Channel Estimation Using Kalman Filter

A Multipath Channel Estimation Algorithm using

the Kalman Filter.

Rupul Safaya

2

OrganizationIntroductionTheoretical BackgroundChannel Estimation AlgorithmConclusionsFuture Work

3

Introduction

4

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.

5

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

6

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.

7

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:

8

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.

9

Theoretical Background

10

Signal Multipath

Base S tation

Tall Build ings

M obile T erm ina l

Path 2

Path 3

D irect Path

Build ing

11

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.

12

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.

13

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

14

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 )

15

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.

16

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.

17

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.

18

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.

19

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

20

Jakes Spectrum at a Doppler spread of 530 Hz.

21

The Channel Estimation Algorithm

22

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

23

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

24

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 .

25

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 .

26

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

27

Tracking a Gauss-Markov Channel

28

A Gauss-Markov tap-gain process has an exponential autocorrelation.

29

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

30

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

31

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

32

Results:Results:

Channel Estimation for a single ray Gauss-Markov channel

33

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

34

Tracking a single Jakes Tap-Gain Process

35

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.

36

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 .

37

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.

38

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 .

39

AR process length comparison

Jakes spectrum for truncated AR processes.

40

Jakes autocorrelation for truncated processes

41

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 .

43

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.