EE4512 Analog and Digital Communications Chapter 2Chapter 2Chapter 2Frequency Domain AnalysisFrequency Domain AnalysisEE4512 Analog and Digital Communications Chapter 2Chapter 2Chapter 2Frequency Domain AnalysisFrequency Domain Analysis••Why Study Frequency DomainWhy Study Frequency DomainAnalysis?Analysis?••Pages 6Pages 6--1313EE4512 Analog and Digital Communications Chapter 2Why 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 6.3500, 1500 and 2500 Hz500, 1500 and 2500 HzEE4512 Analog and Digital Communications Chapter 2Butterworth LPFButterworth LPF1 pole, f1 pole, foo= 1 kHz= 1 kHz500500HzHz15001500HzHz25002500HzHzSVU Fig 6SVU Fig 6--1 modified1 modified••Example 2.1 Input sum of three sinusoidsExample 2.1 Input sum of three sinusoidsEE4512 Analog and Digital Communications Chapter 2••Example 2.1 Input sum of three sinusoidsExample 2.1 Input sum of three sinusoids•Output after Butterworth LPFOutput after Butterworth LPFEE4512 Analog and Digital Communications Chapter 2••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 dB3.81 dB3.81 dBEE4512 Analog and Digital Communications Chapter 2Cursor based Cursor based measurementsmeasurementsdBdBEE4512 Analog and Digital Communications Chapter 2••Example 2.2Example 2.210 MHz sinusoid10 MHz sinusoidwith additive with additive white Gaussianwhite Gaussiannoise (AWGN)noise (AWGN)EE4512 Analog and Digital Communications Chapter 2••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 AWGN10 MHz10 MHzEE4512 Analog and Digital Communications Chapter 2Chapter 2Chapter 2Frequency Domain AnalysisFrequency Domain Analysis••The Fourier SeriesThe Fourier Series••Pages 13Pages 13--3838EE4512 Analog and Digital Communications Chapter 2π∞+∑0n onn=1s(t) = X X cos(2 n f t +φ )••Fourier SeriesFourier SeriesJean Jean BaptisteBaptisteJoseph 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--18301830EE4512 Analog and Digital Communications Chapter 22200n nnnn n nX=a X= a+b| c | = X / 2 X = | 2 c |••Fourier series coefficients:Fourier series coefficients:trignometrictrignometricaan n bbnnpolar polar XXnncomplex complex ccnnSystemVueSystemVuesimulation cansimulation canprovide the magnitude ofprovide the magnitude ofthe complex Fourier seriesthe complex Fourier seriescoefficients for any periodiccoefficients for any periodicwaveform.waveform.EE4512 Analog and Digital Communications Chapter 2••Example 2.3Example 2.3Complex pulseComplex pulseas the addition of as the addition of two periodic pulsestwo periodic pulsesEE4512 Analog and Digital Communications Chapter 2••Example 2.3 SystemVue Design WindowExample 2.3 SystemVue Design WindowEditing Simulate System Time Analysis WindowEditing Simulate System Time Analysis WindowEE4512 Analog and Digital Communications Chapter 2••Example 2.3 SystemVue System TimeExample 2.3 SystemVue System TimeFundamental frequency fFundamental frequency foo= 0.2 Hz, T= 0.2 Hz, To o = 5 sec= 5 secEE4512 Analog and Digital Communications Chapter 2••Example 2.3 SystemVue Analysis WindowExample 2.3 SystemVue Analysis WindowSink calculator | FFT |Sink calculator | FFT |EE4512 Analog and Digital Communications Chapter 2••Example 2.3 SystemVue Analysis WindowExample 2.3 SystemVue Analysis WindowSink calculator Scale DisplaySink calculator Scale DisplayEE4512 Analog and Digital Communications Chapter 2••Example 2.3 Unscaled | FFT |Example 2.3 Unscaled | FFT |••Scaled | FFT |Scaled | FFT |10 Hz10 Hz4 units4 units500 Hz500 HzEE4512 Analog and Digital Communications Chapter 2••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.EE4512 Analog and Digital Communications Chapter 2••Example 2.3 First periodic pulseExample 2.3 First periodic pulse•Second periodic pulseSecond periodic pulseEE4512 Analog and Digital Communications Chapter 2••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 = 0.6FFoo= 0.2 Hz, T= 0.2 Hz, Too= 5 sec= 5 secEE4512 Analog and Digital Communications Chapter 2••Example 2.7Example 2.7Rectangular pulse Rectangular pulse traintrainEE4512 Analog and Digital Communications Chapter 2••Example 2.7 SystemVue System TimeExample 2.7 SystemVue System TimePeriodPeriodTToo≈≈10 msec10 msec, fundamental frequency ffundamental frequency foo= 100 Hz = 100 HzEE4512 Analog and Digital Communications Chapter 2••Example 2.7 One cycle of periodic pulseExample 2.7 One cycle of periodic pulse••Magnitude of the Fast Fourier Transform | FFT |Magnitude of