1. Signals and Systems
EE235
Leo Lam ©

2. Pet
Q: Has the biomedical imaging engineer done anything useful lately? A: No, he's mostly been working on PET projects.
Leo Lam ©

3. Today’s menu System properties examples Time invariance Linearity
Leo Lam ©

4. “System Response is the same no matter when you run the system.”
System properties
Time-invariance: A System is Time-Invariant if it meets this criterion
“System Response is the same no matter when you run the system.”
Leo Lam ©

5. Time invariance The system behaves the same no matter when you use it
Input is delayed by t0 seconds, output is the same but delayed t0 seconds If
then
System T
Delay t0
x(t)
x(t-t0)
y(t)
y(t-t0)
T[x(t-t0)]
System 1st
Delay 1st =
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds
Leo Lam ©

6. Time invariance example
Still you…
T(x(t)) = 3x(t - 5)
y(t) = 3x(t-5)
y(t – t0) = 3x(t-t0-5)
T(x(t – t0)) = 3x(t-t0-5)
y(t-t0)) = T(x(t-t0))
Time invariant!
KEY:
In step 2 you replace t by t-t0.
In step 3 you replace x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
Leo Lam ©

7. Time invariance example
Still you…
T(x(t)) = x(5t)
y(t) = x(5t)
y(t – 3) = x(5(t-3)) = x(5t – 15)
T(x(t-3)) = x(5t- 3)
Oops…
Not time invariant!
Does it make sense?
KEY:
In step 2 you replace t by t-t0.
In step 3 you replace x(t) by x(t-t0).
Shift then scale
Basically taking BOTH routes should come to the same answer.
Leo Lam ©

8. Time invariance example
system output
y(t) = x(5t) 1
Graphically: T(x(t)) = x(5t)
y(t) = x(5t)
y(t – 3) = x(5(t-3)) = x(5t – 15)
T(x(t-3)) = x(5t- 3) t
shifted system output
y(t-3) = x(5(t-3)) t
system input
x(t) 5 t
system output
for shifted system input
T(x(t-3)) = x(5t-3) t
shifted system input
x(t-3)
Basically taking BOTH routes should come to the same answer.
Leo Lam ©

9. Time invariance example
Integral
First:
Second:
Third:
Lastly:
Time invariant!
KEY:
In step 2 you replace t by t-t0.
In step 3 you replace x(t) by x(t-t0).
Basically taking BOTH routes should come to the same answer.
Leo Lam ©

10. System properties
Linearity: A System is Linear if it meets the following two criteria:
Together…superposition If
and
Additivity
Then If
Then
Scaling
“System Response to a linear combination of inputs is the linear combination of the outputs.”
Leo Lam ©

11. Linearity Order of addition and multiplication doesn’t matter.
System T
Linear
combination
System 1st
Combo 1st =
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds
Linear
combination
Leo Lam ©

12. Linearity Positive proof Negative proof
Prove both scaling & additivity separately
Prove them together with combined formula
Negative proof
Show either scaling OR additivity fail (mathematically, or with a counter example)
Show combined formula doesn’t hold
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds
Leo Lam ©

13. Linearity Proof Combo Proof System 1st Combo 1st Step 1: find yi(t)
Step 2: find y_combo
Step 3: find T{x_combo}
Step 4: If y_combo = T{x_combo}
Linear
System T
Linear
combination
System 1st
Combo 1st
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds
Leo Lam ©

14. Linearity Example Is T linear? T x(t) y(t)=cx(t) Equal  Linear
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds
Equal  Linear
Leo Lam ©

15. Linearity Example Is T linear? T x(t) y(t)=(x(t))2
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds
Not equal  non-linear
Leo Lam ©

16. Linearity Example Is T linear? T x(t) y(t)=x(t)+5
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds
Not equal  non-linear
Leo Lam ©

17. Linearity Example Is T linear? =
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds =
Leo Lam ©

18. Linearity unique case How about scaling with 0?
If T{x(t)} is a linear system, then zero input must give a zero output
A great “negative test”
If the input is delayed by t0 seconds, then the output is the same but delayed t0 seconds
Leo Lam ©

19. Spotting non-linearity
multiplying x(t) by another x()
y(t)=g[x(t)] where g() is nonlinear
piecewise definition of y(t) in terms of values of x, e.g.
NOT Formal Proofs!
(although sometimes ok)
Leo Lam ©

20. Summary: System properties
Causal: output does not depend on future input times
Invertible: can uniquely find system input for any output
Stable: bounded input gives bounded output
Time-invariant: Time-shifted input gives a time-shifted output
Linear: response to linear combo of inputs is the linear combo of corresponding outputs
Leo Lam ©

21. Superposition Superposition is…
Weighted sum of inputs  weighted sum of outputs
“Divide & conquer”
Leo Lam ©

22. Superposition example
Graphically
x1(t) T 1
y1(t) 2
x2(t)
y2(t) 3 -1 ?
-y2(t) 22
Leo Lam ©

23. Superposition example
Slightly aside (same system)
Is it time-invariant?
No idea: not enough information
Single input-output pair cannot test positively
x1(t) T 1
y1(t) 2
x2(t)
y2(t) 3
Specific input-output pair (or pairs), you CANNOT test positively for system properties. 23
Leo Lam ©

24. Superposition example
Unique case can be used negatively
x1(t) T 1
y1(t) 2
x2(t) -1
y2(t) -2
NOT Time Invariant:
Shift by 1  shift by 2
x1(t)=u(t) S
y1(t)=tu(t)
NOT Stable:
Bounded input gives unbounded output 24
Leo Lam ©