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Channel Estimation

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Table of Contents

Introduction to Channel EstimationGeneric Pilot Based Channel Estimator

Uplink Channel EstimatorDownlink Channel Estimator

WCDMA Channel Estimation TechniquesSimple AverageWeighted Multi Slot Averaging (WMSA)α-Tracker

InterpolationFast Channel Estimation

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Introduction to Channel Estimation

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IntroductionComplex channel estimation (i.e., estimation of channel gain, which includes phase and amplitude) performed for each individual RAKE fingers is required for coherent detection (Maximal Ratio Combining).Complex channel estimation is performed with the assistance of known transmitted pilot symbols.The accuracy of the channel estimation is crucial for RAKE receiver performance, and it depends on the pilot channel energy, the channel estimation algorithms, and the environment conditions.In particular, mobile speed is required for a variety of channelestimation algorithms.

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Pilot Symbol TransmissionThe pilot symbols can be transmitted in two basic ways:

In the case of dedicate pilot channel scheme, system has one physical channel fully dedicated to pilot symbol transmission.

E.g. Common Pilot Channel, CPICH, in downlink of WCDMA.

Another option is to insert pilot symbols into the data stream(time multiplexed pilot symbols).

E.g. DPDCH/DPCCH in uplink of WCDMA.

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Dedicate Pilot Channel SchemeOne possible phase estimation architecture based on a dedicate pilot channel is shown in the following figure:

The output of the channel estimation is filtered by a low pass filter (LPF), whose bandwidth should be made adjustable to the Doppler frequency.

Correlator(Data Channel)

Correlator(Pilot Channel)

WidebandI/Q signals

LPF

*

NarrowbandSignal

Chip Rate Symbol Rate

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Time Multiplexed Pilot SymbolsIn the case of inserted pilot symbols, there are two well-known channel estimation techniques:

Decision feedbackInterpolation

Decision feedback schemeIn this approach, hard decisions (decision of ±1) are made for the non-pilot (data) symbols.

The hard decisions may be made with the help of pilot symbols.

The data symbols are feedback, together with the pilot symbols, to form a continuous “pilot symbols” for channel estimation.The output of channel is again filtered by a LPF which should be made adaptive to the Doppler frequency.

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Time Multiplexed Pilot SymbolsIf hard decision results in small error rate, decision feedback schemes typically lead to better performance.

InterpolationIn this scheme, since data symbols and pilot symbols are time multiplexed, channel estimations are first obtained for the pilot symbols and are then interpolated to obtain the channel gain of data symbols for coherent combining.If the outputs of hard decision results in a high data error rate, which might due to small signal energy, severe fading effect, or high interference level, the decision feedback performance starts to degrade. Under such scenario, interpolation scheme might perform better since it uses only the pilot symbol energy for estimation.

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Generic Pilot Based Channel Estimator

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Generic Pilot Based Channel EstimatorIn making channel estimation, we assume that channel condition is constant over some time interval.The complex channel estimation can be obtained by multiplying the “recovered pilot symbols” by the known pilot pattern.In uplink direction, channel estimation is made with the help of pilot symbols within the DPCCH.

Channel estimation is typically based on simple block averaging.Interpolation is typically performed between pilot blocks to estimate the channel status.

In downlink direction, channel estimation is made using the dedicated pilot channel, called Common Pilot Channel (CPICH).

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Uplink Channel EstimatorVariables are defined as follows:

( ): Pilot symbols 1 .: Number of Pilot symbols.

: DPDCH symbols at UE transmitter.: DPCCH symbols at UE transmitter.

: Complex scrambling code.

: DPDCH channelization code.: DPCCH channeli

p

d

c

I Q

d

c

PN

DDS jS

CC

±

+

zation code.: Impulse response of complex fading channel.

: Received DPDCH data symbol at BS.: Received DPCCH control symbol at BS.

I Q

d

c

h jh

RR

+

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Uplink Channel EstimatorAt UE, data is BPSK modulated. The spread symbols at the UE transmitter are:

After scrambling:

At base station, the received baseband signal is:

At base station, after descrambling:

d d c c I QD C jD C D jD⋅ + ⋅ ≡ +

( ) ( ) ( ) ( )I Q I Q I I Q Q I Q Q ID jD S jS D S D S j D S D S+ ⋅ + = ⋅ − ⋅ + ⋅ + ⋅

( ) ( )I I Q Q I Q Q I I QD S D S j D S D S h jh⎡ ⎤ ⎡ ⎤⋅ − ⋅ + ⋅ + ⋅ ⋅ +⎣ ⎦⎣ ⎦

( ) ( ) ( )( ) ( ) ( ) ( )

12 I Q I I Q Q I Q Q I I Q

I I Q Q I Q Q I I Q I Q

S jS D S D S j D S D S h jh

D h D h j D h D h D jD h jh

⎡ ⎤ ⎡ ⎤⋅ − ⋅ ⋅ − ⋅ + ⋅ + ⋅ ⋅ +⎣ ⎦⎣ ⎦

= ⋅ − ⋅ + ⋅ + ⋅ = + ⋅ +

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Uplink Channel EstimatorAfter despreading, we obtain:

The channel condition information is estimated from the received pilot symbols:

where average is performed over Np pilot symbols.

( )( )

DPDCH:

DPCCH:

d d I Q

c c Q I

R D h jh

R D h jh

= ⋅ +

= ⋅ − +

{ }

{ }

1 Im

1 Re

p

p

I cNP

Q cNP

h R PN

h R PN

= ⋅ ⋅

−= ⋅ ⋅

∑

∑

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Uplink Channel EstimatorAfter interpolating the channel condition information for the DPCCH and DPDCH, channel compensation is performed by:

{ } ( ){ }{ } { }

( )

{ } ( ){ }{ }

*

*

Re Re

Re Im

Estimated DPDCH

Re Re

Re

d I Q

d d d d I Q

d I d Q

d I I Q Q

c Q I

c c c c Q I

c

h h jh

D R h R h jh

R h R h

D h h h h

h h jh

D R h R h jh

R

≡ +

= ⋅ = ⋅ −

= ⋅ + ⋅

= ⋅ ⋅ + ⋅

≡ − +

= ⋅ = ⋅ − −

= − { }

( )Im

Estimated DPCCH

Q c I

c Q Q I I

h R h

D h h h h

⋅ + ⋅

= ⋅ ⋅ + ⋅

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Downlink Channel EstimatorVariables are defined as follows:

Note that the I and Q branches of either DPCH or CPICH are spread to the chip rate by the same channelization code.

( ): Complex Pilot symbols at BS .

: Number of Pilot symbols (10 Symbols/Slot).

: Complex DPCH symbols at BS transmitter.

: Complex scrambling code.

: Complex channelization co

i q i q

p

i q

I Q

I Q

P P P P P

N

D D

S jS

C jC

+ = ≡

+

+

+ ( )de .

: Impulse response of complex fading channel.

: Received DPCH data symbol at UE.: Received CPICH symbols at UE.

I Q

I Q

d

PI P P

C C

h jh

RR R jR

=

+

= +

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Downlink Channel EstimatorAt BS, data is QPSK modulated. The spread symbols at the BS transmitter are:

After scrambling:

At UE, the received baseband signal is:

At UE, after descrambling:

( ) ( )I Q i I q QD jD D C j D C+ ≡ ⋅ + ⋅

( ) ( ) ( ) ( )I Q I Q I I Q Q I Q Q ID jD S jS D S D S j D S D S+ ⋅ + = ⋅ − ⋅ + ⋅ + ⋅

( ) ( )I I Q Q I Q Q I I QD S D S j D S D S h jh⎡ ⎤ ⎡ ⎤⋅ − ⋅ + ⋅ + ⋅ ⋅ +⎣ ⎦⎣ ⎦

( ) ( ) ( )( ) ( ) ( ) ( )

12 I Q I I Q Q I Q Q I I Q

I I Q Q I Q Q I I Q I Q

S jS D S D S j D S D S h jh

D h D h j D h D h D jD h jh

⎡ ⎤ ⎡ ⎤⋅ − ⋅ ⋅ − ⋅ + ⋅ + ⋅ ⋅ +⎣ ⎦⎣ ⎦

= ⋅ − ⋅ + ⋅ + ⋅ = + ⋅ +

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Downlink Channel EstimatorAfter despreading, we obtain (note that CI=CQ):

The channel condition information is estimated from the received pilot symbols:

where average is performed over Np pilot symbols.

( ) ( )( ) ( )

( ) ( )

DPCH:

CPICH:

d i I q Q i Q q I

i q I Q

PI P I Q P Q I

R D h D h j D h D h

D jD h jh

R R h h jR h h

= ⋅ − ⋅ + ⋅ + ⋅

= + ⋅ +

= − + +

{ } { }{ }

{ } { }{ }

1 Re Im2

1 Im Re2

p

p

I PI PINP

Q PI PINP

h R P R PN

h R P R PN

= ⋅ ⋅ + ⋅

= ⋅ ⋅ − ⋅

∑

∑

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Downlink Channel EstimatorAfter obtaining the channel condition information for the DPCH, channel compensation for the I branch is performed by:

{ } ( ){ }{ } { }

( ) ( )( ) ( )

*

I

Re Re

Re Im

Estimated DPCH

For p

d I Q

i d d d I Q

d I d Q

i I q Q I i Q q I Q

i I I Q Q q I Q Q I

h h jh

D R h R h jh

R h R h

D h D h h D h D h h

D h h h h D h h h h

≡ +

= ⋅ = ⋅ −

= ⋅ + ⋅

= ⋅ − ⋅ ⋅ + ⋅ + ⋅ ⋅

= ⋅ ⋅ + ⋅ + ⋅ ⋅ − ⋅

( )( )2 2

I

erfect channel estimation , :

Estimated DPCH

I I Q Q

i I Q

h h h h

D h h

= =

≈ ⋅ +

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Downlink Channel EstimatorAfter obtaining the channel condition information for the DPCH, channel compensation for the Q branch is performed by:

{ } ( ){ }{ } { }

( ) ( )( ) ( )

*

Q

Im Im

Im Re

Estimated DPCH

For pe

d I Q

q d d d I Q

d I d Q

i Q q I I i I q Q Q

q I I Q Q i Q I I Q

h h jh

D R h R h jh

R h R h

D h D h h D h D h h

D h h h h D h h h h

≡ +

= ⋅ = ⋅ −

= ⋅ − ⋅

= ⋅ + ⋅ ⋅ − ⋅ − ⋅ ⋅

= ⋅ ⋅ + ⋅ + ⋅ ⋅ − ⋅

( )( )2 2

Q

rfect channel estimation , :

Estimated DPCH

I I Q Q

q I Q

h h h h

D h h

= =

≈ ⋅ +

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WCDMA Channel Estimation Techniques

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Channel Estimation TechniquesNumerous approaches can be used for pilot aided channel estimation.We need to take into account at least the following factors:

The performance of the algorithms.The complexity relative to the current DSP processing power.

In WCDMA, channel estimation typically consists of two steps:Periodic channel sampling by virtue of known pilot symbols: the instantaneous channel estimate in each time slot is obtained by averaging the channel estimates calculated from each pilot symbol.Channel tracking (interpolation): the tracking of the time-varying fading channel is realized through a filter to obtain the channel estimates from the data symbols.

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Channel Estimation TechniquesIn general, the channel estimate from each pilot symbol is provided by multiplying the received pilot symbols rl(n,k) by the complex conjugate of the known pilot symbols:

where l is the multi-path index, Np is the number of pilot symbols and time slot index is k.To minimize the effects of noise and interference, an average is taken over the number of pilot symbols within a time slot:

( ) ( )*, , 1, 2,...,l l ph r n k p n k n N= ⋅ =

( ) ( )1

1 ,pN

l lnp

h k h n kN =

= ∑

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Simple AverageThis is a simple but effective method to estimate the channel complex gain.Simple average technique averages the received pilot symbols within one time slot.Simple average technique reduces the sampling rate from symbol rate to slot rate.

DPCCH Symbols

0 Pilot Pattern

Pilot FieldTiming

∫

SlotTiming

ChannelEstimate

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Weighted Multi Slot Average (WMSA)H. Andoh, M. Sawahashi, and F. Adachi, “Channel Estimation Using Time Multiplexed Pilot Symbols for Coherent Rake Combining for DS-CDMA Mobile Radio”, pp. 954—958, IEEE 1997.

( )1lh n −( )2lh n − ( )lh n ( )1lh n + ( )2lh n + ( )3lh n +

( )1α −( )2α − ( )0α ( )1α ( )2α ( )3α

( )lh n

Pilot symbols (n+1)th Slot (n+2)th Slot

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Weighted-Multislot AveragingFirst, instantaneous channel estimation is performed by using the Np pilot symbols belonging to the nth slot:

The channel estimate to be used for the n-th slot data is obtain by using 2K consecutive channel estimates:

where α(i)≤1 is the weighting factor.By properly choosing weighting factors, α(i), accurate channel estimation is possible, particularly in slow fading environments.

( ) ( )1

0

1 ,pN

l lmp

h n r n mN

−

=

= ∑

( ) ( )1

K

l li K

h i h n iα=− +

= ⋅ +∑

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Alpha Tracker

The alpha tracker can be thought of as an IIR low-pass filter. The difference equation describing the alpha track is:

The filter can be described in the z domain:( ) ( ) ( ) ( )1 1 1y n x n y nα α= − ⋅ − + ⋅ −

α D

1-α

X(n-1)

Y(n)Y(n)=(1-α)·X(n-1)+α·Y(n-1)

D

X(n) Y(n-1)

( ) ( )( )

( )1

1

11

−

−

−−

==zz

zXzYzH

αα

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Alpha Tracker

Substituting z with ejω, we can obtain the frequency response:

The cut-off frequency ωc of the IIR filter is given by:

( )( )

12 2

1

1 2 cosjH e ω α

α α ω

−=

+ −

( )ωα

ωω

cossintan 1

−=∠ −jeH

αααω

214cos

21 −+−

= −c

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Other Channel Estimation TechniquesThere are a variety of channel estimation techniques that can be applied:

Adaptive Filter TechniquesSimon Haykin, “Adaptive Filter Theory,” 3rd Edition, Prentice-Hall, 1996.

Wiener FilterB. Lindoff, C. Ostberg, and H. Eriksson, “Channel Estimation for the WCDMA Systems, Performance and Robustness Analysis from a Terminal Perspective,” IEEE Vehicular Technology Conference, Vol. 2, 1999.

Forward Linear PredictionS. Cacopardi, F. Gatti, and G. Reali, “Channel estimation using linear prediction for wireless indoor communications,” IEEE ICC 1995.

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Interpolation

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Newton’s Divided-Difference FormulaAssume values of function, f(x), are known at suitably specialized points, x0, x1, …, xn, which need not be equally spaced or taken in consecutive order.Newton’s divided-difference formula:

( ) ( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( )( )( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( )( )( ) ( )

0 0 0 1 0 1 0 1 2

0 1 1 0 1

0 1 0 1

1

0 0 0 1 0 1 0 1 2

0 1 1 0

, , ,

, , ,

, , , ,

, , ,

,

n n

n n

n n

n

n

f x f x x x f x x x x x x f x x x

x x x x x x f x x x

x x x x x x f x x x x

p x r x

p x f x x x f x x x x x x f x x x

x x x x x x f x

−

+

−

= + − + − − +

+ − − −

+ − − −

≡ +

= + − + − − +

+ − − − ( )( ) ( )( ) ( ) ( )

1

1 0 1 0 1

, ,

, , , , error termn

n n n

x x

r x x x x x x x f x x x x+ = − − − ≡

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Lagrange’s Interpolation FormulaLagrange’s interpolation formula is a polynomial of degree n(or less) since each term on the right hand side is a polynomialof degree n.When x=xi, i=0,1,…n, every fraction except the (i+1)th

vanishes because of the factor (x-xi).

( ) ( )( ) ( )( )( ) ( ) ( )

( )( ) ( )( )( ) ( ) ( )

( )( ) ( )( )( ) ( ) ( )

1 20

0 1 0 2 0

0 20

1 0 1 2 1

0 1 1

0 1 1

n

n

n

n

nn

n n n n

x x x x x xf x f x

x x x x x x

x x x x x xf x

x x x x x x

x x x x x xf x

x x x x x x−

−

− − −=

− − −

− − −+ +

− − −

− − −+

− − −

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Forward Gregory-Newton Interpolation Formula

When the points x0, x1, …, xn on which Newton’s divided-difference formula is based are regularly spaced with tabular interval h, it is generally more convenient to express Newton’s divided-difference formula in terms of ordinary differences.Let x=x0+rh and xk=x0+kh, we have x-xk=h(r-k) k=0,1,…,n.Therefore, (x-x0)(x-x1)…(x-xj)=hj+1r(r-1)…(r-j).Using:

We have, from Newton’s divided difference formula, Forward Gregory-Newton Interpolation formula:

( ) ( )1

00 1 1 1, ,...,

1 !

j

j j

ff x x xj h

+

+ +

∆=

+

( ) ( ) ( ) ( )( )2 30 0 0 0 0

1 1 22! 3!

r r r r rf x f x rh f r f f f

− − −≡ + = + ∆ + ∆ + ∆ +

1

1 11

k k k

n n nk k k

f f f

f f f+

− −+

∆ = −

∆ = ∆ −∆

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Backward Gregory-Newton Interpolation Formula

Forward Gregory-Newton Interpolation formula is especially adapted to interpolation for smaller value of x.Let x=x0+rh and xk=x0-kh, we have x-xk=h(r+k) k=0,1,…,n.Therefore, (x-x0)(x-x1)…(x-xj)=hj+1r(r+1)…(r+j).Using:

We have, from Newton’s divided difference formula, Backward Gregory-Newton Interpolation formula:

( ) ( )

11

0 1 1 1, ,...,1 !

jj

j j

ff x x x

j h

+− −

+ +

∆=

+

( ) ( ) ( ) ( )( )2 30 0 1 2 3

1 1 22! 3!

r r r r rf x f x rh f r f f f− − −

+ + +≡ + = + ∆ + ∆ + ∆ +

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Forward Newton-Gauss Interpolation Formula

Let x0=x0, x1=x0+h, x2=x0-h, x3=x0+2h, x4=x0-2h, …Using:

We have, from Newton’s divided difference formula, Forward Newton-Gauss Interpolation formula:

( ) ( )( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )( )

0

2 3 40 0 1 1 2

2 3 40 1/ 2 0 1/ 2 0

1 1 1 1 1 2...

2! 3! 4!1 1 1 1 1 2

...2! 3! 4!

f x f x rh

r r r r r r r r rf r f f f f

r r r r r r r r rf r f f f fδ δ δ δ

− − −

≡ +

− − + − + −= + ∆ + ∆ + ∆ + ∆ +

− + − + − −= + + + + +

( ) ( )

11

0 1 1 1, ,...,1 !

jj

j j

ff x x x

j h

+− −

+ +

∆=

+

1/ 2 1where .k k kf f fδ + +≡ −

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Backward Newton-Gauss Interpolation Formula

Let x0=x0, x1=x0-h, x2=x0+h, x3=x0-2h, x4=x0+2h, …We have, from Newton’s divided difference formula, Backward Newton-Gauss Interpolation formula:

We can also have Stirling’s interpolation formula:

( ) ( )( ) ( ) ( ) ( )( ) ( )

0

2 3 40 1/ 2 0 1/ 2 0

1 1 1 2 1 1...

2! 3! 4!

f x f x rh

r r r r r r r r rf r f f f fδ δ δ δ− −

≡ +

+ + − + + −= + + + + +

( ) ( )

( ) ( )

221/ 2 1/ 2

0 0 0

2 2 23 341/ 2 1/ 2

0

1! 2 2!1 1

...3! 2 4!

f fr rf x f x rh f f

r r r rf f f

δ δ δ

δ δ δ

−

−

+≡ + = + ⋅ +

− −++ ⋅ + +

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Fast Channel Estimation

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Fast Channel EstimationIn order for the power control to be effective, the TPC symbol must be decoded with minimum possible delay to adjust the transmit power so that the overall power control delay should be one time slot.Similar comment also applies for the SIR measurement since SIR must be calculated and compared to a set point to determine the TPC bits.The only difference between the fast channel estimation and the ordinary (slow) channel estimation is whether interpolation is used or not.As a result, fast channel estimation is obtained by averaging the Np pilot symbols within a time slot.

( ) ( )1

1 ,pN

l lnp

h k h n kN =

= ∑