Extended Kalman Filter and its application in wireless communication

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Extended Kalman Filter and its application in
wireless communication

• Lei You
• Signals Systems
• Dept. of Engineering Sciences
• Uppsala University

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Nonlinear state-space model
fi(.), hi(.) and gi(.) are nonlinear functions of
xi.Linearize the state-space equations by
making the first-order Taylor expansion around
the current estimate xii-1 and xii
Where,
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Linearized state-space model
Now it become a linear state-space model of xi.
we can estimate the state xi recursively using
the solution of a normal linear state-space model
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Extended Kalman Filter

• NOTE
• Pi, Kf,i can not be precomputed due to the
  dependence of Fi , Hi , Gi on the measurement and
  current estimate.
• Pi, Kf,i depend nonlinearly on the measurement y.

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Application DSSS ( direct sequence spread
spectrum) communication

• DSSS communication is widely applied in various
  wireless communication scenarios in which both
  severe frequency selective multi-path and
  narrowband interference may occur simultaneously.
• To receive DSSS signal, we should know
• code delay to obtain pseudo-noise (PN) code
  synchronization
• coefficients of multi-path channel
• narrowband interference.
• Thus, it becomes necessary to jointly estimate
  code delay, multi-path channel and narrowband
  interference parameters in DSSS receiver

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Received signal in DSSS receiver

• sip(t), PNip (t) are deterministic nonlinear
  functions of t.
• f1,f2,,fl,fNf, coefficiencts of multi-path
  channel.
• , code delay.
• j(k), narrowband interferer
• n(k), thermal noise

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Prewhitening the received signal

• To estimate narrowband interference, a
  prewhitening filter is added to prewhiten r(k)
  (,and thus j(k) and n(k) are also be
  prewhitened.).
• With the prewhitening filter, j(k) n(k) can be
  approximated by an N-th order autoregressive (AR)
  process

• e(k) is the white noise process and its
  covariance is known.
• is the tap weights of the prewhitening
  filter and needs to be estimated.

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Prewhitened received signal

• After being prewhitened, received signal can be
  rewritten as

Where,

• Parameters to be estimated
• can be considered as coefficients of
  multi-path channel with prewhitening filter.
• code delay
• the tap weights of the prewhitening filter

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Nonlinear state-space model for joint estimation
Nonlinear function of code delay
Where,
The covariance matrix of the process noise W(k)
is
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Extended kalman filter
After the linearization with respect to the delay
variable, the estimation of x can be computed
recursively as
And,
Depending on the current estimate of code delay
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If there is no prewhitening filter, we can not
estimate narrowband interference, the estimate
error can be very large .
Performance of delay tracking with or without
prewhitening
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Drawbacks solutions

• Drawback Using the predict of delay to
  compute may not be accurate.
• Solution take one or multiple iterations to
  compute using the estimation at time k,
  
• Other problem
• Doppler shift is not considered, which may cause
  a big problem in high speed mobile applications.
• To estimate Doppler shift, we need to estimate
  the velocity of the user which is also a
  nonlinear parameter of the received signal r(k).

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• THANK YOU