Ece 310 Notes

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  • ECE 310

    Discrete and Continuous Signals

    and Systems

  • Outline

    Modulation

    Linearity

    Time and Frequency shifting

    Time scaling

    Time reversal

    Conjugation

  • Modulation

    Vary a carrier signal (ie cos wave) to

    contain information.

    AM/FM Radios.

  • { } { }

    ( ) ( )

    ( ) ( )00

    0

    0

    F2

    1F

    2

    1

    ))((2

    1

    )cos()(

    )()cos()(

    00

    ++=

    +=

    =

    =

    +

    dteetf

    dtettf

    tmttf

    tjtj

    tj

    FF

  • Resulting the original signal to shift by 0

    Allows communication to be more

    efficient.

    High Frequency

    Shorter Antenna (Cellphones)

    Low Frequency

    Under water applications

    fc =

  • Demodulation

    Multiply by the carrier signal again!

    )2cos()(2

    1)(

    2

    12

    )2cos(1)(

    )(cos)()cos()(

    )cos()()(

    0

    0

    02

    0

    ttftf

    ttf

    ttfttm

    ttftm

    +=

    +=

    =

    =

    The result is the original signal plus a modulated signal at twice the frequency.

    We can use a low-pass filter to recover the original signal.

  • Wireless Communication

    Modulation

    Modulation at transmitter

    Multiply by cos (t) Demodulation at receiver

    Multiply by cos (t) again LP filter (chap. 7)

    Requires same frequency

    Additional details of modulation will be covered in ECE311

  • Additional Notes on

    Demodulation

    Demodulation requires the multiplication by cos (t)

    If c Broadcast band

    Tuning an AM radio is selecting the

    cos (t) frequency!

  • Linearity

    )()()()(

    )()()(

    )()(

    )()(

    HbGaFtf

    thbtgatf

    Hth

    Gtg

    +=

    +=

    F

    F

    F

  • )()(

    )()(

    ))()(()()}({

    )()()(

    HbGa

    dtethbdtetga

    dtethbtgaFtf

    thbtgatf

    tjtj

    tj

    +=

    +=

    +==

    +=

    F

  • Time Shift

    )()}({

    )]([2

    1)(

    2

    1)(

    )(2

    1)(

    0

    00

    0

    )(0

    Fettf

    deFedeFttf

    deFtf

    tj

    tjtjttj

    tj

    =

    ==

    =

    F

    Inverse Fourier transform

    Original Fourier Transform with a phase-shift

  • Frequency Shift

    Multiplication of a sinusoid will result a shift

    in frequency spectrum.

    As shown in modulation section

    )()( 00 mFetf tj =

  • Modulation & Frequency Shift

    Example

    )()(

    )(})({

    0)( 0

    00

    +==

    =

    +

    Fdtetf

    dteetfetf

    tj

    tjtjtjF

    )2

    3sinc(}){rect( 3

    += tjetF

    Example }){rect( 3 tjet F

    )()( 00 + Fetf tj)

    2sinc()rect(

    t

  • Time Scaling

    If a signal is expanded in time, the

    frequency spectrum will contract.

    = dteatfatf tj)()}({F at=

    )(1

    )(1

    )()}({)(

    aF

    a

    defa

    adeff

    a

    j

    aj

    =

    =

    =

    F

    Consider the case where aaaa is negative

    We may conclude

    )(1

    )(a

    Fa

    atfF

  • Time Reversal

    Consider the case of a = -1

    )()(

    )()1

    (1

    1)()(

    =

    =

    Ftf

    FFtfatf

    F

    F

  • Conjugation

    )()()()( ** FtfFtfFF

    )(])([

    )()}({

    **

    **

    ==

    =

    Fdtetf

    dtetftf

    tj

    tjF

  • Conjugation

    )()()()( ** FtfFtfFF

    Example

    2)(

    2

    )(2

    1)(

    2

    1

    ))((2

    1)}({

    )(2

    )(2

    00

    0

    0

    0

    0

    )(00

    00

    0

    0

    tjtj

    tjtj

    tj

    tj

    tj

    ed

    e

    dede

    de

    e

    e

    =+=

    +=+=

    +=

    1-F

  • Fourier Transform Properties

    Linearity

    Modulation & Frequency Shift

    Time Shift

    Time scale & Time Reversal

    Conjugation

    )()( 00 mFetf tj

    )()()(

    )()()(

    HbGaFthbtgatf

    +=+=

    )()}({ 00 Fettf tj=F

    ( ) ( )000 F21

    F2

    1)cos()( ++ttf

    )(1

    )(a

    Fa

    atfF )()( Ftf

    F

    )()()()( ** FtfFtfFF

  • Linearity + Time Shift + Scaling

    Example

    Show that the FT of rect(t-0.5)+rect(t+0.5) is the same as the FT of

    rect(t/2)

    ))(2

    sinc()2

    sinc()2

    sinc(

    )}2

    1rect()

    2

    1({

    )2

    rect()2

    1rect()

    2

    1rect(

    2222

    jjjjeeee

    ttrect

    ttt

    +=+=

    ++

    =++

    F

  • xxx

    ee

    eex

    jj

    jxjx

    2sin2

    1cossin

    )sinc(2sin

    2

    2

    )2

    2sin(21

    2

    2

    )2

    cos()2

    sin(2

    )2

    cos()2

    sinc(2))(2

    sinc(

    2)cos(

    22

    =

    ==

    ==

    =+

    +=

  • )(1

    )(a

    Fa

    atfF

    )sinc(2)22

    (sin

    211

    )}2

    {rect(

    2

    1

    )2

    1rect()

    2rect(

    ==

    =

    =

    ct

    a

    tt

    F

  • Linearity + Scaling Example

    })(

    {dt

    tdF

    })(

    {dt

    tdF

    dt

    td )(

    Find

    )(t

  • 25.0

    25.0

    )4

    sinc(2

    1

    5.0

    25.0rect

    )4

    sinc(2

    1

    5.0

    25.0rect

    )4

    sinc(2

    1

    5.0rect

    21

    41

    rect2

    21

    41

    rect2})(

    {

    21

    41

    rect2

    21

    41

    rect2)(

    j

    j

    et

    et

    t

    tt

    dt

    td

    tt

    dt

    td

    +

    +

    +=

    +=

    FF

    )4

    (sinc2

    )4

    sinc(4

    2)

    4sinc(

    )4

    sin(2)4

    sinc()4

    sinc(

    )4

    sinc()4

    sinc(})(

    {

    2

    44

    44

    jj

    jee

    eedt

    td

    jj

    jj

    ==

    =

    =

    =

    F

  • Differentiation

    ==

    ==

    )}({)(2

    )(2

    1)(

    2

    1)(

    )()(

    1

    FjdeFj

    dedt

    dFdeF

    dt

    d

    dt

    tdf

    Fjdt

    tdf

    tj

    tjtj

    F

  • Example

    )()( Fj

    dt

    tdf

    dt

    td )(

    )4

    (sinc2

    1)( 2

    t

    Example: Find the FT of

    )}4

    (sinc2

    1{

    )( 2 jdt

    td Which agrees with the result in p24

  • Cumulative Integral in Time

    )()0()(

    )( F

    j

    Fdf

    t+

    )u()(11

    )( tj

    dt

    +

    Example

    Cumulative Area moving from left to right

  • Cross Correlation

    )()()}({

    )()()(

    *

    *

    YXt

    dtyxt

    xy

    xy

    =

    +=

    F

    2* )()()()(

    )()(

    when

    XXXt

    tytx

    xx =

    = Autocorrelation