# Frequency Modulation - Tutorial - DOC by hcj

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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
(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

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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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