Tài liệu Adaptive WCDMA (P4) doc

Effects of multipath fading on the nor-malized mean time to lose lock MTLL and tracking error versus early–late discriminator offsets /2 are shown in Figures 4.4 and 4.5, respectively..

of Chapter 3 is shown in Figure 4.1 The input signal is correlated with two locally erated, mutually delayed, replicas of the pseudonoise (PN) code After filtering, the useful

gen-component of the control signal e(t) will be proportional to

where Rc(δ) is the auto correlation of the sequence For the analysis of the tracking error

variance, results from the standard phase lock loop theory can be used directly [1]

In Code Division Multiple Access (CDMA) system, the input signal in Delay lock loop(DLL) will be a complete Direct Sequence Spread Spectrum (DSSS) signal In order toget rid of information, a noncoherent structure shown in Figure 4.2(a) may be used withthe simplest form of the input signal



Tc



− R2 c



δ+2

VCO

Loop filter

− + e(t, d)

Spreading waveform generator Spreading waveform clock

Power divider

(a)

Local oscillator

Low-pass filter

Low-pass filter Power

divider

Voltage controlled oscillator

gc

Loop filter

c ( ) 2

direct-sequence spread-spectrum systems IEEE Trans Commun., 46(11), 1516 – 1524, by

permission of IEEE (c) Comparisons of DLL and DLL/IC tracking loops [2].

PN code generator

Noncoherent square-law discriminator V.C.C Loopfilter

+ +

+ + Complex signal flows

(c)

Figure 4.2 (Continued ).

δ for a coherent loop The second term is degradation due to

the noncoherent structure Other modifications of the code-tracking loops like τ -dither

loop or double-dither loop can be seen in Reference [1]

4.1.1 Effects of multipath fading on delay-locked loops

In this section, the effects of a specular multipath fading channel on the performance of

a DLL are discussed For this type of environment, the two-path channel model becomes

h(τ )=√2P {δ(τ − τ1)ej θ1+ g2ej θ2δ(τ − τ1− τd)} ( 4.7) where θ1 is a constant phase shift, and g2 and θ2 are Rayleigh- and uniform-distributed

random variables, respectively When τd= 0, the channel becomes the familiar frequencynonselective Rician-fading model

In order to present some quantitative results, the following important system

param-eters are needed: the power ratio of the main path to the second path R =1/E[g  2

2], the

bit signal-to-noise ratio (SNR) (SNR in data bandwidth) γd = P T  b/N0, the loop SNR

γ L0 = P /N  0BL| = 1 and the ratio ς0 = γ L0/γd where Tb is the duration of an

infor-mation bit, and BL is the closed-loop bandwidth for the case when g2= 0 That is,

BL=−∞∞ |H(f )|2df where H (s) is the closed loop transfer function By using the

standard phase lock loop theory [3], the tracking error variance for this case has beenevaluated and the results are shown in Figure 4.3 Effects of multipath fading on the nor-malized mean time to lose lock (MTLL) and tracking error versus early–late discriminator

offsets /2 are shown in Figures 4.4 and 4.5, respectively.

Figures 4.4 and 4.5 demonstrate performance degradation of DLL due to the ence of multipath components In order to improve the system performance in such anenvironment, some research results are reported in which multipath IC is used

pres-The receiver block diagram is shown in Figure 4.2(b)

For the input signal received through L+ 1 equidistantly modeled paths, the upper half

of the block diagram is used to regenerate multipath interference (MPI) for each path

In the first step, input signal r(t) is correlated with L+ 1 delayed replica of the local

code to separate L + 1 narrowband signal components After processing delay TD, the

wideband components u0(t), , u L (t)are regenerated separately and summed up again

At this point r(t − TD) is created together with all individual components u l (t)available

separately Now in L + 1 branches, signal r(t − TD) − u l (t) = v l (t), representing the

Figure 4.3 Effects of multipath fading on the tracking error performance with various delay

multipath fading on delay locked loops for spread spectrum systems IEEE Trans Commun.,

Early-late discriminator offset ∆

Figure 4.4 Effects of multipath fading on the MTLL performance with various early – late

(1994) Effects of multipath fading on delay locked loops for spread spectrum systems IEEE

Trans Commun., 42(2/3/4), 1947 – 1956, by permission of IEEE.

Figure 4.5 Effects of multipath fading on the tracking error performance with various delay

multipath fading on delay locked loops for spread spectrum systems IEEE Trans Commun.,

IC will be visited again later in the context of multiuser detection in which in addition tothe multipath the multiple access interference (MAI) will be also present at the front end

of the receiver

4.1.2 Identification of channel coefficients

After code synchronization (acquisition and tracking), signal despreading can be

per-formed If the processing gain is large, Tb/Tc≥ 1, after despreading, the received low-passequivalent discrete time signal is

Table 4.1 Simulation parameters

• Sampling rate: 8 samples per chip period

process in each finger of the RAKE receiver If x k is a known training symbol and if the

SNR is high, a good estimate of c k can be easily computed from equation (4.8) as

c k ≈ y k /x k



where y k is the received signal However, most of the received symbols are not training

symbols In these cases, the available information for estimating c k can be based upon

prediction from the past detected data bearing symbols x i (i < k) This scheme will bereferred to as decision feedback adaptive linear predictor (DFALP)

Using a standard linear prediction approach we formulate the predicted fading channel

The block diagram of the receiver is shown in Figure 4.6

Delay DfT

Soft viteberbi decoder

Tentative decision

Adaptive linear predictor

nonselective fading channels using decision feedback and adaptive linear prediction IEEE Trans.

Commun., 43(2), 1484 – 1492, by permission of IEEE.

Figure 4.7 Recommended linear predictor order N and the number of LPF taps for the DFALP

algorithm [4] Reproduced from Liu, Y and Blostein, S (1995) Identification of frequency

nonselective fading channels using decision feedback and adaptive linear prediction IEEE Trans.

Commun., 43(2), 1484 – 1492, by permission of IEEE.

The updating process for the filter coefficients is defined as

Simulation results for predictor order N and the number of taps 2Df+ 1 of the low-passfilter for the minimum bit error rate (BER) are shown in Figure 4.7

4.2 CODE TRACKING IN FADING CHANNELS

The previously presented material on code tracking was based on the assumption thatexcept for the additive white Gaussian noise the channel itself does not introduce anyadditional signal degradation or that only a flat frequency nonselective fading per pathwas present For some applications like land and satellite mobile communications, wehave to take into account the presence of severe fading due to channel dynamics In thissection we will present one possible approach to code tracking in such an environment

where Ts is the Nyquist sampling interval for the transmitted signal, N β is the number

of received signal replicas through different propagation paths and β l (t) represents the

complex-valued time-varying channel coefficients So, for the transmitted signal s(t) the received signal r(k) sampled at t = kTs, will consist of N β mutually delayed replicas thatcan be represented as

signal-to-noise ratio, signal components are weighted with factors β l So the

synchroniza-tion for the RAKE receiver should provide a good estimate of delay τ and all channel intensity coefficients β l l = 0, 1, , N β− 1 The operation of the RAKE receiver will be

elaborated later and within this section we will concentrate on the joint channel (β l) and

code delay (τ ) estimation using the extended Kalman filter (EKF) [5,6].

For these purposes, the channel coefficients and delay are assumed to obey the ing dynamic model equations

follow-β  (k + 1) = α  β  (k) + w l (k) ; l = 0, 1, , N β− 1

where w l (k) and w τ (k)are mutually independent circular white Gaussian processes with

variances σ2

wl and σ2

τ, respectively In statistics, these processes are called autoregressive

(AR) processes of order k, where k shows how many previous samples with indices

(k, k − 1, k − 2, , k − K + 1) are included in modeling a sample with index k + 1 In

equation (4.17), the first-order AR model is used The more the disturbances in signal are

expected due to Doppler, the higher the variance of w l and the lower α l should be used

Variance of w τ will not only depend on Doppler but also on the oscillator stability Acomprehensive discussion of AR modeling of wideband indoor radio propagation can befound in Reference [7]

4.2.2 Joint estimation of PN code delay and multipath using the EKF

From the available signal samples r(k) given by equation (4.15) we are supposed to find the minimum variance estimates of β l and τ These will be denoted by

ˆβ l (k |k) = E{β l (k) |r(k)}

where r (k) is a vector of signal samples

From equation (4.15) one can see that r(k) is linear in the channel coefficients β l (k),

but it is nonlinear in the delay variable τ (k) A practical approximation to the minimum

variance estimator in this case is the EKF This filter utilizes a first-order Taylor’s seriesexpansion of the observation sequence about the predicted value of the state vector, andwill approach the true minimum variance estimate only if the linearization error is small.The basic theory of extended Kalman filtering is available in textbooks [6] Having inmind that in the delay-tracking problem, the state model is linear, while the measurementmodel is nonlinear, we have

3 4

Channel characteristics

Channel zeros

Figure 4.8 Simulation examples – PN code in multipath [5] Reproduced from Iltis, R (1994)

An EKF-based joint estimator for interference, multipath, and code delay in a DS

spread-spectrum receiver IEEE Trans Commun., 42, 1288 – 1299, by permission of IEEE.

By using general results of the EKF theory [6], we have

−0.5

(b)

Figure 4.9 (a) Iteration number tracking error trajectory for E b /N0 = 10 dB – Channel A,

from Iltis, R (1994) An EKF-based joint estimator for interference, multipath, and code delay in

a DS spread-spectrum receiver IEEE Trans Commun., 42, 1288 – 1299, by permission of IEEE.

Tracking error

0.0 0.5 1.0

Figure 4.10 (a) Iteration number tracking error trajectory for E b /N0 = 40 dB – Channel A,

Reproduced from Iltis, R (1994) An EKF-based joint estimator for interference, multipath, and

code delay in a DS spread-spectrum receiver IEEE Trans Commun., 42, 1288 – 1299, by

permission of IEEE.

0.0

0.2 0.3 0.4 0.5

Q = diag [σ2

τ , σ w20, , σ w22, , σ wN2 β−1] ( 4.24)

For the two examples of the channel transfer function shown in Figure 4.8, simulationresults of the tracking error are shown in Figures 4.9 and 4.10

4.3 SIGNAL SUBSPACE-BASED CHANNEL

ESTIMATION FOR CDMA SYSTEMS

In this section we present a multiuser channel estimation problem through a signal

subspace-based approach [8] For these purposes, the received signal for K users will

If phase-shift keying (PSK) is used to modulate the data, then the baseband complex

envelope representation of the kth user’s transmitted signal is given by

s k (t)=2P kej φ k

i

ej ( 2π/M)m (i) k a k (t − iT ) ( 4.27)

where P k is the transmitted power, φ k is the carrier phase relative to the local oscillator

at the receiver, M is the size of the symbol alphabet, m (i) k ∈ {0, 1, , M − 1} is the transmitted symbol, a k (t) is the spreading waveform and T is the symbol duration The

spreading waveform is given by

n = 0, 1, , N − 1 is a signature sequence (possibly complex valued since the signature

alphabet need not be binary) The chip-matched filter can be implemented as an and-dump circuit, and the discrete time signal is given by

Thus, the received signal can be converted into a sequence of wide sense stationary (WSS)

random vectors by buffering r[n] into blocks of length N

y i = [r(iN)r(1 + iN) · · · r(N − 1 + iN)]T∈ CN ( 4.30) where the nth element of the ith observation vector is given by y i,n = r(n + iN) Although

each observation vector corresponds to one symbol interval, this buffering was done out regard to the actual symbol intervals of the users Since the system is asynchronous,

with-each observation vector will contain at least the end of the previous symbol (left) and thebeginning of the current symbol (right) for each user The factors due to the power, phase

and transmitted symbols of the kth user may be collected into a single complex constant

c (i) k , for example, some constant times√

where η i = [η i,0, , η i,N−1]T∈ CN is a Gaussian random vector Its elements are zero

mean with variance σ2= N0/ 2Tc and are mutually independent

Vectors ur

k and ul

k are the right side of the kth user’s code vector followed by zeros, and zeros followed by the left side of the kth user’s code vector, respectively.

In addition, we have defined c i = [c (i −1)

1 c1(i) c K (i −1) c (i) K]T∈ C2K and the signal matrix

A = ur

1ul

1 u r

Kul

K ∈ CN × 2K We will start with the assumption that each user’s

sig-nal goes through a single propagation path with an associated attenuation factor andpropagation delay We assume that these parameters vary slowly with time, so that forsufficiently short intervals the channel is approximately a linear time-invariant (LTI) sys-tem The baseband channel impulse response can then be represented by a Dirac delta

function as h k (t, τ ) = h k (t) = α k δ(t − τ k ), ∀τ where α k is a complex-valued attenuation

weight and τ k is the propagation delay Since there is just a single path, we assume that

α k is incorporated into c (i) k and concentrate solely on the delay

Let us define v ∈ {0, , N − 1} and γ ∈ [0, 1) such that (τ k /Tc) mod N = v + γ If

γ = 0, the received signal is precisely aligned with the chip matched filter and only onechip will contribute to each sample, the signal vectors become

For the more general case of a multipath transmission channel with L distinct propagation

paths, the impulse response becomes a series of delta functions

The signal vectors can be represented as

where the ak’s are as defined in equation (4.32), then the signal vectors may be expressed

as a linear combination of the columns of these matrices

ur k= Ur

khk

ul k= Ul

where hk is the composite impulse response of the channel and the receiver front end,

evaluated modulo, the symbol period Thus, the nth element of the impulse response is

For delay spread T m < T /2, at most two terms in the summation will be nonzero

4.3.1 Estimating the signal subspace

The correlation matrix of the observation vectors is given by

R= E[y iy†i]

where C= E[c i c i†]∈ C2K ×2K is diagonal The correlation matrix can also be expressed

in terms of its eigenvector decomposition

R = VDV†

( 4.40)

where the columns of V∈ CN ×N are the eigenvectors of R, and D is a diagonal matrix

of the corresponding eigenvalues (λ n) Details of eigenvector decomposition are given inthe appendix Furthermore,

λ n=



d n + σ2, if n ≤ 2K

where d n is the variance of the signal vectors along the nth eigenvector and we assume

that 2K < N Since the 2K largest eigenvalues of R correspond to the signal subspace, V

can be partitioned as V = [VSVN], where the columns of VS= [vS,1, , v S, 2K]∈ CN ×2K form a basis for the signal subspace S Y and VN = [vN,1, , v N,N −2K]∈ CN ×N−2K spansthe noise subspace N Y Readers less familiar with eigenvalues decomposition are referred

to the appendix Since we would like to track slowly varying parameters, we form a

moving average or a Bartlett estimate of the correlation matrix based on the J most

It is well known [9] that the maximum-likelihood (ML) estimate of the eigenvalues and

associated eigenvectors of R is just the eigenvector decomposition of ˆ Ri Thus, we perform

an eigenvalue decomposition of ˆRi and select the eigenvectors corresponding to the 2K

largest eigenvalues as a basis for ˆS Y

k=1

λ k (σ2− λ k )2vS,kv†S,k



( 4.47)

Therefore, within an additive constant, the log-likelihood function of ˜ek is

The exact VN and Q are unknown, but we may replace them with their estimates The best

estimates will minimize ˜ek, which will result in the maximum of the likelihood function.Unfortunately, maximizing this likelihood function is prohibitively complex for a gen-eral multipath channel, so we will consider only a single propagation path In this case,

the vector uk is a function of only one unknown parameter: the delay τ k To form thetiming estimate, we must solve

ˆτ k= arg max

Ideally, we would like to differentiate the log-likelihood function with respect to τ However, the desired user’s delay lies within an uncertainty region, τ k ∈ [0, T ], and

uk (τ ) is only piecewise continuous on this interval To deal with these problems, we

divide the uncertainty region into N cells of width Tc and consider a single cell, c ν ≡

[νTc, (ν + 1)Tc) We again define ν ∈ {0, , N − 1} and γ ∈ [0, 1) such that (τ/Tc)

mod N = ν + γ , and for τ ∈ c ν the desired user’s signal vector becomes

Thus, within a given cell, we can differentiate the log-likelihood function and solve for

the maximum in closed form We then choose whichever of the N -solutions that yields

the largest value for equation (4.48) Details can be found in Reference [8]

Under certain conditions, it may be possible to simplify this algorithm Note thatmaximizing the log-likelihood function (4.48) is equivalent to maximizing

(˜ek )≈ −u

†

kVNV†Nuk

This yields a much simpler expression for the stationary points [8]

The MUSIC (multiple signal classification) algorithm is equivalent to equation (4.53)when one only maximizes the numerator and ignores the denominator, that is, one assumes

u†kQuk is equal to one in equation (4.52) or (4.53) This yields an even simpler mation for the log-likelihood function

MUSIC Approx ML

(b)

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Approx ML MUSIC

Figure 4.11 (a) Probability of acquisition for the maximum-likelihood (ML) estimator, the

(b) Root mean-squared error (RMSE) of the delay estimate in chips for the ML estimator, the

Reproduced from Bensley, J S and Aazhang, B (1996) Subspace-based channel estimation for

code division multiple access communications & systems IEEE Trans Commun., 44(8),

1009 – 1020, by permission of IEEE.