Understanding Fourier Analysis Techniques in EMLAB

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Explore Fourier analysis techniques in EMLAB, covering topics such as Fourier series, Fourier transform, quality of approximation, exponential Fourier series, and trigonometric Fourier series. Learn how to determine coefficients, perform integrations, and work with functions exhibiting even symmetry.

  • Fourier Analysis
  • EMLAB
  • Transform Techniques
  • Coefficients
  • Symmetry
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  1. 1 15. Fourier analysis techniques EMLAB

  2. 2 Learning goals Fourier series : General periodic signal. Fourier transform : Arbitrary non-periodic inputs. EMLAB

  3. 3 15.1 Fourier series Periodic function : ? ? = ? ? + ??0,? = 1, 2, 3, Example : ? ? = cos2?? ?, ? = 1, 2, 3, ?0 ? ? + ?0 = cos2?? 2?? ?0 ? + 2?? = cos2?? (? + ?0) = cos ? ?0 ?0 ?? [?? ?? ????0?], ?0=2? ? ? = ?0+ ??cos(??0? + ??) = ?0+ ?0 ?=1 ?=1 ? ? = ?0+ ??cos??0? + ??sin??0? ?=1 ?=1 ??????0? ? ? = ?= EMLAB

  4. 4 Example of quality of approximation Approximation with 4 terms Original Periodic Signal ? ? ??? = ?0+ ??cos??0? + ??sin??0? ?=1 ?=1 Approximation with 2 terms Approximation with 100 terms 2 2 100 100 ?2? = ?0+ ??cos??0? + ??sin??0? ?100? = ?0+ ??cos??0? + ??sin??0? ?=1 ?=1 ?=1 ?=1 EMLAB

  5. 5 Exponential Fourier series Any physically realizable periodic signal, with period To, can be represented by the expression ?0=2? ??????0?, ? ? = , ?1< ? < ?1+ ?0 ?0 ?= How to determine Cn: ?1+?0 ?1+?0 ? ? ? ???0??? = ????(? ?)?0? ?? ?1 ?1 ?= ?1+?0 ??(? ?)?0??? = ?0 (??? ? ?) (?? ??????) 0 ?1+?0 ??=1 ?1 ? ? ? ???0??? ?0 ?1 t1can be set arbitrarily. Only the integration span is important because ? ? is periodic. EMLAB

  6. 6 Example 15.1 Determine the exponential Fourier series ?0=2? ??????0?, ? ? = ?0 ?= ?1+?0 ??=1 ? ? ? ???0??? ?0 ?1 Strategy : 1. Determine ?0and ?0. 2. Select a convenient ?1. 3. Do the integration ?0= ? ?0=2? ? ? ? 4 ? 2 ? ? ???0??? ?/2 ??=1 ? ? ? ???0??? =1 4( ?) ? ???0??? + ?? ???0??? ? ? ? ? ? ?/2 ? ????0 ? ?2??2? 2 4 4 ? ???0 ? 4 ? ???0 ? ? 4 + ? ???0 ? ? 2 ? ???0 ? 4 2 ? ???0 4 + ? ???0 = 2??4sin?? ? ??? 2 2? ??? 2 + ? ??? ????= = 2 2sin?? , ?0? = 2? 2? ??sin?? 0 (?:???) ??= 2 (?:???? 0) EMLAB

  7. 7 Trigonometric Fourier series ?0=2? ? ? ? = ?0+ ??cos??0? + ??sin??0? ?=1 ?=1 ?1+?0 ?1+?0 ? ? cos??0??? = ?0cos??0??? ?1 ?1 ?1+?0 ?1+?0 + ??cos??0? cos??0??? + ??sin??0? cos??0??? ?1 ?1 ?=1 ?=1 ?1+?0 ?0=1 ?0 ? ? ?? ?1 ?1+?0 ??=2 ?0 ? ? cos??0??? ?1 ?1+?0 ??=2 ?0 ? ? sin??0??? ?1 EMLAB

  8. 8 Functions with even symmetry ?0/2 ?0=2 ?0 ? ? ?? ?1+?0 ?0=1 ?0 ? ? ?? 0 ?0/2 ??=4 ?1 ?0 ? ? cos??0??? ?1+?0 ??=2 0 ?0 ? ? cos??0??? ?0 2 ??=2 ?1 ?1+?0 ?0 ? ? sin??0??? = 0 ?0 ??=2 2 ?0 ? ? sin??0??? ?0= 2 ??=4 ?1 1 2 2cos??0??? + 4cos??0??? ?0 0 1 4 = [2sin??0? + 4sin2??0 4sin??0] ??0?0 4 ?? 2sin2?? sin?? = 3 3 ??= 0 Functions with odd symmetry ?0/2 ?0=2 ?0 ? ? ?? = 0 ?0 ?0/2 2 ??=2 ?0 ? ? cos??0??? = 0 ?0 ?0 2? ? sin??0??? 2 ??=4 ?0 0 EMLAB

  9. 9 Fourier coefficients of time shifted signal Time shift in the time domain corresponds to phase shift in the frequency domain. ??????0(? ?0) ??????0? ? ? ?0 = ? ? = ?= ?= (??? ???0?0)????0?= ????0? = ?? ?= ?= = ??? ???0?0 ?? EMLAB

  10. 10 Example 15.6 Let us time-delay the waveform in the figure by a quarter period and compute the Fourier series. ?0=2? ??????0?, ? ? = ?0 ?= 2? ??sin?? 0 (?:???) ??= 2 (?:???? 0) ?0= time shift for ? ? = ??? ???0?0= ??? ??? =2? ? ???0?0= phase shift for ?? ?? 2 ??sin?? = ?2? ?? 2 ? ??? sin?? 2 ? ? = ? ? ?0 2 4 2 = ?2? ?? 2 = cos?? 2 ?sin?? ? ??? 2 = ?2? (?:???) ?? ?? 0 (?:???? 0) EMLAB

  11. 11 Frequency spectrum Definition : The spectrum is a graphical display of the coefficients of the Fourier series. One-sided spectrum is based on the representation ?? [?? ?? ????0?], ? ? = ?0+ ??cos(??0? + ??) = ?0+ ?=1 ?=1 The amplitude spectrum displays ??as the function of the frequency. The phase spectrum displays the angle ??as function of the frequency. The frequency axis is usually drawn in units of fundamental frequency Two-sided spectrum is based on the exponential representation ??????0? ? ? = ?= In the two-sided case, the amplitude spectrum plots |cn| while the phase spectrum plots cn frequency (in units of fundamental frequency) versus Spectrum analyzer EMLAB

  12. 12 Example 15.6 The Fourier series for the triangular-type waveform shown in the figure. Determine and plot the first four terms of the spectrum. (A=5) ? ? ?1? ? 2? ? ???0??? ?/2 ??=1 ?1? ? ???0??? =1 4? ?0 ? ? = ?1 ? +?0 ? ?1? ? 2 0 0 ??? 1? ?[1 1?] ?0 2 ???0 1? 1? 1 ???0 =4? ?0 ? = ?????0?0 2= 1 = ?? 1? 1 2 ??2 ?? 2 1 2 = 2 ??(?:???) = 2? ???+ ??2 ? 2? ? ???0??? = ? 2? ???0? ???0 ?0/2 ?? ???0? ???0 ?? 0 0 0 ? = ?,? = ? ???0? ? = 1,? =? ???0? ???0 EMLAB

  13. 13 Phase spectrum Amplitude spectrum EMLAB

  14. 14 Steady state network response 1. Replace the periodic signal by its Fourier series 2. Determine the steady state response to each harmonic 3. Add the steady state harmonic responses ? ? = ?0? + ?1? + ?2? + ?(??) ? ???1? ? ??1????1? ? cos(?1? + ?) ? ? ??1 cos[?1? + ? + ???1] ?? {|?| ? ???1?} ?? {|?| ? ? ??1 ?(?1)???1?} EMLAB

  15. 15 Example 15.8 Find the out voltage vo(t) for the following network. ? = 5,?0= ? (1) First the input voltage source should be represented by a Fourier series. 1 2 ??= 2? ???+ ??2,(?:???) ??????0? ? ? = ?= ??? =10 ? ??+2 ?2??2?? ?2 ?= ?:??? EMLAB

  16. 16 (2) Find the transfer function of the circuit. 1 1 ?? 2 1 ?? 2 ? ?? =???? 1 2= 1 ?? 2 1 ?? 2 1 2??(??) = ???? = ???? 4 1 + ?? 2 + 2 + (3) Solution ??? =10 ? ??+2 ?2??2?? ?2 ?= ?:??? 5 1 ?2?+1 ? ? ???0??????0?= ?2??2?? ??? = ?2 ?= ?:??? 1 + ?2? ?= EMLAB

  17. 17 Average power In a network with periodic sources (of the same period) the steady state voltage across any element and the current through are all of the form The average power is the sum of the average powers for each harmonic ? ? = ???+ ??cos(??0? + ???) ?=1 ? ? = ???+ ??cos(??0? + ???) ?=1 ?1+?0 Average power ? =1 ? ? ? ? ? ?? ?1 ?1+?0 ?1+?0 ?1+?0 ???????? + ??? ?? cos(??0? + ???)?? + ??? ?? cos(??0? + ???)?? ?1 ?1 ?1 ?=1 ?=1 ?1+?0 + ???? cos(??0? + ???)cos(??0? + ???)?? ?1 ?=1 ?=1 ?1+?0 ?1+?0 ???? 2 cos[(? + ?)?0? + ???+ ???] +cos[(? ?)?0? + ??? ???]?? ???? cos(??0? + ???)cos(??0? + ???)?? = ?1 ?1 ?=1 ?=1 ?=1 ?=1 ?1+?0 ??? ? = ? ??? ? ? cos[(? ?)?0? + ??? ???]?? = ?0cos(??? ???) 0 ?1 ???? 2 Average power ? = ??????+ cos(??? ???) ?=1 EMLAB

  18. 18 Example 15.9 Determine current, the ( i average the and , ) t power absorbed by the network ? ? = 42 + 16cos(377? + 30 ) + 12cos(754? 20 ) [?] ? ?? =? ?? 1 ? ?? , ? ?? = ? + ??? + ? ?? ??? 1 ? = 0 Capacitor acts as open circuit ? ? ?0 = 0 3 ? = 754 ???/? ? ?754 = 12 20 2 ? = 377 ???/? ? ?377 = 16 30 ? ? ?754 = 16 + ?0.020 754 ? 10 4 754 ? ?377 = 16 + ?0.020 377 10 4 377 12 20 ? ?754 = 16 + ?15.08 ?13.26= 0.75 26.49 16 30 ? ?377 = 16 + ?7.54 ?26.53= 0.64 79.88 ? ? = 0.64cos(377? + 79.88 ) + 0.75cos(754? 26.49 ) [?] ???? 2 Average power ? = ??????+ cos(??? ???) ?=1 ? = 42 0 +1 2[16 0.64cos 49.88 + 12 0.75cos 6.49 ] EMLAB

  19. 2. Fourier transform 19 ? ? ? ??t?? ? ? = 1 ? ? ???t?? ?(?) = 2? ? ? = [? ? ] ? ? = 1[? ? ] A heuristic view of the Fourier transform A non-periodic function can be viewed as the limit of a periodic function when the period approaches infinity 1 1 ? ? ? ???t?? ???????0? ?(?) = 2? ??? = ??????0? ? ? = ?0= ?, ??0= ? ?= ?/2 ?= ?/2 ??=1 ? ? ? ??t?? ? ? ? ???0??? ? ? = ??? = ? ? ? ???0??? ? ?/2 ?/2 ? EMLAB

  20. Fourier series vs. Fourier transform 20 ??????0? ? ? = ?= ?/2 ?/2 ??=1 ? ? ? ???0??? =?0 ? ? ? ???0??? ? 2? ?/2 ?/2 Extend the period T to infinity ? 2? ? ? ???0??? ????0? ?0 2? ??? = ? ?= 2 ? 2? ? ? ???0??? ????0??0 1 = 2? ? ?= 2 1 ? ? ? ????? ?????? = 2? ? = ??0, ? = ?0 Fourier transform of ? ? EMLAB

  21. 21 Example 15.9 Determine the Fourier transform for the voltage pulses. For comparison we show the spectrum of a related periodic function sin?? ? 2? ????? = ?? 2 ?(?) = ? sin??0? ? 2? ???0??? =?? ?? 2 ? ??=? 2 2 ?0 ??0? 2 ?0 ? 2 Spectrum for ?0= 5? EMLAB

  22. 22 Example 15.10 Determine the Fourier transform of the unit impulse function ? ? ? ? ??t?? = ? ??? ? ? ? = Sifting or sampling property of the delta function. ? ? ?(? ?) ?? = ?(?) EMLAB

  23. 23 Example 15.11 Determine the Fourier transform of exponential function. ? ? = ???0? ???0?= ???0?? ????? = ??(?0 ?)??? = 2? ?(?0 ?) = 2? ?(? ?0) 1 2? ? ? ?0 ? ????? = ???0? ? ? = 2? ???0?= 2? ?(? ?0) LEARNING EXTENSION Determine the Fourier transform of sine function. ???0? ? ??0? 2? sin?0? = = ??[? ? + ?0 ? ? ?0] EMLAB

  24. 24 Fourier transform of ( ) 1 1 ? ? ? ????? ?????? = ??? ? ??? ?? = ? ? = 2? ? ? 2? ? ? ?(? ?) ?? 1 ??? ? ??? ? ? ? = 2? sin? ? ? ?/2 ?/2 ??? ? ? ? ? ? 2 ??? ? ??? = lim lim ? = lim ? ? = 2?? ? ? ? ? ? 2 ? ?/2 ?/2 sin?? ?? 2 sin?? 2 ? 2 lim ? ? ?? = 2? ?? 2 ? ? =2? ? ? As ? increases to infinity, sinc function becomes similar to delta function. EMLAB

  25. 25 sin? sin? 1 + ?2?? =? 1 ? ???? = lim (sin?)? ?????? = ?? = lim ? 0 ? 0 ? ? 2 0 0 ? 0 0 ??? ? ?? ?? ? ? ? ? +? ?+? ? ? + ? ? ???? =1 =1 1 1 1 (sin?)? ???? = ? ? = 1 + ?2 2? 2? 2? ? + ? 0 0 0 EMLAB

  26. 26 EMLAB

  27. 27 EMLAB

  28. 28 Proof of the convolution property ? ? = ?1? ?2? = ?1? ?2? ? ?? ? ? ? ????? ? ? = ?1? ?2? ? ?? ? ????? = ?2? ? ? ????? ?? = ?1? ?1? ? ????? = ?1? ?2? = EMLAB

  29. 29 Parseval s theorem Think of ? ? as a voltage applied to a one Ohm resistor 2 ? ? = ? ? ? ? = ? ? 1 2?? = 2?? ? ? 2? ? ? 1 2?? = ? ? ?????? ?? ? ? ? ? 2? 1 ? ? ?????? ?? = 2? 1 2? 1 2? ? ? 1 ? ? ? ? ?? = ? ? ? ? ?? = 2? 2?? = ? ? 2= power (or energy density in time) 2= Energy density in frequency domain ? ? ? ? Parseval s theorem permits the determination of the energy of a signal in a given frequency range. Intuitively, if the Fourier transform has a large magnitude over a frequency range then the signal has significant energy over that range EMLAB

  30. 30 Example 15.12 Find the output voltage ??? when ? ???? = 5? 2?? ? ? ???? = 5cos2? ? ? ? = 1? + + ???? ? = 10 ??? ? ???? = ??? ? ???? = ???? ? + ? ? ? ??? = ? ?(?) ??? = ??? + ????? 10 ?? + 10???? 10 ?? + 10 + ? ? ? ?? ??? = ? ?(?) = 5 ? ? = ???? = 2 + ?? 5?[? ? 2 + ? ? + 2 ] 10 2 + ??=50 5 1 1 50 8(? 2? ? 10?)?(?) 25? 52(5cos2? + 2sin2?) ?? + 10 2 + ?? 8 10 + ?? ??? = ??? = 10 ?? + 10 5?[? ? 2 + ? ? + 2 ] EMLAB

  31. 31 Example 15.13 Examine the effect of this low-pass filter in the quality of the input signal 1 ???? = 1 +?? 20 1 1 1 ??? 1 ???+ ? ? ? = = 1 + ????= 1 +?? 5 EMLAB

  32. 32 1 =20 1 1 ??? = ? ? ???? = 5 + ?? 1 +?? 1 +?? 3 20 + ?? 20 5 ??? =20 ? 5? ? 20??(?) 3 High frequencies in the input signal are attenuated in the output. The effect is clearly visible in the time domain. The output signal is slower and with less energy than the input signal. ? EMLAB

  33. 33 Effect of ideal filters Effect of band-pass filter Effect of low-pass filter Effect of band-stop filter Effect of high-pass filter EMLAB

  34. 34 Example 15.14 Audio signals do not propagate well in atmosphere they get attenuated very quickly. Original Solution: Move the audio signals to a different frequency range for broadcasting. The frequency range 540kHz 1700kHz is reserved forAM modulated broadcasting Audio signal Carrier signals ? ? = cos(2????) ??? = cos(2????) ? ? = [? + ? ? ]??(?) Broadcasted signal ?????+ ? ???? 2 ? ? = cos??? = = ? ? ? ?? + ? ? + ?? ??? = cos??? = ? ? ? ?? + ? ? + ?? EMLAB

  35. 35 cos??? = ? cos??? +1 2 ? ? ? ????+ ? ? ?+???? ? ? = ? + ? ? = ??(?) +1 2? ? ?? + ? ? + ?? = ?? ? ? + ?? + ? ? ?? +? 4[? ? ?? ?? + ? ? ??+ ?? + ? ? ??+ ?? + ? ? + ??+ ??] am modulation ?? ?? ? ? ?? ?? ?? ? ?? EMLAB

  36. 36 Example 15.17 Tuning-out an AM radio station Signal from station 1 : ?1? = 1 + ?1? Signal from station 2 : ?2? = 1 + ?2? For simplicity, assume ?1? = ?2? ?1= 900 ???,?2= 960 ??? cos?1? cos?2? Assuming both stations reach the receiver with equal strength; Signal at antenna : ??? = ?[?1? + ?2? ] Fourier transform of signal broadcast by two AM stations Proposed tuning circuit Ideal filter to tune out one AM station ??? =?0? ? = ? + ?? +1 1 ??? +1 ??? Fourier transform of received signal ?? 1 ? ?0+?0 = = Next we show how to design the tuning circuit by selecting suitable R,L,C 1 +1 1 + ? ?? ? EMLAB

  37. 37 Designing the tuning circuit Ideal filter to tune out one AM station ?? ??? =?0? ? ?+1 = ?2+ ?? ??? ?? Design equations 1 Center frequency : ?0= Bandwidth : ?? = 2? ?? 2? ? 1 Frequency response of circuit tuned to 960kHz ? Design specifications ?? 60 ??? ?0= 900 ???,?? 960 ??? More unknowns than equations. Let us arbitrarily choose ?? = 10 ???,? = 10? 1 ? ? ? = 159.2?? ?? = 2? 1 2? ?? ? = 196.4 ?? ? = 172.6 ?? ?0= 900 ??? ?0= 960 ??? ?0= EMLAB

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