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Frequency Domain Characterization of Signals

Frequency Domain Characterization of Signals . Prof. Hongang Wang. Signal Representation. Signal? Time-domain description Waveform Periodic vs. non-periodic signals Frequency-domain description Periodic signals Sinusoidal signals Fourier series for periodic signals. Signal?.

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Frequency Domain Characterization of Signals

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  1. Frequency DomainCharacterization ofSignals Prof. Hongang Wang

  2. Signal Representation • Signal? • Time-domain description • Waveform • Periodic vs. non-periodic signals • Frequency-domain description • Periodic signals • Sinusoidal signals • Fourier series for periodic signals

  3. Signal? • A variable (or multiple variables) that changes in time • Speech or audio signal: A sound amplitude that varies in time • Temperature readings at different hours of a day • Stock price changes over days • More generally, a signal may vary in 2-D space and/or time • A picture: the color varies in a 2-D space • A video sequence: the color varies in 2-D space and in time • Continuous vs. Discrete • The value can vary continuously or take from a discrete set • The time and space can also be continuous or discrete

  4. Waveform Representation • Waveform representation • Plot of the variable value (sound amplitude, temperature reading, stock price) vs. time • Mathematical representation: s(t)

  5. Sample Speech Waveform »[y,fs]=wavread(‘hello.wav'); »figure;plot(y); »x=y(10:45000);plot(x); Blown-upofasection. »figure;plot(x); »axis([2000,8000,-0.1,0.08]); Download: http://www.faculty.umassd.edu/honggang.wang/ece591_web/lecture.html

  6. Ring Waveform Sample • [y,fs]=wavread(’ringin.wav'); • figure; plot(y);

  7. Sinusoidal Signals • s(t ) =A cos(2πf0t + f ) • f0 : frequency(cycles/second) • T0= 1/f0: period • A : Amplitude • f : Phase (time shift) Sinusoidal signalsare important because theycanbeused tosynthesize any signal – An arbitrary signal can be expressed as a sum of manysinusoidal signalswith different frequencies, amplitudes and phases

  8. Frequency of An ArbitrarySignal? • Sinusoidal signals have a distinct (unique) frequency • An arbitrary signal does not have a unique frequency, but can be decomposed into many sinusoidal signals with different frequencies, each with different magnitude and phase • Fourier series and Fourier transform are ways to find spectrums for periodic and aperiodic signals, respectively

  9. Approximation of Periodic Signalsby Sum of Sinusoids Example: http://en.wikipedia.org/wiki/Square_wave

  10. Period Signal • Period T: The minimum interval on which a signal repeats • Fundamental frequency: f0=1/T • Harmonic frequencies: kf0

  11. Approximation of Periodic Signals bySinusoids • Any periodic signal can be approximated by a sum of many sinusoids at harmonic frequencies of the signal (kf0 ) with appropriate amplitude and phase. • The more harmonic components are added, the more accurate the approximation becomes. • Instead of using sinusoidal signals, mathematically, we can use the complex exponential functions with both positive and negative harmonic frequencies

  12. Complex Exponential Signals • Complex number • Complex exponential signal • Euler formula

  13. Fourier Series Representation ofPeriodic Signals

  14. Fourier Series Representation of Square Wave • Apply the Fourier series analysis formula to the square wave, we get

  15. Fourier Transform for Non-Periodic Signals • A periodic signal=uncountable number of harmonics->integral instead of sum

  16. Advantage of Frequency DomainRepresentation • Clearly shows the frequency composition of the signal • One can change the magnitude of any frequency component arbitrarily by a filtering operation • Lowpass-> smoothing, noise removal • Highpass-> edge/transition detection • High emphasis -> edge enhancement • One can also shift the central frequency by modulation • A core technique for communication, which uses modulation to multiplex many signals into a single composite signal, to be carried over the same physical medium.

  17. Typical Filter • Lowpass-> smoothing, noise removal • Highpass-> edge/transition detection • Bandpass-> Retain only a certain frequency range

  18. Low Pass Filtering • Remove high freq, make signal smoother

  19. High Pass Filtering • Remove low freq, detect edges

  20. A Image Representation • From Time to Frequency Domain (DWT) • DWT functions in Matlab • x=imread(‘barbara.jpg’) • ???? • Imshow(x);

  21. Hands-On Problem (Homework) Step 1: Record your voice and save it as .wav file or download a .wav file from internet Step 2: Display the waveform Step 3: Read a .jpg image file Step 4: Display the image after DWT transformation Step 3: Submit your work including code and results (email or paper) at 5:00 pm next Thursday.

  22. Any Questions?

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