E85.2607: Lecture 10 -- Modulation - Columbia ronw/adst-spring2010/lectures/ modulation Ring...

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Transcript of E85.2607: Lecture 10 -- Modulation - Columbia ronw/adst-spring2010/lectures/ modulation Ring...

  • 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