1 joint channel and doppler offset estimation in dynamic

1

Joint Channel and Doppler Offset Estimation in

Dynamic Cooperative Relay NetworksIdo Nevat1, Gareth W. Peters 2,3, Arnaud Doucet 4 and Jinhong Yuan5

1 Wireless & Networking Tech. Lab, CSIRO, Sydney, Australia.2 Department of Statistical Science, University College London, London, UK.

3 CSIRO Mathematical and Information Sciences, Sydney, Australia.4 Department of Statistics, Oxford, UK.

5 School of Electrical Engineering,University of NSW, Sydney, Australia.

Abstract

We propose new algorithms to perform joint channel and Doppler offset estimation in dynamic

cooperative wireless relay networks. We first formulate the problem as a Bayesian non-linear state

space model, then develop two algorithms that are based on different sampling methodologies to jointly

estimate the dynamic channels and static Doppler offsets. The first is based on a MCMC-within-Gibbs

sampling methodology and the second is obtained via a novel development of a Particle Adaptive

Marginal Metropolis-Hastings method. We show that the Particle Adaptive Marginal Metropolis-Hastings

algorithm requires only a fraction of the computational complexity of the MCMC-within-Gibbs sampler.

In addition, we develop a version of the recursive marginal Cramer-Rao lower bound on the path space

which involves marginalistion of the static parameters. Simulation results demonstrate that the proposed

algorithm performs close to the Marginal Cramer-Rao Lower Bound.

Index Terms

Channel Estimation, Doppler Offset Estimation, Cooperative Relay Networks, Particle filter, MCMC.

I. INTRODUCTION

The relay channel, first introduced by van der Meulen [1], has recently received considerable attention

due to its potential in wireless applications. Relaying techniques have the potential to provide spatial

diversity, improve energy efficiency, and reduce the interference level of wireless channels, see [2], [3].

This work was presented in part at the 2011 IEEE VTC Conference in San Francisco, USA.

November 15, 2012 DRAFT

2

In order to utilise the relay channel, an accurate channel state information (CSI) is required at the

destination. When the communicating terminals are mobile, channels form a cascade of mobile-to-mobile

channels, and can change rapidly with time. Therefore, a dynamic time-varying channel model is required.

However, most current works concentrate on performing relay-based channel estimation (and assuming

no Doppler offset is present) in a static environment, where the channels are assumed to be block fading:

in [4], the authors designed two linear estimators, that is the Least Squares (LS) and Linear Minimum

Mean Squared Error (LMMSE) for Amplify and Forward (AF) based relay networks; in [5], an algorithm

for LMMSE channel estimation in OFDM based relay systems was derived; and in [6], a training based

LMMSE channel estimator for time division multiplex AF relay networks was proposed. A non-linear

state-space model was developed in [7] to perform a sub-optimal (linear) estimation (via Kalman filter)

under the assumption that the model parameters are known. The authors also reported that the relay

speed has a significant impact on the BER performance, and therefore it is important to estimate those

parameters accurately in cases where the Doppler offset is a-priori unknown.

The problem of joint channel tracking and Doppler offset estimation is of practical importance in

wireless communications [8], [9] [10], where in many networks the relays may be mobile. In these cases,

high user mobility leads to large and time varying Doppler frequencies at the destination. In relay wireless

links this problem has not been addressed yet and it involves jointly estimating the Doppler offset for

each link in the relay system, which we refer to as static parameters, and the channel tracking problem

in non-linear state-space model.

To solve this problem, we develop an efficient framework to perform Bayesian inference with respect

to filtering and parameter estimation in the resulting high-dimensional space. This involves development

of advanced Monte Carlo techniques which combine the strength of non-linear filtering frameworks,

such as Sequential Monte Carlo (SMC), with adaptive versions of Markov chain Monte Carlo (MCMC)

techniques [11].

In particular, we make the following contributions:

1) We propose a statistical model to address the inference problem of channel tracking and Doppler

offset estimation in relay networks, such that all quantities are estimated at the destination.

2) We develop two novel algorithms for joint channel tracking and Doppler estimation which are

based on the following methodologies:

∙ We develop a Rao-Blackwellised adaptive Particle MCMC (PMCMC) algorithm. The adaptive

MCMC algorithm we design provides an improved mixing rate, thus making the algorithm


3

more efficient (presented in Section V).

∙ We design a Gibbs sampler by splitting the high dimensional posterior distribution into sub-

blocks of parameters and then running a blockwise MCMC-within-Gibbs sampling framework

(presented in Section IV).

3) We perform complexity analisys and show that the PMCMC algorithm is much more efficient

and requires only a fraction of the computational complexity of the Gibbs sampler for the same

estimation accuracy.

4) We derive the marginal Cramer-Rao Lower Bound (CRLB) for the dynamic state-space model and

demonstrate that the proposed algorithm performs close to the bound.

Notation: random variables are denoted by upper case letters and their realizations by lower case letters;

bold face to denote vectors and non-bold for scalars; super script will be used to refer to the index for

a particular relay in the network; sub-script will denote discrete time, where ℎ1:T denotes ℎ1, . . . , ℎT ;

and in the sampling methodology combining MCMC and particle filtering we use the following notation

[⋅](j, i) to denote the j-th state of the Markov chain for the i-th particle in the particle filter. In addition,

we denote the proposed Markov chain state by [⋅]∗(j, i), and � (⋅) denotes the delta of Dirac.

II. SYSTEM DESCRIPTION

We consider the case where one mobile station is transmitting to a Base Station (BS) via L relays,

which may be mobile or stationary, see Fig. 1. We consider frequency-flat fading channels, and the

extension to multi-path channels is straight forward, for example via the use of OFDM modulation.

A. Statistical Model of Relay Network

i. Assume a wireless relay network with one mobile source node, transmitting symbols in frames of

length T to a Base Station (BS) via L mobile or stationary relays.

ii. We consider a half duplex transmission model in which the data for a given frame are transmitted

via a two step procedure. In the first step, the source node broadcasts a frame to all the relay

nodes. In the second step, the relay nodes transmit the processed frame, termed the relay signals,

to the destination node in orthogonal fashion, i.e. non-interfering channels via time division or

frequency devision multiplex, see for example [12], [13].

vi. The time-varying channels are parametrized under a Gauss-Markov model [14], [15]. The l-th

relay channel is modeled as a two stage latent stochastic process, in which at time n we denote


4

the realization of the two channel stages by ℎ(l)n and g

(l)n .

H(l)n = �(l)H

(l)n−1 +

√1−

(�(l))2Υ(l)

n

G(l)n = �(l)G

(l)n−1 +

√1−

(�(l))2Ω(l)n ,

(1)

where Υ(l)n ∼ CN (0, 1), Ω(l)

n ∼ CN (0, 1).

vii. The velocities of both the user and the relays are assumed constant over a frame and follow a

uniform distribution V(l)M→R ∼ U [0, vmax] and V

(l)R→D ∼ U [0, vmax], respectively, where vmax is a

practical upper bound.

viii. The channel coefficients �(l) and �(l) are modelled according to Jakes model as [16]

�(l) = J0

(2�

V(l)M→Rfc

cTs

)= J0

(2�F

(l)M→RTs

),

�(l) = J0

(2�

V(l)M→Rfc

cTs

)J0

(2�

V(l)R→Dfcc

Ts

)= J0

(2�F

(l)M→RTs

)J0

(2�F

(l)R→DTs

),

(2)

where J0 is the zeroth-order Bessel function of the first kind, fc is the carrier frequency, c is the

speed of light and Ts is the symbol duration [17], [18].

viii. The received signal at the l-th relay is a random variable given by

R(l)n = snH

(l)n +W (l)

n , l ∈ {1, . . . , L} , (3)

where at time n, H(l)n is the channel coefficient between the transmitter and the l-th relay, sn is

the transmitted pilot symbol and W(l)n is the noise realization associated with the relay receiver.

ix. The received signals at the destination is a random variable given by

Y (l)n = f (l)

(R(l)

n , n)G(l)

n + V (l)n , l ∈ {1, . . . , L} , (4)

where at time n, G(l)n is the channel coefficient between the l-th relay and the receiver. Generically,

the model we develop is general enough to allow for many different possible relay functions,

generally any mapping of the form f (l)(R(l), n

): ℝ 7→ ℝ, for any continous d-differentiable

functions f (l)(R(l), n

)∈ ℂd[ℝ] on the real line. As an example consider a relay function given by

the popular Amplify-and-Forward with constant gain [4], [6], with function

f (l)(R(l), n

)=

√√√⎷1

�2ℎ + �2

g +�2v

E[∣sn∣2]

(5)

where E

[∣sn∣2

]is the average symbols power.


5

x. All received signals are corrupted by i.i.d. zero-mean additive white complex Gaussian noise

(AWGN). At the l-th relay at the n-th transmitted symbol is denoted by W(l)n ∼ CN

(0, �2

w

).

Then at the receiver this is denoted by V(l)n ∼ CN

(0, �2

v

).

Based on the aforementioned assumptions, the following state-space model is presented by the graphical

model structure in Fig. 2, and is expressed as⎧⎨⎩

H(l)n = �(l)H

(l)n−1 +

√1−

(�(l))2Υ

(l)n , (State equation I)

G(l)n = �(l)G

(l)n−1 +

√1−

(�(l))2Ω(l)n , (State equation II)

Y(l)n = f (l)

(snH

(l)n +W

(l)n

)G

(l)n + V

(l)n (Observation equation) .

(6)

B. Channel and Doppler Offset estimation formulation

In order to formulate the problem, the distribution of transformed random variables J0

(2�F

(l)M→RTs

)

and J0

(2�F

(l)R→DTs

)need to be derived. To this end we recall that we used a uniform prior distribution

for the velocities of both mobile and relay terminals (point vii. of Section II-A). To accommodate for a

wide range of velocities, carrier frequencies and symbol rates, we set vmax = c�fcTs

.

Next, in the following Lemma we present the distribution functions of J0(2�F

(l)M→RTs

)and J0

(2�F

(l)R→DTs

).

Lemma 1: Under a uniform prior distribution of the mobile velocity, V ∼ U [0, vmax], its Doppler

offset follows a Beta distribution as follows:

J0

(2�

V fcc

Ts

)∼ Beta (1, 1/2) . (7)

Proof: We expend J0

(2� V fc

cTs

)via Taylor Series expansion around 0:

J0

(2�V fcTs

c

)=

∞∑

m=0

(−1)m

m!Γ (m+ 1)

(1

2

2�V fcTs

c

)2m

≈K∑

m=0

(−1)m

m!Γ (m+ 1)

(�V fcTs

c

)2m

≈︸︷︷︸K=1

1

Γ (1)− 1

Γ (2)

(�V fcTs

c

)2

= 1−(�V fcTs

c

)2

.

(8)

Next, we note that(�V fcTs

c

)∼ U [0, 1], and therefore, by utilising the power function distribution, we

obtain that(�V fcTs

c

)2∼ Beta (1/2, 1). Then, by using the mirror-image symmetry of the Beta distribution,

we obtain that(1−

(�V fcTs

c

)2)∼ Beta (1, 1/2).


6

Next, the posterior distribution can be expressed as:

p (�,�,g1:T ,h1:T ∣y1:T )

∝ p (y1:T ∣�,�,g1:T ,h1:T ) p (g1:T ,h1:T ∣�,�) p (�,�)

=

L∏

l=1

[T∏

n=1

∫p(y(l)n ∣�(l), �(l), g(l)n , ℎ(l)n , w(l)

n

)p(w(l)n

)dw(l)

n p(ℎ(l)n ∣ℎ(l)n−1

)p(g(l)n ∣g(l)n−1

)]

×L∏

l=1

p(g(l)1

)p(ℎ(l)1

)p(�(l))p(�(l))

=

L∏

l=1

⎡⎢⎣

T∏

n=1

1√2��2

v

∫exp

⎧⎨⎩−

∣∣∣y(l)n − f(snℎ

(l)n + w

(l)n

)g(l)n

∣∣∣2

2�2v

⎫⎬⎭

1√2��2

w

exp

⎧⎨⎩−

∣∣∣w(l)n

∣∣∣2

2�2w

⎫⎬⎭

dw(l)n

× 1√2�(1−

(�(l))2)

exp

⎧⎨⎩−

∣∣∣g(l)n − �(l)g(l)n−1

∣∣∣2

2(1−

(�(l))2)

⎫⎬⎭

1√2�(1−

(�(l))2)

exp

⎧⎨⎩−

∣∣∣ℎ(l)n − �(l)ℎ(l)n−1

∣∣∣2

2(1−

(�(l))2)

⎫⎬⎭

⎤⎥⎥⎦

× 1√2�

L∏

l=1

exp

{−1

2

∣∣∣g(l)1

∣∣∣2}

1√2�

L∏

l=1

exp

{−1

2

∣∣∣ℎ(l)1∣∣∣2}

×L∏

l=1

1

B (a, b)

(�(l))a−1 (

1− �(l))b−1

L∏

l=1

1

B (c, d)

(�(l))c−1 (

1− �(l))d−1

(9)

where �(1:L), �(1:L), g(1:L)1:T , ℎ

(1:L)1:T , y

(l:L)1:T ≜ �,�,g1:T ,h1:T ,y1:T .

Next we define the quantities of interest. Based on the marginal posterior in (9) we have:

MAP estimation(�MAP, �MAP, hMAP

1:T , gMAP1:T

)= argmax

�,�,g1:T ,h1:T

p (�,�,g1:T ,h1:T ∣y1:T ) . (10)

MMSE estimation(�MMSE, �MMSE, hMMSE

1:T , gMMSE1:T

)=

∫. . .

∫p (�,�,g1:T ,h1:T ∣y1:T ) � � g1:T h1:T d� d� dh1:T dg1:T .

(11)

Since the posterior model in (9) is very high dimensional with L(3T+2) parameters and highly nonlinear,

solving (10-11) requires advanced computational tools that will be the scope of the proposed algorithms.

Corollary 1: The posterior distribution in (9) factorises according to the following independence

structure

p (�,�,g1:T ,h1:T ∣y1:T ) =

L∏

l=1

p(�(l), �(l), g

(l)1:T , ℎ

(l)1:T ∣y

(l)1:T

), (12)


7

with respect to the number of parallel relay transmission paths.

Corollary 1 enables us to exploit the structure of the model in the design of the estimation framework.

In particular, the problem admits a block factorisation structure in which the particle filters per block may

be run independently. Thus, we are able to estimate the proposal distribution for the channels trajectories,

via parallel independent particle filters. Therefore, the variance of the incremental important weights used

in the estimation of the proposal can be reduced leading to an increased PMCMC acceptance probability.

Therefore, under the model and estimation procedure proposed, increasing the number of relays

will not degrade the solution.

C. The need for augmented Bayesian posterior

In order to perform inference, the likelihood function in (9), p(y(l)n ∣�(l), �(l), g

(l)n , ℎ

(l)n

), needs to be

evaluated. To achieve that we write the marginal distribution at the l-th relay as

pR

(l)n

(r∣sn, g(l)n , ℎ(l)n

)= p

(snℎ

(l)n + w(l)

n ∣sn, ℎ(l)n , g(l)n

)= CN

(snℎ

(l)n , �2

w

).

Then, finding the distribution of the random variable after the relay function is applied i.e. the distribution

of f(r(l)n

)≜ f

(r(l)n

)G

(l)n given sn, ℎ

(l)n , g

(l)n , involves the following change of variable formula

p(f(r(l)n

)∣sn, ℎ(l)n , g(l)n

)= p

R(l)n

((f (l))−1

(r(l)n

)∣sn, ℎ(l)n , g(l)n

) ∣∣∣∣∣∂f (l)

∂r(l)n

∣∣∣∣∣

−1

,

which can not always be written down analytically for an arbitrary relay function. The second complication

is that even if f(r(l)n

)is known, one must then solve the following convolution to obtain the likelihood:

p(y(l)n ∣sn, g(l)n , ℎ(l)n

)= p

(f(r(l)n

)∣sn, g(l)n , ℎ(l)n

)⊗ p

V(l)n

=

∫ ∞

−∞p(f(z∣sn, g(l)n , ℎ(l)n

))pV

(l)n

(y(l)n − z

)dz,

where ⊗ denotes the convolution operation. Typically this will be intractable to evaluate pointwise.

To circumvent this difficulty we utilise an augmented Bayesian posterior p (�,�,g1:T ,h1:T ,w1:T ∣y1:T ),

containing auxiliary variables W1:T which we marginalise out numerically in our sampling algorithm to

obtain the posterior corresponding to (9).

Under the augmented model, the likelihood function can now be expressed as

p(y(l)n ∣sn, g(l)n , ℎ(l)n , w(l)

n

)∼ CN

(f (l)

(snℎ

(l)n + w(l)

n

)g(l)n , �2

v

).


8

III. OVERVIEW OF MCMC METHODS

We now present a short overview of Markov Chain Monte Carlo (MCMC) methods that will be used

in the development of the two algorithms in Sections IV-V. MCMC is a class of Monte Carlo methods

for obtaining a sequence of random samples from a probability distribution for which direct sampling is

difficult. MCMC methods generate samples by running a reversible chain whose equilibrium distribution

is the desired target distribution [11]. An aperiodic and irreducible Markov chain has the property of

converging to a unique stationary distribution, �(x∣y), which does not depend on the initial sample. After

convergence, a MCMC method produces samples from the target distribution, �(x∣y), so that :

�(x∣y) = 1

J

J∑

j=1

� (x− x (j)) , (13)

where J is the number of samples generated. The most popular MCMC algorithm is based on Metropolis-

Hastings (MH) method. It uses a Markov transition kernel (proposal density), q(x∗;x (j − 1)), which

depends on the current state x (j − 1), to generate a new proposed sample x∗ and then accepts it

probabilistically via a rejection step. The generic MH algorithm is described by the following steps:

Algorithm 1 Generic Metropolis-Hastings MCMC Algorithm

1: Initialise x (1) to an arbitrary starting point.

2: Repeat steps (3− 6) for j = 2, . . . , J

3: Sample x∗ ∼ q (x∗;x (j − 1)) via a proposal density.

4: Compute the acceptance probability: A (x∗,x (j − 1)) = min{1, p(x∗∣y)

p(x(j−1)∣y) ×q(x(j−1);x∗)q(x∗;x(j−1))

}.

5: Generate a uniform random variable u ∼ U [0, 1] .

6: If u ≤ A (x∗,x (j − 1)) then x (j) = x∗ else x (j) = x (j − 1) end if.

The accuracy of the estimate in (10-11) depends on the choice of the Markov transition kernel

q(x∗;x (j − 1)). Choosing an appropriate proposal is particularly challenging in high dimensions and

when strong posterior correlation is present, as in our case, see (9).

IV. ALGORITHM I: JOINT CHANNEL AND DOPPLER ESTIMATION VIA MCMC-WITHIN-GIBBS

We develop a novel algorithm to estimate the MAP and MMSE estimates in (10-11) via a Gibbs

sampler framework [19]. The Gibbs sampler we develop involves splitting the high dimensional posterior

distribution into subblocks of parameters and then running a blockwise MCMC-within-Gibbs sampling


9

framework. In the proposed Gibbs sampling framework with the posterior p (�,�,g1:T ,h1:T ,w1:T ∣y1:T ),

sampling is obtained by splitting the vector of latent states into k sub-blocks of length � , where k� = T .

Then we run a blockwise MCMC-within-Gibbs sampling framework, where each iteration of the Markov

chain updates each sub-block of the states until a Markov chain of length J is obtained.

The MCMC-within-Gibbs sampler can be summarised in Algorithm 2, where all required posterior

distributions including the full conditional distributions are derived in Appendix I:

Algorithm 2 Joint Channel and Doppler Estimation via MCMC-within-Gibbs

1: Initialise [�,�,g1:T ,h1:T ,w1:T ](1) to an arbitrary starting point.

2: for j = 2, . . . , J do

3: Sample [�,�](j + 1) ∼ p(�,�∣[g1:T ,h1:T ,w1:T ](j), y

(l:L)1:T

)via a MH proposal and evaluate

acceptance probability according to (36-37) [Appendix I].

4: Repeat k = 1 to K:

Sample [G(k−1)�+1:k� ](j+1) ∼ p(g(k−1)�+1:k� ∣[�,�,g1:(k−1)� ](j + 1), [gk�+1:T ,h1:T ,w1:T ](j), y

(l:L)1:T

)

via a MH proposal and evaluate acceptance probability in (38-39) [Appendix I].


Sample [h(k−1)�+1:k� ](j+1) ∼ p(h(k−1)�+1:k� ∣[�,�,h1:(k−1)� ,g1:T ](j + 1), [hk�+1:T ,w1:T ](j), y

(l:L)1:T

)



Sample [w(k−1)�+1:k� ](j+1) ∼ p(w(k−1)�+1:k� ∣[�,�,w1:(k−1)� ,h1:T ,g1:T ](j + 1), [wk�+1:T ](j), y

(l:L)1:T

)


7: end for

The efficiency of the block Gibbs sampling algorithm is dependent on the choice of blocking of the

posterior parameters, the size of the blocks updated at each stage and the sampling mechanism for each

block. It is a significant challenge in practice to design algorithms which are efficient in such block Gibbs

settings when correlation is present between the parameters of the posterior distribution, as occurs in (9).

As a result for moderate sized values of � the MH-within-Gibbs framework can be poorly mixing, due

to low acceptance probabilities, even when carefully designed proposals are implemented. This leads to

requirements for very long Markov chain lengths which is not practical.


10

V. ALGORITHM II: JOINT CHANNEL AND DOPPLER ESTIMATION VIA PMCMC

To overcome the shortcomings of the MCMC-within-Gibbs based algorithm developed in the previous

section, we develop a novel algorithm which is based on adaptive Particle MCMC (PMCMC) methodology

[20]. This will lead to a much more efficient sampling algorithm from the posterior distribution given

in (9). In particular, we consider the Marginal Metropolis-Hastings within Rao-Blackwellised particle

filter. This filter targets the full conditional posterior in (9) by operating on the factorisation of the joint

space (�,�,g1:T ,h1:T ) in such a way that would allow for an efficient sampling from this posterior

distribution. The key advantage of the PMCMC algorithm is that it allows one to jointly update the entire

set of posterior parameters at each iteration and only requires calculation of the marginal acceptance

probability in the Metropolis-Hastings algorithm. PMCMC achieves this by embedding a particle filter

estimate of the optimal proposal distribution for the latent process into the MCMC algorithm. This allows

the Markov chain to mix efficiently in the high dimensional posterior parameter space due to the particle

filter approximation of the optimal proposal distribution in the MCMC algorithm, thereby allowing high-

dimensional parameter block updates even in the presence of strong posterior parameter dependence.

To develop the algorithm we begin by writing the MH acceptance probability for the augmented model

(�,�,h1:T ,g1:T ,w1:T ). This corresponds to Step 4 in the MCMC algorithm presented in Section III.

A([�,�,h1:T ,g1:T ,w1:T ]

∗

; [�,�,h1:T ,g1:T ,w1:T ] (j))

=min

(1,

p([h1:T ,g1:T ,w1:T ,�,�]

∗ ∣y1:T

)

p ([h1:T ,g1:T ,w1:T ,�,�] (j) ∣y1:T )× q

([h1:T ,g1:T ,w1:T ,�,�] (j) ; [h1:T ,g1:T ,w1:T ,�,�]

∗)

q([h1:T ,g1:T ,w1:T ,�,�]

∗

; [h1:T ,g1:T ,w1:T ,�,�] (j)))

=min

(1,

p([h1:T ,g1:T ,w1:T ]

∗ ∣ [�,�]∗

,y1:T

)p([�,�]

∗ ∣y1:T

)

p ([h1:T ,g1:T ,w1:T ] (j) ∣ [�,�] (j) ,y1:T ) p ([�,�] (j) ∣y1:T )

×q([h1:T ,g1:T ,w1:T ,�,�] (j) ; [h1:T ,g1:T ,w1:T ,�,�]

∗)

q([h1:T ,g1:T ,w1:T ,�,�]

∗

; [h1:T ,g1:T ,w1:T ,�,�] (j))).

(14)

Next we specify the Markov transition kernel, q (x∗;x (j − 1)). We propose a particular choice of proposal

that will provide a significant dimension reduction in evaluation of the acceptance probability. Our choice

to move from a state at iteration j to a new state at iteration (j + 1) is split into two components:

q([h1:T ,g1:T ,w1:T ,�,�]

∗

; [h1:T ,g1:T ,w1:T ,�,�] (j))= p

([h1:T ,g1:T ,w1:T ]

∗ ∣ [�,�]∗

,y1:T

)q([�,�]

∗ ∣ [�,�] (j)).

(15)

The first component involves the sampling of a trajectory for g1:T ,h1:T ,w1:T , while the second component

involves a Markov transition kernel to sample the static parameters �,�. Plugging (15) into the acceptance

probability in (14) results in the dimension reduction in the acceptance probability over the latent path


11

space as follows

A ([�,�,h1:T ,g1:T ,w1:T ]∗ ; [�,�,h1:T ,g1:T ,w1:T ] (j))

= min

(1,

p ([h1:T ,g1:T ,w1:T ]∗ ∣ [�,�]∗ ,y1:T ) p ([�,�]∗ ∣y1:T ) p ([h1:T ,g1:T ,w1:T ] (j) ∣ [�,�] (j) ,y1:T )

p ([h1:T ,g1:T ,w1:T ] (j) ∣ [�,�] (j) ,y1:T ) p ([�,�] (j) ∣y1:T ) p ([h1:T ,g1:T ,w1:T ]∗ ∣ [�,�]∗ ,y1:T )

× q ([�,�] (j) ∣ [�,�]∗))

q ([�,�]∗ ∣ [�,�] (j))

)

= min

(1,

p ([�,�]∗ ∣y1:T )

p ([�,�] (j) ∣y1:T )× q ([�,�] (j) ∣ [�,�]∗))

q ([�,�]∗ ∣ [�,�] (j))

)

= min

(1,

p (y1:T ∣ [�,�]∗) p ([�,�]∗)

p (y1:T ∣ [�,�] (j)) p ([�,�] (j))× q ([�,�] (j) ∣ [�,�]∗))

q ([�,�]∗ ∣ [�,�] (j))

).

(16)

This solution involves marginalisation over the path space g1:T ,h1:T ,w1:T to obtain the marginal like-

lihood required to evaluate the dimension reduced marginal MH acceptance probability. In other words,

the acceptance probability is only evaluated only on the parameter space and not on the full parameter

space and the latent space.

However, to utilise this MH algorithm, a solution to the following two fundamental problems is required:

Problem I:

The marginal likelihood p(y1:T ∣�,�) which is used in the evaluation of (16) can not be obtained

analytically. This is due to the fact that marginalization of the joint likelihood over the path space

involves the following integration

p(y1:T ∣�,�) =

T∏

n=1

p(yn∣y1:n−1,�,�)

=

∫. . .

∫ [ T∏

n=1

p(yn∣hn,gn,wn,�,�)p(hn,gn,wn∣y1:n−1,�,�)

]dh1:Tdg1:Tdw1:T

(17)

which can not be performed analytically.

Problem II:

We need to evaluate and sample from the distribution of the latent path space

p(h1:T ,g1:T ,w1:T ∣y1:T ,�,�). This can be achieved by constructing the sequence of distributions recur-

sively, over the path space given by {p(h1,g1,w1∣y1, [�,�] (j)), p(h1:2,g1:2,w1:2∣y1:2, [�,�] (j)), . . . ,

p(h1:T ,g1:T ,w1:T ∣y1:T , [�,�] (j))} via a two step filter recursion involving the Chapman-Kolmogorov

equation:


12

Stage I (Prediction):

p(hn,gn,wn∣y1:n−1, [�,�] (j))

=

∫p(hn,gn,wn∣hn−1,gn−1,wn−1, [�,�] (j))p(hn−1,gn−1,wn−1∣y1:n−1, [�,�] (j))dhn−1dgn−1dwn−1.

Stage II (Update):

p(hn,gn,wn∣y1:n, [�,�] (j)) =p(yn∣hn,gn,wn, [�,�] (j))p(hn,gn,wn∣y1:n−1, [�,�] (j))

p(yn∣y1:n−1, [�,�] (j)),

which can not be performed analytically.

A. Solution for Problem I and Problem II

The solution for Problem I and Problem II involve decomposing the Markov transition kernel

according to (14). We approximate the first component in (14) via a Rao-Blackwellised particle filter and

the second component in (14) is constructed as an adaptive MH Markov transition kernel. These solutions

detailed below provide the particle filter based estimates which are utilised in the PMCMC algorithm, to

approximate the acceptance probability of the ideal choice, given in (16).

p (h1:T ,g1:T ,w1:T ∣y1:T , [�,�]∗ (j + 1)) =

N∑

i=1

[Ξ1:T ] (j, i)�[h1:T ,g1:T ,w1:T ](j,i) (h1:T ,g1:T ,w1:T ) (18a)

p (y1:T ∣ [�,�]∗ (j + 1)) =

T∏

n=1

(1

N

N∑

i=1

[�n] (j, i)

), (18b)

where [Ξ1:T ] (j, i) and [�n] (j, i) are the normalised importance weight on the path space and the incre-

mental importance weight at time n, respectively, for the i-th particle at the j-th iteration.

We now present how to construct the Markov transition kernel in (15) for the PMCMC algorithm. We

first present the proposal for the latent states (Rao-Blackwellised particle filter) followed by the proposal

for the static parameters (adaptive MCMC).

Development of the proposal p ([h1:T ,g1:T ,w1:T ]∗ ∣ [�,�]∗ ,y1:T ) q ([�,�]∗ ∣ [�,�] (j)) :

Obtaining the approximation given in (18a) involves a Rao-Blackwellised particle filter detailed below.

1) Rao-Blackwellised particle filter for p(h1:T ,g1:T ,w1:T ∣y1:T ,�,�):

To recursively approximate the sequence of distributions, detailed in Problem II, we utilise a

specialised version of SMC.

The proposal kernel for p (g1:T ,h1:T ,w1:T ∣y1:T , [�,�] (j)) can be decomposed as

p (g1:T ,h1:T ,w1:T ∣y1:T , [�,�] (j)) = p (g1:T ∣h1:T ,w1:T ,y1:T , [�,�] (j))︸︷︷︸Kalman filter

× p (h1:T ,w1:T ∣y1:T , [�,�] (j))︸︷︷︸Particle filter

,

(19)


13

which involves a particle filter with a conditional Rao-Blackwellisation achieved via a Kalman

filter recursion conditional on each particles state realisation. The Rao-Blackwellised particle filter

estimate of (19) is given by the importance weighted particle approximation

p (h1:T ,g1:T ,w1:T ∣y1:T , [�,�] (j)) =

N∑

i=1

[Ξ](j, i) p (g1:T ∣ [h1:T ,w1:T ,�,�] (j, i),y1:T ) . (20)

We note that one can then conditionally perform a Kalman filter recursion to obtain

p (g1:T ∣ [h1:T ,w1:T ,�,�] (j, i),y1:T ) for each particle, which is optimal for this conditionally

linear Gaussian state structure. The recursive construction of the SIR particle filter under Rao-

Blackwellised scheme estimate in (20) therefore proceeds according to the following recursive

steps involving unnormalised importance weights, given by[Ξn

](j, i) ∝ [Ξn−1] (j, i)p(yn∣y1:n−1, [h1:n,w1:n,�,�] (j, i)), (21)

where the incremental importance weight is given by the marginal evidence at time n,

p(yn∣y1:n−1, [h1:n,w1:n,�,�] (j, i)) and it is directly obtained as an output in each stage of the

Kalman filter. The conditional Kalman filter recursion at time n, for the i-th particle and the j-th

marginal Metropolis proposed static parameters involves obtaining recursively the sufficient statistics

for the conditional mean (MMSE) and covariance matrix of p (gt∣ [h1:t,w1:t,�,�] (j, i),y1:T )

according to the following Kalman filter recursion:

y = yn − f (sn [hn] (j, i), n)[�n∣n−1

](j, i)

S = f (sn [hn] (j, i), n)[�n∣n−1

](j, i)

[Σn∣n−1

](j, i)f (sn [hn] (j, i), n)

[�n∣n−1

](j, i) + �2

wI

K =[Σn∣n−1

](j, i)f (sn [hn] (j, i), n)

[�n∣n−1

](j, i)⊤S−1

[�n∣n

](j, i) =

[�n−1∣n−1

](j, i) +Ky

[Σn∣n

](j, i) =

(I−Kf (sn [hn] (j, i), n)

[�n∣n−1

](j, i)

) [Σn∣n−1

](j, i).

2) Adaptive MCMC for static parameters �, �:

We now present the proposal kernel, q ([�,�]∗ ∣ [�,�] (j)), for the static parameters in (15).

The static parameters are updated via an adaptive MH proposal comprised of a mixture of Gaussians.

Adaptive MCMC attempts to improve the mixing rate by automatically learning better parameter

values of the MCMC algorithm while it is running. In particular, one of the mixture components

has a covariance structure which is adaptively learnt on-line. The mixture proposal distribution for


14

parameters [�,�] at iteration j of the Markov chain is given by,

q ([�,�] (j); [�,�] (j + 1)) = w1N

([�,�] (j + 1); [�,�] (j),

(2.38)2

dΣj

)

+ (1− w1)N

([�,�] (j + 1); [�,�] (j),

(0.1)2

dI2L,2L

),

(22)

where I2L,2L is the identity matrix of size 2L. Here, Σj is the current empirical estimate of the

covariance between the parameters of �,� estimated using samples from the PMCMC chain up

to time j, and w1 is a mixture proposals weight which we set according to the recommendation of

[21] and are based on optimality conditions presented in [22].

The PMCMC joint channel estimation and Doppler offset estimation algorithm is presented in Algorithm

3.

VI. CRAMER-RAO LOWER BOUND FOR CHANNEL COEFFICIENTS AND DOPPLER OFFSETS

We derive the Bayesian Cramer-Rao Lower Bound (BCRLB) for the channel coefficients and the

Doppler offset parameters.

A. Bayesian CRLB of Channel Coefficients g1:T ,h1:T

The BCRLB provides a lower bound on the MSE matrix for estimation of the path space parameters

which correspond in our model to the estimation of the latent process states x1:T ≜ g1:T ,h1:T ,w1:T . We

denote the Fisher Information Matrix (FIM), used in the CRLB, on the path space by [F1:T (x1:T )] (j) and

marginally by [Fn (xn)] (j) for time n in the path space, conditional on the proposed static parameters at

iteration j of the algorithm. Assuming regularity holds for the probability density functions, the BCRLB

inequality states that the mean squared error (MSE) of any estimator is bounded as:

[F−1n (Xn)

](j) ≤ Ep(xn,y1:n∣�,�)

[(Xn − Xn

)(Xn − Xn

)H]. (23)

However, this formulation assumes prior knowledge of the static parameters � and �. We show how the

BCRLB can be calculated recursively on the path space at each iteration of the algorithm by marginalising


15

Algorithm 3 Joint Channel and Doppler Estimation via Particle MCMC

1: Initialise [�,�,h1:T ,g1:T ,w1:T ] (1) by sampling each value from the corresponding priors.

2: for j = 2, . . . , J do

Sample [�,�]∗ (j + 1) ∼ q ([�,�] (j); [�,�] (j + 1)) according to (22):

3: Sample a realisation u1 of random variable U1 ∼ U [0, 1]

4: if u1 ≥ w1 then sample [�,�] (j + 1) then

Sample [�,�] (j + 1) from the adaptive component as follows:

5: Estimate Σj , the empirical covariance of �, �, using samples {[�,�](i)}i=1:j .

6: Sample [�,�] (j + 1) ∼ N(�,�; [�,�] (j), (2.38)

2

dΣj

);

7: else

Sample [�,�] (j + 1) from the non-adaptive component as follows:

8: Sample [�,�] (j + 1) ∼ N(�,�; [�,�] (j), (0.1)

2

dId,d

)

9: end if

10: Run the Rao-Blackwellised particle filter with N particles to obtain:

p (h1:T ,g1:T ,w1:T ∣y1:T , [�,�]∗ (j + 1)) =

N∑

i=1

[Ξ1:T ] (j, i)�[h1:T ,g1:T ,w1:T ](j,i) (h1:T ,g1:T ,w1:T )

p (y1:T ∣ [�,�]∗ (j + 1)) =

T∏

n=1

(1

N

N∑

i=1

[�n] (j, i)

),

11: Compute (16): A ([�,�,h1:T ,g1:T ,w1:T ] (j); [�,�,h1:T ,g1:T ,w1:T ]∗ (j + 1))

12: Sample a realisation u1 of random variable U1 ∼ U [0, 1]

13: if u1 < A ([�,�,h1:T ,g1:T ,w1:T ] (j) [�,�,h1:T ,g1:T ,w1:T ]∗ (j + 1)) then

14: [�,�,h1:T ,g1:T ,w1:T ] (j + 1) = [�,�,h1:T ,g1:T ,w1:T ]∗ (j + 1)

15: else

16: [�,�,h1:T ,g1:T ,w1:T ] (j + 1) = [�,�,h1:T ,g1:T ,w1:T ] (j)

17: end if

18: end for


16

numerically over the static parameters. Thus, we numerically evaluate

Ep(xn,y1:n)

[(Xn − Xn

)(Xn − Xn

)H]

=

∫⋅ ⋅ ⋅∫ {[

Xn − Xn

] [Xn − Xn

]H}p (xn,y1:n,�,�) dxndy1:nd�d�

=

∫⋅ ⋅ ⋅∫ {[

Xn − Xn

] [Xn − Xn

]H}p (xn,y1:n∣�,�) p (�,�) dxndy1:nd�d�

=

∫⋅ ⋅ ⋅∫

Ep(xn,y1:n∣�,�)

{[Xn − Xn

] [Xn − Xn

]H∣�,�

}p (�,�) d�d�

≈ 1

J

J∑

j=1


{[Xn − Xn

] [Xn − Xn

]H∣ [�,�] (j)

}

=1

J

J∑

j=1

⎧⎨⎩

[Xn − 1

N

N∑

i=1

[Ξ] (j, i)

][Xn − 1

N

N∑

i=1

[Ξ] (j, i)

]H∣ [�,�] (j)

⎫⎬⎭

(24)

where xn ≜ [gn,hn,wn]. Thus, we numerically marginalise over the static parameters.

Next, conditional on the previous Markov chain state [�,�,X1:T ] (j − 1) and the new sampled Markov

chain proposal for the static parameters at iteration j, [�,�] (j), we can utilise the following recursive

expression for the FIM from [23], denoted generically for state-space models as:

[Jn(Xn)

](j) =

[D22

n−1(xn)](j)−

[D21

n−1(xn)](j)([Jn−1(xn)] (j) +

[D11

n−1(xn)](j))−1 [

D12n−1(xn)

](j) (25)

with components:

[J1(x1)] (j) = −E

[∇x1

{∇x1log p (x1)}T

];

[D11

n−1

](j) = −E

[∇xn−1

{∇xn−1

log p (xn∣xn−1)}T ]

= E

{[∇xn−1

f (xn−1;�,�)]Q−1

n−1

[∇xn−1

f (xn−1;�,�)]T}

;

[D12

n−1

](j) =

[D21

n−1

](j) = −E

[∇xn

{∇xn−1


= −E[∇xn−1

f (xn−1;�,�)]Q−1

n−1;

[D22

n−1

](j) = −E

[∇xn

{∇xnlog p (xn∣xn−1)}T

]− E

[∇xn

{∇xnlog p (yn∣xn)}T

]

= Q−1n−1 + E

{[∇xn

ℎ (xn;�,�)]R−1n [∇xn

ℎ (xn;�,�)]T}

(26)

where f (xn−1;�,�) is the state model with process noise covariance Qn and ℎ (xn;�,�) is the

observation model with observation noise covariance Rn.


17

Utilising this recursion, we obtain the following solution for the system model we consider, given by[Jn(Xn)

](j)

=

⎡⎢⎢⎢⎣

11−[�](j)2 0 0

0 11−[�](j)2 0

0 0 1�2w

⎤⎥⎥⎥⎦

− 1

�2v

⎡⎢⎢⎢⎢⎣

f2 (rn) gn

(f (rn)

∂f(rn)∂ℎn

+ ∂f2(rn)∂ℎn

)gn

(f (rn)

∂f(rn)∂wn

+ ∂f2(rn)∂wn

)

gn

(f (rn)

∂f(rn)∂ℎn

+ ∂f2(rn)∂ℎn

)−g2n

(∂f(rn)∂ℎn

)2∂2f(rn)∂ℎ2

ng2n

∂f(rn)∂ℎn

∂f(rn)∂wn

gnf (rn)∂f(rn)∂wn

g2n∂f(rn)∂wn

∂f(rn)∂ℎn

g2n

(∂f(rn)∂wn

)2

⎤⎥⎥⎥⎥⎦

−

⎡⎢⎢⎢⎣

[�](j)1−[�](j)2 0 0

0 [�](j)1−[�](j)2 0

0 0 0

⎤⎥⎥⎥⎦

⎛⎜⎜⎜⎝[Jn−1(Xn−1)

](j) +

⎡⎢⎢⎢⎣

[�](j)2

1−[�](j)2 0 0

0 [�]2(j)1−[�](j)2 0

0 0 0

⎤⎥⎥⎥⎦

⎞⎟⎟⎟⎠

−1 ⎡⎢⎢⎢⎣

[�](j)1−[�](j)2 0 0

0 [�](j)1−[�](j)2 0

0 0 0

⎤⎥⎥⎥⎦ .

(27)

The full derivation can be found in the Appendix II.

In our model framework we get analytic expressions for the BCRLB recursion. However, a key point

about utilising this recursive evaluation for the FIM matrix is that in the majority of cases one clearly can

not evaluate the required expectations analytically over the joint distribution of the data and latent states.

However, since we are constructing a particle filter proposal distribution for the PMCMC algorithm

to target the filtering distribution p (xn∣y1:n, [�](j)) we can use this particle estimate to evaluate the

expectations at each iteration t for each data set. It is important to note that this recursion avoids ever

calculating the expectations using the entire path space empirical estimate, only requiring marginal filter

density estimates, which wont suffer from degeneracy as a path space empirical estimate would.

When developing the approximation stage introduced in (24) it is important to understand the accuracy

of this approximation. The following properties can be stated from the perspective of estimation accuracy:

1) Asymptotic Consistency (in J for all N ∈ ℕ+): it was proven in [20] that for any generic

PMCMC methodology, for any number of particles, the estimate of an integral such as

ℐ = Ep(xn,y1:n)

[(Xn − Xn

)(Xn − Xn

)H](28)

will be consistently estimated with the PMCMC samples obtained according to

ℐ(J,N) =1

J

J∑

j=1


{[Xn − Xn

] [Xn − Xn

]H∣ [�,�] (j)

}(29)


18

such that the following statement holds for all N (number of particles) as the number of Markov

chain draws (J) is taken asymptotically to J → ∞,

limJ→∞

Pr{∣∣∣ℐ(J,N)− ℐ

∣∣∣ < �}= 1, ∀� ≥ 0, ∀N ∈ ℕ

+. (30)

This is known as weak consistency, the rate of this convergence will depend on the algorithmic

choices for the SMC algorithm and the MCMC proposal on the static parameters. In addition, it can

be shown that the Strong Law of Large Numbers (SLLN) can apply to this sequence of estimators.

The optimal choice would be the Kalman filter where the filtering is exact and the MCMC proposal

kernel selected to be geometrically or uniformly ergodic. Of course in practice we dont have these

conditions and so these properties of the mean squared error are studied numerically as demonstrated

in the simulation results.

2) Asymptotic Bias (in J for all N ∈ ℕ+): the estimates obtained from utilising the samples from

the PMCMC algorithm to approximate expectations with respect to a target measures defined over

a state space structure can also be shown to be asymptotically unbiased in J (the length of the

Markov chain), for any number of particles N .

B. Bayesian CRLB of Doppler Offset Parameters �, �

The CRLB for the Doppler offset parameters can not be obtained in closed form. This is because it

would require calculation of the Fisher Information matrix for the following marginal likelihood and then

evaluation of the second order partial derivatives with respect to the static parameters given by solving

∇∇�

∫p(y1:T ∣x1:T , �)p(x1:T ∣�)dx1:T (31)

where � = [�, �]. Firstly, it is clear one can not solve this integral analytically, however an unbiased

particle estimate is obtained in each stage of the PMCMC algorithm resulting in a set of J evaluations

of this marginal likelihood from the PMCMC algorithm giving {p (y1:T ∣[�](j))}j=1:J according to the

unbiased estimator described previously in presenting the PMCMC algorithm and given for each � by

p(y1:T ∣�) =∫

p(y1:T ∣x1:T , �)p(x1:T ∣�)dx1:T ≈ p (y1:T ∣�) =T∏

n=1

(1

N

N∑

i=1

[�n] (j, i)

), (32)

with incremental particle weight [�n] (j, i) given at iteration j of the PMCMC for the i-th particle a

function of parameter �. Then we note that the estimates for the marginal likelihood utilising the particle

filter as described in (32) can be shown to satisfy the properties that the estimator for the marginal


19

likelihood is unbiased as studied in [24] and furthermore, according to [24, Proposition 9.4.1, page 301]

the following central limit theorem holds,

√N

(p (y1:T ∣�)p(y1:T ∣�)

− 1

)d→ N

(0, 2(�)

)(33)

for a 2(�) which is problem specific and finite. Therefore, we can confidently utilise these estimates

{p (y1:T ∣[�](j))}j=1:J from each iteration of the PMCMC algorithm to construct a continous and differ-

entiable kernel density estimator given generically for all � by

pK(y1:T ∣�) =1

Jℎ

J∑

j=1

K

(� − [�](j)

ℎ

), (34)

where K(⋅) is the kernel which is a symmetric but not necessarily positive function that integrates to one

and ℎ > 0 is a smoothing parameter called the bandwidth. Given the smooth kernel density estimator

which can be differentiated to obtain the Hessian around the mode.

∇∇�

∫p(y1:T ∣x1:T , �)p(x1:T ∣�)dx1:T ≈ ∇∇� pK(y1:T ∣�)∣�=mode(�) (35)

VII. COMPLEXITY ANALYSIS

We now derive the computational complexity comparison between each of the algorithms. The compu-

tational cost of each of these algorithms can be split into three parts: the first cost involves constructing

and sampling from the proposal; the second significant computational cost comes from the evaluation

of the acceptance probability for the proposed new Markov chain state; and the third is related to the

mixing rate of the overall MCMC algorithm as affected by the length of the Markov chain required to

obtain estimators of a desired accuracy. We define the following building blocks for a single MCMC

iteration and their associated complexity, measured by O (⋅) as order of magnitude and Cm and Ca are

the complex multiplications and complex additions, respectively:

1) Sampling a random variable using exact sampling ≈ O (1).

2) Likelihood evaluation of∏T

n=1

∏Ll=1 p

(y(l)n ∣�(l), �(l), g

(l)n , ℎ

(l)n

)≈ TL (Cm + Ca) +O (1).

3) Prior evaluations of∏T

n=1

∏Ll=1 p

(ℎ(l)n ∣ℎ(l)n−1

)p(g(l)n ∣g(l)n−1

)p(g(l)1

)p(ℎ(l)1

)p(�(l))p(�(l))≈ 6TLO (1).

Based on these building blocks we estimate the overall complexity of the proposed algorithms as follows.

A. Computational Complexity of Algorithm I

The total computational complexity of one iteration of the block MCMC-within-Gibb sampler, in

Section IV, involves L (3T + 2) parameters and requires updating and accepting each proposed move.

This produces L (3T + 2)× (2TL+ 2)O (1) =(6T 2L2 + 10TL+ 4L

)O (1) operations.


20

B. Computational Complexity of Algorithm II

1) Adaptive MCMC component in (22): Complexity (2L× 2L+ 2L)O (1).

2) Rao-Blackwellised SIR filter component (19):

∙ Kalman filter component : TLO (1).

∙ SIR filter component : 2NLTO (1).

∙ Evaluation of marginal likelihood: NTO (1).

∙ Sampling SIR filter path space proposal: NO (1).

3) Evaluation of acceptance probability in (16): Complexity (NT + 4L)O (1).

Therefore, the total cost of a single PMCMC iteration can be approximated as(2L2 + TL+NT (2L+ 2) +N

)O (1).

Now that we have obtained the computational complexity of each MCMC iteration for both methods,

we are able to perform a fair comparison with respect to algorithmic complexity. This will be presented

in the next section.

VIII. SIMULATION RESULTS

We study the performance of the proposed algorithms in comparison to the BCRLB. This is presented

in two parts:

1) Analysing the properties of the proposed algorithms. This involves analysis of Markov chain paths,

the acceptance probability and the SIR filter performance. These aspects are studied under three

different settings: the dimension of the posterior is increased by increasing the length of the frame

T ; the number of particles in the Rao-Blackwellised SIR filter, N , is increased; and the SNR is

varied. In addition, we demonstrate the significant improvement in computation efficiency that our

algorithm has over MCMC-within-Gibbs sampler.

2) We address the question of how well the proposed algorithm solves the problem of joint chan-

nel tracking and Doppler estimation by considering the estimated MSE of the channels and the

distribution of the MMSE for the parameters of the non-linear state space model.

Simulation Set-Up: the channels are generated according to Rayleigh flat-fading channel model with

Jakess Doppler spectrum [25]. We consider a carrier frequency of 6GHz and a bandwidth of 10kHz,

which is suitable for IEEE 802.16e [26]. The velocities of the mobile terminal and the relay were set to

80km/h and 100km/h, respectively, which correspond to � = 0.98 and � = 0.95.


21

A. Analysis of algorithm performance versus T , N and SNR

The network topology used in the simulations involved a single relay network, K = L = 1, and

the relay processing function f (l)(R(l), n

)is a constant gain Amplify and Forward. In all simulations

we ran a Markov chain of length 20, 000 iterations, discarding the first 5, 000 samples as burnin. We

then systematically varied each of the three variables (T,N ,SNR) and assessed the performance of the

algorithm. We took values of N ranging from 10, 20, 50, 100, 200, 500, 1, 000 and 5, 000 particles. The

length of frame considered involved T ranging from 50, 100 and 200 symbols per frame, leading to

posteriors to be sampled from in dimensions 152, 302 and 602 respectively. Finally, we consider SNR

levels from 0dB through to 25dB which covers a wide range of possible operating environments.

1) Analysis of the number of particles N versus acceptance rate: we study the performance of the

PMCMC algorithm average acceptance probability as a function of the number of particles N . This is

meaningful as it allows us to recommend a setting for N . In Fig. 3 we present the average acceptance

probability, corresponding to a posterior distribution in 302 dimensions. The key finding of this study

is that we only require a very small number of particles to obtain accurate estimation and efficient

performance. In particular we see that as expected, when the number of particles increases, the average

acceptance probability of the joint proposal in the Markov chain increases. Secondly, we note that even

for a relatively small number of particles, N = 100 we obtain average acceptance rates around 20%. This

is a very good indication that it is suitable to work with the Rao-Blackwellised SIR filter for our proposal.

It is typical in the MCMC literature to tune a proposal mechanism in a standard MCMC algorithm to

produce acceptance rates between [0.2, 0.5] and in some cases it is provably optimal to use 0.234 [22].

2) Analysis of the estimated MMSE versus SNR: In Figs. 4-5 we present the MMSE estimates for the

channels h1:T and g1:T as well as the 95% posterior confidence intervals. These results demonstrate that

the PMCMC algorithm performs much better than the MCMC-within-Gibbs algorithm. These results were

obtained using the MH-within-Gibbs sampler with identical Markov chain initialisation to that used in the

proposed algorithm. In addition, we pre-tuned the MCMC-within-Gibbs sampler to have an acceptance

rate of approximately 20%. Clearly, the lower mixing rate of the Gibbs sampler results in sub-optimal

performance, requiring significantly longer chain to obtain the same accuracy as the PMCMC sampler.

B. Analysis of estimated MSE versus SNR

In this section we compare the MSE performance of the proposed PMCMC algorithm with the Gibbs

sampler, where the number of iterations of the Gibbs sampler is set such its overall computational


22

complexity is the same as the PCMCM algorithm. In addition we present the BCRLB results. The results

of this analysis are presented in Fig. 6. Clearly, there is a decrease in the Total MSE as the SNR increases.

In addition we note that as expected, since the estimate of the channels g1:T is performed using the Rao-

Blackwellising conditionally optimal Kalman Filter, this is reflected in the level of the total MSE. The

estimates for g1:T are more accurate than those for the particle filter sampled estimates h1:T . We note

that the PMCMC algorithm achieves lower MSE than the Gibbs sampler at every SNR. This indicates

the superiority of the PMCMC algorithm over the Gibbs sampler. In addition we present the BCRLB

after marginalizing out the model parameters according to (24). The results demonstrate that the PMCMC

methodology performs close to optimally.

In Fig. 7 we present the MSE for the Doppler offsets � and � as a function of the SNR. We compare

these with the BCRLB which was developed in Section VI-B. Clearly as the SNR increases, the estimates

converge to the true parameter values. The results show that the estimation of the Doppler offset � is

more accurate than of �.

IX. CONCLUSIONS

We introduced a new framework to perform joint channel and Doppler offset estimation in dynamic

cooperative wireless relay networks. We developed two algorithms to estimate the associated high dimen-

sional posterior via MCMC-within-Gibbs and Particle Adaptive Marginal Metropolis-Hastings methods.

We compared the PMCMC algorithm to the MCMC-within-Gibbs sampler and showed that the PMCMC

algorithm is considerably more computationally efficient than the Gibbs sampler. We also developed a

recursive marginal Cramer-Rao lower bound and demonstrate that the proposed algorithm performs close

to the lower bound. Simulation results show the ability of the proposed algorithm to jointly estimate the

dynamic channels as well as the static frequency offsets. Future research includes extending this work

to blind estimation.

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APPENDIX A

MCMC-WITHIN-GIBBS FULL CONDITIONAL POSTERIOR DISTRIBUTIONS AND ACCEPTANCE

PROBABILITY

Full conditional posterior for [�,�](j + 1):

[�,�](j + 1) ∼ p(�,�∣[g1:T ,h1:T ,w1:T ](j), y

(l:L)1:T

)

∝ p (g1:T ,h1:T ,w1:T ∣�,�) p (�) p (�)

=

L∏

l=1

⎡⎢⎢⎣

T∏

n=1

1√2�(1−

(�(l))2)

exp

⎧⎨⎩−

∣∣∣g(l)n − �(l)g(l)n−1

∣∣∣2

2(1−

(�(l))2)

⎫⎬⎭

1√2�(1−

(�(l))2)

exp

⎧⎨⎩−

∣∣∣ℎ(l)n − �(l)ℎ

(l)n−1

∣∣∣2

2(1−

(�(l))2)

⎫⎬⎭

⎤⎥⎥⎦

×L∏

l=1

1

B (a, b)

(�(l))a−1 (

1− �(l))b−1 L∏

l=1

1

B (c, d)

(�(l))c−1 (

1− �(l))d−1

=

L∏

l=1

⎡⎢⎣(�(l))a−1 (

1− �(l))b−1 (

�(l))c−1 (

1− �(l))d−1

2�B (a, b)B (c, d)(1−

(�(l))2)(

1−(�(l))2) exp

⎧⎨⎩−1

2

T∑

n=1

⎛⎜⎝

∣∣∣g(l)n − �(l)g(l)n−1

∣∣∣2

(1−

(�(l))2) +

∣∣∣ℎ(l)n − �(l)ℎ

(l)n−1

∣∣∣2

(1−

(�(l))2)

⎞⎟⎠

⎫⎬⎭

⎤⎥⎦ .

(36)

Acceptance probability for [�,�](j + 1):

A ([�,�]∗ , [�,�] (j)) = min

{1,

p ([�,�]∗ ∣y, [h1:T ,g1:T ,w1:T ] (j))

p ([�,�] (j) ∣y, [h1:T ,g1:T ,w1:T ] (j))× q([�,�] (j) ; [�,�]∗)

q([�,�]∗ ; [�,�] (j))

}.

(37)


25

Full conditional posterior for [G(k−1)�+1:k� ](j + 1):

[G(k−1)�+1:k� ](j + 1) ∼ p(g(k−1)�+1:k� ∣[�,�,g1:(k−1)� ](j + 1), [gk�+1:T� ,h1:T ,w1:T ](j), y

(l:L)1:T

)

∝ p(y(l:L)(k−1)�+1:k� ∣g(k−1)�+1:k� ,h(k−1)�+1:k� ,w(k−1)�+1:k� ,�,�

)p(g(k−1)�+1:k� ∣g(k−1)� ,�

)

=L∏

l=1

⎡⎢⎣

k�∏

n=(k−1)�

1√2��2

v

exp

⎧⎨⎩−


(l)n + w

(l)n

)g(l)n

∣∣∣2

2�2v

⎫⎬⎭

× 1√2�(1−

(�(l))2)

exp

⎧⎨⎩−

∣∣∣g(l)n − �(l)g(l)n−1

∣∣∣2

2(1−

(�(l))2)

⎫⎬⎭

⎤⎥⎥⎦

1√2�

L∏

l=1

exp

{−1

2

∣∣∣g(l)1

∣∣∣2}

=1

(2�)3/2

�v

L∏

l=1

⎡⎢⎢⎣

1√(1−

(�(l))2)

exp

⎧⎨⎩−1

2

k�∑

n=(k−1)�

⎛⎜⎝


(l)n + w

(l)n

)g(l)n

∣∣∣2

�2v

+

∣∣∣g(l)n − �(l)g(l)n−1

∣∣∣2

(1−

(�(l))2) +

∣∣∣g(l)1

∣∣∣2

⎞⎟⎠

⎫⎬⎭

⎤⎥⎥⎦ .

(38)

Acceptance probability for [G(k−1)�+1:k� ](j + 1):

A([

G(k−1)�+1:k�

]∗

,[G(k−1)�+1:k�

](j))

= min

⎧⎨⎩1,

p([

G(k−1)�+1:k�

]∗ ∣y, [h1:T ,w1:T ,�,�] (j)

)

p([G(k−1)�+1:k�

](j − 1) ∣y, [h1:T ,w1:T ,�,�] (j)

) × q([G(k−1)�+1:k�

](j) ;

[G(k−1)�+1:k�

]∗

)

q([G(k−1)�+1:k�

]∗

;[G(k−1)�+1:k�

](j))

⎫⎬⎭ .

(39)

Full conditional posterior for [h(k−1)�+1:k� ](j + 1):

[H(k−1)�+1:k� ](j + 1) ∼ p(h(k−1)�+1:k� ∣[�,�,h1:(k−1)� ](j + 1), [hk�+1:T� ,g1:T ,w1:T ](j), y

(l:L)1:T

)


)p(h(k−1)�+1:k� ∣h(k−1)� ,�

)

=L∏

l=1

⎡⎢⎣

k�∏

n=(k−1)�

1√2��2

v

exp

⎧⎨⎩−


(l)n + w

(l)n

)g(l)n

∣∣∣2

2�2v

⎫⎬⎭

× 1√2�(1−

(�(l))2)

exp

⎧⎨⎩−

∣∣∣ℎ(l)n − �(l)ℎ

(l)n−1

∣∣∣2

2(1−

(�(l))2)

⎫⎬⎭

⎤⎥⎥⎦

1√2�

L∏

l=1

exp

{−1

2

∣∣∣ℎ(l)1

∣∣∣2}.

(40)

Acceptance probability for [h(k−1)�+1:k� ](j + 1):

A([

H(k−1)�+1:k�

]∗

,[H(k−1)�+1:k�

](j))

= min

⎧⎨⎩1,

p([

H(k−1)�+1:k�

]∗ ∣y, [g1:T ,w1:T ,�,�] (j − 1)

)

p([H(k−1)�+1:k�

](j) ∣y, [g1:T ,w1:T ,�,�] (j)

) × q([H(k−1)�+1:k�

](j) ;

[H(k−1)�+1:k�

]∗

)

q([H(k−1)�+1:k�

]∗

;[H(k−1)�+1:k�

](j))

⎫⎬⎭ .

(41)


26

Full conditional posterior for [w(k−1)�+1:k� ](j + 1):

[w(k−1)�+1:k� ](j + 1) ∼ p(w(k−1)�+1:k� ∣[�,�,w1:(k−1)� ,h1:T ,g1:T ](j + 1), [wk�+1:T� ](j), y

(l:L)1:T

)


)p(w(k−1)�+1:k�

)

=

L∏

l=1

⎡⎢⎣

k�∏

n=(k−1)�

1√2��2

v

exp

⎧⎨⎩−


(l)n + w

(l)n

)g(l)n

∣∣∣2

2�2v

⎫⎬⎭

1√2��2

w

exp

⎧⎨⎩−

∣∣∣w(l)n

∣∣∣2

2�2w

⎫⎬⎭

⎤⎥⎦ .

(42)

Acceptance probability for [w(k−1)�+1:k� ](j + 1):

A([

W(k−1)�+1:k�

]∗

,[W(k−1)�+1:k�

](j))

= min

⎧⎨⎩1,

p([

W(k−1)�+1:k�

]∗ ∣y, [g1:T ,h1:T ,�,�] (j)

)

p([W(k−1)�+1:k�

](j) ∣y, [g1:T ,h1:T ,�,�] (j)

) × q([W(k−1)�+1:k�

](j − 1) ;

[W(k−1)�+1:k�

]∗

)

q([W(k−1)�+1:k�

]∗

;[W(k−1)�+1:k�

](j))

⎫⎬⎭ .

(43)

APPENDIX B

BAYESIAN CRLB ELEMENTS OF (26)

:

[J1(x1)] (j) = −E

[∇x1

{∇x1log p (x1)}T

]= −E

[∇x1

{∇x1

[g212�2

g

,ℎ21

2�2ℎ

,w2

1

2�2w

]}T]=

⎡⎢⎢⎢⎣

1�2g

0 0

0 1�2h

0

0 0 1�2w

⎤⎥⎥⎥⎦ .

[D11

n−1

](j) = −E

[∇xn−1

{∇xn−1


= −E

⎡⎢⎣∇xn−1

⎧⎨⎩∇xn−1

⎡⎣ (gn − �gn−1)

2

2(1− (�)

2) ,

(ℎn − �ℎn−1)2

2(1− (�)

2) ,

w2n

2�2w

⎤⎦⎫⎬⎭

T⎤⎥⎦ =

⎡⎢⎢⎢⎣

[�](j)2

1−[�](j)2 0 0

0 [�](j)2

1−[�](j)2 0

0 0 0

⎤⎥⎥⎥⎦ .

[D12

n−1(xn)](j) =

[D21

n−1(xn)](j) = −E

[∇xn

{∇xn−1


= −E

⎡⎢⎣∇xn

⎧⎨⎩∇xn−1

⎡⎣ (gn − �gn−1)

2

2(1− (�)

2) ,

(ℎn − �ℎn−1)2

2(1− (�)

2) ,

w2n

2�2w

⎤⎦⎫⎬⎭

T⎤⎥⎦ =

⎡⎢⎢⎢⎣

[�](j)1−[�](j)2 0 0

0 [�](j)1−[�](j)2 0

0 0 0

⎤⎥⎥⎥⎦ .


27

[D22

n−1(xn)](j) = −E

[∇xn

{∇xnlog p (xn∣xn−1)}T

]− E

[∇xn

{∇xnlog p (yn∣xn)}T

]

= −E

⎡⎢⎣∇xn

⎧⎨⎩∇xn

⎡⎣ (gn − �gn−1)

2

2(1− (�)

2) ,

(ℎn − �ℎn−1)2

2(1− (�)

2) ,

w2n

2�2w

⎤⎦⎫⎬⎭

T⎤⎥⎦− E

⎡⎣∇xn

{∇xn

[(yn − f (sℎn + wn) gn)

2

2�2v

]}T⎤⎦

=

⎡⎢⎢⎢⎣

11−[�](j)2 0 0

0 11−[�](j)2 0

0 0 1�2w

⎤⎥⎥⎥⎦

+ E

⎡⎢⎢⎢⎢⎢⎣

−f2(rn)�2v

−yn∂f(rn)∂ℎn

+∂f2(rn)

∂ℎngn

�2v

−yn∂f(rn)∂wn

+∂f2(rn)

∂wngn

�2v

−yn∂f(rn)∂ℎn

+∂f2(rn)

∂ℎngn

�2v

−yngn

∂2f(rn)

∂ℎ2n

−g2n(

∂f(rn)∂ℎn

)2−g2

nf(rn)∂2f(rn)

∂ℎ2n

�2v

yngn∂2f(rn)∂ℎn∂ℎw

−g2n

∂f(rn)∂ℎn

∂f(rn)∂wn

−g2nf(rn)

∂2f(rn)∂ℎn∂ℎw

�2v

(yn−2f(rn)gn)∂f(rn)∂wn

�2v

yngn∂2f(rn)∂wn∂ℎn

−g2n

∂f(rn)∂wn

∂f(rn)∂ℎn

−f(rn)g2n

∂2f(rn)∂wn∂ℎn

�2v

yngn∂2f(rn)

∂w2n

−g2n(

∂f(rn)∂wn

)2−f(rn)g

2n

∂2f(rn)

∂w2n

�2v

⎤⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎣

11−[�](j)2 0 0

0 11−[�](j)2 0

0 0 1�2w

⎤⎥⎥⎥⎦

−

⎡⎢⎢⎢⎢⎣

f2(rn)�2v

gnf(rn)∂f(rn)∂ℎn

+∂f2(rn)

∂ℎngn

�2v

gnf(rn)∂f(rn)∂wn

+∂f2(rn)

∂wngn

�2v

gnf(rn)∂f(rn)∂ℎn

+∂f2(rn)

∂ℎngn

�2v

−g2n(

∂f(rn)∂ℎn

)2 ∂2f(rn)

∂ℎ2n

�2v

g2n

∂f(rn)∂ℎn

∂f(rn)∂wn

�2v

gnf(rn)∂f(rn)∂wn

�2v

g2n

∂f(rn)∂wn

∂f(rn)∂ℎn

�2v

g2n(

∂f(rn)∂wn

)2

�2v

⎤⎥⎥⎥⎥⎦.


28

DestinationRelay 1

( )1h

( )1W

( )1g( )1V

Relay 2( )2h

( )2W

( )2g( )2V

Relay L

( )Lh ( )LW( )Lg

( )LV

Source

+

+

+

Fig. 1. System model of the relay network with L relays and a dual hop

…….

…….

)(lα

)(1

lh

)(1

ly )(2ly

)(lTy

)(lβ

)(lTh)(

2lh

)(1

lg )(2lg

)(lTg

)(1

lW )(2

lW )(lTW

)(1

lV )(2

lV )(lTV

},...,1{ Ll =

Fig. 2. Graphical model of the relay system.

101 102 103 1040

0.05

0.1

0.15

0.2

0.25

0.3

Number of particles

Mea

n ac

cept

ance

pro

babil

ity

Fig. 3. PMCMC: Acceptance probability for different number of particles


29

10 20 30 40 50 60 70 80 90 100−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Symbol index

h

Algorithm I: MCMC−within−Gibbs

True channel hMMSE estimate of hMMSE confidence interval

10 20 30 40 50 60 70 80 90 100−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Symbol index

h

Algorithm II: PMCMC

MMSE estimate of hTrue channel hMMSE confidence interval

Fig. 4. MSE for h for Algorithm I: MCMC-within-Gibbs (left panel) and Algorithm II: PMCMC (right panel).

10 20 30 40 50 60 70 80 90 100−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Symbol index

g

Algorithm I: MCMC−within−Gibbs

MMSE estimate of gTrue channel gMMSE confidence interval

10 20 30 40 50 60 70 80 90 100−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Symbol index

g

Algorithm II: PMCMC

MMSE estimate of gTrue channel gMMSE confidence interval

Fig. 5. MSE for g for Algorithm I: MCMC-within-Gibbs sampler (left panel) and Algorithm II: PMCMC (right panel).


30

0 5 10 15 20 250.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

SNR [dB]

MSE

Algorithm II: PMCMCAlgorithm I: GibbsBCRLB

0.04

0.02

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

SNR [dB]

MSE

Algorithm II: PMCMCAlgorithm I: GibbsBCRLB

0.03

Fig. 6. Channel estimation performance for h (upper panel) and g (lower panel) of the PMCMC compared with Gibbs-within-

MCMC algorithms and the BCRLB.

0 5 10 15 20

0.05

0.1

0.15

0.2

0.25

SNR [dB]

MSE

of α

Algorithm II: PMCMCBCRLB, Eq. (35)Algorithm I: Gibs−within−MCMC

0.05

0.03

0 5 10 15 20

0.04

0.06

0.08

0.1

0.12

0.14

SNR [dB]

MS

E o

f β

Algorithm II: PMCMCBCRLB, Eq. (35)Algorithm I: Gibs−within−MCMC

0.02

0.03

Fig. 7. Estimation of Doppler offsets � (left panel) and � (right panel) of the PMCMC compared with Gibbs-within-MCMC

algorithms and the BCRLB vs. SNR.


1 joint channel and doppler offset estimation in dynamic

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