Bandpass Signalling

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    Chapter4

    Bandpass Signalling Bandpass Filtering and Linear Distortion

    Bandpass Sampling Theorem Bandpass Dimensionality Theorem

    Amplifiers and Nonlinear Distortion

    Total Harmonic Distortion (THD)

    Intermodulation Distortion (IMD)

    Huseyin Bilgekul

    Eeng360 Communication Systems IDepartment of Electrical and Electronic Engineering

    Eastern Mediterranean University

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    Bandpass Filtering and Linear DistortionEquivalent Low-pass filter: Modeling a bandpass filter by using an equivalent lowpass filter (complex impulse response)

    ])(Re[)( 11tjwcetgtv ])(Re[)( 22

    tjwcetgtv

    ])(Re[)( 11tjwcetkth

    *1 1( ) ( ) ( )

    2 2

    c cH f K f f K f f

    cc ffGffGfV *2

    1)( tj cetgtv )(Re

    )(1 tv

    )(2 tv

    )(1 th

    )( fH

    Input bandpass waveform

    Output bandpass waveform

    Impulse response of the bandpass filter

    Frequency response of the bandpass filter

    H(f) = Y(f)/X(f)

    Bandpass filter

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    Bandpass Filtering

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    Bandpass Filtering

    ;2

    1

    2

    1

    2

    112 tktgtg

    fKfGfG2

    1

    2

    1

    2

    112

    fHfVfV 12

    cc ffGffG *222

    1

    cccc ffKffKffGffG **

    112

    1

    2

    1

    cccc

    cccc

    ffKffGffKffG

    ffKffGffKffG

    **

    1

    *

    1

    *

    11

    4

    1

    ,0*1 cc ffKffG

    Theorem:

    g1(t)complex envelope of inputk(t)complex envelope of impulse response

    Also,

    Proof: Spectrum of the output is

    Spectra of bandpass waveforms are related to that of their complex enveloped

    But

    .0*1

    cc ffKffG

    cccccc ffKffGffKffGffGffG

    **

    11

    *

    22

    2

    1

    2

    1

    2

    1

    2

    1

    2

    1

    2

    1

    The complex envelopes for the input, output, and impulse response of

    a bandpass filter are related by

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    Bandpass Filtering

    fKfGfG2

    1

    2

    1

    2

    112

    Taking inverse fourier transform on both sides

    ;21

    2

    1

    2

    112 tktgtg

    Thus, we see that

    Any bandpass filter may be described and analyzed by using an equivalent low-pass

    filter.

    Equations for equivalent LPF are usually much less complicated than those for

    bandpass filters & so the equivalent LPF system model is very useful.

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    Linear Distortion

    AfH

    gT

    df

    fd

    2

    1

    0

    2 gfTf

    For distortionless transmission of bandpass signals, the channel transfer function

    H(f) should satisfy the following requirements: fjefHfH

    The amplitude response is constant

    A- positive constant

    The derivative of the phase response is constant

    Tgcomplex envelope delay dfTfHf 2)( )(fHf

    Integrating the above equation, we get

    Are these requirements sufficient for distortionless transmission?

    constantshiftphase0

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    Linear Distortion

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    gg fTjjfTj eAeAefH 22 00

    ttyttxtv cc sincos1 gfTje

    2

    002 sincos gcggcg TtTtAyTtTtAxtv

    ccgccg ftTtAyftTtAxtv sincos2

    dcgcc TfTf 20

    dcgdcg TtTtAyTtTtAxtv sincos2

    delayphaseTshiftphasecarrier;2 d cdcc fTff

    Linear Distortion

    The channel transfer function is

    fjefHfH

    If the input to the bandpass channel is

    Then the output to the channel (considering the delay Tg due to ) is

    Using 00

    2 gg TfTf

    dfTfHf 2)(

    Modulation on the carrier is delayed by Tg & carrier by Td

    Bandpass filterdelays input info by

    Tg , whereas the

    carrier by Td

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    Bandpass Sampling Theorem

    Ts Bf 2

    ttyttxtvcc

    sincos

    212 fffc

    n

    n bb

    bbc

    b

    c

    b fntf

    fntft

    f

    nyt

    f

    nxtv

    sinsincos

    bfnx bfny

    If a waveform has a non-zero spectrum only over the interval , where

    the transmission bandwidth BT

    is taken to be same as absolute BW, BT

    =f2

    -f1

    , then

    the waveform may be reproduced by its sample values if the sampling rate is

    21 fff

    Theorem:

    Quadrature bandpass representation

    Let fc be center of the bandpass:

    x(t) andy(t) are absolutely bandlimited to B=BT/2

    The sampling rate required to represent the baseband signal is Tb BBf 2

    Quadrature bandpass representation now becomes

    Where and samples are independent , two sample values

    are obtained for each value of n

    Overall sampling rate for v(t): Tbs Bff 22

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    Bandpass Dimensionality Theorem

    02 TN B T

    Assume that a bandpass waveform has a nonzero spectrum only over a frequency

    interval , where the transmission bandwidth BT is taken to be the absolute

    bandwidth given by BT=f2-f1 and BT

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    Received Signal Pulse

    tj cetgts Re

    tnthtstr

    tneTtAgtr cc ftjg Re

    tnetgtr tj c Re

    The signal out of the transmitter

    Transmission

    medium(Channel)

    Carrier

    circuits

    Signal

    processing

    Carrier

    circuits

    Signal

    processing

    Information

    minput m~)(

    ~ tg)(tr)(ts)(tg

    g(t)Complex envelope ofv(t)

    If the channel is LTI , then received signal + noise

    n(t)Noise at the receiver input

    Signal + noise at the receiver input

    Signal + noise at the receiver input

    - carrier phase shift caused by the channel, Tgchannel group delay.)( cf

    Again of the channel

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    Amplifiers

    Non-linear Linear

    Circuits with memory and circuits with no memory

    Memory - Present output value ~ function of present input + previous input values

    - contain L & C

    No memory - Present output values ~ function only of its present input values.

    Circuits : linear + no memoryresistive ciruits

    - linear + memoryRLC ciruits (Transfer function)

    Nonlinear Distortion

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    Nonlinear DistortionAssume no memoryPresent output as a function of present input in t domain

    tKvtv i0 K- voltage gain of the amplifier

    If the amplifier is linear

    In practice, amplifier output becomes saturated as the amplitude of

    the input signal is increased.

    0

    2

    2100

    n

    n

    inii vKvKvKKv

    output-to-input characteristic (Taylors expansion):

    Where

    0

    0

    !

    1

    iv

    n

    i

    n

    ndv

    vd

    nK

    0K

    ivK1

    2

    2 ivK

    - output dc offset level

    - 1st order (linear) term

    - 2nd order (square law) term

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    tAtvi 00

    sin

    tAK

    tAK 0

    2

    022

    002 2cos1

    2

    sin

    3032021010 3cos2cos)cos( tVtVtVVtvout

    100V

    THD(%)1

    2n

    2

    nV

    Nonlinear Distortion

    Let the input test tone be represented by

    Harmonic Distortion associated with the amplifier output:

    Then the second-order output term is

    In general, for a single-tone input, the output will be

    Vnpeak value of the output at the frequency nf0

    2

    2 ivK =

    To the amplifier input

    The Percentage Total Harmonic Distortion (THD) of an amplifier is defined by

    2nd Harmonic

    Distortion with2

    2

    02AK

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    Nonlinear Distortion

    Intermodulation distortion (IMD) of the amplifier:

    If the input (tone) signals are tAtAtvi 2211 sinsin

    Then the second-order output term is

    tAKttAAKtAK

    tAttAAtAKtsnAtAK

    222

    2221212122

    12

    2

    22

    221211

    22

    12

    2

    22112

    sinsinsin2sin

    sinsinsin2sinsin

    IMDHarmonic distortion at 2f1 & 2f2

    Second-order IMD is:

    212121221212 coscossinsin2 tAAKttAAK

    l

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    Nonlinear Distortion

    Third order term is

    )sinsinsin3

    sinsin3sin(

    sinsin

    2

    33

    22

    2

    1

    2

    21

    21

    2

    2

    2

    11

    33

    13

    3

    22113

    3

    3

    tAttAA

    ttAAtAK

    tAtAKvK i

    ttAAKttAAK 1222

    1321

    2

    2

    2

    13 2cos1sin2

    3sinsin3

    tttAAK 212122213 2sin2sin

    21sin

    23

    tttAAK

    ttAAK

    12121

    2

    213

    2

    2

    1

    2

    213

    2sin2sin2

    1sin

    2

    3

    sinsin3

    The third term is

    The second term (cross-product) is

    Intermodulation terms at nonharmonic frequencies

    For bandpass amplifiers, wheref1 &f2 are within the pasband,f1 close tof2,

    the distortion products at 2f1+f2 and 2f2+f1 ~ outside the passband

    Main Distortion Products