ENGG 330

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ENGG 330

• Class 2
• Concepts, Definitions, and Basic Properties

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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 e3n

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Remember

• Real world signals are very complex
• Cant 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

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Transformations of the Independent Variable

• Example Transformations
• Periodic Signals
• Even and Odd Signals

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Transformations of Signals

• A central concept is transforming a signal by the
  system
• An audio system transforms the signal from a tape
  deck

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Example Transformations

• Time Shift Radar, Sonar, Seismic
• xn-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
• xn becomes x-n by reflection about n 0
• Time Scaling tape played slower/faster
• x(t), x(2t), x(t/2)

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Time Shift
t0 lt 0 so x(t-t0) is an advanced version of x(t)
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Time Reversal
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Time Scaling
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?
What does x(t1) look like?
Th e other way t 1 1 advanced
in time
When t -2 t1 -1 what is x(t) at 1? 0 When
t -1 t1 0 what is x(t) at 0? 1 When t
0 t1 1 what is x(t) at 1? 1 When t 1 t1
2 what is x(t) at 2? 0
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Given x(t) what would x(t-1) look like?
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?
What does x(-t1) look like?
When t -1 -t1 2 what is x(t) at 2? 0 When
t 0 -t1 1 what is x(t) at 1? 1 When t
1 -t1 0 what is x(t) at 0? 1 When t 2
-t1 -1 what is x(t) at 1? 0
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The other wayx(-t 1)
Apply the 1 time shift
Apply the t reflection about the y axis
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?
What does x(3 /2 t) look like?
When t -1 3t/2 -3/2 what is x(t) at -3/2?
0 When t 0 3t/2 0 what is x(t) at 0?
1 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?
1 Why 2/3? What is the next t that should
be evaluated? 4/3 why?
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?
What does look like?
First apply the 1 and advance the signal
Next apply the 3t/2 and compress the signal
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Signal Transformations

• X(at b) where a and b are given numbers
• Linearly Stretched if a lt 1
• Linearly Compressed if a gt 1
• Reversed if a lt 0
• Shifted in time if b is nonzero
• Advanced in time if b gt 0
• Delayed in time if b lt 0
• But watch out for x(-2t/3 1)

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Periodic Signals

• x(t) x(t T) x(t) periodic with period T
• xn xn N periodic with period N
• Fundamental period T or N
• Aperiodic

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Even and Odd Signals

• Even signals
• x(-t) x(t)
• x-n xn
• Odd signals
• x(-t) -x(t)
• x-n -xn
• Must be 0 at t 0 or n 0

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• Any signal can be broken into a sum of two
  signals on even and one odd
• Evx(t) ½x(t) x(-t)
• Odx(t) ½x(t) x(-t)

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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

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C-T Complex Exponential and Sinusoidal Signals

• x(t) Ceat where C and a are complex numbers
• Complex number
• a jb rectangular form
• Rej? 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

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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

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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

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General Complex Exponential Signals

• Most general case of complex exponential
• Can be expressed in terms of the two cases we
  have examined so far

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Periodicity Properties of D-T Complex Exponentials
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Unit Impulse and Unit Step Functions

• D-T Unit Impulse and Unit Step Functions
• C-T Unit Impulse and Unit Step Functions

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C-T D-T Systems

• Simple Examples

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Basic System Properties

• Memory
• Inverse
• Causality
• Stability
• Time Invariance
• Linearity

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Memory

• Memoryless output for each value of independent
  variable is dependent on the input at only that
  same time
• Memoryless
• y(t) x(t), yn 2xn x22n
• Memory
• Yn Sxk, yn xn-1

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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 isnt invertible

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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

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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

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Time Invariance

• See Fowler Note Set 5 System Properties

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Linearity

• See Fowler Note Set 5 System Properties

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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