E85.2607: Lecture 10 – Modulation E85.2607: Lecture 10 – Modulation 20100415 1 / 19
E85.2607: Lecture 10 Modulation
E85.2607: Lecture 10 Modulation 20100415 1 / 19
Modulation
Modulation Use an audio signal to vary theparameters of a sinusoid
ymod[n] = m[n] cos (2fcn + [n])
m[n], [n] modulating signals
cos(fcn) carrier signal with carrier freq. fc
Used for:
Transmitting radio signals
Tremolo, vibrato, other effects
Synthesizing complex harmonic series
Tremolo? When the modulating frequency is less than 20Hz this
produces a well known music effect called tremolo In musical terms: A regular and repetitive variation in
amplitude for the duration of a single note.
However when the modulation frequency is in the audible range, new sound textures can be created
FM synthesis Like in the case of AM, when the frequency variation is in the audible range
a timbre change is produced.
This modulation may be controlled to produce varied dynamic spectra with relative little computational overheads.
FM was well developed for radio applications in the 1930s, and is nowadays the most widely used broadcast signal format for radio
Introduced as a tool for sound synthesis by John Chowning (Stanford U.) in the early 1970s
1980s: Used by Yamaha to develop its DX series (The DX7 was to become one of the most popular synthesisers of all times) and the OPL chip series (soundblaster sound cards, mobile phones)
E85.2607: Lecture 10 Modulation 20100415 2 / 19
Ring modulation
yRM[n] = m[n] cos(2fcn)
Shifts spectrum of modulating signal to be centered around fc
e.g. let m[n] = cos(2fmn):
yRM[n] = cos(2fmn) cos(2fcn)
=1
2cos (2(fc fm)) + cos (2(fc + fm))
Using trigonometric identity: cos(a b) = cos(a) cos(b) sin(a) sin(b)
x(n) y(n)
m(n)
USBLSB
0
X(f)
cf
(a)
f
c
(b)
USBLSB
fcf
Y(f)c
E85.2607: Lecture 10 Modulation 20100415 3 / 19
Ring modulationRing Modulation
freq
amp
fc
fc  fm fc + fm
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Amplitude modulation
Like ring modulation, but with DC offset added to modulating signal
yAM[n] = (1 + m[n]) cos(2fcn)
Receiver (demodulator) is easier to build
e.g. let m[n] = cos(2fmn):
yAM[n] = (1 + cos(2fmn)) cos(2fcn)
= cos(2fcn) +
2cos (2(fc fm)) + cos (2(fc + fm))
Thus its Fourier Transform is defined as:
Amplitude Modulation
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Amplitude modulation
Amplitude Modulation
freq
amp
!c
!c  !m !c + !m
E85.2607: Lecture 10 Modulation 20100415 6 / 19
Amplitude modulation in the time domain
Demodulate using an envelope detector
= rectifier + LPF
or product detector
= coherent ring modulation + LPF
yAM [t] cos(2fc)
= (1 + m[n]) cos(2fcn) cos(2fcn)
= (1 + m[n])
(1
2+
1
2cos(22fcn)
)
Also works for ring modulation
Amplitude modulation Amplitude modulation (AM) is a form of modulation in which the
amplitude of a carrier wave is varied in direct proportion to that of a modulating signal.
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Effect of modulation index ()
Modulation index
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Single Sideband (SSB) modulation
AM and RM waste bandwidth (and power) in redundant sidelobes
Singlesideband modulation
m(t)
sin!ct
cos!ct
s(t)
H(j!)
!
1
"H(j!)
!
#/2
#/2
90 phaseshift s1(t)
s2(t)
Singlesideband modulation M(!)
!
1
S1(!)
!
1/2
!c !c S2(!)
!
1/2
!c !c
S (!)
!
1
!c !c
With changes of !c the spectrum of m(t) will be shifted accordingly, so SSB modulation is also known as frequency shifting
E85.2607: Lecture 10 Modulation 20100415 9 / 19
Angle modulation
yPM/FM[n] = cos(2fcn + PM/FM[n])
PM[n] = m[n]
FM[n] = 2
n
m[ ] d
Looks like phase is being modulated, but theyre really the same
instantaneous frequency = n (2fcn + [n])(FM often used to refer to phase modulation)
0 200 400 600 800 10002
1
0
1
2
x FM
(n) a
nd m
(n)
(a) FM
n 0 200 400 600 800 1000
2
1
0
1
2
x PM
(n) a
nd m
(n)
(b) PM
n
0 200 400 600 800 10002
1
0
1
2
x FM
(n) a
nd m
(n)
(c) FM
n 0 200 400 600 800 1000
2
1
0
1
2
x PM
(n) a
nd m
(n)
(d) PM
n
0 100 200 300 400 500 600 700 800 900 10002
1
0
1
2
x FM
(n) a
nd m
(n)
(e) FM
n
0 100 200 300 400 500 600 700 800 900 10002
1
0
1
2
x PM
(n) a
nd m
(n)
(f) PM
n
E85.2607: Lecture 10 Modulation 20100415 10 / 19
FM vs PM0 200 400 600 800 10002
1
0
1
2
x FM
(n) a
nd m
(n)
(a) FM
n 0 200 400 600 800 1000
2
1
0
1
2
x PM
(n) a
nd m
(n)
(b) PM
n
0 200 400 600 800 10002
1
0
1
2
x FM
(n) a
nd m
(n)
(c) FM
n 0 200 400 600 800 1000
2
1
0
1
2
x PM
(n) a
nd m
(n)
(d) PM
n
0 100 200 300 400 500 600 700 800 900 10002
1
0
1
2
x FM
(n) a
nd m
(n)
(e) FM
n
0 100 200 300 400 500 600 700 800 900 10002
1
0
1
2
x PM
(n) a
nd m
(n)
(f) PM
n
0 200 400 600 800 10002
1
0
1
2
x FM
(n) a
nd m
(n)
(a) FM
n 0 200 400 600 800 1000
2
1
0
1
2
x PM
(n) a
nd m
(n)
(b) PM
n
0 200 400 600 800 10002
1
0
1
2
x FM
(n) a
nd m
(n)
(c) FM
n 0 200 400 600 800 1000
2
1
0
1
2
x PM
(n) a
nd m
(n)
(d) PM
n
0 100 200 300 400 500 600 700 800 900 10002
1
0
1
2
x FM
(n) a
nd m
(n)
(e) FM
n
0 100 200 300 400 500 600 700 800 900 10002
1
0
1
2
x PM
(n) a
nd m
(n)
(f) PM
n
E85.2607: Lecture 10 Modulation 20100415 11 / 19
Implementing angle modulation
Mx(n)
z (M + frac)
symbol
m(n)
y(n)x(n)
m(n)=M + fracInterpolation
y(n)
frac
x(nM) x(n(M+1) x(n(M+2)
Just index into carrier using timevarying delay
Interpolate as necessary
E85.2607: Lecture 10 Modulation 20100415 12 / 19
Effects: Tremolo
Modulate amplitude of audio signal with low frequency sinusoid
0 1000 2000 3000 4000 5000 6000 7000 8000 90000.5
0
0.5x(
n)Lowfrequency Amplitude Modulation (fc=20 Hz)
0 1000 2000 3000 4000 5000 6000 7000 8000 90001
0
1
2
y(n)
and
m(n
)
0 1000 2000 3000 4000 5000 6000 7000 8000 90000.5
0
0.5
1
1.5
y(n)
and
m(n
)
n
90
90
x(n)
m(n)
LSB(n)
USB(n)
fc
USB
f f
fc
LSB
f f
x(n)
m(n)
USB(f)
LSB(f)
CF
CF
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More effects
Vibrato modulate phase of audio signal with low frequency sinusoid
x(n) y(n)
m(n)=M+DEPTH.sin( nT)!
z(M+frac)
90
x(n)
cos( n)!m
y (n)R
y (n)LAllpass 1
Allpass 2

Detuning SSB modulation to shift spectrum up or down in frequency
E85.2607: Lecture 10 Modulation 20100415 14 / 19
Applications: synthesizing notes
AM synthesis change carrier frequency to change pitch
e.g. simple synthesizer with 3 harmonics by modulatingsinusoidal carrier with sinusoidal signal:
(1 + cos(2fmn)) cos(2fcn)
easy to implementbut, limited timbral possibilities . . .
FM synthesis produce spectrally rich sounds with minimal effort
cos(2fcn + sin(2fmn))
need integer fcfm to make harmonic sounds
sidebands at fc kfmintroduced by John Chowning at Stanford in early 1970scommercialized by Yamaha in the 1980s (DX7)
E85.2607: Lecture 10 Modulation 20100415 15 / 19
FM modulation index
y [n] = cos(2 220 n + sin(2 440 n))
Modulation index
Timedomain Frequencydomain FM signals theoretically have infinite bandwidth 2( + 1) audible sidebands
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http://www.allsciencefairprojects.com/science_fair_projects_encyclopedia/upload/1/12/Frequencymodulationdemocf220mf440.ogg
Note dynamics
Real notes are timelimited
struck/plucked vs. bowed/blown
E4896 Music Signal Processing (Dan Ellis) 20100208  /16
3. Envelopes Notes need to be limited in time
simple gating not enoughamplitude envelope
Different (real) instruments have clear variations in envelopestruck/plucked vs. bowed/blown
10
simulate using ADSR envelope
E4896 Music Signal Processing (Dan Ellis) 20100208  /16
ADSR 4parameter classic envelope model
Attack  initial rise time
Decay  fall time immediately following initial attack
Sustain  amplitude of asymptote of decaywhile key is held down
Release  decay from sustain to zero after key released
11
Tobi
as R
.  M
etoc
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Toward more realistic synthesis
Amplitude modulation alone is not enough
real instruments have timevarying spectrae.g. plucked string
E4896 Music Signal Processing (Dan Ellis) 20100208  /16
4. Filtering Amplitude modulation alone is not enough
real instruments have timevarying spectrae.g. plucked string
Generally just LPF (+ resonance)high frequencies die away after initial transientresonance can give some BPF effect
13
Model using LPF
high frequencies die away after initial transient
Or just model the physics...
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Reading
DAFX Chapter 4  Modulators and Demodulators
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