous-time system with memory 10 Future input 𝑦(n)= 𝑥 2 (n)+ 𝑥(2n) n=0, 𝑦(0)= 𝑥 2 (0)+ 𝑥(0) n=1, 𝑦(1)= 𝑥 2 (1)+ 𝑥(2) n=-1, 𝑦(-1)= 𝑥 2 (-1)+ 𝑥(-2) Past input

11. Static and Dynamic system( contd ) Determine whether the following system is static or dynamic 𝒚(𝒕) =𝒙(𝟐𝒕)+𝟐𝒙(𝒕) t=0 , 𝑦 (0) =𝑥 (0) +2𝑥 (0) ⇒ present inputs t=-1 , 𝑦(−1) =𝑥 (−2) +2𝑥(−1) ⇒ past and present inputs t=1 , 𝑦( 1) =𝑥(2) +2𝑥( 1 )⇒ future and present inputs Since output depends on past and future inputs the given system is dynamic system. 11

12. Static and Dynamic system( contd ) Determine whether the following systems are static or dynamic 𝒚(𝒏)=𝒔𝒊𝒏𝒙(𝒏) 𝑦(0) = 𝑠𝑖𝑛𝑥(0) ⇒ present input 𝑦(−1) = 𝑠𝑖𝑛𝑥(−1) ⇒ present input 𝑦(1) = 𝑠𝑖𝑛𝑥(1) ⇒ present input Since output depends on present input the given system is Static system 12

13. Causal and Non-Causal system Causal system (Non-Anticipative): If the response of a system at any instant of time depends only on the present input, past input and past output but does not depends upon the future input and future output. Examples: The motion of an automobile is causal, since it does not anticipate future actions of the driver 𝑦(𝑡)=3𝑥(𝑡)+𝑥(𝑡−1) 𝑦(0)=3𝑥(0)+𝑥(0−1) 𝑦(1)=3𝑥(1)+𝑥(1−1) 𝑦(-1)=3𝑥(-1)+𝑥(-1−1) 13 𝑦(0)=3𝑥(0)+𝑥(−1) 𝑦(1)=3𝑥(1)+𝑥(0) 𝑦(-1)=3𝑥(-1)+𝑥(-2) At any instant of time, output depends on only present and past input

14. Causal and Non-Causal system( contd ) Non-Causal system (Anticipative) : If the response of a system at any instant of time depends on the future inputs also. Example 𝑦(𝑡)=𝑥(𝑡+2)+𝑥(𝑡−1) 𝑦(0)=𝑥(0+2)+𝑥(0−1) 𝑦(1)=𝑥(1+2)+𝑥(1−1) 𝑦(-1)=𝑥(-1+2)+𝑥(-1−1) 14 𝑦(1)= 𝑥(3) +𝑥(0) 𝑦(0)= 𝑥(2) +𝑥(−1) 𝑦(-1)= 𝑥(1) +𝑥(-2) At any instant of time, output depends on future values also

15. Causal and Non-Causal system( contd ) Determine whether the following systems are causal or not y[n]=x[n-2]+x[n]+x[n-1] y[0]=x[0-2]+x[0]+x[0-1] y[1]=x[1-2]+x[1]+x[1-1] y[-1]=x[-1-2]+x[-1]+x[-1-1] Given system is Causal 15 y[0]=x[-2]+x[0]+x[-1] y[1]=x[-1]+x[1]+x[0] y[-1]=x[-3]+x[-1]+x[-2] At any instant of time, output depends on only present and past inputs

16. Causal and Non-Causal system( contd ) Determine whether the following systems are causal or not 𝑦(n)= 𝑥 2 (n)+ 2𝑥(n+3) 𝑦(0)= 𝑥 2 (0)+ 2𝑥(0+3) 𝑦(1)= 𝑥 2 (1)+ 2𝑥(1+3) 𝑦(-1)= 𝑥 2 (-1)+ 2𝑥(-1+3) Given system is Non-Causal 16 𝑦(0)= 𝑥 2 (0)+ 2 𝑥(3) 𝑦(1)= 𝑥 2 (1)+ 2 𝑥(4) 𝑦(-1)= 𝑥 2 (-1)+ 2 𝑥(2) At any instant of time, output depends on Present and Future inputs

17. lnvertibility and Inverse Systems If a system is invertible it has an Inverse System Eg : Encoder Examples of noninvertible systems are y[n] = 0 , the system that produces the zero output sequence for any input sequence. 17 System 𝑥(𝑡) y(𝑡) Inverse System w(𝑡)=x(t) 2 𝑥(𝑡) y(𝑡)=2𝑥(𝑡) 1/2 w(𝑡)=1/2 [y(t)] =(1/2) [2𝑥(𝑡)] =x(t)

18. Time invariant (Shift invariant) and Time variant (Shift variant) system Time invariant system (Shift invariant) : If the relationship between the input and output does not change with time . A system is called time invariant if a time shift in the input signal 𝑥(t-t ) causes the same time shift in the output signal 𝑦(t-t ) Condition: 18 CT System: If 𝑦(t)=𝑇[𝑥(t)], then 𝑇[𝑥(t-t )]=𝑦(t-t ) must be satisfied System T 𝑥(t) 𝑦(t) 𝑦(t-t ) System T 𝑥(t-t ) DT System: If 𝑦(n)=𝑇[𝑥(n)], then 𝑇[𝑥(n-k)]=𝑦(n-k) must be satisfied Where k represent delay

19. Time invariant (Shift invariant) and Time variant (Shift variant) system Time Variant system (Shift variant) : If the relationship between the input and output changes with time. Condition: 19 CT System: If 𝑦(t)=𝑇[𝑥(t)], then 𝑇[𝑥(t-t )] ≠ 𝑦(t-t ) DT System: If 𝑦(n)=𝑇[𝑥(n)], then 𝑇[𝑥(n-k)] ≠ 𝑦(n-k)

20. TEST FOR TIME INVARIANCE (CT) 20 https://www.youtube.com/watch?v=LezLNMznZm4

21. TEST FOR TIME INVARIANCE(CT SYSTEM) Apply a delay to the input 𝑥(t) Compute the output 𝑦(t, t ) Apply the same delay to original output 𝑦(t) Compare the results from steps 2 and 3 If 𝑦(t- t )= 𝑦(t, t ) then System is Time InVarient 21 𝑦(t- t ) 𝑥(t- t )

22. Problem-TIME VARIANCE(CT SYSTEM) Determine whether the following system is time invariant or not 𝑦(t) = t 𝑥(t) Apply a delay to the input 𝑥(t) 𝑥(t- t ) Compute the output 𝑦(t, t ): 𝑦 (t, t )=t 𝑥(t- t ) Apply the same delay t to original output 𝑦(t) and compute 𝑦(t- t ) : 𝑦(t- t )= (t- t ) 𝑥(t- t ) 𝑦(t, t ) ≠ 𝑦(t- t ). Hence the given system is time variant 22

23. TEST FOR TIME INVARIANCE(DT SYSTEM) Apply a delay to the input 𝑥(n) Recompute the output 𝑦( n,k ) Apply the same delay to original output 𝑦(n) Compare the results from steps 2 and 3 If 𝑦(n-k)= 𝑦( n,k ) , then System is Time InVarient 23 𝑥(n-k) 𝑦(n-k)

24. TEST FOR TIME INVARIANCE(DT SYSTEM) Determine whether the following systems are time invariant or not 𝒚(𝒏) =𝒙(−𝒏+𝟐) Apply a delay to the input 𝒙(−𝒏+𝟐) Output due to input delayed by k seconds 𝑦( n,k ) =𝑥(−𝑛−𝑘+2) Output delayed by k seconds 𝑦(n-k) =𝑥 (−(𝑛−𝑘)+2)=𝑥(−𝑛+𝑘+2) ∵ 𝑦( n,k ) ≠ 𝑦(n-k) The given system is time variant 24 𝒙(−𝒏− k +𝟐)

25. Problem-TIME INVARIANCE(CT SYSTEM) 25