ENGG 330
Class 2
Concepts, Definitions, and Basic
Properties
Quiz
• What is the difference between
– Stem & Plot
– How do I specify a discrete sample space from
0 to 10
– How do I multiply a scalar times a matrix
– How do I express e3[n]
Remember
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•
•
•
Real world signals are very complex
Can’t hope to model them
Can model simple signals
Can tell a lot about systems with simple
signals
• Can model complex signals with, dare I say,
transformations of simple signals
Transformations of the
Independent Variable
• Example Transformations
• Periodic Signals
• Even and Odd Signals
Transformations of Signals
• A central concept is transforming a signal
by the system
– An audio system transforms the signal from a
tape deck
Example Transformations
• Time Shift – Radar, Sonar, Seismic
– x[n-n0] & x(t-t0)
• Notice a difference? n for D-T, t for C-T
– Delayed if t0 positive, Advanced if t0 negative
• Time Reversal – tape played backwards
– x[n] becomes x[-n] by reflection about n = 0
• Time Scaling – tape played slower/faster
– x(t), x(2t), x(t/2)
Time Shift
t0 < 0 so x(t-t0) is an
advanced version of
x(t)
Time Reversal
Time Scaling
?
What does x(t+1) look like?
When t = -2 t+1 = -1 what is x(t) at –1? 0
When t = -1 t+1 = 0 what is x(t) at 0? 1
When t = 0 t+1 = 1 what is x(t) at 1? 1
When t = 1 t+1 = 2 what is x(t) at 2? 0
Th e other way –
t+1
+1 advanced in time
Given x(t) what would x(t-1)
look like?
?
What does x(-t+1) look like?
When t = -1 -t+1 = 2 what is x(t) at 2? 0
When t = 0 -t+1 = 1 what is x(t) at 1? 1
When t = 1 -t+1 = 0 what is x(t) at 0? 1
When t = 2 -t+1 = -1 what is x(t) at –1? 0
The other way
x(-t + 1)
Apply the +1 time shift
Apply the –t reflection about the y
axis
?
What does x(3 /2 t)
look like?
When t = -1 3t/2 = -3/2 what is x(t) at -3/2?
When t = 0 3t/2 = 0 what is x(t) at 0?
When t = 1 3t/2 = 3/2 what is x(t) at 3/2?
When t = 2/3 3t/2 = 1 what is x(t) at 1?
Why 2/3?
What is the next t that should be evaluated?
4/3 why?
0
1
?
1
?
What does
look like?
First apply the +1 and advance the signal
Next apply the 3t/2 and compress the signal
Signal Transformations
• X(at + b) where a and b are given numbers
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–
–
–
Linearly Stretched if |a| < 1
Linearly Compressed if |a| > 1
Reversed if a < 0
Shifted in time if b is nonzero
• Advanced in time if b > 0
• Delayed in time if b < 0
• But watch out for x(-2t/3 + 1)
Periodic Signals
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•
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x(t) = x(t + T) x(t) periodic with period T
x[n] = x[n + N] periodic with period N
Fundamental period T or N
Aperiodic
Even and Odd Signals
• Even signals
– x(-t) = x(t)
– x[-n] = x[n]
• Odd signals
– x(-t) = -x(t)
– x[-n] = -x[n]
– Must be 0 at t = 0 or n = 0
• Any signal can be broken into a sum of two
signals on even and one odd
– Ev{x(t)} = ½[x(t) + x(-t)]
– Od{x(t)} = ½[x(t) – x(-t)]
Exponential and Sinusoidal
Signals
• C-T Complex Exponential and Sinusoidal
Signals
• D-T Complex Exponential and Sinusoidal
Signals
• Periodicity Properties of D-T Complex
Exponentials
C-T Complex Exponential and
Sinusoidal Signals
• x(t) = Ceat where C and a are complex numbers
– Complex number
• a + jb – rectangular form
• Rejθ – polar form
• Depending on Values of C and a Complex
Exponentials exhibit different characteristics
– Real Exponential Signals
– Periodic Complex Exponential and Sinusoidal Signals
– General Complex Exponential Signals
Real Exponential Signals
• If C and a are real
– x(t) = Ceat then called real exponential
• If a is positive x(t) is a growing exponential
• If a is negative x(t) is a decaying exponential
• If a 0 x(t) is a constant
– That depends upon the value of C
• Use MATLAB to plot
– e2n, e-2n , e0n , 3e0n
Periodic Complex Exponential
and Sinusoidal Signals
• If a is purely imaginary
– x(t) is then periodic
• x(t) = ejw0t – Plot via MATLAB
• ? j is needed to make a imaginary
• a closely related signal is Sinusoid
General Complex Exponential
Signals
• Most general case of complex exponential
– Can be expressed in terms of the two cases we
have examined so far
Periodicity Properties of D-T
Complex Exponentials
Unit Impulse and Unit Step
Functions
• D-T Unit Impulse and Unit Step Functions
• C-T Unit Impulse and Unit Step Functions
C-T & D-T Systems
• Simple Examples
Basic System Properties
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Memory
Inverse
Causality
Stability
Time Invariance
Linearity
Memory
• Memoryless output for each value of
independent variable is dependent on the
input at only that same time
• Memoryless
– y(t) = x(t), y[n]= 2x[n] – x2[2n]
• Memory
– Y[n] = Σx[k], y[n] = x[n-1]
Inverse
• Invertible if distinct inputs lead to distinct
outputs
• Think of an encoding system
– It must be invertible
• Think of a JPEG compression system
– It isn’t invertible
Causality
• A system is causal if the output at any time
depends on values of the input at only
present and past times.
• See Fowler Note Set 5 System Properties
Stability
• If the input to a stable system is bounded
the the output must also be bounded
– Balanced stick
• Slight push is bounded
• Is the output bounded
Time Invariance
• See Fowler Note Set 5 System Properties
Linearity
• See Fowler Note Set 5 System Properties
Assignment
• Read Chapter 1 of Oppenheim
– Generate math questions for Dr. Olson
• Buck
– Section 1.2 a, b, c, d
– Section 1.3 a, b, c
– Section 1.4 a, b
• Turn in .m files
– All plots/stems need titles and xy labels
– Answers to questions documented in .m file with
references to plots/stems