chapter 2 frequency domain analysis - temple …silage/chapter2svu.pdfchapter 2 frequency domain...

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 SVU Figure 6.2 and Figure 6.3SVU Figure 6.2 and Figure 6.3

500, 1500 and 2500 Hz500, 1500 and 2500 Hz


Butterworth LPFButterworth LPF1 pole, f1 pole, foo= 1 kHz= 1 kHz

500500HzHz

15001500HzHz

25002500HzHz SVU Fig 6SVU Fig 6--1 modified1 modified

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


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

• Output after Butterworth LPFOutput after Butterworth LPF


•• 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 LPFAttenuation (decibel dB)Attenuation (decibel dB)

12.42 dB12.42 dB

3.81 dB3.81 dB


Cursor based Cursor based measurementsmeasurements

dB

dB


•• Example 2.2Example 2.2

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


•• 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


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

SystemVueSystemVue simulation cansimulation canprovide the magnitude ofprovide the magnitude ofthe complex Fourier seriesthe complex Fourier seriescoefficients for any periodiccoefficients for any periodicwaveform.waveform.



Complex pulseComplex pulseas the addition of as the addition of two periodic pulsestwo periodic pulses


•• Example 2.3 SystemVue Design WindowExample 2.3 SystemVue Design Window

Editing Simulate System Time Analysis WindowEditing Simulate System Time Analysis Window


•• Example 2.3 SystemVue System TimeExample 2.3 SystemVue System Time

Fundamental frequency fFundamental frequency foo= 0.2 Hz, T= 0.2 Hz, To o = 5 sec= 5 sec


•• Example 2.3 SystemVue Analysis WindowExample 2.3 SystemVue Analysis Window

Sink calculator | FFT |Sink calculator | FFT |



Sink calculator Scale DisplaySink calculator Scale Display


•• Example 2.3 Unscaled | FFT |Example 2.3 Unscaled | FFT |

•• Scaled | FFT |Scaled | FFT |

10 Hz10 Hz

4 units4 units

500 Hz500 Hz


•• Example 2.3 Scaled | FFT |Example 2.3 Scaled | FFT |The Fourier series components are The Fourier series components are discrete. discrete. In theIn theSystemVue SystemVue Analysis Window the connection between data Analysis Window the connection between data points can be eliminated if warranted.points can be eliminated if warranted.


•• Example 2.3 First periodic pulseExample 2.3 First periodic pulse

• Second periodic pulseSecond periodic pulse


•• Example 2.3 Sum of first and second periodic pulsesExample 2.3 Sum of first and second periodic pulses

• Magnitude of the Fourier Transform | FFT |Magnitude of the Fourier Transform | FFT |

Mean (DC level) = 3 / 5 = 0.6Mean (DC level) = 3 / 5 = 0.6

FFoo = 0.2 Hz, T= 0.2 Hz, Too = 5 sec= 5 sec



Rectangular pulse Rectangular pulse traintrain



PeriodPeriod TToo ≈≈ 10 msec10 msec, fundamental frequency ffundamental frequency foo= 100 Hz = 100 Hz


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

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


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

564/1000 = 0.564 (0.561)564/1000 = 0.564 (0.561)

455/1000 = 0.455 (0.454)455/1000 = 0.455 (0.454)

302/1000 = 0.302 (0.303)302/1000 = 0.302 (0.303)

139/1000 = 0.139 (0.140)139/1000 = 0.139 (0.140)

2.4/1000 = 0.0024 (0)2.4/1000 = 0.0024 (0)

cf. S&M p. 33cf. S&M p. 33--3535

ττ = 0.0625 sec= 0.0625 sec

600/1000 = 0.6 (0.6)600/1000 = 0.6 (0.6)

500 Hz500 Hz

Complex Fourier series Complex Fourier series components ccomponents cnn



Rectangular pulseRectangular pulsetrains with pulse trains with pulse period of 0.5 sec 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



Fundamental frequency fFundamental frequency foo= 2 Hz, T= 2 Hz, To o = 0.5 sec= 0.5 sec


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--373716 Hz16 Hz

8 Hz8 Hz

4 Hz4 Hz

Complex Fourier series Complex Fourier series components ccomponents cnn


Chapter 2Chapter 2

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



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 (R = 1Average Normalized (R = 1ΩΩ) ) PowerPower

Periodic signal as a frequencyPeriodic signal as a frequencydomain representationdomain representation

Average normalizedAverage normalizedpower in the signal aspower in the signal asa time domain ora time domain orfrequency domainfrequency 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.



Normalized power Normalized power spectrum of a spectrum of a periodic rectangular periodic rectangular pulse trainpulse train



Period TPeriod Too ≈≈ 10 msec, frequency resolution = 100 Hz 10 msec, frequency resolution = 100 Hz



Sink calculator Power Spectral Density dBm/Hz 1Sink calculator Power Spectral Density dBm/Hz 1ΩΩ


•• Example 2.9 PSD dBm/Hz 1Example 2.9 PSD dBm/Hz 1ΩΩ

5.60 dBm/Hz5.60 dBm/Hz




S&M p. 43S&M p. 43--4444

Power Spectral Density Power Spectral Density (PSD)(PSD)


•• Power Spectral Density dBm/Hz 1Power Spectral Density dBm/Hz 1ΩΩ

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

However, for aperiodic signals the PSD is conceptually However, for aperiodic 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.


o-3

5.6/10 -3 -3 2

o-3 2

2 2

5.6 dBm/Hz f = 100 Hz5.6 = 10 log (Power/Hz / 10 )Power/Hz = 10 (10 ) = 3.63 x 10 V /HzPower = (Power/Hz)(f Hz)Power = (3.63 x 10 V /Hz)(100 Hz)Power = 0.36 V (0.36 V , S&M p. 43)

•• Power Spectral Density dBm/Hz 1Power Spectral Density dBm/Hz 1ΩΩ

dBm is decibel (dB) referenced to 1 normalized dBm is decibel (dB) referenced to 1 normalized milliwattmilliwatt (mW = 10(mW = 10--33 W, normalized VW, normalized V22/R, R = 1/R, R = 1ΩΩ))

dBm = 10 log (Power/ 10dBm = 10 log (Power/ 10--33 VV22) normalized R = 1) normalized R = 1ΩΩ


• 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


•• Example 2.10 PSD dBm/Hz 1Example 2.10 PSD dBm/Hz 1ΩΩ





S&M p. 44S&M p. 44--4646


•• Example 2.10 PSD dBm/Hz 1Example 2.10 PSD dBm/Hz 1ΩΩ Converted to VConverted to V22

0.363 V0.363 V22

0.633 V0.633 V22

0.414 V0.414 V22

0.182 V0.182 V22

S&M p. 44S&M p. 44--4646

Bandwidth = 300 HzBandwidth = 300 Hz


•• 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 must alwayscommunication engineers must alwaysdefine what if meant by define what if meant by ““bandwidthbandwidth””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:

The total power in the signal then is 1.8 VThe total power in the signal then is 1.8 V22 and the and the percentage of the total power in the signal in a bandwidth percentage of the total power in the signal in a bandwidth of 300 Hz then is approximately 88% (S&M p. 45)of 300 Hz then is 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



Period TPeriod To o ≈≈ 1 sec1 sec, fundamental frequency fundamental frequency foo= 1 Hz = 1 Hz


•• Example 2.12 Simulated single pulseExample 2.12 Simulated single pulse


pulse width = 1 msecpulse width = 1 msec

pulse period = 1 secpulse period = 1 sec

Duty Cycle = 10-3/1 = 0.001 = 0.1%


•• Example 2.12 Simulated single pulseExample 2.12 Simulated single pulse

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 SystemVueSystemVue simulation the simulation the data points are very dense and virtually display a data points are very dense and virtually display a continuous plot.continuous plot.


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

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

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 1000 2000 3000 4000 5000 5000

HzHz


•• Example 2.12a Simulated single pulseExample 2.12a Simulated single pulse


pulse width = 10 msecpulse width = 10 msec

pulse period = 1 secpulse period = 1 sec

Duty Cycle = 10-2/1 = 0.01 = 1%


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

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

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

AAττ = 1(10= 1(10--22) = 10) = 10--22

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

100 200 100 200

HzHz


•• 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 squared of by the magnitude squared of the Fast Fourier Transform (FFT) in a the Fast Fourier Transform (FFT) in a SystemVueSystemVue simulation simulation as described in Chapter 3.as described in Chapter 3.

ψψ(f) (f) ≈≈ | FFT || FFT |22


End of Chapter 2End of Chapter 2

Frequency Domain AnalysisFrequency Domain Analysis

chapter 2 frequency domain analysis - temple …silage/chapter2svu.pdfchapter 2 frequency domain...

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