(section 1471x) discrete-time signals and systems · discrete-time signals and systems lecture 2...
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EEL 3135EEL 3135(Section 1471X)(Section 1471X)
DiscreteDiscrete--Time Signals and SystemsTime Signals and Systems
Lecture 2Lecture 2
Prof. Jian Li
Dept. of Electrical and Computer EngineeringUniversity of Florida,Gainesville, FL 32611
Office: 437 EBPhone: 352-392-2642 Email: [email protected]
Dept. of ECE 2
Topics
Signals
Transformations
Even and Odd
Complex Numbers
Cartesian and Polar Forms
Sinusoids
Properties
Phasors
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Signals
Continuous-Time Signal x(t)
with t being a continuous variable.
Discrete-Time Signals x[n] = x(nTs)
with n being a discrete value.
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Transformations of Time Variable
x(t-t0) means shifting to right t0 units
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More Transformations (2)
x(-t) means reflection about t = 0
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More Transformations (3)
x(2t) means compressing
x(t/2) means stretching
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More Transformations (4)
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Periodic Signals
x(t) = x(t-T)
T = period, so is 2T, 3T, ….
Fundamental period T0 is the smallest possible period.
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Even and Odd SignalsEven and Odd Signals
Even Signal:
x(t) = x(-t)
Odd Signal:
x(-t) = -x(t)
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More Even and Odd SignalsMore Even and Odd Signals
Any signal can be
broken into:
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SinusoidsSinusoids
Sinusoids occur in: music, phone, radar, communication, ….
(a) Before (b) After
ISAR autofocusing example
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SinusoidsSinusoids
AmplitudeRadian freq
Phase
Cycles per second (Hz)
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Complex NumbersComplex Numbers
Cartesian form:
Polar form:
Relationships:
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Euler Euler Formulas and Formulas and PhasorsPhasors
Euler Formula:
Inverse Euler Formulas:
Sinusoid:
Phasor (Complex Amplitude):
Positive freq Negative freq
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Phasor Phasor Addition RuleAddition Rule
Sum of sinusoids with frequency
is another sinusoid with same frequency
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Phasor Phasor Addition ExampleAddition Example
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DiscreteDiscrete--Time SinusoidsTime Sinusoids
Larger does not mean faster oscillation for
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DiscreteDiscrete--Time SinusoidsTime Sinusoids
Discrete-Time Sinusoids are not always periodic!
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AssignmentsAssignments
Finish Lecture 1 Assignments
Read Chapter 1
Read Appendix A on Complex Numbers
Read Chapter 2 as much as possible
Implement Lab. C.2 in Appendix C as much as possible
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THANK YOU
And
Happy Learning!!