Channel Estimation Using Kalman Filter
Transcript of Channel Estimation Using Kalman Filter
A Multipath Channel Estimation Algorithm using
the Kalman Filter.
Rupul Safaya
2
OrganizationIntroductionTheoretical BackgroundChannel Estimation AlgorithmConclusionsFuture Work
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Introduction
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Definitions: Channel: In its most General sense can describe everything
from the source to the sink of the radio signal. Including the physical medium. In this work “Channel” refers to the physical medium.
Channel Model: Is a mathematical representation of the transfer characteristics of the physical medium. Channel models are formulated by observing the characteristics of the
received signal. The one that best explains the received signal behavior is used to
model the channel. Channel Estimation: The process of characterizing the
effect of the physical medium on the input sequence.
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General Channel Estimation Procedure
ErrorS igna le (n)
ActualR eceived S igna l
C hannel
Estim atedC hannel
M odel
Estim ation A lgorithm
+
Estim atedS ignal
)(ˆ nY
)(nY
Transm itted sequence
+
-
)(nx
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Aim of any channel estimation procedure: Minimize some sort of criteria, e.g. MSE. Utilize as little computational resources as possible
allowing easier implementation. A channel estimate is only a mathematical
estimation of what is truly happening in nature. Why Channel Estimation?
Allows the receiver to approximate the effect of the channel on the signal.
The channel estimate is essential for removing inter symbol interference, noise rejection techniques etc.
Also used in diversity combining, ML detection, angle of arrival estimation etc.
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Training Sequences vs. Blind MethodsTraining Sequence methods: Sequences known to the
receiver are embedded into the frame and sent over the channel.
Easily applied to any communications system.
Most popular method used today.
Not too computationally intense.
Has a major drawback: It is wasteful of the information bandwidth.
Blind Methods: No Training sequences
required Uses certain underlying
mathematical properties of the data being sent.
Excellent for applications where bandwidth is scarce.
Has the drawback of being extremely computationally intensive
Thus hard to implement on real time systems.
There Are two Basic types of Channel Estimation Methods:
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Algorithm Overview Consider a radio communications system using training
sequences to do channel estimation. This thesis presents a method of improving on the training
sequence based estimate without anymore bandwidth wastage. Jakes Model: Under certain assumptions we can adopt the Jakes
model for the channel. This allows us to have a second estimate independent of the data
based (training sequence) estimate. The Kalman estimation algorithm uses these two independent
estimates of the channel to produce a LMMSE estimate. Performance improvement: As a result of using the Jakes model
in conjunction with the data based estimates there is a significant gain in the channel estimate.
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Theoretical Background
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Signal Multipath
Base S tation
Tall Build ings
M obile T erm ina l
Path 2
Path 3
D irect Path
Build ing
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Multipath Signal multipath occurs when the transmitted signal arrives
at the receiver via multiple propagation paths. Each path can have a separate phase, attenuation, delay
and doppler shift associated with it. Due to signal multipath the received signal has certain
undesirable properties like Signal Fading, Inter-Symbol-Interference, distortion etc.
Two types of Multipath: Discrete: When the signal arrives at the receiver from a
limited number of paths. Diffuse: The received signal is better modeled as being
received from a very large number of scatterers.
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Diffuse Multipath The Signal arrives via a continuum of multipaths.
Thus the received signal is given by:
τ)dτ(tst(τα(t)y ~);~~ where ),(~ tα is the complex
attenuation at delay and time t and ts~ is thesignal sent
The Low Pass time variant impulse response is:
Cfjetth
2);();(~
where Cf is the carrier frequency. If the signal is bandlimited then the channel can
be represented as a tap-delayed line with time-varying coefficients and fixed tap spacing.
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Tap Delayed Line Channel Model
Attenuator
Attenuator
A ttenuator Attenuator
Attenuator
Attenuator
)()()(~ tjststs sc
(t)g M~
2M
21M
21
20
21M 2
M
)(~ty
sT Delay
(t)g 1~ (t)g0
~ (t)gM 1~ (t)gM
~(t)g M 1~
sT Delay
sT Delay
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Tap-Delay Line Model T h e r e c e i v e d s i g n a l c a n b e w r i t t e n a s :
m W
mtstmgty )(~)(~)(~ w h e r e :
);(~1)(~ tW
mhW
tmg i s t h e s a m p l e d ( i n t h e d o m a i n ) c o m p l e x
l o w - p a s s e q u i v a l e n t i m p u l s e r e s p o n s e . W : i s t h e B a n d w i d t h o ft h e B a n d p a s s S i g n a l
: W S S U S ( W i d e S e n s e S t a t i o n a r y U n c o r r e l a t e d
S c a t t e r i n g ) : A s s u m i n g W S S U S t h e d e l a y p r o f i l e a n d s c a t t e r i n gf u n c t i o n a r e a s f o l l o w s : M u l t i p a t h I n t e n s i t y P r o f i l e :
),(~),(*~
21)( ththECR T h i s
d e f i n e s t h e v a r i a t i o n o f a v e r a g e r e c e i v e d p o w e r w i t h d e l a y .D e l a y s p r e a d i s t h e r a n g e ( i n d e l a y ) f o r w h i c h t h e a v e r a g ep o w e r i s n o n - z e r o .
S c a t t e r i n g f u n c t i o n : )]([);( CRFvS T h i s d e s c r i b e s t h e p o w e rs p e c t r a l d e n s i t y a s a f u n c t i o n o f D o p p l e r f r e q u e n c y ( f o r f i x e dd e l a y )
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Tap gain functions Complex Gaussian process: Assuming infinite
scatterers, as a consequence of the Central Limit
Theorem, we can model the impulse response as a
complex Gaussian process. Rayleigh: If there is no one single dominant path, then the
process is zero mean and the channel is Rayleigh fading.
Ricean: If there is asingle dominant path then the process is non-
zero-mean and the channel is Ricean.
The tap gain functions are then sampled complex
gaussian processes.
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Model Parameters The tap-delay model requires the following
information. Number of taps are TMW+1. Where TM is the delay spread and W
is the information bandwidth. Tap Spacing is: 1/W. Tap gain functions are discrete time complex Gaussian processes
with variance given by the Multipth spread and PSD given by thescattering function.
Tap gain functions as key: Once having specified thetap spacing and the number, it only remains to trackthe time varying tap gain functions in order tocharacterize the channel as modeled by the tap-delayline.
Jakes model: The Jakes model (under certainassumptions) assigns the spectrum and autocorrelationto the tap-gain processes.
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Jakes Model
Base S tation
Veh ic le Ve loc ity
PropagationPath
M obile
a
Assume plane waves are incident upon an omni-directional antenna from stationary scatterers.
There will be a doppler shift induced in every wave. Function of angle of arrival, carrier frequency and
the receiver velocity.
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Bounded Doppler Shift: The doppler shift is given by cosvfd (where v is Vehicle velocity, , the
wavelength of the carrier and is the angle of arrival)and is thus bounded.
Narrowband process: Since the Doppler spectrum isbounded, the electric field at the receiver is anarrowband process.
Complex Gaussian Process: Assuming infinitescatterers and using the Central Limit theorem fornarrowband processes, the received electric field isapproximately a complex gaussian process.
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Jakes Spectrum
U n i f o r m a n g l e o f a r r i v a l : A s s u m e t h a t t h e r e c e i v e dp o w e r i s u n i f o r m l y d i s t r i b u t e d o v e r t h e a n g l e o f a r r i v a l .
C o n s t a n t v e h i c l e v e l o c i t y : F i n a l l y t h e a s s u m p t i o n i sm a d e t h a t t h e v e h i c l e i s n o t a c c e l e r a t i n g .
P o w e r s p e c t r u m e x p r e s s i o n :
2/12
1)(
mfCff
fS w h e r e Cf i s t h e c a r r i e r f r e q u e n c y a n d mf i s
t h e d o p p l e r s p r e a d . S h a p i n g F i l t e r : T h e J a k e s s p e c t r u m c a n b e s y n t h e s i z e d
b y a s h a p i n g f i l t e r w i t h t h e f o l l o w i n g i m p u l s e r e s p o n s e : )2(4/1
4/1)2()43(4/12)( tmfJtmfmftjh
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Jakes Spectrum at a Doppler spread of 530 Hz.
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The Channel Estimation Algorithm
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Introduction Aim: To improve on the data-only estimate. Jakes model: We have adopted the Jakes model for the radio
channel. Tap-gains as auto-regressive processes: The Jakes power
spectrum is used to represent the tap-gains as AR processes. State-Space Representation: We have two independent estimates
of the process from the data-based estimate and the Jakes model. These are used to formulate a State-Space representation for the tap-
gain processes. An appropriate Kalman filter is derived from the state-space
representation. Derivation: The algorithm is developed first for a Gauss-Markov
Channel and then for the Jakes Multipath channel
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AR representation of the Tap-gains
General form: Any stationary random process can be represented as an infinite tap AR process.
The current value is a weighted sum of previous values and the plant noise.
U nitD e lay
U n itD elay
U n itD e lay
.
.
.
. . .
1 h
2 h
Nh
)(nS)1( nS
)( NnS
)(nw
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D i f f e r e n c e E q u a t i o n f o r m : )()()(
1
nwinSnSN
ii
W h e r e S ( n ) : i s t h e c o m p l e x g a u s s i a n
p r o c e s s , i a r e t h e p a r a m e t e r s o f t h e m o d e l . T h e m o d e l i s d r i v e n
b y w ( n ) : A s e q u e n c e o f i d e n t i c a l l y d i s t r i b u t e d z e r o - m e a n
C o m p l e x G a u s s i a n r a n d o m v a r i a b l e s .
S o l v i n g f o r t h e A R p a r a m e t e r s : T h i s c a n b e d o n e i n t w o w a y s
Y u l e - W a l k e r e q u a t i o n s o l u t i o n : T h e a u t o c o r r e l a t i o n c o e f f i c i e n t
o f t h e p r o c e s s , i s a n N ( n u m b e r o f t a p s i n t h e A R m o d e l ) o r d e r
d i f f e r e n c e e q u a t i o n t h a t c a n b e s o l v e d .
P S D o f t h e p r o c e s s : T h i s m e t h o d i s m o r e c o m p l e x a n d i n v o l v e s
f i n d i n g t h e f o r m o f t h e s h a p i n g f i l t e r f o r t h e p r o c e s s P S D .
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Data based estimator A s s u m p t i o n : T h e c h a n n e l r e m a i n s c o n s t a n t o v e r
t h e s p a n o f t h e t r a i n i n g s e q u e n c e . S p e c i fi c s :
T r a i n i n g s e q u e n c e : L e t t h e ‘ M ’ l e n g t h t r a i n i n g s e q u e n c e b e : TMxxxx 110 .................
C h a n n e l I m p u l s e R e s p o n s e : L e t t h e ‘ L ’ l e n g t h i m p u l s e
r e s p o n s e b e TLh......hhhh 1210~~~~
R e c e i v e d s i g n a l : F o r c h a n n e l n o i s e Cn o f v a r i a n c e 2C t h e
r e c e i v e d s i g n a l i n v e c t o r f o r m i s g i v e n b y : cnhXY . W h e r e X i s
a n LNL 1 T o e p l i t z m a t r i x c o n t a i n i n g d e l a y e d v e r s i o n s o f t h e
t r a i n i n g s e q u e n c e s e n t .
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Channel estimate: The data based estimate is given by
correlating the received signal with the training sequence:
)()(ˆ 1 YXXXh TT
Estimation error: As expected the estimation error is a
function of the channel noise. It is given by )()(~
1c
TT nXXXh
Error Covariance: The performance of the data based
estimator depends on the length of the training sequence and
its autocorrelation: 12 )( XXP T
cD
For an ideal training sequence autocorrelation the error
covariance is given by M
P cD
2
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Tracking a Gauss-Markov Channel
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A Gauss-Markov tap-gain process has an exponential autocorrelation.
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Kalman Filter Derivation System M odel: Since the auto-correlation is
exponential, the tap gain is a simple first order AR
process: )()1()( 1 nwnSnS W here
)(nS : is the complex gaussian tap-gain process.
1 is the parameter of the AR model assumed.
Observation M odel: Since the data based estimator
produces “noisy” estimates of the process. The
following model emerges:
)()()( nvnSnX . W here X(n) is the data based estimate of S(n) v(n) is the error of the data based estimate and
Mσ
σ cv
22 is the error variance
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Kalman filter equations S c a l a r K a l m a n f i l t e r : G i v e n t h e s t a t e s p a c e
r e p r e s e n t a t i o n , a s t a n d a r d s c a l a r K a l m a n f i l t e r i s u s e dt o t r a c k t h e p r o c e s s .
E q u a t i o n s : T h e i n i t i a l c o n d i t i o n s a r e :
00ˆ S(n))=E(S
22)1( vandwP
T h e K a l m a n g a i n i s g i v e n b y :
)()()()( 2 nnP
nPnkv
T h e c u r r e n t e s t i m a t e o f t h e p r o c e s s , a f t e r r e c e i v i n g t h e d a t a e s t i m a t e i sg i v e n b y :
)]ˆ[ˆˆ (nSX(n)-k(n)(n)S(n)=S curr
T h e p r e d i c t e d e s t i m a t e o f t h e p r o c e s s i s g i v e n b y :
(n)S)=(nS cu rrˆ1ˆ
1 T h e c u r r e n t e r r o r c o v a r i a n c e i s g i v e n b y :
)()(1)( nPnknP cu rr T h e p r e d i c t i o n e r r o r c o v a r i a n c e ( M S E i n t h i s c a s e ) i s g i v e n b y :
221 )()1( wcu rr nPnP
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Simulation Parameters S y s t e m e q u a t i o n : w(n))S (n.S (n) 19 O b s e r v a t i o n E q u a t i o n : )()()( nvnSnX T h e f r a m e r a t e i s : sec105 4 Frames/R F . T h e s i m u l a t i o n i s r u n a t t h e
f r a m e r a t e .
8M : T h e l e n g t h o f t h e t r a i n i n g s e q u e n c e . T h e c h a n n e l i s a s s u m e d
t o b e i n v a r i a n t f o r t h e s e M b i t s .
T h e s i g n a l t o n o i s e r a t i o o f t h e c h a n n e l , f o r 1BE i s :
dBENE
c
B
O
B 62 2
T h u s 1256.2 c
T h e d a t a e s t i m a t o r v a r i a n c e 0.0157σ2
2V
Mc
T h e v a r i a n c e o f t h e t a p g a i n p l a n t n o i s e i s 0314.02σ 22w V
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Results:Results:
Channel Estimation for a single ray Gauss-Markov channel
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MSE for the estimator: D e f i n i t i o n :
T h e c u r r e n t M S E i s :
H
currcurrcurr (n)SS(n)(n)SS(n)EP ˆˆ
T h e p r e d i c t i o n M S E i s :
H(n)SS(n)(n)SS(n)EP ˆˆ
S t e a d y S t a t e :
T h e s t e a d y s t a t e c u r r e n t e r r o r c o v a r i a n c e i s :0113.
SScurrP T h e s t e a d y s t a t e p r e d i c t i o n e r r o r c o v a r i a n c e i s :
0406.SSP
S i m u l a t i o n R e s u l t s : T h e c u r r e n t e r r o r c o v a r i a n c e i s :
0113.SimcurrP
T h e p r e d i c t i o n e r r o r c o v a r i a n c e i s :0406.SimP
P e r f o r m a n c e I m p r o v e m e n t : W e c a n s e e t h a t c o m p a r e d t o t h e d a t a o n l y
e s t i m a t o r , t h e r e i s a s i g n i f i c a n t i m p r o v e m e n t :
%282
2
V
currV SSP
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Tracking a single Jakes Tap-Gain Process
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Single ray Jakes Channel Consider a single ray line of sight radio channel. More Complex Channel: The underlying channel model is no
longer a single tap AR process. AR representation: The tap-gain process with the Jakes
spectrum is a stationary process. We can represent it as an AR process. Parameters: We derive the co-efficients for the process from the
closed form expression of the Jakes channel-shaping filter. State-Space representation: Using the AR model and the data
based estimator, a state-space representation is derived. Kalman tracking filter: Similar to the Gauss-Markov case, a
Kalman filter to track the process is derived from the State-Space representation.
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AR representation of the Jakes Process
J a k e s S h a p i n g F i l t e r : T h e c l o s e d f o r m e x p r e s s i o n o f t h e
J a k e s f i l t e r i s g i v e n b y :
)2(4/14/1)2()
43(4/12)( tmfJtmfmftjh . W h e r e i s t h e G a m m a
f u n c t i o n a n d 4/1J i s t h e f r a c t i o n a l B e s s e l f u n c t i o n .
F I R f i l t e r : T h i s e x p r e s s i o n i s s a m p l e d t o p r o d u c e t h e F I R
J a k e s c h a n n e l s h a p i n g f i l t e r .
O u t p u t : I f t h e i n p u t t o t h i s f i l t e r i s g a u s s i a n w h i t e n o i s e ,
t h e n t h e o u t p u t i s s i m p l y t h e c o n v o l u t i o n s u m :
1
0
)()()(M
mj mnwmhnS . W h e r e M i s t h e l i e n g t h o f t h e F I R f i l t e r .
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UnitDelay
UnitDelay
UnitDelay
.
.
.
. . .
1 h
Nh
)1( nw)1( Mnw
)(nw
0 h
)2( nw
2 h
1M- nh
)(nS
Convolution as a MA sum: The convolution is nothing but a
weighted moving average of the white noise inputs.
Partial Fraction Inversion: A finite order MA model can be represented as an infinite order AR series
by the method of partial fractions as described by Box and Jenkins.
Order Truncation: Obviously for our purposes an infinite order AR model
is impractical, so the infinite order AR model is truncated to order N.
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Model Validation
T r u n c a t e d A R m o d e l : T h e a c c u r a c y o f th e t r u n c a t e d m o d e l s i n
r e p r e s e n t i n g t h e J a k e s p r o c e s s i s s t u d i e d .
P a r a m e te r s : Hzf m 50 . T h i s i s t h e D o p p l e r b a n d w id t h o f t h e c h a n n e l . mS fF 16 . T h i s i s t h e J a k e s - s h a p in g - f i l t e r s a m p l in g r a t e . HzsamplesF
PSDS /105 3 . T h i s i s t h e J a k e s s p e c t r u m s a m p l i n g f r e q u e n c y . 64MAN . T h i s i s t h e F I R s h a p in g f i l t e r l e n g t h . N i s t h e l e n g th o f t h e t r u n c a t e d A R m o d e l .
P S D a n d a u to c o r r e l a t i o n :
P S D i s g iv e n b y th e a n a ly t i c a l e x p r e s s io n : 2
1
2
2
1
)(
N
i
fiji
WSS
e
fS
A u to c o r r e l a t i o n w a s e s t im a te d b y a c t u a l l y c r e a t i n g t h e p r o c e s s e s a n d
f in d in g t h e i r a u to c o r r e l a t io n s .
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AR process length comparison
Jakes spectrum for truncated AR processes.
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Jakes autocorrelation for truncated processes
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Kalman filter Derivation S t a t e S p a c e r e p r e s e n t a t i o n : A s i n t h e p r e v i o u s c a s e ,
w e a r e g o i n g t o r e p r e s e n t t h e s y s t e m u s i n g t h e S t a t e -
S p a c e f o r m a n d d e r i v e t h e a p p r o p r i a t e K a l m a n f i l t e r .
S y s t e m m o d e l : T h e A R f o r m o f t h e J a k e s p r o c e s s i s
)(1
)()( nwN
iinSinS
. W h e r e i a r e t h e A R c o e f f i c i e n t s
c a l c u l a t e d a n d N i s t h e o r d e r o f t h e m o d e l u s e d .
I t s t w o e q u i v a l e n t f o r m s a r e :
0
021
010000...000.1000..1
..21
1
1
.
.
w(n)
S(n-N ).
. )S(n-
)S(n-
N
)S(n-N..
)S(n-S(n)
i n m a t r i x f o r m
a n d )()1()( nWnSAnS i n v e c t o r f o r m .
42
T h e s y s t e m m a t r i x i s d e f i n e d a s :
010000...000.1000..1
..21 N
A
T h e P l a n t n o i s e c o v a r i a n c e m a t r i x i s
0000000
0000.002
......... ........
. .......... ... σ
(n)W(n)WEQ
wH__
O b s e r v a t i o n E q u a t i o n : )()()( nvnSnX W h e r e v ( n ) i s t h e e r r o r o f t h e d a t a b a s e d
e s t i m a t e w i t h a v a r i a n c e o f M
cvσ
22
E x p r e s s e d i n m a t r i x a n d v e c t o r f o r m :
)v(n-N..
)v(n-v(n)
)S(n-N..
)S(n-S(n)
)X(n-N..
)X(n-X(n)
1
1
1
1
1
1o r )()()( nvnSHnX
w h e r e H i s t h e i d e n t i t y m a t r i x .
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O b s e r v a t i o n n o i s e c o v a r i a n c e m a t r i x :
1. .000. .100
0.100.. . .001
2
. . . . . . . . . . . . . . . . .
.. . . . . . . . . . . . .
vσH
(n )_v(n )
_vER
K a l m a n f i l t e r E q u a t i o n s : G i v e n t h e a b o v e S t a t e S p a c ef o r m u l a t i o n , w e c a n u s e a v e c t o r K a l m a n f i l t e r t o t r a c kt h e t a p g a i n p r o c e s s . T h e i n i t i a l c o n d i t i o n s a r e :
S (n ))= E(S 0ˆ = z e r o m a t r i x o f l e n g t h L ) (N
Iva n dIwP 22)1(
T h e K a l m a n g a i n i s g i v e n b y :1])([)()( RHnHPHnPnK TT
T h e c u r r e n t e s t i m a t e o f t h e p r o c e s s , g i v e n t h e d a t a e s t i m a t e i s g i v e nb y :
]ˆ[ˆˆ (n )SX (n )-HK(n )(n )S(n )=S cu r r
T h e p r e d i c t e d e s t i m a t e o f t h e p r o c e s s , i s g i v e n b y :
(n )S)= A(nS cu r rˆ1ˆ
T h e c u r r e n t e r r o r c o v a r i a n c e i s g i v e n b y : )()()( nPHnKInP cu r r
T h e p r e d i c t e d e r r o r c o v a r i a n c e i s g i v e n b y : QAnPAnP T
cu r r )()1(
44
Simulation Parameters S y s t e m e q u a t i o n :
)()1()( nWnSAnS
N = 5 . T h i s i s t h e n u m b e r o f t a p s i n t h e A R m o d e l .
040900486005480059009086 .,.,.,.,.
010000010000010000010409.00486.00548.00590.09086.
A
O b s e r v a t i o n E q u a t i o n :
)()()( nvnSHnX
T h e D o p p l e r b a n d w i d t h i s Hzf m 500
T h e f r a m e r a t e i s : sec4105 Frames/FR . T h e s i m u l a t i o n i s r u n a t t h e f r a m er a t e .
45
8M : T h e l e n g t h o f t h e t r a i n i n g s e q u e n c e .
T h e s i g n a l t o n o i s e r a t i o o f t h e c h a n n e l , f o r 1BE i s :
dBENE
c
B
O
B 62 2
T h u s 1256.2 c
T h e v a r i a n c e o f t h e t a p g a i n p l a n t n o i s e i s 0314.0222wσ V
000000
000001
03140
......... ........ .
... ..............
.Q
T h e d a t a e s t i m a t o r v a r i a n c e 0.01572
2Vσ
Mc
1000100010
0001
.0157.
......... ........
. .......... ...
R
46
Results
Channel Estimation of a single ray Jakes channel
47
Error covariance: T h e e r r o r c o v a r i a n c e i s d e f i n e d a s :
HnSnSEP 1)(n
^S)1(1)(n
^S)1(
W e c a n t h e n i n t e r p r e t t h e d i a g o n a l e l e m e n t s a s f o l l o w s :
)/2( .......................* *..
*....... * *)/1( * *
*....... * *)/( *
*....... * *)/1(
nNnSMSE
nnMSEnnMSE
nnMSE
P
W h e r e )/1( nnMSE i s t h e M S E o f t h e p r e d i c t i o n .
)/( nnMSE i s t h e M S E o f t h e c u r r e n t s t a t e s e s t i m a t e
48
T h e s te a d y s ta te a n d s im u la te d e r r o r c o v a r ia n c em a t r i c e s a r e :
0.0296 0.0388 0.0565 0.1117 1.0921
)( SSPdiag
0.1144 0.1118 0.1081 0.1125 1.0777
)( SimPdiag
P e r f o r m a n c e I m p r o v e m e n t : T h e r e i s a s ig n i f i c a n tp e r f o r m a n c e g a in c o m p a r e d to th e d a ta o n ly e s t im a to r %29
2)1,2)((2
V
SimPdiagV
49
Multipath Channel Estimation
50
Multipath Jakes Channel Consider a multipath radio channel. Assume the Jakes model on each path. AR representation: For the Multipath case, a modification of
the single ray AR system model is presented. State-Space representation: Using the AR model and the
data based estimator, a state-space representation is derived.
Kalman tracking filter:Once again a vector Kalman filter is used to track the tap-gain functions.
51
System model A s s u m p tio n s : T h e ta p g a in p r o c e s s e s a r e in d e p e n d e n t
b u t h a v e th e s a m e J a k e s s p e c t r u m . A R R e p r e s e n ta t io n :
)(1
)()(
.
.
)(21)(2)(2
)(11)(1)(1
nLwN
iinLSinLS
nwN
iinSinS
nwN
iinSinS
w h e r e )( nlS i s th e thl p ro c e s s to b e tr a c k e d
i a r e th e A R m o d e l p a r a m e te r s . T h e s e a r e th e s a m e f o r e a c hp ro c e s s .
)( nw l i s th e p la n t n o is e d r iv in g th e ta p g a in f u n c t io n . T h e r e la t iv ev a r ia n c e is d e te r m in e d b y th e p o w e r d e la y p ro f i le o f th e c h a n n e l .
L i s th e n u m b e r o f p ro c e s s e s b e in g t r a c k e d
52
T h e m a t r i x f o r m o f th e s y s t e m e q u a t i o n f o r ‘ L ’ p r o c e s s e s i s :
0................... 0.
. .0..........0.........
(n)L..w(n).......1w
N)-(nL....S N)-(n1S.
. .
2)-(nLS2)........-(n1S
1)-(nLS1)........-(n1S
0 .......1 0 00 ................0 .1......... 0
0......0 0 1N........21
1)N-(nL...S 1)N-(n1S.
. .
1)-(nL..S1)........-(n1S
(n)L.......S(n).......1S
I n v e c to r f o r m : )()1()( nWnSAnS , w h e r e
010000100001
21
....... ...... ..........
.......... ......
N........
A
i s t h e s y s t e m m a t r i x .
0..0
0.000
01
2
........ ..........
......... .........
.......... .........
........L
l(l)wσ
H
(n)_
W(n)_
WEQ i s t h e p l a n t n o i s e c o v a r i a n c e m a t r i x .
53
Observation Model Assuming that the data based estimates are path-wise
independent, we have the following model:
)()()(..
)(2)(2)(2
)(1)(1)(1
nLvnLSnLX
nvnSnX
nvnSnX
Where, )(nSl : The l’th process at time n.
)(nXl : The l’th data based estimate of S(n) )(nvl : Error of the l’th data based estimate.
Mσ c
v
22 is assumed to be the same for all paths
54
The Observation Equation can be written in matrix and vector form asfollows:
1)N-(nLv..
1)-(nLv
(n)Lv
.
.
.
.
.
.
.
.
.
.
1)N-(n1v..
1)-(n1v
(n)1v
1)N-(nLS..
1)-(nLS
(n)LS
.
.
.
.
.
.
.
.
.
.
1)N-(n1S..
1)-(n1S
(n)1S
1)N-(nLX..
1)-(nLX
(n)LX
.
.
.
.
.
.
.
.
.
.
1)N-(n1X..
1)-(n1X
(n)1X
)()()(___
nvnSHnX where H is an identity matrix. The observation noise covariance matrix is given by:
][)2(])()([ IvLHnvnvER where ][I is an )( NN identity matrix.
55
Simulation parameters N=5. This is the number of taps in the AR model.
L=3: This is the number of tap-gain processes being tracked.
040900486005480059009086 .,.,.,.,.
The System Equation
010000010000010000010409.00486.00548.00590.09086.
A
The Doppler bandwidth is Hz500fm
The frame rate is : sec105 4 Frames/RF . The simulation is run at the
frame rate.
8M : The length of the training sequence. The channel is assumed to
be invariant for these M bits.
56
T h e s i g n a l t o n o i s e r a t i o o f t h e c h a n n e l , f o r 1BE i s :
dBENE
c
B
O
B 62 2
T h u s 1256.2 c
T h e p o w e r d e l a y p r o f i l e ]81.0,9.0,1[0314.0)(2wσ l
T h e c o v a r i a n c e o f t h e p l a n t n o i s e i s :
0.0...
0.0.0851 Q
T h e d a t a e s t i m a t o r v a r i a n c e 0.0157σ2
2V
Mc . H e r e t h e a s s u m p t i o n i s
m a d e t h a t t h e t r a i n i n g s e q u e n c e h a s a n i d e a l a u t o c o r r e l a t i o n . I .0157.03 R
57
Results
Channel estimation for the first process.
58
Channel estimation for the second process.
59
Channel estimation for the third process.
60
Error Covariance T h e E r r o r c o v a r i a n c e i s d e f i n e d a s :
H)(nS)S(n)(nS)S(nEP 1ˆ11ˆ1
W e c a n t h e n i n t e r p r e t t h e d i a g o n a l e l e m e n t s a s f o l l o w s :
l /n)N(n
(l)MSE....... ...........* *.......
.** * ......l /n)(n
(l)MSE* *
.** * ......l (n/n)
(l)MSE*
.** * ......l /n)(n
(l)MSE
P
2
1
1
w h e r e
l nn
lMSE)/1(
)( : t h e s u m o f t h e M S E o f t h e p r e d i c t e d s t a t e e s t i m a t e
o n a l l p r o c e s s e s .
l (n/n)
(l)MSE : t h e M S E o f t h e c u r r e n t s t a t e s e s t i m a t e o n a l l
p r o c e s s e s .
61
T h e s i m u l a t e d e r r o r c o v a r i a n c e d i a g o n a l i s g i v e n b y :
1 2 30 . 0 3 8 4 0 . 0 3 5 1 0 . 0 3 2 1 0 . 1 0 5 6 0 . 1 1 1 10 . 0 1 1 2 0 . 0 1 0 9 0 . 0 1 0 6 0 . 0 3 2 7 0 . 0 3 2 10 . 0 0 9 8 0 . 0 0 9 7 0 . 0 0 9 7 0 . 0 2 9 2 0 . 0 1 6 90 . 0 1 0 5 0 . 0 1 0 5 0 . 0 1 0 5 0 . 0 3 1 5 0 . 0 1 2 20 . 0 1 1 2 0 . 0 1 1 1 0 . 0 1 1 2 0 . 0 3 3 5 0 . 0 0 9 6
0 . 0 1 1 2 0 . 0 1 0 9 0 . 0 1 0 6 0 . 0 3 2 7 0 . 0 3 2 10 . 0 0 9 8 0 . 0 0 9 7 0 . 0 0 9 7 0 . 0 2 9 2 0 . 0 1 6 90 . 0 1 0 5 0 . 0 1 0 5 0 . 0 1 0 5 0 . 0 3 1 5 0 . 0 1 2 20 . 0 1 1 2 0 . 0 1 1 1 0 . 0 1 1 2 0 . 0 3 3 5 0 . 0 0 9 60 . 0 1 1 7 0 . 0 1 1 7 0 . 0 1 1 7 0 . 0 3 5 1 0 . 0 0 8
)ocess # (lPr
)(SSCurrPdiag)(
SimCu rrPdiag))(( lPdiagS imCurr
))(( lPdiag S im )( S imPdiag )( SSPdiag
P e r f o r m a n c e i m p r o v e m e n t o n e a c h p a t h : T h e d a t a b a s e d e s t i m a t e h a s a M S E o f 0157.0σ 2
V o n e a c h p a t h .
P a t h 1 i m p r o v e m e n t : %66.281000157.
0112.0157.)1,1)((2
2
V
currV simPdiag
P a t h 2 i m p r o v e m e n t : %57.301000157.
0109.0157.
P a t h 3 i m p r o v e m e n t : %48.321000157.
0106.0157.
A l m o s t a 3 0 % i m p r o v e m e n t o n e a c h p a t h
62
Conclusions Developed a Kalman filter based channel
estimation algorithm for the Multipath radio channel.
Significant gain in performance over a training sequence based estimator.
This improvement is obtained without wasting any more bandwidth.
Also allows us to predict the channel state without having to wait for data.
63
Future Work Use Of Multiple Sampling Rates:
Instead of waiting for the data to arrive at the end of every frame we can run the Kalman filter at a higher rate than the frame rate.
In the absence of a data based estimate perform the time-update portion of the algorithm and do a measurement update when data is received.
Allows estimates to be available as required. Different process models on each path:
In case the process model varies with path, we can still use the Kalman filter but with some modifications to the system matrix.
Correlated paths: For correlated paths the Kalman filter needs to be modified.