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  • Chapter 2 Continuous-Wave Modulation 2 1 Introd ction2.1 Introduction

    1

  • 2.2 Consider a carrier (2.1) )2cos()( tfAtc cc π=

    1+k m(t) S(t) X

    1+kam(t) S(t)

    The output of the modulator Accos(2πfct)

    Where m(t) is the baseband signal , ka is the amplitude sensitivity. [ ] (2.2) )2cos()(1)( tftmkAts cac π+=

    ( ) g a p y

    (2 4)2 (2.3) t allfor ,1 )( .1

    f tmka <

    )( offreqency hightest theis where (2.4) .2

    tmW Wfc >>

    2

  • Recall (2.2) )2cos()()2cos()( tftmkAtfAts caccc ππ +=

    1 [ ]

    [ ])()(1)2cos()(

    )()( 2 1)2cos(

    ffMffMtftm

    fffftf ccc

    ++⇔

    ++−⇔

    π

    δδπ

    [ ]

    [ ] [ ] (2.5) )()( 2

    )()( 2

    )(

    )()( 2

    )2cos()(

    ffMffMAkffffAfs

    ffMffMtftm

    cc ca

    cc c

    ccc

    ++−+++−=

    ++−⇔

    δδ

    π

    [ ] [ ] )( of Transform Fourier theis )( where

    22 tmfM

    1 Negative frequency component of m(t) becomes visible1.Negative frequency component of m(t) becomes visible. 2.fc-W < M(f) < fc lower sideband

    f < M(f) < f +W upper sideband 3

    fc < M(f) < fc+W upper sideband 3.Transmission bandwidth BT=2W

  • Virtues and Limitations of Amplitude Modulation Transmitter

    Receiver

    Major limitations

    4

    Major limitations 1.AM is wasteful of power. 2.AM is wasteful of bandwidth.

  • 2 3 Linear Modulation Schemes2.3 Linear Modulation Schemes

    Linear modulation is defined by

    (2.7) )2sin()()2cos()()( −= tftstftsts cQcI ππ

    componentQuadrature)( component phase-In)(

    = =

    ts ts

    Q

    I

    Three types of linear modulation:

    componentQuadrature)(tsQ

    1.Double sideband-suppressed carrier (DSB-SC) modulation 2 Single sideband (SSB) modulation2.Single sideband (SSB) modulation 3.Vestigial sideband (VSB) modulation

    5

  • Notes: 1 sI(t) is solely dependent on m(t)1.sI(t) is solely dependent on m(t) 2.sQ(t)is a filtered version of m(t).

    Th l difi i f ( ) i l l d ( ) 6

    The spectral modification of s(t) is solely due to sQ(t).

  • Double Sideband-Suppressed Carrier (DSB-SC) Modulation (2 8))2cos()()( tftmAts π= (2.8) )2cos()()( tftmAts cc π=

    The Fourier transform of S(t) is 7

    ( ) (2.9) )()(

    2 1)( ⎥⎦⎤⎢⎣⎡ ++−= ccc ffMffMAfs

  • Coherent Detection (Synchronous Detection)

    The product modulator output is )()2cos(' )( tstfAtv cc ϕπ +=

    (2 10))()('1)()4('1 )()2cos()2cos('

    tAAttfAA

    tmtftfAA cccc

    φφ

    φππ

    ++

    +=

    Let V(f) be the Fourier transform of v(t)

    (2.10) )()cos(' 2

    )()4cos(' 2

    tmAAtmtfAA ccccc φφπ ++=

    1

    filtered out

    (2.11) )( cos' 2 1)(0 tmAAtv cc φ= (Low pass filtered)

    8

  • Costas Receiver I-channel and Q-channel are coupled together toI channel and Q channel are coupled together to form a negative feedback system to maintain synchronization

    0φ ≈

    2 2 2 21 1cos sin ( ) ( )sin(2 ) 4 8

    c cA m t A m t

    φ

    φ φ φ=

    The phase control signal ceases with modulation

    1 4

    2 2( ) (sin2 2 )cA m t φ φ φ≈ ≈ The phase control signal ceases with modulation.

    (multiplier +(multiplier + very narrow band LF)

    9

  • Quadrature-Carrier Multiplexing (or QAM) T DSB SC i l h h lTwo DSB-SC signals occupy the same channel bandwidth, where pilot signal (tone ) may be

    d dneeded.

    )2sin()()2cos()()( 21 tftmAtftmAts cccc ππ += )2sin()()2cos()()( 21 tftmAtftmAts cccc ππ +

    10

  • Single-Sideband Modulation (SSB) The lower sideband and upper sideband of AM signal contain same information . The frequency-discrimination method consists of a

    d d l (DSB SC) d b d filproduct modulator (DSB-SC) and a band-pass filter. The filter must meet the following requirements: a The desired sideband lies inside the passbanda.The desired sideband lies inside the passband. b.The unwanted sideband lies inside the stopband. c.The transition band is twice the lowest frequency of

    the message.

    11 To recover the signal at the receiver, a pilot carrier or a stable oscillator is needed (Donald Duck effect ).

  • Vestigial Sideband Modulation (VSB) When the message contains near DC component

    The transition must satisfy

    liihb h 1)()(.a ffHffH cc =++−

    (2.13) for 1)()( :linearisresponsephase b.The

    W fWffHffH cc ≤≤−=++−

    12 (2.14) fWB νT +=

  • Consider the negative frequency response:g q y p ( )H f

    c vf f− cf c vf f+ cf W+c v f f− +cf−c vf f− −cf W− −

    Here, the shift response │H(f-fc)│ is ( )cH f f−

    f0 2 c vf f− 2 cf 2 c vf f+ 2 cf W+v f0vf−W

    13

  • and │H(f+fc)│ is ( )H f f+( )cH f f+

    vf− 0 vf W 2 c vf f− +2 cf−2 c vf f− −2 cf W− −

    14

  • So, we get │H(f-fc)│ +│ H(f+fc)│ is ( )H f f( )cH f f−

    2 c vf f− 2 cf 2 c vf f+v f0vf−W−

    ( )H f f+( )cH f f+

    vf− 0 vf W 2 c vf f− +2 cf−2 c vf f− −

    15

  • Consider –W

  • (2.15) )2sin()(' 2 1)2cos()(

    2 1)( tftmAtftmAts cccc ππ ±=

    ± corresponds to upper or lower sideband 22

    ff

    HQ(f) m(t) m’(t)

    [ ] (2.16) for )()( )( W f WffHffHjfH ccQ ≤≤−+−−=

    17

  • Television Signals (NTSC)

    18

  • 2.4 Frequency Translation

    cos(2 )A f tπl l

    Up conversion f2=f1+fl , fl=f2-f1

    Down conversion f f f f f f

    19

    f2=f1-fl , fl=f1-f2

  • 2.5 Frequency-Division Multiplexing (FDM)

    20

  • 2.6 Angle Modulation Basic Definitions: Better discrimination against noise and interference g (expense of bandwidth).

    [ ] (2.19))(cos)( tAts ic θ= The instantaneous frequency is

    [ ] (2.19) )(cos)( tAts ic θ

    )(lim)( Δti tftf =

    2 )()(lim

    )()(

    Δ0Δ

    ii

    t

    tti

    t ttt

    ff

    π θθ

    ⎥⎦ ⎤

    ⎢⎣ ⎡

    Δ −Δ+

    = →

    (2.21) )( 2 1

    20Δ

    i

    t

    dt td

    t θ

    π

    π

    =

    ⎥⎦⎢⎣ Δ→

    (2 22)2)( is )( carrier, dunmodulate anFor

    2 i

    tft t

    dt

    φπθ θ

    π

    += 21constant is where

    (2.22) 2)(

    c

    cci tft φ

    φπθ +=

  • 1. Phase modulation (PM) )(2)( tmktft pci += πθ

    [ ] (2 23))(2cos modulator theofy sensitivit phase :

    )()(

    tmktfAs(t) k

    f p

    pci

    += π

    πθ

    2. Frequency Modulation (FM) [ ] (2.23) )(2cos tmktfAs(t) pcc += π

    (2 24)( ) ( )f t f k m t= + (2.24)

    π π 0

    ( ) ( )

    ( ) 2 2 ( )

    i c f

    t

    i c f

    f t f k m t

    t f t k m dθ τ τ

    = +

    = + ∫ (2.25)

    π π (2.26)

    :frequency sensitivity of the modulator 0

    cos 2 2 ( ) t

    c c fs(t) A f t k m d

    k

    τ τ⎡ ⎤= +⎢ ⎥⎣ ⎦∫ :frequency sensitivity of the modulator

    compare (2.23) and (2.26)

    f

    p

    k

    k m'⇒ π 0

    2 ( ) t

    f(t) k m dτ τ= ∫

    22 generating FM signal generating PM signal

  • 2.7 Frequency Modulation FM is a nonlinear modulation process , we can not apply Fourier transform to have spectral analysis directly.Fourier transform to have spectral analysis directly.

    1.Consider a single-tone modulation which produces a b d FM (k i ll)narrowband FM (kf is small)

    2.Next consider a single-tone and wideband FM (kf is large)

    (2 27))2()(l t tfAt (d t i i ti ) )2cos( )(

    (2.27) )2cos()(let

    mmfci

    mm

    tfAkftf tfAtm

    += =

    π π (deterministic)

    d i tifΔ (2.28) )2cos( mc

    f

    Akf tfff Δ+= π

    23 deviationfrequency :Δ mf Akf =

  • )(2)( (2.25), Recall 0

    ττπθ dft t

    ii = ∫ (2.30) )2sin(2 π tf

    f ftπf mc

    Δ +=

    (2.31) index Modulation β f fm

    Δ =

    (2.32) )2sin(2)(

    ( )

    πβθ

    β

    tftπft f

    mci

    m

    += [ ]

    dill hiFMN b d (2.33) )2sin(2cos)( ( ))()(

    β πβπ

    β tftfAts

    ff

    mcc

    mi

    +=(2.19) =>

    radian. onenlarger thais, FM Wideband ra