Classification of Systems of the I unit in Signals and Systems

1. Systems and its
Classification:Part-1
Dr.K.G.SHANTHI
Professor/ECE
shanthiece@rmkcet.ac.in

2. Systems:
✘ 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
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Or
Excitation
Or
Response
System
Input Signal Output Signal

3. System - An entity that responds to a signal
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Examples
 Circuit
systeminput output

4. Systems-Examples
 The Camera
 The Speech Recognition System
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Identified
ImageLight

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

6. Systems-Examples
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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
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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)}
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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)= R𝑥(t)
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𝑦(n)= 𝑥2(n)+ 𝑥(n)
 n=0, 𝑦(0)= 𝑥2(0)+ 𝑥(0)
 n=1, 𝑦(1)= 𝑥2(1)+ 𝑥(1)
 n=-1, 𝑦(-1)= 𝑥2(-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
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Future
input
𝑦(n)= 𝑥2(n)+ 𝑥(2n)
 n=0, 𝑦(0)= 𝑥2(0)+ 𝑥(0)
 n=1, 𝑦(1)= 𝑥2(1)+ 𝑥(2)
 n=-1, 𝑦(-1)= 𝑥2(-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.
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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
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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)
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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)
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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
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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)= 𝑥2(n)+ 2𝑥(n+3)
✘ 𝑦(0)= 𝑥2(0)+ 2𝑥(0+3)
✘ 𝑦(1)= 𝑥2(1)+ 2𝑥(1+3)
✘ 𝑦(-1)= 𝑥2(-1)+ 2𝑥(-1+3)
✘ Given system is Non-Causal
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𝑦(0)= 𝑥2(0)+ 2𝑥(3)
𝑦(1)= 𝑥2(1)+ 2𝑥(4)
𝑦(-1)= 𝑥2(-1)+ 2𝑥(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.
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System
𝑥(𝑡) y(𝑡)
Inverse
System w(𝑡)=x(t)
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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-t0)
causes the same time shift in the output signal 𝑦(t-t0)
Condition:
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CT System: If 𝑦(t)=𝑇[𝑥(t)], then 𝑇[𝑥(t-t0)]=𝑦(t-t0) must be satisfied
System
T𝑥(t) 𝑦(t) 𝑦(t-t0)
System
T𝑥(t-t0)
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:
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CT System: If 𝑦(t)=𝑇[𝑥(t)], then 𝑇[𝑥(t-t0)] ≠ 𝑦(t-t0)
DT System: If 𝑦(n)=𝑇[𝑥(n)], then 𝑇[𝑥(n-k)] ≠ 𝑦(n-k)

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

21. TEST FOR TIME INVARIANCE(CT SYSTEM)
1. Apply a delay to the input
𝑥(t)
2. Compute the output 𝑦(t, t0)
3. Apply the same delay to original output
𝑦(t)
4. Compare the results from steps 2 and 3
5. If 𝑦(t-t0)= 𝑦(t, t0) then System is Time InVarient
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𝑦(t-t0)
𝑥(t-t0)

22. Problem-TIME VARIANCE(CT SYSTEM)
✘ Determine whether the following system is time
invariant or not 𝑦(t) = t 𝑥(t)
1. Apply a delay to the input 𝑥(t) 𝑥(t-t0)
2. Compute the output 𝑦(t, t0):
𝑦(t, t0)=t 𝑥(t-t0)
4. Apply the same delay t0to original output 𝑦(t)and compute 𝑦(t-t0) :
𝑦(t-t0)= (t-t0) 𝑥(t-t0)
5. 𝑦(t, t0) ≠ 𝑦(t-t0).Hence the given system is time variant
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23. TEST FOR TIME INVARIANCE(DT SYSTEM)
1. Apply a delay to the input
𝑥(n)
2. Recompute the output 𝑦(n,k)
3. Apply the same delay to original output
𝑦(n)
4. Compare the results from steps 2 and 3
If 𝑦(n-k)= 𝑦(n,k) ,
then System is Time InVarient
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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
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𝒙(−𝒏−k +𝟐)

25. Problem-TIME INVARIANCE(CT SYSTEM)
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