113.8 Signal Processing ExamplesEx. 13.3 Time-Varying Channel EstimationTxRxDirect PathMulti Pathv(t)y(t)∫−=Ttdtvhty0)()()(τττT is the maximum delayModel using a time-varying D-T FIR systemChannel changes with time if:• Relative motion between Rx, Tx• Reflectors move/change with time∑=−=pknknvkhny0][][][Coefficients change at each nto model time-varying channel2In 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)3Need 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…<“uncorrelated scattering”> A, Q , Ch , are Diagonalcov{h[-1]} = M[-1|-1]cov{u[n]}4Have measurement model from convolution view:][][][][0nwknvkhnxpkn+−=∑=zero-mean, WGN, σ2Known training signalNeed Observation Equation:][][ nwnxnT+= hvObservation “Matrix”is made up of the samples of the known transmitted signalState Vectoris the filter coefficients5Simple Specific Example: p = 2 (1 Direct Path, 1 Multipath)==0001.0000001.0999.00099.0QA][]1[][ nnn uAhh +−=Q =cov{u[n]}Typical Realization of Channel CoefficientsBook doesn’t state how the initial coefficients were chosen for this realizationNote: hn[0] decays fasterand that the random perturbation is small6Known Transmitted SignalNoise-Free Received Signal<It is a bit odd that the received signal is larger than the transmitted signal>Noisy Received SignalThe variance of the noise in the measurement model is σ2= 0.17Estimation Results Using Standard Kalman FilterInitialization:1.0100]1|1[]00[]1|1[ˆ2==−−=−−σIMhTChosen to reflect that little prior knowledge is knownIn theory we said that we initialize to the a priorimean… but in practice it is common to just pick some arbitrary initial value and set the initial covariance quite high… this forces the filter to start out trusting the data a lot!Transient due to wrong ICEventually Tracks Well!!hn[0]hn[1]8Kalman Filter GainsDecay down… relies more on modelGain is zero when signal is noise onlyKalman Filter MMSEFilter Performance improves with time9Example: Radar Target TrackingState Model: Constant-Velocity A/C Model!"!#$!"!#$!!!"!!!#$!"!#$][]1[][][][00]1[]1[]1[]1[10000100010001][][][][nyxnyxyxnyxyxnununvnvnrnrnvnvnrnrusAs+−−−−∆∆=−==2200000000000000}cov{uuσσuQ++=−][][][][tan][][][122nwnwnrnrnrnrnRxyyxβxObservation Model: Noisy Range/Bearing Radar Measurements For this simple example…. assume:==2200}cov{βσσRwCVelocity perturbations due to wind, slight speed corrections, etc.Velocity perturbations due to wind, slight speed corrections, etc.in radians10Extended Kalman Filter Issues1. Linearization of the observation model (see book for details)• Calculate by hand, program into the EKF to be evaluated each iteration2. Covariance of State Driving Noise• Assume wind gusts, etc. are as likely to occur in any direction w/ same magnitude ! model as indep. w/ common varianceNeed the following:==2200000000000000}cov{uuσσuQσu= what??? Note: ux[n]/ ∆ = acceleration from n-1 to nSo choose σu in m/s so that σu/ ∆ gives a reasonable range of accelerations for the type of target expected to track113. 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{σR2[n], σβ2[n]}• But best to derive joint CRLB to see if they are correlated124. Initialization Issues• Typically… Convert first range/bearing into initial rx& ryvalues• If radar provides no velocity info (i.e. does not measure Doppler) can assume 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 for rx& rybased 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 use those as ICs for the initializing the new radar" The other radar/sensor would likely hand-off a MSE measure of the quality its lasttrack… so use that as M[-1|-1]13State Model Example Trajectories: Constant-Velocity A/C Model-20 -15 -10 -5 0 5 10 15-10-50510152025X p o s it ion ( m)Y position (m)RadarRed Line is Non-Random Constant Velocity Trajectorym/s 2.0]1[2.0]1[m 5]1[10]1[)/sm 001.0( m/s 0.0316sec 1222=−−=−−=−=−===∆yxyxuuvvrrσσ140 10 20 30 40 50 60 70 80 90 10005101520Sample Index nRange R (meters)0 10 20 30 40 50 60 70 80 90 100-50050100Bearing β (degree s)Sample Index nObservation Model Example MeasurementsRed Lines are Noise-Free Measurements)rad 01.0( deg 5.7 rad 0.1)m 1.0( m 0.31622222=====RRRσσσσβIn reality, these would get worse when the target is far away due to a weaker returned signal15If we tried to directly convert the noisy range and bearing measurements into a track… this is what we’d get.Not a very accurate track!!!! ! Need a Kalman Filter!!!But… Nonlinear Observation Model… so use Extended KF! RadarNote how the track gets worse when far from the radar (angle accuracy converts into position accuracy in a way that depends