Characterization Of Signals Frequency Domain
Frequency DomainCharacterization of SignalsYao WangPolytechnic University, Brooklyn, NY11201http: //eeweb.poly.edu/ yao
Signal Representation What is a signal Time-domain description– Waveform representation– Periodic vs. non-periodic signals Frequency-domain description–––––––Periodic signalsSinusoidal signalsFourier series for periodic signalsFourier transform for non-periodic signalsConcepts of frequency, bandwidth, filteringNumerical calculation: FFT, spectrogramDemo: real sounds and their spectrogram (from DSP First) Yao Wang, 2006EE3414: Signal Characterization2
What is a signal A variable (or multiple variables) that changes in time––––Speech or audio signal: A sound amplitude that varies in timeTemperature readings at different hours of a dayStock price changes over daysEtc 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– We will look at continuous-time signal only in this lecture Yao Wang, 2006EE3414: Signal Characterization3
Waveform Representation Waveform representation– Plot of the variable value (sound amplitude, temperaturereading, stock price) vs. time– Mathematical representation: s(t) Yao Wang, 2006EE3414: Signal Characterization4
Sample Speech ire waveformBlown-up of a section.»»»»» figure; plot(x);» axis([2000,3000,-0.1,0.08]);[y,fs] wavread('morning.wav');sound(y,fs);figure; plot(y);x y(10000:25000);plot(x); Yao Wang, 20063000Signal within each short time interval is periodicPeriod depends on the vowel being spokenEE3414: Signal Characterization5
Sample Music WaveformEntire waveformBlown-up of a section» [y,fs] wavread(’sc01 L.wav');» v axis;» sound(y,fs);» axis([1.1e4,1.2e4,-.2,.2])» figure; plot(y);Music typically has more periodic structure than speechStructure depends on the note being played Yao Wang, 2006EE3414: Signal Characterization6
Sinusoidal Signals2.5s (t ) A cos(2πf 0t φ )f 0 : frequency(cycles/second)T0 1 / f 0 : periodA : Amplitudeφ : Phase (time shift)21.510.50-0.5-1-1.5-2-2.5-1.5-1-0.500.511.5 Sinusoidal signals are important because they can be used tosynthesize any signal– An arbitrary signal can be expressed as a sum of many sinusoidalsignals with different frequencies, amplitudes and phases Music notes are essentially sinusoids at different frequencies Yao Wang, 2006EE3414: Signal Characterization7
What is frequency of an arbitrarysignal? Sinusoidal signals have a distinct (unique) frequency An arbitrary signal does not have a unique frequency, but canbe decomposed into many sinusoidal signals with differentfrequencies, each with different magnitude and phase The spectrum of a signal refers to the plot of the magnitudesand phases of different frequency components The bandwidth of a signal is the spread of the frequencycomponents with significant energy existing in a signal Fourier series and Fourier transform are ways to find spectrumsfor periodic and aperiodic signals, respectively Yao Wang, 2006EE3414: Signal Characterization8
Approximation of Periodic Signalsby Sum of 2 Sinusoidssinusoids: 1 and 3d harmonicsst10.5 s (t ) Ak cos(2πkf 0t )0k 0-0.5-1-1.5-1-0.500.5View note for matlab code11.54 sinusoids: 1,3,5,7 harmonicsWith many more sinusoids with appropriate magnitude, we will get the square wave exactly Yao Wang, 2006EE3414: Signal Characterization9
Line Spectrum of Square WaveMagnitude Spectrum for Square Wave1.4Each line corresponds to oneharmonic frequency. The linemagnitude (height) indicatesthe contribution of thatfrequency to the signal.1.2 4 Ak πk 0Amplitude1k 1,3,5,.k 0,2,4,.The line magnitude dropsexponentially, which is notvery fast. The very sharptransition in square wavescalls for very high frequencysinusoids to synthesize.0.80.60.40.2013 Yao Wang, 2006579k,fk f0*k111315EE3414: Signal Characterization10
Period Signal Period T: The minimum interval on which a signalrepeats– Sketch on board Fundamental frequency: f0 1/T Harmonic frequencies: kf0 Yao Wang, 2006EE3414: Signal Characterization11
Approximation of Periodic Signals bySinusoids Any periodic signal can be approximated by a sum ofmany sinusoids at harmonic frequencies of the signal(kf0 ) with appropriate amplitude and phase. The more harmonic components are added, the moreaccurate the approximation becomes. Instead of using sinusoidal signals, mathematically,we can use the complex exponential functions withboth positive and negative harmonic frequencies Yao Wang, 2006EE3414: Signal Characterization12
Complex Exponential Signals Complex number:A A exp( jφ ) A cos φ j A sin φ Re j Im Complex exponential signals (t ) A exp( j 2πf 0t ) A cos(2πf 0t φ ) j A sin( 2πf 0t φ ) Euler formulaexp( jωt ) exp( jωt ) 2 cos(ωt )exp( jωt ) exp( jωt ) j 2 sin(ωt ) Yao Wang, 2006EE3414: Signal Characterization13
Fourier Series Representation ofPeriodic SignalsFourier Series Synthesis (inverse transform) : s (t ) A0 Ak cos(2πkf 0t φ k ) (single sided, for real signal only)k 1 Sk kexp( j 2πkf 0t ) (double sided, for both real and complex)Fourier series analysis (forward transform) :S k 1T0 T00s (t ) exp( j 2πkf 0t ) dt ; k 0, 1,2,.S k is in general a complex numberFor real signals, Sk S*-k Sk S-k (Symmetric spectrum) Yao Wang, 2006EE3414: Signal Characterization14
Fourier Series Representation ofSquare Wave Applying the Fourier series analysis formula to thesquare wave, we get 2 S k jπk 0k 1,3,5,.k 0, 2,4,. Do the derivation on the board Yao Wang, 2006EE3414: Signal Characterization15
Line Spectrum of Square WaveMagnitude Spectrum for Square Wave1.41.2 4 Ak πk 0Amplitude1k 1,3,5,.k 0,2,4,.0.80.60.40.2013 Yao Wang, 2006579k,fk f0*k111315EE3414: Signal CharacterizationOnly the positive frequencyside is drawn on the left(single sided spectrum), withtwice the magnitude of thedouble sided spectrum.16
Fourier Transform for Non-PeriodicSignalsAperiodic signal T 0 f 0 0 uncountable number of harmonics integral instead of sumFourier synthesis (inverse transform) :s (t ) S ( f ) exp( j 2πft )df Fourier analysis (forward transform) :S(f) s(t ) exp( j 2πft )dt For real signals, S(f) S(-f) (Symmetric magnitude spectrum) Yao Wang, 2006EE3414: Signal Characterization17
Pulse Function: Time DomainA Rectangular Pulse 0.20.4Derive Fourier transform on the board Yao Wang, 2006EE3414: Signal Characterization0.60.81 1 T / 2 t T / 2s (t ) otherwise 018
Pulse Function: SpectrumMagnitude Spectrum of Rectangular Pulse1.210.8The peaks of the FTmagnitude drops slowly.This is because the pulsefunction has sharptransition, whichcontributes to very highfrequency in the signal. S(f) 0.60.40.20-0.2-0.4-10-8-6-4-20f2 1 T / 2 t T / 2s (t ) 0otherwise Yao Wang, 200646S( f ) T810sin(πTf ) T sinc(Tf )πTfEE3414: Signal Characterization19
Exponential Decay: Time Domain1s(t) exp(-α t), t 0); α 10.90.80.7s(t)0.60.50.40.30.20.1000.511.52t 0 exp( αt ) s (t ) 0otherwise Yao Wang, 20062.5t3S( f ) 3.544.5511; S( f ) α j 2πfα 2 4π 2 f 2EE3414: Signal Characterization20
Exponential Decay: Spectrum1S(f) 1/(α j 2π f), α 10.90.80.7 S(f) 0.60.5The FT magnitude dropsmuch faster than for thepulse function. This isbecause the exponentialdecay function does nothas sharp transition.0.40.30.20.10-10-8-6-4-20ft 0 exp( αt ) s (t ) 0otherwise Yao Wang, 20062S( f ) 4681011; S( f ) α j 2πfα 2 4π 2 f 2EE3414: Signal Characterization21
(Effective) Bandwidth1 0.90.80.70.6 0.50.4 0.30.2B0.10-10 -8-6-4-20246810 fmin Yao Wang, 2006fmaxEE3414: Signal Characterizationfmin (fma): lowest(highest)frequency wherethe FT magnitudeis above athresholdBandwidth:B fmax-fminThe threshold is oftenchosen with respect tothe peak magnitude,expressed in dBdB 10 log10(ratio)10 dB below peak 1/10 of the peak value3 dB below 1/2 of thepeak22
More on Bandwidth Bandwidth of a signal is a critical feature whendealing with the transmission of this signal A communication channel usually operates only atcertain frequency range (called channel bandwidth)– The signal will be severely attenuated if it containsfrequencies outside the range of the channel bandwidth– To carry a signal in a channel, the signal needed to bemodulated from its baseband to the channel bandwidth– Multiple narrowband signals may be multiplexed to use asingle wideband channel Yao Wang, 2006EE3414: Signal Characterization23
How to Observe Frequency Contentfrom Waveforms? A constant - only zero frequency component (DC compoent) A sinusoid - Contain only a single frequency component Periodic signals - Contain the fundamental frequency andharmonics - Line spectrum Slowly varying - contain low frequency only Fast varying - contain very high frequency Sharp transition - contain from low to high frequency Music: contain both slowly varying and fast varying components,wide bandwidth Highest frequency estimation?– Find the shortest interval between peak and valleys Go through examples on the board Yao Wang, 2006EE3414: Signal Characterization24
Estimation of Maximum FrequencyBlown-Up of the 4802490250025102520253025402550Time index Yao Wang, 2006EE3414: Signal Characterization25
Numerical Calculation of FT The original signal is digitized, and then a FastFourier Transform (FFT) algorithm is applied, whichyields samples of the FT at equally spaced intervals. For a signal that is very long, e.g. a speech signal ora music piece, spectrogram is used.– Fourier transforms over successive overlapping shortintervals Yao Wang, 2006EE3414: Signal Characterization26
FFTFFT-0.08FFTFFTFFT-0.1200022002400260028003000t Yao Wang, 2006EE3414: Signal Characterization27
Sample Speech Waveform(click to hear the 0Entire 8003000Blown-up of a section.Signal within each short time interval is periodic. The period T is called “pitch”.The pitch depends on the vowel being spoken, changes in time. T 70 samples in this ex.f0 1/T is the fundamental frequency (also known as formant frequency). f0 1/70fs 315 Hz.k*f0 (k integers) are the harmonic frequencies. Yao Wang, 2006EE3414: Signal Characterization28
Sample Speech SpectrogramPower Power Spectrum Magnitude (dB)fs 0» figure;» psd(x,256,fs);200030004000Time500060007000» figure;» specgram(x,256,fs);Signal power drops sharply at about 4KHzLine spectra at multiple of f0,maximum frequency about 4 KHzWhat determines the maximum freq? Yao Wang, 2006EE3414: Signal Characterization29
Another Sample Speech WaveformEntire waveformBlown-up of a section.“In the course of a December tour in Yorkshire” Yao Wang, 2006EE3414: Signal Characterization30
Speech Spectrogram» figure;» psd(x,256,fs);» figure;» specgram(x,256,fs);Signal power drops sharply at about 4KHz Yao Wang, 2006Line spectra at multiple of f0,maximum frequency about 4 KHzEE3414: Signal Characterization31
Sample Music WaveformEntire waveformBlown-up of a section» [y,fs] wavread(’sc01 L.wav');» v axis;» sound(y,fs);» axis([1.1e4,1.2e4,-.2,.2])» figure; plot(y);Music typically has more periodic structure than speechStructure depends on the note being played Yao Wang, 2006EE3414: Signal Characterization32
Sample Music Spectrogram» figure; » psd(y,256,fs);Signal power drops gradually in the entirefrequency range Yao Wang, 2006» figure; » specgram(y,256,fs);Line spectra are more stationary,Frequencies above 4 KHz, more than20KHz in this ex.EE3414: Signal Characterization33
Summary of Characteristicsof Speech & Music Typical speech and music waveforms are semi-periodic– The fundamental period is called pitch period– The inverse of the pitch period is the fundamental frequency (f0) Spectral content– Within each short segment, a speech or music signal can bedecomposed into a pure sinusoidal component with frequency f0,and additional harmonic components with frequencies that aremultiples of f0.– The maximum frequency is usually several multiples of thefundamental frequency– Speech has a frequency span up to 4 KHz– Audio has a much wider spectrum, up to 22KHz Yao Wang, 2006EE3414: Signal Characterization34
Demo Demo in DSP First, Chapter 3, Sounds andSpectrograms– Look at the waveform and spectrogram of sample signals,while listening to the actual sound– Simple sounds– Real sounds Yao Wang, 2006EE3414: Signal Characterization35
Advantage of Frequency DomainRepresentation Clearly shows the frequency composition of thesignal One can change the magnitude of any frequencycomponent arbitrarily by a filtering operation– Lowpass - smoothing, noise removal– Highpass - edge/transition detection– High emphasis - edge enhancement One can also shift the central frequency bymodulation– A core technique for communication, which uses modulationto multiplex many signals into a single composite signal, tobe carried over the same physical medium. Yao Wang, 2006EE3414: Signal Characterization36
Typical Filters Lowpass - smoothing, noise removal Highpass - edge/transition detection Bandpass - Retain only a certain frequency rangeLow-passHigh-passH(f)H(f)0 Yao Wang, 2006f0Band-passH(f)fEE3414: Signal Characterization0f37
Low Pass Filtering(Remove high freq, make signal smoother)Magnitude Spectrum of Rectangular PulseFiltering is done by asimple multiplification:1.21Ideallowpassfilter0.8Y(f) X(f) H(f)H(f) is designed tomagnify or reduce themagnitude (andpossibly changephase) of the originalsignal at differentfrequencies. S(f) 0.60.40.20-0.2-0.4-10Spectrum of the pulse signal-8 Yao Wang, 2006-6-4-20f246EE3414: Signal Characterization8A pulse signal afterlow pass filtering (left)will have roundedcorners.1038
The original pulse function and its low-passed versionsS(t)1.210.80.6originalaveraging over 11 samplesfilter .60.81t
Impulse Response of the Filters0.30.25h(t)0.2averaging over 11 t Yao Wang, 2006EE3414: Signal Characterization40
Frequency Response of the FiltersAveragingMagnitude ed Angular Frequency ( π ormalized Angular Frequency ( π rads/sample)0.20.30.40.50.60.70.8Normalized Angular Frequency ( π rads/sample)0.91fir11(10,0.25)MagnitudePhase (degrees)(dB)10000-50-100-100-200-150 Yao Wang, 20060EE3414: Signal Characterization41
High Pass Filtering(remove low freq, detect edges)Magnitude Spectrum of Rectangular Pulse1.21Idealhigh-passfilter0.8 S(f) 0.60.40.20-0.2-0.4-10 Yao Wang, 2006Spectrum of the pulse signal-8-6-4-20fEE3414: Signal Characterization24681042
The original pulse function and its high-passed versionS(t)10.8originalhigh-pass 40.60.81t Yao Wang, 2006EE3414: Signal Characterization43
The High Pass lse response:Current sample –neighboring 0 Yao Wang, 200620Magnitude Normalized Angular Frequency ( π rads/sample)EE3414: Signal Characterization0.9144
Filtering in Temporal Domain(Convolution) Convolution theoremX ( f )H ( f ) x(t ) * h(t ) x(t ) * h(t ) x(t τ )h(τ )dτ Interpretation of convolution operation– replacing each pixel by a weighted sum of its neighbors– Low-pass: the weights sum weighted average– High-pass: the weighted sum left neighbors –rightneighbors Yao Wang, 2006EE3414: Signal Characterization45
Implementation of Filtering Frequency Domain– FT - Filtering by multiplication with H(f) - Inverse FT Time Domain– Convolution using a filter h(t) (inverse FT of H(f)) You should understand how to perform filtering infrequency domain, given a filter specified infrequency domain Should know the function of the filter given H(f) Computation of convolution is not required for thislecture Filter design is not required. Yao Wang, 2006EE3414: Signal Characterization46
What Should You Know (I) Sinusoid signals:– Can determine the period, frequency, magnitude and phase of asinusoid signal from a given formula or plot Fourier series for periodic signals– Understand the meaning of Fourier series representation– Can calculate the Fourier series coefficients for simple signals (onlyrequire double sided)– Can sketch the line spectrum from the Fourier series coefficients Fourier transform for non-periodic signals–––––Understand the meaning of the inverse Fourier transformCan calculate the Fourier transform for simple signalsCan sketch the spectrumCan determine the bandwidth of the signal from its spectrumKnow how to interpret a spectrogram plot Yao Wang, 2006EE3414: Signal Characterization47
What Should You Know (II) Speech and music signals– Typical bandwidth for both– Different patterns in the spectrogram– Understand the connection between music notes and sinusoidalsignals Filtering concept– Know how to apply filtering in the frequency domain– Can interpret the function of a filter based on its frequencyresponse Lowpass - smoothing, noise removal Highpass - edge detection, differentiator Bandpass - retain certain frequency band, useful for demodulation Yao Wang, 2006EE3414: Signal Characterization48
References Oppenheim and Wilsky, Signals and Systems, Sec. 4.2-4.3(Fourier series and Fourier transform) McClellan, Schafer and Yoder, DSP First, Sec. 2.2,2.3,2.5(review of sinusoidal signals, complex number, complexexponentials) Yao Wang, 2006EE3414: Signal Characterization49
Characterization of Signals Yao Wang, 2006 EE3414: Signal Characterization 2 Signal Representation What is a signal Time-domain description – Waveform representation – Periodic vs. non-periodic signals Frequency-domain description – Periodic signals – Sinusoidal signals
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Signals And Systems by Alan V. Oppenheim and Alan S. Willsky with S. Hamid Nawab. John L. Weatherwax January 19, 2006 wax@alum.mit.edu 1. Chapter 1: Signals and Systems Problem Solutions Problem 1.3 (computing P and E for some sample signals)File Size: 203KBPage Count: 39Explore further(PDF) Oppenheim Signals and Systems 2nd Edition Solutions .www.academia.eduOppenheim signals and systems solution manualuploads.strikinglycdn.comAlan V. Oppenheim, Alan S. Willsky, with S. Hamid Signals .www.academia.eduSolved Problems signals and systemshome.npru.ac.thRecommended to you based on what's popular Feedback
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