eece 522 notes_32 ch_13d

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  • 8/8/2019 EECE 522 Notes_32 Ch_13D

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    13.8 Signal Processing Examples

    Ex. 13.3 Time-Varying Channel Estimation

    Tx RxDirect Path

    Multi Path

    v(t)y(t)

    =T

    t dtvhty

    0

    )()()( Tis the maximum delay

    Model using a time-varying D-T FIR system

    Channel changes with time if:

    Relative motion between Rx, Tx Reflectors move/change with time

    ==

    p

    kn knvkhny 0 ][][][ Coefficients change at each n

    to model time-varying channel

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    In communication systems, multipath channels degrade performance

    (Inter-symbol interference (ISI), flat fading, frequency-selective

    fading, etc.)

    Need To: First estimate the channel coefficients

    Second Build an Inverse Filter or Equalizer

    2 Broad Scenarios:

    1. Signal v(t) being sent is known (Training Data)2. Signal v(t) being sent is not known (Blind Channel Est.)

    One method for scenario #1 is to use a Kalman Filter:

    State to be estimated is h[n] = [hn[0] hn[p]]T

    (Note: h here is no longer used to notate the observation model here)

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    Need State Equation:

    Assume FIR tap coefficients change slowly

    ][]1[][ nnn uAhh +=

    Assumed Known

    That is a weakness!!

    Assume FIR taps are uncorrelated with each other

    A, Q , Ch , are Diagonal

    cov{h[-1]} = M[-1|-1]

    cov{u[n]}

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    Have measurement model from convolution view:

    ][][][][

    0

    nwknvkhnxp

    k

    n += =

    zero-mean,WGN, 2

    Knowntraining signal

    Need Observation Equation:

    ][][ nwnx nT += hv

    Observation Matrix

    is made up of thesamples of the known

    transmitted signal

    State Vector

    is the filtercoefficients

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    Simple Specific Example:p = 2 (1 Direct Path, 1 Multipath)

    =

    = 0001.00

    00001.0

    999.00

    099.0

    QA][]1[][ nnn uAhh +=

    Q = cov{u[n]}Typical Realization of Channel Coefficients

    Book doesnt state how

    the initial coefficients

    were chosen for thisrealization

    Note: hn[0] decays faster

    and that the random

    perturbation is small

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    Known Transmitted Signal

    Noise-Free Received Signal

    Noisy Received SignalThe variance of the noise in the

    measurement model is 2 = 0.1

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    Estimation Results Using Standard Kalman Filter

    Initialization: 1.0100]1|1[]00[]1|1[ 2 === IMh T

    Chosen to reflect that little

    prior knowledge is known

    In theory we said that we

    initialize to the a priori

    mean but in practice it is

    common to just pick somearbitrary initial value and

    set the initial covariance

    quite high this forces the

    filter to start out trustingthe data a lot!

    Transient due to wrong IC

    Eventually Tracks Well!!hn[0]

    hn[1]

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    Kalman Filter Gains

    Decay down relies more on model

    Gain is zero when signal is noise only

    Kalman Filter MMSE

    Filter Performance

    improves with time

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    Example: Radar Target Tracking

    State Model: Constant-Velocity A/C Model

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

    ][]1[][

    ][

    ][

    0

    0

    ]1[

    ]1[

    ]1[

    ]1[

    1000

    0100

    010

    001

    ][

    ][

    ][

    ][

    n

    y

    x

    n

    y

    x

    y

    x

    n

    y

    x

    y

    x

    nu

    nu

    nv

    nv

    nr

    nr

    nv

    nv

    nr

    nr

    usAs

    +

    =

    ==

    2

    2

    000

    000

    0000

    0000

    }cov{

    u

    u

    uQ

    +

    +

    =

    ][

    ][

    ][

    ][

    tan

    ][][

    ][1

    22

    nw

    nw

    nr

    nr

    nrnr

    nR

    x

    y

    yx

    x

    Observation Model: Noisy Range/Bearing Radar Measurements

    For this simple example. assume:

    ==

    2

    2

    0

    0}cov{

    RwC

    Velocity perturbations due to wind, slight speed corrections, etc.Velocity perturbations due to wind, slight speed corrections, etc.

    in radians

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    Extended Kalman Filter Issues

    1. Linearization of the observation model (see book for details) Calculate by hand, program into the EKF to be evaluated each iteration

    2. Covariance of State Driving Noise

    Assume wind gusts, etc. are as likely to occur in any direction w/ samemagnitude ! model as indep. w/ common variance

    Need the following:

    ==

    2

    2

    000

    000

    0000

    0000

    }cov{

    u

    u

    uQ

    u = what??? Note: ux[n]/ = acceleration from n-1 to n

    So choose u in m/s so that u/ gives areasonable range of accelerations for thetype of target expected to track

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    3. Covariance of Measurement Noise The DSP engineers working on the radar usually specify this or build

    routines into the radar to provide time updated assessments of

    range/bearing accuracy

    Usually assume to be white and zero-mean

    Can use CRLBs for Range & Bearing

    " Note: The CRLBs depend on SNR so the Range & Bearing measurement

    accuracy should get worse when the target is farther away Often assume Range Error to be Uncorrelated with Bearing Error

    " So use C[n] = diag{R2[n],

    2[n]}

    But best to derive joint CRLB to see if they are correlated

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    4. Initialization Issues

    Typically Convert first range/bearing into initial rx & ry values

    If radar provides no velocity info (i.e. does not measure Doppler) canassume zero velocities

    Pick a large initial MSE to force KF to be unbiased

    " If we follow the above two ideas, then we might pick the MSE forrx

    & ry

    based on

    statistical analysis of conversion of range/bearing accuracy into rx & ryaccuracies

    Sometimes one radar gets a hand-off from some other radar or sensor

    "The other radar/sensor would likely hand-off its last track values so usethose as ICs for the initializing the new radar

    " The other radar/sensor would likely hand-off a MSE measure of the quality its

    last track so use that as M[-1|-1]

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    State Model Example Trajectories: Constant-Velocity A/C Model

    -20 -15 -10 -5 0 5 10 15-10

    -5

    0

    5

    10

    15

    20

    25

    X position (m)

    Yposition(m)

    Radar

    Red Line is Non-Random

    Constant Velocity Trajectory

    m/s2.0]1[2.0]1[

    m5]1[10]1[

    )/sm001.0(m/s0.0316

    sec1

    222

    ==

    ==

    ==

    =

    yx

    yx

    uu

    vv

    rr

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    0 10 20 30 40 50 60 70 80 90 1000

    5

    10

    15

    20

    Sample Index n

    RangeR(meters)

    0 10 20 30 40 50 60 70 80 90 100-50

    0

    50

    100

    B

    earing(

    degrees)

    Sample Index n

    Observation Model Example Measurements

    Red Lines are Noise-Free Measurements

    )rad01.0(deg5.7rad0.1

    )m1.0(m0.3162

    22

    22

    ===

    ==

    R

    RR

    In reality, these would get

    worse when the target is

    far away due to a weaker

    returned signal

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    If we tried to directly convert the noisy range and bearing

    measurements into a track this is what wed get.Not a very accurate track!!!! ! Need a Kalman Filter!!!But Nonlinear Observation Model so use Extended KF!

    Radar

    Note how the track gets worse whenfar from the radar (angle accuracy

    converts into position accuracy in a way

    that depends on range)

    Measurements Directly Give a Poor Track

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    Extended Kalman Filter Gives Better Track

    Note: The EKF was run with the correctvalues forQ and C

    (i.e., the Q and C used to simulate the trajectory and measurementswas used to implement the Kalman Filter)

    Initialization: s[-1|-1] = [5 5 0 0]T M[-1|-1] = 100I

    Picked Arbitrarily

    Radar

    Initialization

    Set large to assert that

    little is known a priori

    After about 20 samplesthe EKF attains trackeven with poor ICs andthe linearization.

    Track gets worse nearend where measurementsare worse MSE show obtain track

    and show that things getworse at the end

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    MSE Plots Show Performance

    First a transient where

    things get worse

    Next the EKF seems to

    obtain track

    Finally the accuracy degrades

    due to range magnification of

    bearing errors