chapter 2 - frequency domain analysissilage/chapter2ms.pdf · chapter 2 frequency domain analysis...

EE4512 Analog and Digital Communications Chapter 2

Chapter 2Chapter 2

Frequency Domain AnalysisFrequency Domain Analysis


Chapter 2Chapter 2

Frequency Domain AnalysisFrequency Domain Analysis•• Why Study Frequency DomainWhy Study Frequency DomainAnalysis?Analysis?

•• Pages 6Pages 6--1313


Why frequency domainWhy frequency domainanalysis? analysis?

•• Allows simple algebraAllows simple algebrarather than timerather than time--domaindomaindifferential equationsdifferential equationsto be usedto be used

• Transfer functions canTransfer functions canbe applied to transmitter, be applied to transmitter, communication channelcommunication channeland receiverand receiver

•• Channel bandwidth,Channel bandwidth,noise and power arenoise and power areeasier to evaluate easier to evaluate

MS Figure 4.2 and Figure 4.3MS Figure 4.2 and Figure 4.3

500, 1500 and 2500 Hz500, 1500 and 2500 Hz


Why frequency domainWhy frequency domainanalysis? analysis?

•• A complex signalA complex signalconsisting of the sum ofconsisting of the sum ofthree sinusoids isthree sinusoids isdifficult to discern in thedifficult to discern in thetemporal domaintemporal domainbut easy to identify in thebut easy to identify in thespectral domainspectral domain

MS Figure 4.2 and Figure 4.3MS Figure 4.2 and Figure 4.3

500, 1500 and 500, 1500 and 2500 Hz2500 Hz


Butterworth LPFButterworth LPF9 pole,9 pole, ffcutoff cutoff = 1 kHz= 1 kHz

500500HzHz

15001500HzHz

25002500HzHz

MS Fig 4.8 modifiedMS Fig 4.8 modified

•• Example 2.1 Input sum of three sinusoids, temporal Example 2.1 Input sum of three sinusoids, temporal displaydisplay

EX21.mdlEX21.mdl


•• Example 2.1 Input sum of three sinusoidsExample 2.1 Input sum of three sinusoids

• Output after Butterworth LPFOutput after Butterworth LPF


Butterworth LPFButterworth LPF9 pole, 9 pole, ffcutoffcutoff = 1 kHz= 1 kHz


•• Example 2.1 Input sum of three sinusoids, temporal Example 2.1 Input sum of three sinusoids, temporal displaydisplay

Since this is a Since this is a MMATLABATLAB andand Simulink timeSimulink time--based based simulationsimulation, an , an analog low pass filteranalog low pass filter is used hereis used here

EX21.mdlEX21.mdl


Butterworth LPFButterworth LPF9 pole,9 pole, ffcutoffcutoff = 1 kHz= 1 kHz

500500HzHz

15001500HzHz

25002500HzHz


•• Example 2.1 Input sum of three sinusoids, spectral Example 2.1 Input sum of three sinusoids, spectral displaydisplay

EX21spectral.mdlEX21spectral.mdl


•• Input power spectral density of the sum of three sinusoidsInput power spectral density of the sum of three sinusoids

• Output power spectral density after Butterworth LPFOutput power spectral density after Butterworth LPF

Attenuation (decibel dB)Attenuation (decibel dB)

34 dB34 dB

––36 dB36 dB


•• Butterworth FiltersButterworth Filters

Stephen Butterworth was aStephen Butterworth was aBritishBritish physicistphysicist who inventedwho inventedthe the Butterworth filterButterworth filter, a class, a classof circuits that are used toof circuits that are used tofilterfilter electrical signals. Heelectrical signals. Heworked for several yearsworked for several yearsat the National Physicalat the National PhysicalLaboratory (UK), where he didLaboratory (UK), where he didtheoretical and experimental theoretical and experimental work for the determination ofwork for the determination ofstandards of electrical standards of electrical inductance and analyzed theinductance and analyzed theelectromagnetic field around submarine cables.electromagnetic field around submarine cables.

18851885--19581958


•• Example 2.2Example 2.2

10 MHz sinusoid10 MHz sinusoidwith additive with additive white Gaussianwhite Gaussiannoise (AWGN)noise (AWGN)

EX22.mdlEX22.mdl

Communications channelCommunications channel


•• Example 2.2 Example 2.2 SimulinkSimulink design windowdesign window

Configure Simulation SimulateConfigure Simulation Simulate

EX22.mdlEX22.mdl

T = 0.01 secT = 0.01 sec


•• Example 2.2 Example 2.2 SimulinkSimulink design windowdesign window

Configure Simulation Configure Simulation TTsimulationsimulation = 10= 10--88 = 10 nsec= 10 nsecffsimulationsimulation = 1/= 1/TTsimulationsimulation = 10= 1088 = 100 MHz = 100 MHz TT = 0.01 sec= 0.01 sec


•• Example 2.2 10 MHz sinusoid with AWGNExample 2.2 10 MHz sinusoid with AWGN

• Power spectral density of 10 MHz sinusoid with AWGNPower spectral density of 10 MHz sinusoid with AWGN

10 MHz10 MHz HereHere’’s the signal!s the signal!


Chapter 2Chapter 2

Frequency Domain AnalysisFrequency Domain Analysis•• The Fourier SeriesThe Fourier Series



π∞

+∑0 n o nn=1

s(t) = X X cos(2 n f t +φ )

•• Fourier SeriesFourier Series

Jean Jean BaptisteBaptiste Joseph Fourier wasJoseph Fourier wasa French mathematician and physicista French mathematician and physicistwho is best known for initiating thewho is best known for initiating theinvestigation of Fourier Series and itsinvestigation of Fourier Series and itsapplication to problems of heat flow.application to problems of heat flow.The Fourier transform is also namedThe Fourier transform is also namedin his honor.in his honor.

17681768--18301830


2 20 0 n n n

n n n n

X = a X = a +b| c | = X / 2 X = | 2 c |

•• Fourier series coefficients:Fourier series coefficients:trignometrictrignometric aan n bbnnpolar polar XXnncomplex complex ccnn

MMATLABATLAB and and Simulink Simulink simulationsimulationcan provide the magnitude ofcan provide the magnitude ofthe complex Fourier seriesthe complex Fourier seriescoefficients for any periodiccoefficients for any periodicwaveformwaveform

Xo = co


•• Example 2.3 modifiedExample 2.3 modified

Temporal display Temporal display of a complex pulseof a complex pulseas the addition of as the addition of two periodic pulsestwo periodic pulses

EX23.mdlEX23.mdl


•• Example 2.3 modified Example 2.3 modified SimulinkSimulink design windowdesign window

Configure Simulation SimulateConfigure Simulation Simulate

EX23.mdlEX23.mdl

T = 4 secT = 4 sec


Example 2.3 modified Example 2.3 modified SimulinkSimulink design windowdesign window

Configure Simulation Configure Simulation ffsimulationsimulation = 1024 Hz= 1024 HzTTsimulationsimulation = 1/= 1/ffsimulationsimulation = 0.976562 msec = 0.976562 msec TT = 4 sec= 4 sec



Simulink Simulink discrete discrete pulse generators pulse generators

EX23.mdlEX23.mdl

Temporal display scopesTemporal display scopes


•• Example 2.3 modified Example 2.3 modified

ffSS == 1024 Hz1024 Hz

TTsimulationsimulation = 1/= 1/ffsimulation simulation = = 0.976562 msec 0.976562 msec

EX23.mdlEX23.mdl

First PulseFirst Pulse


EX23.mdlEX23.mdl

SecondSecondPulsePulse

•• Example 2.3 modified Example 2.3 modified

ffSS == 1024 Hz1024 Hz

TTsimulationsimulation = 1/= 1/ffsimulation simulation = = 0.976562 msec 0.976562 msec



PeriodPeriod TToo == 4 sec4 secFundamental frequencyFundamental frequency

ffoo= 0.25 Hz = 0.25 Hz

Sample based simulation:Sample based simulation:Period = 4096 samplesPeriod = 4096 samples

must be 2must be 2N N for FFTfor FFT

Pulse width = 1024Pulse width = 1024samplessamples

Sample time = 0.978562Sample time = 0.978562msecmsec

4/4096 = 9.78562 x 104/4096 = 9.78562 x 10--44



The default display or The default display or autoscalingautoscaling for the temporal scope for the temporal scope does not have a uniform amplitude. does not have a uniform amplitude.

EX23.mdlEX23.mdl

55

11



RightRight--ClickClick ononthe verticalthe verticalaxis and useaxis and useAxes propertiesAxes propertiesto change theto change theamplitudeamplitudescaling scaling


•• Example 2.3 modified First periodic pulseExample 2.3 modified First periodic pulse

•• Second periodic pulseSecond periodic pulse

T = 4 secT = 4 secT = 1 secT = 1 sec

T = 2 secT = 2 sec


•• Example 2.3 modified Sum of periodic pulsesExample 2.3 modified Sum of periodic pulses

T = 1 secT = 1 secT = 2 secT = 2 sec

T = 4 secT = 4 sec

EX23.mdlEX23.mdl



Spectral display Spectral display of a complex pulseof a complex pulseas the addition of as the addition of two periodic pulsestwo periodic pulses

Spectral display scopeSpectral display scope



ffSS == 1024 Hz1024 Hz

T T = 4 sec = 4096= 4 sec = 4096samplessamples

ffSS == ffsimulationsimulation forforconveninenceconveninence

Input signal isInput signal isnonnon--framed basedframed basedso buffer input isso buffer input isrequired required



ffSS == 1024 Hz1024 Hz

T T = 4 sec = 4096= 4 sec = 4096samplessamples

Amplitude scalingAmplitude scalingmagnitudemagnitude--squaredsquared



Scaled | FFT |Scaled | FFT |2 2 using using SimulinkSimulink Plot Tools. In Plot Tools. In Command Command WindowWindow::

>>plottools on>>plottools on Default spectral displayDefault spectral display

plottools



Add data point markers (o)Add data point markers (o)and change the frequencyand change the frequencyaxis to 5 Hzaxis to 5 Hz


Chapter 2Chapter 2

Frequency Domain AnalysisFrequency Domain Analysis•• Power in the Frequency Domain Power in the Frequency Domain



•• Example 2.3 modified Unscaled | FFT |Example 2.3 modified Unscaled | FFT |22

•• Scaled | FFT |Scaled | FFT |22

500 Hz500 Hzffss / 2/ 2 = 512 Hz= 512 Hz

5 Hz5 Hz

Spectral Resolution MS Eq 1.1Spectral Resolution MS Eq 1.1∆∆f f = = ffss / / NN = 1024/4096 = 0.25 Hz= 1024/4096 = 0.25 Hz


•• Example 2.3 modified Scaled | FFT |Example 2.3 modified Scaled | FFT |22

The Fourier series components are The Fourier series components are discrete. discrete. However, theHowever, theSpectrum Scope does not read out the values. AnSpectrum Scope does not read out the values. Aninteractive cursorinteractive cursoris available in theis available in theFigures Figures windowwindow..

CopyCopy and and PastePastethe Spectrumthe SpectrumScope plot asScope plot asFigure 1Figure 1



The interactive cursor can The interactive cursor can locklock--onon to the markers and read to the markers and read out the data values sequentially. The interactive cursor is out the data values sequentially. The interactive cursor is evoked from the toolbar for a Figure plot. evoked from the toolbar for a Figure plot.

x = 0 (x = 0 (f = f = 0 Hz)0 Hz)y = 2362 = | y = 2362 = | FFT(FFT(ff = 0) |= 0) |22



The magnitude squared data values are divided by the The magnitude squared data values are divided by the number of points in the FFT number of points in the FFT NN = 4096 to get the value. The = 4096 to get the value. The magnitude of the DC (0 Hz) value is then:magnitude of the DC (0 Hz) value is then:

(2362/4096)(2362/4096)0.50.5 = (0.56767)= (0.56767)0.5 0.5 = 0.7594 = c= 0.7594 = c00 = X= X00

The DC value of the complex pulse is calculated as 0.75The DC value of the complex pulse is calculated as 0.75



The first five spectral components can be read, computedThe first five spectral components can be read, computedand then compared to a direct calculation of and then compared to a direct calculation of ccnn..

f f HzHz DataData nn Component Component Spectral ResolutionSpectral Resolution0 2362 0 0.7594 MS Eq 1.10 2362 0 0.7594 MS Eq 1.10.25 1038 1 0.5034 0.25 1038 1 0.5034 ∆∆f f = = ffss / / NN0.5 103.8 2 0.1592 0.5 103.8 2 0.1592 ∆∆ff == 1024/40961024/40960.75 115.3 3 0.1678 0.75 115.3 3 0.1678 ∆∆f f = = 0.25 Hz0.25 Hz1 0 4 01 0 4 0



Rectangular pulse Rectangular pulse traintrain



Amplitude = 3Amplitude = 3PeriodPeriod TToo == 10 msec10 msecFundamental frequencyFundamental frequency

ffoo= 100 Hz= 100 HzPulse width = 2 msecPulse width = 2 msecDuty cycle = 2/10 = 0.2Duty cycle = 2/10 = 0.2

Sample based simulation:Sample based simulation:Period = 4096 samplesPeriod = 4096 samplesPulse width = 819 samplesPulse width = 819 samples

819/4096 819/4096 ≈≈ 0.20.2Sample time = 2.441 Sample time = 2.441 µµsecsec

1010--22/4096 = 2.441 x 10/4096 = 2.441 x 10--66



PeriodPeriod TToo == 10 msec10 msec

Since the Spectrum Since the Spectrum Scope requires 2Scope requires 2NN ==4096 samples, the4096 samples, thesimulation time = simulation time = 10 msec divided by10 msec divided bythe sample the sample time =time =2.441 2.441 µµsec must besec must be≥≥ 40964096

1010--22/2.441 x 10/2.441 x 10--66 ==4096.684096.68


•• Example 2.7 One cycle of periodic pulseExample 2.7 One cycle of periodic pulse

•• Magnitude squared of the Fast Fourier Transform | FFT |Magnitude squared of the Fast Fourier Transform | FFT |22

TToo = 10 msec= 10 msecTTpulsepulse = 2 msec= 2 msec

DC value = 0.6DC value = 0.6

33

| FFT (f = 0) || FFT (f = 0) |2 2 = 1447= 1447

(1447/4096)(1447/4096)0.50.5 = (0.353271)= (0.353271)0.50.5 = 0.5944= 0.5944


•• Example 2.7 | FFT |Example 2.7 | FFT |22

First First spectral nullspectral null at 1/at 1/TTpulse pulse = 1/2 msec = 500 Hz = 1/2 msec = 500 Hz Successive Successive nullsnulls at at n/n/TTpulsepulse = n x 500 Hz= n x 500 Hz

Hard to discern when the nulls are when plotted on a Hard to discern when the nulls are when plotted on a linear scalelinear scale

500 Hz500 Hz 1 kHz1 kHz 1.5 kHz1.5 kHz



First First spectral nullspectral null at 1/at 1/TTpulse pulse = 1/2 msec = 500 Hz = 1/2 msec = 500 Hz Successive Successive nullsnulls at at n/n/TTpulsepulse = n x 500 Hz= n x 500 Hz

Easier to see when the nulls are plotted on a Easier to see when the nulls are plotted on a decibeldecibel (dB) (dB) scalscale where e where ––40 dB 40 dB ≈≈ 00

500 Hz500 Hz 1 kHz1 kHz 1.5 kHz1.5 kHz

-40 dB


• dede··cici··belbel ((ddĕĕss''əə--bbəəll, , --bbĕĕll') ') n.n. ((Abbr.Abbr. dB) dB) A unit used to express relative difference in power or A unit used to express relative difference in power or intensity, usually between two acoustic or electric signals, intensity, usually between two acoustic or electric signals, equal to ten times the common logarithm of the ratio of the equal to ten times the common logarithm of the ratio of the two levels.two levels.

The The belbel (B) as a unit of measurement(B) as a unit of measurementwas originally proposed in 1929 bywas originally proposed in 1929 byW. H. Martin of Bell Labs. The belW. H. Martin of Bell Labs. The belwas too large for everyday use, sowas too large for everyday use, sothe decibel (dB), equal to 0.1 B,the decibel (dB), equal to 0.1 B,became more commonly used.became more commonly used. Alexander Graham Bell

1847-1922



Spectrum Scope axisSpectrum Scope axispropertiesproperties


•• Example 2.7 | FFT |Example 2.7 | FFT |2 2 ((DataData/4096)/4096)0.50.5

f f HzHz DataData Component n Calculated Component n Calculated ccnn0 1447 0.5944 0 0.6 0 1447 0.5944 0 0.6 100 1290 0.5612 1 0.561 100 1290 0.5612 1 0.561 200 844 0.4540 2 0.454 200 844 0.4540 2 0.454 cfcf S&MS&M300 376 0.3030 3 0.303 pp 300 376 0.3030 3 0.303 pp 3333--3535400 81 0.1406 4 0.140400 81 0.1406 4 0.140500 500 ≈≈ 0 0 5 00 0 5 0

2 20 0 n n n

n n n n

X = a X = a +b| c | = X / 2 X = | 2 c |

cn

Xo = co



Rectangular pulse trains Rectangular pulse trains with pulse period of 0.5 sec with pulse period of 0.5 sec and pulse widths of and pulse widths of

0.0625 sec 0.0625 sec 0.125 sec0.125 sec0.250 sec0.250 sec

Sample based simulation:Sample based simulation:Period = 2Period = 21616 = 65536 = 65536

samplessamplesPulse widths = 8192, Pulse widths = 8192,

16384, 32768 16384, 32768 Sample time = 7.629 Sample time = 7.629 µµsecsec

0.5/65536 = 7.629 x 100.5/65536 = 7.629 x 10--66

EX28.mdlEX28.mdl



Spectrum Scope hasSpectrum Scope hasinherent amplitude andinherent amplitude andfrequency axes scalingfrequency axes scaling(but (but not not cursor readout) cursor readout)

EX28.mdlEX28.mdl


ττ = 0.0625 sec= 0.0625 sec

ττ = 0.250 sec= 0.250 sec

ττ = 0.125 sec= 0.125 sec

S&M p. 36S&M p. 36--3737

First null 4 Hz = 1/First null 4 Hz = 1/ττ



2 20 0 n n n

n n n n

X = a X = a +b| c | = X / 2 X = | 2 c |


π∞

∞

+

= +

∑

∑∫o

o

0 n o nn=1

t +T 22 2 n

S 0n=1t

s(t) = X X cos(2 nf t +φ )

X1P s (t) dt = XT 2

•• Average Normalized (RAverage Normalized (RLL = 1= 1ΩΩ) ) PowerPower

Fourier series componentsFourier series componentsccnn from simulationfrom simulation

Periodic signal as aPeriodic signal as afrequency domainfrequency domainrepresentationrepresentation

Average (RAverage (RL L = 1= 1ΩΩ))power in the signal Ppower in the signal PSS asasa time domain ora time domain orfrequency domainfrequency domainrepresentationrepresentation

ParsevalParseval’’ss TheoremTheorem

Xo = co


π∞

∞

+

= +

∑

∑∫o

o

0 n o nn=1

t +T 22 2 n

S 0n=1L Lt

s(t) = X X cos(2 nf t +φ )

X1 1P s (t) dt = XT R R 2

•• Average (RAverage (RLL ≠≠ 11ΩΩ) ) PowerPower

The average The average nonnon--normalizednormalized (R(RL L ≠≠ 11ΩΩ)) power in the power in the signal Psignal PSS as a time domain or frequency domainas a time domain or frequency domainrepresentationrepresentation

ParsevalParseval’’ss TheoremTheorem


∞

= +∑∫o

o

t +T 22 2 n

S 0n=1t

X1P s (t) dt = XT 2

•• ParsevalParseval’’ss TheoremTheorem

MarcMarc--Antoine Antoine ParsevalParseval des des ChênesChênes waswasa French mathematician, most famousa French mathematician, most famousfor what is now known as for what is now known as ParsevalParseval’’ssTheorem, which presaged theTheorem, which presaged theequivalence of the Fourier Transform.equivalence of the Fourier Transform.A monarchist opposed to the French A monarchist opposed to the French Revolution, Revolution, ParsevalParseval fled the country fled the country 17551755--18361836after being imprisoned in 1792 by Napoleon for after being imprisoned in 1792 by Napoleon for publishing tracts critical of the government.publishing tracts critical of the government.


•• Examples 2.9 and 2.10Examples 2.9 and 2.10

Normalized power Normalized power spectrum of a spectrum of a rectangular pulserectangular pulseand a Butterworthand a Butterworthlow pass filteredlow pass filtered(LPF) rectangular(LPF) rectangularpulsepulse

EX29.mdlEX29.mdl



Amplitude = 3Amplitude = 3PeriodPeriod TToo = 10 msec= 10 msecFundamental frequencyFundamental frequency

ffoo= 100 Hz= 100 HzPulse width = 2 msecPulse width = 2 msecDuty cycle = 2/10 = 0.2Duty cycle = 2/10 = 0.2Sample based simulation:Sample based simulation:Period = 4096 samplesPeriod = 4096 samplesPulse width = 819 samplesPulse width = 819 samples

819/4096 819/4096 ≈≈ 0.20.2Sample time = 10Sample time = 10--22/4096 = 2.441 x 10/4096 = 2.441 x 10--66 = 2.441 = 2.441 µµsec sec Sampling rate = 1/ 2.441 x 10Sampling rate = 1/ 2.441 x 10--6 6 ≈≈ 409.6 kHz 409.6 kHz

EX29.mdlEX29.mdl



Since this is a Since this is a MATLAB MATLAB andand SimulinkSimulink sample based sample based simulationsimulation (period = 4096 samples, pulse width = 819 (period = 4096 samples, pulse width = 819 samples) a samples) a digital low pass filterdigital low pass filter design is used here (a design is used here (a timetime--based simulationbased simulation requires an requires an analog filteranalog filter design)design)

EX29.mdlEX29.mdl


•• Example 2.9 and 2.10 Digital Low Pass Filter DesignExample 2.9 and 2.10 Digital Low Pass Filter Design

ffcuttoffcuttoff = 300 Hz= 300 Hz

ffss = 409600= 409600


•• Power Spectral DensityPower Spectral Density

The normalized (RThe normalized (RL L = 1 = 1 ΩΩ)) power spectral density power spectral density (PSD) for periodic signals is (PSD) for periodic signals is discretediscrete because of the because of the fundamental frequency ffundamental frequency fo o = 1/T= 1/Too = 100 Hz here.= 100 Hz here.

However, for However, for aperiodicaperiodic signals the PSD is conceptually signals the PSD is conceptually continuouscontinuous. Periodic signals contain . Periodic signals contain no informationno informationand only aperiodic signals are, in fact, communicated.and only aperiodic signals are, in fact, communicated.


•• Power Spectral DensityPower Spectral Density

Simulated by | FFT |Simulated by | FFT |22 ininMATLAB MATLAB and and Simulink Simulink totoobtain cobtain cn n fromfrom which wewhich wecan derive can derive XXnn

| FFT || FFT |2 2 ≈≈ PSDPSD

2 20 0 n n n

n n n n

X = a X = a +b| c | = X / 2 X = | 2 c |Xo = co


•• Power Spectral Density Simulated by | FFT |Power Spectral Density Simulated by | FFT |22

The dB scale for PSD is more prevalent for analysisThe dB scale for PSD is more prevalent for analysis

Spectral nulls

dBdB

LinearLinear


•• Power Spectral Density Low Pass Filtered Power Spectral Density Low Pass Filtered

dBdB

dBdB

ffcutoffcutoff = 300 Hz= 300 Hz LPFLPF

UnfilteredUnfiltered


•• BandwidthBandwidth

The The bandwidth bandwidth of a signal is the width of the frequency of a signal is the width of the frequency band in Hertz that contains a sufficient number of the band in Hertz that contains a sufficient number of the signalsignal’’s frequency components to reproduce the signal s frequency components to reproduce the signal with an acceptable amount of distortion.with an acceptable amount of distortion.

Bandwidth is a nebulous term andBandwidth is a nebulous term andcommunication engineers mustcommunication engineers mustalways define what if meant byalways define what if meant bybandwidthbandwidth in the context of use.in the context of use.


•• Total Power in the SignalTotal Power in the Signal

ParsevalParseval’’ss Theorem allows us to determine the Theorem allows us to determine the total total normalized powernormalized power in the signal without the infinite sum of in the signal without the infinite sum of Fourier series components by integrating in the temporal Fourier series components by integrating in the temporal domain:domain:

From Example 2.10 the totalFrom Example 2.10 the totalnormalized (Rnormalized (RLL = 1= 1ΩΩ)) power in thepower in thesignal is 1.8 Vsignal is 1.8 V22 ((not not W) and the percentage of the total W) and the percentage of the total power in the signal in a power in the signal in a bandwidthbandwidth of 300 Hz is of 300 Hz is approximately 88% (S&M p. 45). approximately 88% (S&M p. 45).

∞

= +∑∫o

o

t +T 22 2 n

S 0n=1t

X1P s (t) dt = XT 2


Chapter 2Chapter 2

Frequency Domain AnalysisFrequency Domain Analysis•• The Fourier TransformThe Fourier Transform




Spectrum of a Spectrum of a simulated singlesimulated singlepulse from a verypulse from a verylow duty cycle low duty cycle rectangular pulserectangular pulsetraintrain EX212.mdlEX212.mdl


•• Example 2.12 Simulated single pulseExample 2.12 Simulated single pulse

• | FFT || FFT |22

pulse width = 1 msecpulse width = 1 msec

pulse period = 1 secpulse period = 1 sec

duty cycle = 10duty cycle = 10--33/1 = 0.001 = 0.1%/1 = 0.001 = 0.1%



TheThe magnitude of the Fourier Transform of a single pulse magnitude of the Fourier Transform of a single pulse is is continuous continuous andand not discretenot discrete since there is no Fourier since there is no Fourier series representation. In the series representation. In the MATLAB MATLAB and and SimulinkSimulinksimulation here the data points are very dense and simulation here the data points are very dense and virtually display a continuous plot.virtually display a continuous plot.


•• Example 2.12 Simulated single pulse | FFT |Example 2.12 Simulated single pulse | FFT |22

S&M p. 56S&M p. 56--6060

S(fS(f) = A) = Aττ sinc(sinc(ππ f f ττ))

AAττ = 1(10= 1(10--33) = 10) = 10--33

(c(coo/65536)/65536)0.5 0.5 = 10= 10--33

cco o = 0.065536 = 0.065536

cco o = 0.065536= 0.065536



S&M p. 56S&M p. 56--6060


ZeroZero--crossing at integral multiples of 1/crossing at integral multiples of 1/ττ = 1/10= 1/10--33 = 1000 Hz= 1000 Hz

1000 2000 3000 4000 5000 6000 70001000 2000 3000 4000 5000 6000 7000HzHz

0.005



• | FFT || FFT |22

pulse width = 10 msecpulse width = 10 msec

pulse period = 1 secpulse period = 1 sec

duty cycle = 10duty cycle = 10--22/1 = 0.01 = 1%/1 = 0.01 = 1%



S&M p. 56S&M p. 56--6060


ZeroZero--crossing at integral multiples of 1/crossing at integral multiples of 1/ττ = 1/10= 1/10--22 = 100 Hz= 100 Hz

100 200 300 400 500 600 700100 200 300 400 500 600 700 HzHz

0.05


•• PropertiesPropertiesof theof theFourierFourierTransformTransform


•• PropertiesPropertiesof theof theFourierFourierTransformTransform

ModulationModulationprincipleprinciple


Chapter 2Chapter 2

Frequency Domain AnalysisFrequency Domain Analysis•• Normalized Energy Spectral DensityNormalized Energy Spectral Density



•• Normalized Energy Normalized Energy

If If s(ts(t) is a ) is a nonnon--periodic, finite energy signalperiodic, finite energy signal (a single (a single pulse) then the average normalized power Ppulse) then the average normalized power PSS is 0:is 0:

However, the normalized energy EHowever, the normalized energy ESS for the same for the same s(ts(t) is ) is nonnon--zero by definitionzero by definition (S&M p. 60(S&M p. 60--61). 61).

→∞ →∞

∞

−∞

=

=

∫

∫

o

o

t +T2 2

S T Tt

2 2S

1 finite valueP lim s (t) dt = lim = 0 VT T

E s (t) dt V - sec


•• ParsevalParseval’’ss Energy Theorem Energy Theorem

ParsevalParseval’’ss energy theorem follows directly then from the energy theorem follows directly then from the discussion of power in a periodic signal:discussion of power in a periodic signal:

•• Energy Spectral DensityEnergy Spectral Density

Analogous to the power spectral density is the energy Analogous to the power spectral density is the energy spectral density (ESD) spectral density (ESD) ψψ(f). For a linear, time(f). For a linear, time--invariant (LTI) invariant (LTI) system with a transfer function system with a transfer function H(fH(f), the output ESD which is ), the output ESD which is the energy flow through the system is:the energy flow through the system is:

ψψOUTOUT(f(f)) = = ψψININ(f(f) | ) | H(fH(f) |) |22

∞ ∞

−∞ −∞

= ∫ ∫2 2 2SE s (t) dt = | S (f) | df V - sec


•• Energy Spectral DensityEnergy Spectral Density

The energy spectral density (ESD) The energy spectral density (ESD) ψψ(f) is the magnitude (f) is the magnitude squared of the Fourier transform squared of the Fourier transform S(fS(f) of a pulse signal ) of a pulse signal s(ts(t):):

ψψ(f)(f) = | = | S(fS(f) |) |22

The ESD can be The ESD can be approximatedapproximated by the magnitude squaredby the magnitude squaredof the Fast Fourier Transform (FFT) in a of the Fast Fourier Transform (FFT) in a MATLAB MATLAB and and SimulinkSimulink simulation as described in Chapter 3.simulation as described in Chapter 3.

ψψ(f) (f) ≈≈ | FFT || FFT |22


End of Chapter 2End of Chapter 2

Frequency Domain AnalysisFrequency Domain Analysis

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