# EC05032Notes-33.pdf

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www.jntuworld.com www.jwjobs.net ANALOG COMMUNICATIONS LECTURE NOTES-33 UNIT-IV UNIT-IV ANGLE MODULATION    Notes-33 Demodulation of FM Waves : Frequency demodulation is one in which the original modulating wave is recovered from the frequency modulated wave. In the fm demodulation, the amplitude of the output signal must be directly proportional to the instantaneous frequency of the frequency modulated wave used as input signal. There are two method. i. Direct Method and ii. Indirect M
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ANALOG COMMUNICATIONS UNIT-IV LECTURE NOTES-33 D.TIRUMALA RAO ECE GMRIT Page 1 of 5 Ref: Analog & Digital Communications Simon Haykins UNIT-IV ANGLE MODULATION Notes-33 Demodulation of FM Waves :    Frequency demodulation is one in which the srcinal modulating wave is recovered from the frequency modulated wave.    In the fm demodulation, the amplitude of the output signal must be directly proportional to the instantaneous frequency of the frequency modulated wave used as input signal.    There are two method. i. Direct Method and ii. Indirect Method. Balanced Frequency Discriminator:    Consider an ideal slope circuit which is characterized by a purely imaginary transfer function, varying linearly with frequency function, varying linearly with frequency in the prescribed interval.    Consider the following transfer function as per the diagram fig a)  +−≤≤−−      −++≤≤−      +−= otherwise B f  f  B f  B f  f a j  B f  f  B f  B f  f a j  f  H  cccccc 022,2222,22)( 1  π  π        When a  is constant Consider the f  m  S(t) with carrier frequency f  c  and transmission Bandwidth B. Now output S1(t) of slope circuit is to be determined, www.jntuworld.com www.jntuworld.com www.jwjobs.net   ANALOG COMMUNICATIONS UNIT-IV LECTURE NOTES-33 D.TIRUMALA RAO ECE GMRIT Page 2 of 5 Ref: Analog & Digital Communications Simon Haykins    Now replace the slope circuit by an equivalent low-pass filter as shown in fig. to which the complex envelope of FM input S(t) is applied.    Let the complex transfer function of the slope circuit be )(~ 1  f  H   as shown in fig(a).    ( ) )2(0)( ~ 11  −>=−  f  f  H  f  f  H  c      ∴ From (1) and (2), we have ( )  −≤≤−+ = )3(0 2222)(~ 1 elsewhere B f  B f a j f  H  π    As shown in fig b) The incoming fm wave S(t) is given by www.jntuworld.com www.jntuworld.com www.jwjobs.net   ANALOG COMMUNICATIONS UNIT-IV LECTURE NOTES-33 D.TIRUMALA RAO ECE GMRIT Page 3 of 5 Ref: Analog & Digital Communications Simon Haykins )4()(22cos)( 0 −  += ∫ t  f c  dt t mk t  f  AC t S   π  π    The complex envelope of this fm wave is )5()(2exp)( ~ 0 −  = ∫ t  f   dt t mt k  j AC t S   π    Now let )(~ 1  t S  be the complex envelope of the response of the slope circuit. Fourier transform of this is  −≤≤−      +== elsewhere B f  B f S  B f a jt S t  H t S  0)6(22)(~22)(~)(~)(~ 11 π    Where )(~  f S   is the Fourier transform of )(~ t S   Now by the relation g(t) →  G(f) And [ ] ).(2)(  f  fG jt g dt d  π   ↔  i.e., from this multiplication of a Fourier transform of a signal by a factor j2 π f is equivalent to differentiating the signal in the time domain. )7()( ~)(~)(~ 1  −  +=∴  t S  B jt S  dt d at S   π    Now substituting (5) in (7), we have )8()(2exp)( 21)(~ 01  −  += ∫ t  f  f C   t mK  jt m  BK  BaA jt S   π  π    The response of the slope circuit is ( ) )9()(2cos)( 212exp)( ~Re)(~ 011 −  +  +== ∫ t  f c f C c t mK t  f t m  BK  BaAt  f  jt S t S  π  π  π        The above signal S1(t) is a hybrid-modulated wave in which both the amplitude and frequency of the carrier wave vary with the message signal m(t). And also .1)( 2 t all for t  M   BKf  <      Now an envelope detector can be used to recover the amplitude variations in turn srcinal message signal.    Hence envelope detector output is )10()( 21)(~ 1  −  +=  t m BKf  BaAC t S   π     www.jntuworld.com www.jntuworld.com www.jwjobs.net   ANALOG COMMUNICATIONS UNIT-IV LECTURE NOTES-33 D.TIRUMALA RAO ECE GMRIT Page 4 of 5 Ref: Analog & Digital Communications Simon Haykins    In the above, bias term π BaAC is proportional to the slope a of the transfer function of the slope circuit.    This bias term can be removed by subtracting the envelope detector output from the second envelope detector output preceded by the complementary slope current with transfer function H2(f).    In such case complex transfer function of the complementary slope circuit is given by )11()(1 ~)(2~ −−=  f  H  f  H   )12()( 21)(~ 2  −  −=  t m BK  BaAt S   f C  π        ∴ The difference between the two envelopes is )13()(4)( ~)(~)(0 21  −=−=  t maAK t S t S t S  C  f  π        The above procedure can be explained in following scheme which is called back-to-back frequency detector.    The above scheme can be realized by the following    Here the upper and lower resonant filter sections are tuned to frequencies above and below the unmodulated carrier frequency f  c .    The amplitude responses of the two tuned filters separately and the total response can be observed in the fig(c). www.jntuworld.com www.jntuworld.com www.jwjobs.net

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