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EEL 3135EEL 3135(Section 1471X)(Section 1471X)
DiscreteDiscrete--Time Signals and SystemsTime Signals and Systems
Lecture 3Lecture 3
Prof. Jian Li
Dept. of Electrical and Computer EngineeringUniversity of Florida,Gainesville, FL 32611
Office: 437 EBPhone: 352-392-2642 Email: [email protected]
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Dept. of ECE 2
Topics
! General Discrete Spectrum
! Two-Sided Spectrum
! Amplitude Modulation
! Periodic Signals
! Harmonically related frequencies
! Fourier Series
! Aperiodic Signals
! Continuous-Time Fourier Transform
! Properties
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General Discrete SpectrumGeneral Discrete Spectrum
! If x(t) has the form (arbitrary frequencies – x(t) may or may not be periodic):
Two-Sided Spectrum:
! Beat Notes:
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Amplitude ModulationAmplitude Modulation
Message Carrier
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Periodic Signals
! x(t) = x(t-T0)
T0 = smallest period = Fundamental Period
! f0 = 1/T0 = Fundamental Frequency
! Example:
Sum of sinusoids with harmonically related frequencies:
Harmonic frequencies:
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Harmonically Related Sinusoids
! x(t) is still periodic with period T0
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FourierFourier
! Almost all periodic
signals can be represented
by sum of harmonically
related Sinusoids
! First discovered by
Fourier almost
200 years ago
Periodic Signal "#"#"#"# Harmonically Related Discrete Spectra
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Fourier Series Fourier Series
! Fourier Synthesis
! Fourier Analysis
! Alternatively
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Spectrum of Periodic SignalsSpectrum of Periodic Signals
! Fourier Series
! Two-Sided Spectrum
! Ex:
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ExampleExample
Magnitude of
Phase of
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Pulse TrainPulse Train
Or
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Pulse Train SpectrumPulse Train Spectrum
T fixed
Smaller T1 yields wider spectra
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Pulse Train SynthesisPulse Train Synthesis
Gibb’s Phenomenon(9% overshoot)
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Pulse Train => PulsePulse Train => Pulse
T1 fixed
Larger T yields smaller gaps
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ContinuousContinuous--Time Fourier Transform (CTFT)Time Fourier Transform (CTFT)
! Fourier Synthesis (Inverse Fourier Transform)
! Fourier Analysis (Fourier Transform)
Fourier’s Most Important Contribution!
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Fourier Transform of PulseFourier Transform of Pulse
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Interesting PropertiesInteresting Properties
Finite Duration "# Infinite Duration
Real-Valued Even "# Real-Valued Even
Real -Valued Odd "# Imaginary -Valued Odd
Compressed "# Stretched
Spectra: Magnitude Even, Phase Odd
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Important Fourier Transform PropertiesImportant Fourier Transform Properties
UseProperties whenDetermining Fourier Transforms!
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Example Example
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Duality Duality
"#"#"#"#
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Unit Impulse Unit Impulse
Even function:
Dirac Delta
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Unit Impulse related SpectraUnit Impulse related Spectra
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AssignmentsAssignments
$ Finish Lectures 1 and 2 Assignments
$ Read Chapter 2
$ Read Chapter 3 and 13 as much as possible
$ Implement Lab. C.2 in Appendix C as much as possible
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THANK YOU
And
Happy Learning!!