EEL 3135EEL 3135(Section 1471X)(Section 1471X)

DiscreteDiscrete--Time Signals and SystemsTime Signals and Systems

Lecture 9Lecture 9

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

DFT

FFT

Spectrogram

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DTFT ReviewDTFT Review

Fourier Synthesis (Inverse Fourier Transform)

Fourier Analysis (Fourier Transform)

Periodicity

Discrete Signal Periodic Spectra

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Discrete Fourier Transform (DFT)Discrete Fourier Transform (DFT)

Periodic Discrete Signal Discrete Periodic Spectra

Sampling will make x[n] periodic!

N-Point DFT Pair

Synthesis

Analysis

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DFT ExampleDFT Example

DTFT

8-point DFT

16-point DFT

32-point DFT

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MATLAB for DFTMATLAB for DFT

Number of frequency samples

XX = XX = fftfft((xxxx, N);, N);

Input sequence of length L < N

---- xx xx first zerofirst zero--padded to length Npadded to length N---- NN--pointpoint DFT is next computedDFT is next computed---- FFT is a fast way of computing DFTFFT is a fast way of computing DFT

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FFTFFT

Computations ofComputations ofDFT ( )DFT ( )vs. FFTvs. FFT( ( ))

N = # of PointsN = # of PointsN = Power of 2N = Power of 2

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Spectrum AnalysisSpectrum Analysis

Frequency content Frequency content of a signal is often a of a signal is often a function of timefunction of time

Not very informative

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SpectrogramSpectrogram

--ShortShort--time time (windowed)(windowed)Fourier Fourier transformtransform

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Spectrogram ExampleSpectrogram Example

--Frequency vs. time Frequency vs. time --GrayGray--level for level for intensityintensity

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Resolution in SpectrogramResolution in Spectrogram

OriginalResolution in time Resolution in time and frequency and frequency always in conflictalways in conflict

Good freq resolution

Good time resolution

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AssignmentsAssignments

Finish Lectures 1 – 8 Assignments

Read Chapter 9 as much as possible

Play with demos etc. on your CD

Enjoy your new knowledge on signals and systems

Apply your new knowledge

Update me with your achievements in life!

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THANK YOU

And

Happy Learning!!