1. Signal Processing First
Lecture 19
Continuous-Time Signals and Systems
9/3/2018
© 2003, JH McClellan & RW Schafer

2. © 2003, JH McClellan & RW Schafer
READING ASSIGNMENTS
This Lecture:
Chapter 9, Sects 9-1 to 9-5
Other Reading:
Recitation: Ch. 9, all
Next Lecture: Chapter 9, Sects 9-6 to 9-8
9/3/2018
© 2003, JH McClellan & RW Schafer

3. © 2003, JH McClellan & RW Schafer
LECTURE OBJECTIVES
Bye bye to D-T Systems for a while
The UNIT IMPULSE signal
Definition
Properties
Continuous-time signals and systems
Example systems
Review: Linearity and Time-Invariance
Convolution integral: impulse response
9/3/2018
© 2003, JH McClellan & RW Schafer

4. D-T Filtering of C-T Signals
C-to-D
D-to-C
LTI System
H(z)
LTI ANALOG
System
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© 2003, JH McClellan & RW Schafer

5. © 2003, JH McClellan & RW Schafer
ANALOG SIGNALS x(t)
INFINITE LENGTH
SINUSOIDS: (t = time in secs)
PERIODIC SIGNALS
ONE-SIDED, e.g., for t>0
UNIT STEP: u(t)
FINITE LENGTH
SQUARE PULSE
IMPULSE SIGNAL: d(t)
DISCRETE-TIME: x[n] is list of numbers
9/3/2018
© 2003, JH McClellan & RW Schafer

6. © 2003, JH McClellan & RW Schafer
CT Signals: PERIODIC
Sinusoidal signal
Square Wave
INFINITE DURATION
9/3/2018
© 2003, JH McClellan & RW Schafer

7. © 2003, JH McClellan & RW Schafer
CT Signals: ONE-SIDED
Unit step signal
One-Sided
Sinusoid
“Suddenly applied”
Exponential
9/3/2018
© 2003, JH McClellan & RW Schafer

8. CT Signals: FINITE LENGTH
Square Pulse signal
Sinusoid multiplied
by a square pulse
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© 2003, JH McClellan & RW Schafer

9. © 2003, JH McClellan & RW Schafer
What is an Impulse?
A signal that is “concentrated” at one point.
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© 2003, JH McClellan & RW Schafer

10. © 2003, JH McClellan & RW Schafer
Defining the Impulse
Assume the properties apply to the limit:
One “INTUITIVE” definition is:
Concentrated at t=0
Unit area
9/3/2018
© 2003, JH McClellan & RW Schafer

11. © 2003, JH McClellan & RW Schafer
Sampling Property
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© 2003, JH McClellan & RW Schafer

12. General Sampling Property
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© 2003, JH McClellan & RW Schafer

13. Properties of the Impulse
Concentrated at one time
Unit area
Sampling Property
Extract one value of f(t)
Derivative of unit step
9/3/2018
© 2003, JH McClellan & RW Schafer

14. Continuous-Time Systems
Examples:
Delay
Modulator
Integrator
Output
Input
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© 2003, JH McClellan & RW Schafer

15. © 2003, JH McClellan & RW Schafer
CT BUILDING BLOCKS
INTEGRATOR (CIRCUITS)
DIFFERENTIATOR
DELAY by to
MODULATOR (e.g., AM Radio)
MULTIPLIER & ADDER
9/3/2018
© 2003, JH McClellan & RW Schafer

16. © 2003, JH McClellan & RW Schafer
Ideal Delay:
Mathematical Definition:
To find the IMPULSE RESPONSE, h(t), let x(t) be an impulse, so
9/3/2018
© 2003, JH McClellan & RW Schafer

17. Output of Ideal Delay of 1 sec
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© 2003, JH McClellan & RW Schafer

18. © 2003, JH McClellan & RW Schafer
Integrator:
Mathematical Definition:
To find the IMPULSE RESPONSE, h(t), let x(t) be an impulse, so
Running Integral
9/3/2018
© 2003, JH McClellan & RW Schafer

19. © 2003, JH McClellan & RW Schafer
Integrator:
Integrate the impulse
IF t<0, we get zero
IF t>0, we get one
Thus we have h(t) = u(t) for the integrator
9/3/2018
© 2003, JH McClellan & RW Schafer

20. Graphical Representation
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21. © 2003, JH McClellan & RW Schafer
Output of Integrator
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© 2003, JH McClellan & RW Schafer

22. © 2003, JH McClellan & RW Schafer
Differentiator:
Mathematical Definition:
To find h(t), let x(t) be an impulse, so
Doublet
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© 2003, JH McClellan & RW Schafer

23. Differentiator Output:
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© 2003, JH McClellan & RW Schafer

24. Linear and Time-Invariant (LTI) Systems
If a continuous-time system is both linear and time-invariant, then the output y(t) is related to the input x(t) by a convolution integral
where h(t) is the impulse response of the system.
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© 2003, JH McClellan & RW Schafer

25. © 2003, JH McClellan & RW Schafer
Testing for Linearity
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© 2003, JH McClellan & RW Schafer

26. Testing Time-Invariance
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27. © 2003, JH McClellan & RW Schafer
Integrator:
Linear
And Time-Invariant
9/3/2018
© 2003, JH McClellan & RW Schafer