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Frequency Modulation - Tutorial - DOC

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					Acoustics of Music, Semester 1


Frequency Modulation – Tutorial Solutions
1.
       (a)    Explain what is meant by Simple Frequency Modulation (in the
              context of electronic music) and give an equation for the instantaneous
              output voltage of a Simple FM synthesiser using sinusoids.
                                                                           (3 Marks)
       (b)    Calculate (table of Bessel functions supplied) the seven terms centred
              around the carrier in the frequency spectrum which result from a 2000
              Hz carrier sine wave being frequency-modulated with a modulation
              index of 2 by:
              (i)     a      200 Hz sine wave;
              (ii)    a      210 Hz sine wave;
              (iii)   a      800 Hz sine wave.
              In each case give the frequency and amplitude in decibels referenced to
              the carrier amplitude.
                                                                           (9 Marks)


       (c)    Use your answers to part (b) to derive a rule for generating harmonic
              (as opposed to inharmonic) spectra with FM synthesis.
                                                                           (2 Marks)


       (d)    Consider the variant of FM synthesis called Parallel Multiple-
              Modulator FM.
              (i)     Design a synthesiser to perform Parallel Multiple-Modulator
                      FM with 2 sinusoidal modulators and 1 sinusoidal carrier. Give
                      a block diagram of your synthesiser.
                                                                           (5 Marks)
              (ii)    Give an equation for the instantaneous output voltage of your
                      synthesiser.
                                                                           (3 Marks)
              (iii)   Derive a formula to predict the frequencies present in the
                      output spectrum of your synthesiser.
                                                                           (3 Marks)




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Acoustics of Music, Semester 1


Solutions

1)
     (a)    Explain what is meant by Simple Frequency Modulation (in the context of
            electronic music) and give an equation for the instantaneous output
            voltage of a Simple FM synthesiser using sinusoids.
                                                                                                     (3 marks)
            Simple FM involves one signal modulating the frequency of another at a
            rate large enough to have an audible effect.
                                                                                  [1]
            The output from simple FM is
            v(t)= A sin (  c t + I sin (  m t))
                                                                                  [1]
                         f
            where I =
                         fm
            where f is the desired frequency deviation from the carrier frequency fc.
                                                                                    [1]

     (b)    Calculate (table of Bessel functions supplied) the seven terms centred
            around the carrier in the frequency spectrum which result from a 2000 Hz
            carrier sine wave being frequency-modulated with a modulation index of
            2 by:
            (i)                a 200 Hz sine wave;
            (ii)               a 210 Hz sine wave;
            (iii)              a 800 Hz sine wave.
            In each case give the frequency and amplitude in decibels referenced to
            the carrier amplitude.
                                                                                                     (9 marks)
            The spectrum from simple FM is given by
                         
             v(t)=  A J n (I) sin ((  c  n  m ) t)
                        n= 0

            Applying this gives us the frequency and amplitude of each component.
            Components at negative frequencies appear as inverted components at the
            mirror image positive frequency:


            fm:         200                             210                            800
            n        f (Hz) Jn(I)       amp (dB)     f (Hz) Jn(I)      amp (dB)     f (Hz)   Jn(I)        amp      amp (dB)
                -9                                                                   -5200     2.49E-06
                -8                                                                   -4400     2.22E-05
                -7                                                                   -3600       0.0002
                -6                                                                   -2800       0.0012
                -5                                                                   -2000       0.0070
                -4                                                                   -1200       0.0340
                -3    1400     0.1289       -17.79    1370    0.1289       -17.79     -400       0.1289




                                                                                                          Page 2
Acoustics of Music, Semester 1

            -2   1600   0.3528    -9.05   1580   0.3528    -9.05    400    0.3528     0.482    -6.34
            -1   1800   0.5767    -4.78   1790   0.5767    -4.78   1200    0.5767      0.61    -4.29
            0    2000   0.2239   -13.00   2000   0.2239   -13.00   2000    0.2239     0.231   -12.72
            1    2200   0.5767    -4.78   2210   0.5767    -4.78   2800    0.5767    0.5768    -4.78
            2    2400   0.3528    -9.05   2420   0.3528    -9.05   3600    0.3528    0.3528    -9.05
            3    2600   0.1289   -17.79   2630   0.1289   -17.79   4400    0.1289    0.1289   -17.79
            4                                                      5200    0.0340    0.0340   -29.37
            5                                                      6000    0.0070
            6                                                      6800    0.0012
            7                                                      7600     0.0002
            8                                                      8400   2.22E-05
            9                                                      9200   2.49E-06




     (c)   Use your answers to part (b) to derive a rule for generating harmonic (as
           opposed to inharmonic) spectra with FM synthesis.
                                                                               (2 marks)
           For a harmonic spectrum,
            fc
               = integer
            fm




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