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: li@dsp.ufl.edu

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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:

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Beat Notes ExampleBeat Notes Example

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Amplitude Modulation ExampleAmplitude Modulation Example

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:

f0 = ?

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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 X0

X1/2

X2/2

kth harmonic

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

Spectra: Magnitude Even, Phase Odd

Real-Valued Even Real-Valued Even

Real -Valued Odd Imaginary -Valued Odd

Compressed Stretched

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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

Dirac Delta

Even function:

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Unit Impulse related SpectraUnit Impulse related Spectra

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ConvolutionConvolution

• Definition:

• Commutative:

• Associative:

• Distributive:

Convolution with a delta function results in a shift!

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Amplitude ModulationAmplitude Modulation

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Amplitude DemodulationAmplitude Demodulation

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AssignmentsAssignments

Finish Lectures 1 and 2 Assignments

Read Chapter 2

Read Chapter 3 as much as possible

Implement Lab. C.2 in Appendix C as much as possible

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THANK YOU

And

Happy Learning!!