E85.2607: Lecture 10 -- Modulation - Columbia Universityronw/adst-spring2010/lectures/lecture10.pdfRing modulation Ring Modulation freq amp f c f c -f m f c + f m E85.2607: Lecture

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 (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 2010-04-15 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 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 + 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

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(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.

E85.2607: Lecture 10 Modulation 2010-04-15 7 / 19

Effect of modulation index ()

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(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 2010-04-15 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 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 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-

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(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 2010-04-15 15 / 19

FM modulation index

y [n] = cos(2 220 n + sin(2 440 n))

Modulation index

Time-domain Frequency-domain FM signals theoretically have infinite bandwidth 2( + 1) audible sidebands

E85.2607: Lecture 10 Modulation 2010-04-15 16 / 19

http://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/upload/1/12/Frequencymodulationdemo-cf220mf440.ogg

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