e85.2607: lecture 10 -- modulation - columbia...
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E85.2607: Lecture 10 – Modulation
E85.2607: Lecture 10 – Modulation 2010-04-15 1 / 19
Modulation
Modulation Use an audio signal to vary theparameters of a sinusoid
ymod[n] = m[n] cos (2πfcn + φ[n])
m[n], φ[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 1930’s, 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 1970’s
• 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 2010-04-15 2 / 19
Ring modulation
yRM[n] = m[n] cos(2πfcn)
Shifts spectrum of modulating signal to be centered around fc
e.g. let m[n] = cos(2πfmn):
yRM[n] = cos(2πfmn) cos(2πfcn)
=1
2cos (2π(fc − fm)) + cos (2π(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 2010-04-15 3 / 19
Ring modulationRing Modulation
freq
amp
fc
fc - fm fc + fm
E85.2607: Lecture 10 – Modulation 2010-04-15 4 / 19
Amplitude modulation
Like ring modulation, but with DC offset added to modulating signal
yAM[n] = (1 + αm[n]) cos(2πfcn)
Receiver (demodulator) is easier to build
e.g. let m[n] = cos(2πfmn):
yAM[n] = (1 + α cos(2πfmn)) cos(2πfcn)
= cos(2πfcn) +α
2cos (2π(fc − fm)) + cos (2π(fc + fm))
• Thus its Fourier Transform is defined as:
Amplitude Modulation
E85.2607: Lecture 10 – Modulation 2010-04-15 5 / 19
Amplitude modulation
Amplitude Modulation
freq
amp
!c
!c - !m !c + !m
E85.2607: Lecture 10 – Modulation 2010-04-15 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(2πfc)
= (1 + αm[n]) cos(2πfcn) cos(2πfcn)
= (1 + αm[n])
(1
2+
1
2cos(2π2fcn)
)
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.
E85.2607: Lecture 10 – Modulation 2010-04-15 7 / 19
Effect of modulation index (α)
Modulation index
E85.2607: Lecture 10 – Modulation 2010-04-15 8 / 19
Single Sideband (SSB) modulation
AM and RM waste bandwidth (and power) in redundant sidelobes
Single-sideband modulation
m(t)
sin!ct
cos!ct
s(t)
|H(j!)|
!
1
"H(j!)
!
#/2
-#/2
90° phase-shift s1(t)
s2(t)
Single-sideband 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 2010-04-15 9 / 19
Angle modulation
yPM/FM[n] = cos(2πfcn + β φPM/FM[n])
φPM[n] = m[n]
φFM[n] = 2π
∫ n
−∞m[τ ] dτ
Looks like phase is being modulated, but they’re really the same
instantaneous frequency = ∂∂n (2πfcn + βφ[n])
(“FM” often used to refer to phase modulation)
0 200 400 600 800 1000−2
−1
0
1
2
x FM(n
) and
m(n
)
(a) FM
n !0 200 400 600 800 1000
−2
−1
0
1
2
x PM(n
) and
m(n
)
(b) PM
n !
0 200 400 600 800 1000−2
−1
0
1
2
x FM(n
) and
m(n
)
(c) FM
n !0 200 400 600 800 1000
−2
−1
0
1
2
x PM(n
) and
m(n
)
(d) PM
n !
0 100 200 300 400 500 600 700 800 900 1000−2
−1
0
1
2
x FM(n
) and
m(n
)
(e) FM
n !
0 100 200 300 400 500 600 700 800 900 1000−2
−1
0
1
2
x PM(n
) and
m(n
)
(f) PM
n !
E85.2607: Lecture 10 – Modulation 2010-04-15 10 / 19
FM vs PM0 200 400 600 800 1000−2
−1
0
1
2
x FM(n
) and
m(n
)
(a) FM
n !0 200 400 600 800 1000
−2
−1
0
1
2
x PM(n
) and
m(n
)
(b) PM
n !
0 200 400 600 800 1000−2
−1
0
1
2
x FM(n
) and
m(n
)
(c) FM
n !0 200 400 600 800 1000
−2
−1
0
1
2
x PM(n
) and
m(n
)
(d) PM
n !
0 100 200 300 400 500 600 700 800 900 1000−2
−1
0
1
2
x FM(n
) and
m(n
)
(e) FM
n !
0 100 200 300 400 500 600 700 800 900 1000−2
−1
0
1
2
x PM(n
) and
m(n
)
(f) PM
n !
0 200 400 600 800 1000−2
−1
0
1
2
x FM(n
) and
m(n
)
(a) FM
n !0 200 400 600 800 1000
−2
−1
0
1
2
x PM(n
) and
m(n
)
(b) PM
n !
0 200 400 600 800 1000−2
−1
0
1
2
x FM(n
) and
m(n
)
(c) FM
n !0 200 400 600 800 1000
−2
−1
0
1
2
x PM(n
) and
m(n
)
(d) PM
n !
0 100 200 300 400 500 600 700 800 900 1000−2
−1
0
1
2
x FM(n
) and
m(n
)
(e) FM
n !
0 100 200 300 400 500 600 700 800 900 1000−2
−1
0
1
2
x PM(n
) and
m(n
)
(f) PM
n !
E85.2607: Lecture 10 – Modulation 2010-04-15 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(n-M) x(n-(M+1) x(n-(M+2)
Just index into carrier using time-varying delay
Interpolate as necessary
E85.2607: Lecture 10 – Modulation 2010-04-15 12 / 19
Effects: Tremolo
Modulate amplitude of audio signal with low frequency sinusoid
0 1000 2000 3000 4000 5000 6000 7000 8000 9000−0.5
0
0.5x(
n)Low−frequency Amplitude Modulation (fc=20 Hz)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000−1
0
1
2
y(n)
and
m(n
)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000−0.5
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
x̂(n)
m̂(n)
|USB(f)|
|LSB(f)|
CF
CF-
E85.2607: Lecture 10 – Modulation 2010-04-15 13 / 19
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)LAll-pass 1
All-pass 2
-
Detuning SSB modulation to shift spectrum up or down in frequency
E85.2607: Lecture 10 – Modulation 2010-04-15 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(2πfmn)) cos(2πfcn)
easy to implementbut, limited timbral possibilities . . .
FM synthesis produce spectrally rich sounds with minimal effort
cos(2πfcn + β sin(2πfmn))
need integer fcfm
to make harmonic sounds
sidebands at fc ± kfm
introduced by John Chowning at Stanford in early 1970scommercialized by Yamaha in the 1980s (DX7)
E85.2607: Lecture 10 – Modulation 2010-04-15 15 / 19
FM modulation index
y [n] = cos(2π 220 n + β sin(2π 440 n))
Modulation index
Time-domain Frequency-domain FM signals theoretically have infinite bandwidth∼ 2(β + 1) audible sidebands
E85.2607: Lecture 10 – Modulation 2010-04-15 16 / 19
Note dynamics
Real notes are time-limited
struck/plucked vs. bowed/blown
E4896 Music Signal Processing (Dan Ellis) 2010-02-08 - /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) 2010-02-08 - /16
ADSR• 4-parameter 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
E85.2607: Lecture 10 – Modulation 2010-04-15 17 / 19
Toward more realistic synthesis
Amplitude modulation alone is not enough
real instruments have time-varying spectrae.g. plucked string
E4896 Music Signal Processing (Dan Ellis) 2010-02-08 - /16
4. Filtering• Amplitude modulation alone is not enough
real instruments have time-varying 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...
E85.2607: Lecture 10 – Modulation 2010-04-15 18 / 19
Reading
DAFX Chapter 4 - Modulators and Demodulators
E85.2607: Lecture 10 – Modulation 2010-04-15 19 / 19