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Egali Satie 1

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    1

    TELIN

    1

    Equalization in digital communication

    Prof. Marc Moeneclaey

    TELIN

    2

    Intersymbol interference (1)

    Dispersive (frequency-selective) channel

    Hch(f) : channel transfer function (often unknown)

    w(t) : AWGN

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    TELIN

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    Intersymbol interference (2)

    Linear modulation

    r t E a p(t mT w ts mm

    ( ) ) ( )= +!

    s t E a p t kTs k trk

    ( ) ( )= !

    P(f) = Hch(f)Ptr(f)

    TELIN

    4

    Intersymbol interference (3)

    Receiver

    z(kT E a g E a g mT n kTs k s k mm

    ) ( ) ( ) ( )= + +

    !00useful term

    intersymbol interferenceterm (ISI)

    noise term! "# $#

    ! "### $###

    !"$

    G(f) = H(f)P(f) =H(f)Hch(f)Ptr(f)

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    5

    Intersymbol interference (4)

    Receiver (cont.)

    P

    P

    N

    E gH f df noise

    useful s

    =

    +

    "0 22

    0| ( ) || ( )|

    is minimum for H(f) = Ptr*(f)Hch*(f) (matched filter)

    is zero when G(f) fulfills the Nyquist criterion

    P

    P g g mTISI

    useful m=

    !

    1

    0 22

    0| ( ) || ( )|

    In general, the matched filter does not yield zero ISI

    TELIN

    6

    Intersymbol interference (5)

    Eye diagram

    Eye diagram shows Re ( )a g t mTmm

    #

    $%%

    &

    '((

    !

    as a function of time for all possible data sequences

    Eye diagram is periodic in t with period T

    M-PAM : g(t) has duration LT

    )eye diagram has ML lines per period

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    TELIN

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    Intersymbol interference (6)

    Eye diagram 20 % cosine rolloff pulse

    1 2

    2

    1

    0

    -1

    -20

    t/T

    TELIN

    8

    Intersymbol interference (7)

    Eye diagram 50 % cosine rolloff pulse

    2

    1

    0

    -1

    -20 1 2

    t/T

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    5

    TELIN

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    Intersymbol interference (8)

    Eye diagram 100 % cosine rolloff pulse

    0 1 2

    2

    1

    0

    -1

    -2

    t/T

    TELIN

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    Intersymbol interference (9)

    Scatter diagram

    For a given sampling instant kT, scatter diagram shows

    in the complex plane for all possible data sequences

    Constellation of M points : g(t) has duration LT

    )scatter diagram has ML points

    a g kT mT a g a g mTm km

    k mm

    ( ) ( ) ( ) = +! !

    0

    0

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    Intersymbol interference (10)

    Scatter diagram (cont.)

    50 % cosine rolloff pulse, T/10 sampling error, 8-PSK

    -1 0 1

    1

    0

    -1

    Im[zk]

    Re[zk]

    TELIN

    12

    Intersymbol interference (11)

    Intersymbol interference power

    ( )g mT j fmTT

    G f k

    TG e

    m

    fldj fT

    k

    ( ) exp( ) = +*+, -

    ./ =

    =

    +

    =

    +

    ! !2 1 2

    folding of G(f) yields Gfld(exp(j2fT)) which is

    periodic in f with period 1/T

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    TELIN

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    Intersymbol interference (12)

    Notation :

    ( ) ( )X e T X e df j fT j fT

    T

    T2 2

    1

    2

    1

    2

    =

    "

    Intersymbol interference power (cont.)

    (averaging over interval (-1/(2T), 1/(2T))

    TELIN

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    Intersymbol interference (13)

    Intersymbol interference power (cont.)

    g mT G e j fmTfldj fT( ) ( ) exp( )= 2 2

    ( )g G efld j fT( )0 2=

    ( )| ( )|g mT G em

    fldj fT2 2 2! =

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    TELIN

    15

    Intersymbol interference (14)

    Intersymbol interference power (cont.)

    ( ) ( )

    ( )

    ( )( )

    P

    P

    G e G e

    G e

    e

    e

    ISI

    useful

    fldj fT

    fldj fT

    fldj fT

    fld

    j fT

    fldj fT

    =

    =

    2 2 2

    2 2

    2

    2 2

    power of fluctuation of G about its mean

    mean of G

    TELIN

    16

    Intersymbol interference (15)

    Intersymbol interference power (cont.)

    Zero ISI is obtained when

    ( )G efldj fT2

    =constant (Nyquist criterion)

    Equalization : elimination (or reduction) of ISI

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    TELIN

    17

    Intersymbol interference (16)

    Example

    Transmit pulse : P(f) = 50% square-root cosine rolloff

    Channel : direct path + reflection

    H f b j f ch ( ) exp( )= + 1 2

    reflection : delay , magnitude b (0

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    TELIN

    19

    Intersymbol interference (18)

    Example (cont.)

    0.01

    0.1

    1

    10

    -0.5 -0.3 -0.1 0.1 0.3 0.5

    fT

    Grec,f

    ld(exp(j2fT))

    0 dB ripple

    5 dB ripple10 dB ripple

    20 dB ripple

    Transmitter :50% sqrt cosinerolloff pulse

    Channel : reflectionwith 3.5T delay

    Hrec(f) : matched to

    transmit pulse +channel

    Folded Fourier transformmatched filter output pulse

    TELIN

    20

    Intersymbol interference (19)

    Example (cont.)

    -0.20

    0.00

    0.20

    0.40

    0.60

    0.80

    1.00

    -10 -5 0 5 10

    t/T

    grec

    (t)

    0 dB ripple

    5 dB ripple

    10 dB ripple

    20 dB ripple

    Transmitter :50% sqrt cosinerolloff pulse

    Channel : reflectionwith 3.5T delay

    Hrec(f) : matched totransmit pulse +channel

    Matched filteroutput pulse

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    TELIN

    21

    Intersymbol interference (20)

    Example (cont.)

    Eye diagram at matched filter output (0 dB ripple)

    2

    1

    0

    -1

    -20 1 2

    t/T

    TELIN

    22

    Intersymbol interference (21)

    Example (cont.)

    Eye diagram at matched filter output (5 dB ripple)

    2

    0

    -20 1 2

    t/T

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    TELIN

    23

    Intersymbol interference (22)

    Example (cont.)

    Eye diagram at matched filter output (10 dB ripple)

    0 1 2

    t/T

    0

    -3

    3

    TELIN

    24

    Intersymbol interference (23)

    Example (cont.)

    Ripple [dB] P

    PISI

    useful

    [dB]

    0 -

    5 -9.96

    10 -6.03

    15 -4.84

    20 -4.47

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    Zero-forcing linear equalizer (ZF-LE) (1)

    Decompose the receive filter as

    ( )H f H f C j fTrec( ) ( ) exp( )=

    analogprefilter

    linear equalizer!"$ ! "## $##

    2

    ( )C e c j fmTj fT mm

    22

    =

    =

    +! exp( )

    cm: m-th equalizer tap

    Equalizer structure

    TELIN

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    Zero-forcing linear equalizer (2)

    z(t c x(t mTmm

    ) )=

    =

    +

    ! z(kT c x(kT mTmm

    ) )=

    =

    +

    !

    Equalizer structure (cont.)

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    TELIN

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    Zero-forcing linear equalizer (3)

    z(kT c x(kT mTmm

    ) )=

    =

    +

    !

    Equalizer structure (cont.)

    TELIN

    28

    Zero-forcing linear equalizer (4)

    Equalizer structure (cont.)

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    Zero-forcing linear equalizer (5)

    ( )G f P f H f H f C j fTtr ch recf

    ( ) ( ) ( ) ( ) exp( )

    ( )

    =

    G periodic in f,period 1/T

    rec

    ! "### $### ! "## $##

    2

    ( ) ( ) ( )G e G e C efld j fT rec fld j fT j fT2 2 2 = ,

    The zero-forcing linear equalizer should be selected such

    that the resulting Gfld(exp(j2fT)) is constant (equalized),

    in order to eliminate all ISI

    Equalizer transfer function

    TELIN

    30

    Zero-forcing linear equalizer (6)

    The ZF-LE is not unique : different choice of Hrec(f)

    yields a different equalizer and a different Pnoise/Puseful

    ( ) ( )) =C e

    G e

    j fT

    rec fld j fT

    2

    2

    1

    ,

    Equalizer transfer function (cont.)

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    Zero-forcing linear equalizer (7)

    Optimum zero-forcing linear equalizer

    H(f) minimizes Pnoise/Puseful, under the restriction

    of zero ISI

    Solution : ( )H f P f H f C j fTtr ch( ) ( ) ( ) exp( )* *

    =

    matched filter linear equalizer! "## $## ! "## $##

    2

    G f P f H f rec tr ch( ) ( ) ( )= 2

    ( )( )

    C eG e

    j fT

    rec fldj fT

    22

    1

    =

    ,

    TELIN

    32

    Zero-forcing linear equalizer (8)

    Noise enhancement

    As H(f) for the optimum ZF-LE is different from the

    matched filter, the equalizer eliminates ISI at the

    expense of an increased noise level

    P

    P

    P

    Pnoise

    useful matchedfilter

    noise

    useful ZF LE

    *

    +,

    -

    ./

    ) = ) =

    2 2 0

    G z g mT z Gz

    rec fld recm

    m

    rec fld, ,*

    ( ) ( )*

    = = *+, -

    ./

    =

    +

    ! 1

    poles and zeroes of Grec,fld(z) occur in pairs :

    zpand 1/zp* (poles), znand 1/zn* (zeroes)

  • 8/12/2019 Eg