Time-Varying Frequency Fading Channel Tracking In
OFDM-PLNC System, Using Kalman Filter
S. Khosroazad *, N. Neda * and H. Farrokhi *
Abstract: Physical-layer network coding (PLNC) has the ability to drastically improve the throughput of multi-source wireless communication systems. In this paper, we focus on the problem of channel tracking in a Decode-and-Forward (DF) OFDM PLNC system. We proposed a Kalman Filter-based algorithm for tracking the frequency/time fading channel in this system. Tracking of the channel is performed in the time domain while data detection is implemented in the frequency domain. As an important advantage, this approach does not need for training of some subcarriers in every OFDM symbols and this, results in higher throughput, compared to other methods. High accuracy, no phase ambiguity, and stability in fast fading conditions are some other advantages of this approach.
Keywords: Physical Layer Network Coding; Decode-and-forward; Orthogonal Frequency Division Multiplexing (OFDM); channel tracking; Kalman Filter.
1 Introduction
1
Recently, the research community has shown growing interest in PLNC (Physical Layer Network Coding) as a way to exploit the network coding operation that occurs naturally in superimposed electromagnetic (EM) waves [1], [2]. In many wireless communication networks today, interference is treated as a destructive phenomenon. When multiple transmitters in the same environment send radio waves to their respective receivers, each receiver gets signals from its dedicated transmitter as well as from other transmitters. The radio waves, coming from other transmitters, are often treated as interference that corrupts the intended signal. Now, PLNC by exploiting the network coding operation, performed by nature, has changed the situation [3].
In a TWRC (two-way relay channel), for example, by allowing the two end nodes (ଵǡ ଶ) to transmit simultaneously to the relay and not treating this as collision, PLNC can boost the system throughput by 100% [1]. This throughput advantage of PLNC, however, is predicated on accurate estimation of the channels between the end nodes and the relay. This estimation is typically done using known training
Iranian Journal of Electrical & Electronic Engineering, 2016. Paper first received 25 April 2016 and accepted 2 October 2016. * The Authors are with the Department of Electrical and Computer Engineering, University of Birjand, Birjand, Iran.
E-mails: [email protected], [email protected], [email protected].
Corresponding Author: Naaser Neda.
symbols and/or pilots embedded in the packets. Channel estimation in PLNC systems is an active area of research since unlike the point to point communication systems, at least two channel gains need to be estimated based on the simultaneously received signals from multi sources.
Broadband wireless communications, on the other hand, experience frequency/time fading which distorts the transmitted signal. Orthogonal frequency division multiplexing (OFDM) is a well-known transmission technique which is robust against the frequency selectivity of the channel. In addition, OFDM provides a natural way to deal with relative symbol offset between nodes 1 and 2 in PLNC. In particular, any time-domain symbol offset (less than cyclic prefix length) will be treated as a phase term in the channel gain in the frequency domain, which can simplify the system. Because of these advantageous potentials, OFDM PLNC is now, a popular system under investigation by many researchers [4]-[11]. Most of these papers assume perfect knowledge of channel at the receiver side [7], [8], [11] while the lack of channel state information (CSI) has a significant impact on system performance. On the other hand, estimation of the channel gains on the subcarriers of an OFDM-PLNC system is a new challenge that has not addressed by the traditional point to point OFDM researches. As an example, the training symbols and pilots need to be redesigned for OFDM PLNC.
In general, two types of a training sequence for channel estimation (CE) in OFDM systems were proposed; block-based and pilot-tone-based (PT)
Iranian Journal of Electrical & Electronic Engineering, Vol. 12, No. 3, September 2016 187
training [4]. In block-based training method, one
complete OFDM block is dedicated to training. However, the channel estimation obtained by this
method remains static between successive training blocks, while the channel may be changing,
continuously. In pilot-tone (PT) training,some carriers inevery OFDM blockareselectedas pilots to track the
channel. Clearly, this could greatly reduce the system throughput.
In this paper, we propound the problem of
estimation and tracking of the time-varying channel in an OFDM-PLNC system, where, the channel with both
frequency and timeselectivity are considered. With the help of the state space model of the communication
channel, we offer a method for the channel estimation and tracking in the time domain, anddatadetection in
the frequency domain. Channel tracking in the time domain is more stable forestimation errors, owing to
the very goodextractionof the frequency correlationof the channel coefficients. In addition, the estimation errorsareextendedon theentire transmission frequency band,andit will not centralizeonly onaspecific set of
subcarriers. Channel tracking in the frequency domain,
without considering the state-space model for it, the
channel tracking is done separately on any subcarrier.
Thus, theerrorin channel estimationoneachsub-carrier directly affects the information estimation of that s ub-carrier, and this error will continuously affect tracking the channel anddatarelated to that sub-carrier. Tracking
of the channel in time-domain anddatadetectionin the
frequency domainis well knownin the traditional point to point OFDM systems [12]-[16].Here, we propose a
time-domain algorithm for an OFDM PLNC channel tracing derived from KalmanFilter (KF).Joint decode
-and-forward (DF) PLNC scheme [17] is considered,
where the relay decodes both packets from ଵ and ଶ simultaneously, and re-encodes by modulo-2
summation, before broadcasting. This method removes
the adverse effect of fading and noise in the source
-relay channels. For this purpose, we have to jointly
estimate the source-relay channels and jointly decode
the superposed packets. Also, using this proposed KF algorithm, wedon’t need PTs for training whichoccupy
at least ଵ ଶ ʹ subcarriers in each OFDM block
[4] (where ୨ǡ ൌ ͳǡʹ represents the delay spreadof
the corresponding discrete channel model).
Therest of the paperisorganizedas follows. In thenext
section, the system and signal models are presented.
Proposed KF-CE (Kalman Filter-Channel Estimation)
algorithm for decode-and-forward TWRC system in a
time/frequency selective fading channel isintroducedin Section 3.Some computersimulationresults,as well as relateddiscussions, comeinSection 4.Finally,Section 5 concludes the paper.
2 SystemModel
We consider a PLNC-based two-way relay system,
with twosourcenodes,ଵandଶ,and therelay node R
as shown in Fig. 1. The source nodes and the relay communicate using a two-phase PLNC transmission
consist of uplinkanddownlink phase.
2.1 UplinkPhase
In the uplink phase, ଵandଶ transmit their packets
to R simultaneously. The data-modulated symbol of
each node is represented by {୨ሺሻ ൌ ࣧ ቀ୨ሺሻቁ ǡ
̱Ͳǣ െ ͳǡ ൌ ͳǡʹǡ ሽ, where୨ሺሻis the binary data
of eachnode, and ቂห୨ሺሻห ଶ
ቃ ൌ ͳ. In each node, the
symbol sequences (܆୨ሺ୩ሻൌ
ሾ୨ሺ୩ሻሺͳሻǡ ୨ሺ୩ሻሺʹሻǡ ǥ ǡ ୨ሺ୩ሻሺሻሿ) in time instant are
given to an -point normalized Inverse Fast Fourier
Transform (IFFT) to generate an OFDM signal waveform asܠ୨ሺ୩ሻin the୲୦ OFDM symbol time. Thena
cyclic prefix (CP) vector with the length of େǡis insertedandsignal from eachnodeis transmittedovera
time varying frequency selective fading channel. The
baseband channel responses betweenଵ and R,and ଶ and R at symbol time , are denoted by ܐሺ୩ሻൌ
ሾሺ୩ሻǡ ଵሺ୩ሻǡ ǥ ǡ భିଵሺ୩ሻ ሿ, ሺ୩ሻൌ ሾ
ሺ୩ሻ
ǡ ଵሺ୩ሻǡ ǥ ǡ మିଵሺ୩ሻ ሿ
Fig. 1:PLNC-based two-way relay system. Twosourcenodesܖandܖ,and therelay node R
respectively, where ୨ǡ ൌ ͳǡʹ represents the delay spread of the corresponding discrete channel model in sample. Theelements of ܐ andare assumed aszero -mean circularly symmetric complex Gaussian random variablesandareindependent from eachother.
Relay Operation: the normalized Discrete Fourier
Transform (DFT) of the୲୦ OFDM block,receivedat R
(afterremoving the CP), can beexpressedas:
ܚሺ୩ሻൌ ൣ
ሺ୩ሻǡ
ଵ
ሺ୩ሻǡ ǥ ǡ
ିଵ ሺ୩ሻ ൧
ൌ ۶ሺ୩ሻ܆ ሺ୩ሻ
۵ሺ୩ሻ܆
ሺ୩ሻ
ܖሺ୩ሻ, (1)
where,۶ሺ୩ሻand۵ሺ୩ሻare ൈ diagonal matrices with diagonal elements given by -point DFT of ܐሺ୩ሻ andሺ୩ሻ, respectively. ܆
୨ ሺ୩ሻ
, j=1,2 is an ൈ ͳ OFDM block transmittedat time from ୨ to therelay R,and ܖሺܓሻ is an ൈ ͳ zero-mean additive white Gaussian noise (AWGN) vector, each of its elements with
variance ɐ୬ଶ , i.e.ܖሺ୩ሻ̱घ
ԧሺǡ ɐ୬ଶ۷ሻ. In time instant ,
relay estimates۶ሺ୩ሻǡ ۵ሺ୩ሻ,܆ ሺ୩ሻ
and ܆ሺ୩ሻ using the KF -CE algorithm, proposedinsection 3.
2.2 DownlinkPhase
For network coding the relay performs estimation
withrespect to the XOR of the binary informationof the nodes ଵ and ଶ ܊୰ሺ୩ሻൌ ܊መଵሺ୩ሻْ ܊መଶሺ୩ሻ based on ܚሺ୩ሻ.
This estimated relay codeword is again modulated
(܆୰ሺ୩ሻൌ गሺ܊୰ሺ୩ሻሻ) and the relay broadcasts OFDM
modulation of ܆୰ሺ୩ሻ to both nodes ଵǡ ଶ. In fact the system in the downlink phase is like a SIMO (Single
Input Multi Output) system (Lookat downlink phasein Fig. 1) and the received signal in each end node experiences a simple point to point communications
from the relay to the end node. Without loss of generality we focus on the process at ଵ. The ୲୦
OFDM demodulated blockreceivedat ଵ,isrepresented as:
ܡሺ୩ሻൌ ቂ
ሺ୩ሻ
ǡ ଵሺ୩ሻǡ ǥ ǡ ିଵሺ୩ሻ ቃൌ ۶୰ሺ୩ሻ܆୰ሺ୩ሻ ܖ୰
ሺ୩ሻ
, (2)
ܖ୰ሺ୩ሻ̱घԧሺǡ ɐ୬౨ଶ ۷ሻ,
where, ۶୰ሺ୩ሻ is an ൈ diagonal matrix with the diagonal elements given by -point normalized DFT of
ܐ୰ሺ୩ሻ. Here, the vector ܐ୰ሺ୩ሻൌ ሾ୰ሺ୩ሻబ ǡ ୰ሺ୩ሻభǡ ǥ ǡ ୰ ై౨షభ ሺ୩ሻ ሿ
shows the baseband downlink channel between R and ଵat time where୰represents thedelay spreadof the
corresponding channel in sample. The elements of ܐ୰ are assumed as zero-mean circularly symmetric complex Gaussian random variables and ܐ୰ is independent from uplink channel ܐ (However,identical channelsareoftenassumedin both up anddownlink [4] which can make the process muchsimpler). ܖ୰ሺ୩ሻis the zero-mean AWGN,eachelement with varianceɐ୬౨ଶ . In
thisstage,ଵ can also estimateܐ୰ ሺ୩ሻ
and܊୰ሺ୩ሻby using KF-CE algorithm (details comein thenext section),and estimates ܊ଶሺ୩ሻ, using its known information܊
ሺ୩ሻ , by
applying: ܊
መܚሺܓሻْ ܊
ሺܓሻ
ൌ ሺ܊መሺܓሻْ ܊መሺܓሻሻْ ܊ሺܓሻൌ ܊መሺܓሻ (3)
3 KF-CE Algorithm forOFDM-PLNCSystem
3.1 KF-CE Algorithm
In mobile communications, the channels are time
varying andneed to be trackedover the time. Equation
(1) (that isin frequency domain) can then be writtenin
timedomainas:
ܚሺ୩ሻൌ ൣ܆෩ሺ୩ሻ܆෩ሺ୩ሻ ൧ൣܐሺ୩ሻሺ୩ሻ൧ ܖሺ୩ሻ
ൌ ܆෩ሺ୩ሻ۱ሺ୩ሻ ܖሺ୩ሻ (4)
where,܆෩ܒሺ୩ሻǡ ൌ ͳǡʹisa ൈ ୨ circulant matrix (i.e.its
columns are circularly shifted versions of eachother),
formed with the modulated block ܠ୨ሺ୩ሻൌ
ሾ୨ሺ୩ሻሺͲሻǡ ୨ሺ୩ሻሺͳሻǡ ǥ ǡ ୨ሺ୩ሻሺ െ ͳሻሿat timeinstant ,and
۱ሺ୩ሻൌ ൣܐሺ୩ሻሺ୩ሻ൧is the ሺ
ଵ ଶሻ ൈ1 channel state
vector obtained from channel vectors ܐሺ୩ሻ and ሺ୩ሻ
which introduced earlier. In fact, Eq. (4) shows the
convolution process by a matrix product which makes
the notation much easier. Also, we defined the new parameter ۱ to express the equation according to appropriate structure for Kalman Filter (Eq. (A1) in
Appendix).
To model the time variationsof wireless channel,on
the other hand, we note that according to the Bello
model [18], the fading process from the transmitter to receiver can be modeledasa complex Gaussian process.
A widely accepted model, to represents the local behavior of the time-varying wireless channel is an auto-regressive (AR) model [19]. Therefore, using the
first-orderassumption, the channel stateat time can be evaluated, using a first-orderautoregressive process of the form
Iranian Journal of Electrical & Electronic Engineering, Vol. 12, No. 3, September 2016 189
۱ሺ୩ሻൌ Ⱦ۱ሺ୩ିଵሻ ܄ሺ୩ሻ (5-a)
where, Ⱦ is the static AR coefficient, and
܄ሺ୩ሻ̱घ
ԧሺǡ ɐ୴ଶ۷ሻ is the complex state noise of the
model. Given therequirement to track the channel state as accurately as possible, there are different ways to determine valuesof Ⱦandɐଶ୴, [20], [21]. Inour paper, due to the frequency fading and time varying natureof the wireless channel, we used thedynamic AR channel model proposed in [20], which is robust for realistic wireless situations. This paper rewrites the state
-equation (5-a) as۱ሺ୩ሻൌ ۱ሺ୩ିଵሻ ሺȾ െ ͳሻ۱ሺ୩ିଵሻ ܄ሺ୩ሻ.
Then,it definesanew term ૄሺ୩ሻൌ ሺȾ െ ͳሻ۱ሺ୩ିଵሻanda
primitive equation ૄሺ୩ሻൌ ૄሺ୩ିଵሻ ܟሺ୩ሻ, where
ܟሺ୩ሻ̱घ
ԧሺǡ ɐ୵ଶ۷ሻ,ૄሺ୩ሻ̱घԧሺܕሺ୩ሻǡ ۾ሺ୩ሻሻ. Then, using a
Kalman filter formulation, the estimate of
ૄሺ୩ሻ̱घ
ԧሺܕሺ୩ሻǡ ۾ሺ୩ሻሻ can be drawn corresponding to
the changein the wireless channel estimate (see Table
I). Werefer thereader to [20] fordetails. Equations (4)
and (5-a) orits modified version,
۱ሺ୩ሻൌ ۱ሺ୩ିଵሻ ૄሺ୩ሻ ܄ሺ୩ሻ (5-b)
form the state space model to describe the system in
uplink phase (equivalent to Eqs. (A1) and (A2) in
Appendix). These equations are our signal model for running KF algorithm [22]. Whereas the system model
is linear and the channel noise is Gaussian, we can design a Kalman Filter algorithm to continuously
estimate thestate vector۱. Theoverall structureof relay
operationis illustrated inFig. 2. As this figure shows,
the relay operation is divided to two basic parts; Data estimation and Channel tracking. Data estimation is done in frequency domain, while channel tracking is donein timedomain. DiscreteFourier Transform (DFT)
and Inverse DFT (IDFT) blocksare used toswitch from time to frequency domain and vice versa,respectively. According toFig. 2 the following stepsarerealized:
x Step 1: At ୲୦ time instant, the superimposed signal ܚሺ୩ሻisreceivedat therelay, the predictionof
the channel ۱ሺ୩ȁ୩ିଵሻisalso propagated from symbol
time െ ͳ to.
x Step 2: ܐሺ୩ȁ୩ିଵሻand ሺ୩ȁ୩ିଵሻare extracted from ۱ሺ୩ȁ୩ିଵሻusing separator block, easily. The normalized DFT is used to transfer signals to
frequency domain (computing
ܚሺ୩ሻǡ ۶ሺ୩ȁ୩ିଵሻ,۵ሺ୩ȁ୩ିଵሻ). Then, using Eq. (1) and
Maximum Likelihood (ML) detection, the primitive
estimationof ܆୨ሺ୩ሻ,say ܆ෙ୨ሺ୩ሻǡ ൌ ͳǡʹ isobtained.
This step is done with Data Estimator block, surrounded part by thedottedinFig. 2.
x Step 3: Building ܆෩ෙሺ୩ሻ, a primitive estimation of timedomain circulant matrix ܆෩ሺ୩ሻ presentedin Eq.
(4), by taking the normalized IDFT of ܆ෙ୨ሺ୩ሻ. Now,
܆ ෩
ෙሺ୩ሻand ۱ሺ୩ȁ୩ିଵሻ run the KF-CE Algorithm to
obtain an updated channel estimation۱ሺ୩ȁ୩ሻ, anda new prediction۱ሺ୩ାଵȁ୩ሻrespectively.
x Step 4: Another Data Estimator block, thesameas
the first oneinstep 2, producesanew estimationof
܆୨ሺ୩ሻ using ۶ሺ୩ȁ୩ሻand ۵ሺ୩ȁ୩ሻ extracted from ۱ሺ୩ȁ୩ሻ.
The inputs of this Data Estimator block are the
updated channelsestimation, therefore theoutput is a goodestimationof data,܆୨ሺ୩ሻ.
The implemented Kalman filter process is illustrated in Table I,indetails, using thestandardnotations. The readerisreferred to Appendix forreminder the general
structureof KalmanFilter.
TableI: KF-CE Algorithm
Initialization: Getting:σȁିଵ,ɐ
ܖ ,ɐ
୴
ଶ,۱,۾ǡ ܕǡ Ɂ
For୲୦OFDMsymbol,
1. Input:ܚሺ୩ሻ,۱ሺ୩ȁ୩ିଵሻ,܆෩ෙሺ୩ሻǡ ۾ሺ୩ିଵሻǡ ܕሺ୩ିଵሻ
2. Compute theKalman gain۹ሺ୩ሻ
۹ሺ୩ሻൌ σ୩ȁ୩ିଵ܆෩ෙሺ୩ሻౄሾ܆෩ෙሺ୩ሻσ୩ȁ୩ିଵ܆෩ෙሺ୩ሻౄ ɐ ܖ ۷ሿିଵ
3. Estimate channel state۱ሺ୩ȁ୩ሻ,
۱ሺ୩ȁ୩ሻൌ ۱ሺ୩ȁ୩ିଵሻ ۹ሺ୩ሻሾܚሺ୩ሻെ ܆෩ෙሺ୩ሻ۱൫ห െ ͳ൯ሿ
4. σሺ୩ȁ୩ሻൌ σሺ୩ȁ୩ିଵሻെ ۹ሺ୩ሻ܆෩ෙሺ୩ሻσ୩ȁ୩ିଵ
5. Calculate primitive posteriorestimate
ૄሺ୩ሻ̱घ
ԧሺܕሺ୩ሻǡ ۾ሺ୩ሻሻ , [20]
ɐ୵ଶ۷ ൌͳ െ Ɂ
Ɂ ۾
ሺ୩ିଵሻ
Fig. 2 Overall structureof relay operation
܈ሺ୩ሻൌ ۾ሺ୩ିଵሻ ɐ୵ଶ۷ ۿሺ୩ሻൌ ܈ሺ୩ሻ ɐ
୴ ଶ۷
ܕሺ୩ሻൌ ܕሺ୩ିଵሻ ܈ሺ୩ሻۿሺ୩ሻିଵቂ۱ሺ୩ȁ୩ሻ
െ ሺ۱൫ห െ ͳ൯ ܕሺ୩ିଵሻሻቃ
۾ሺ୩ሻൌ ɐ
୴
ଶ܈ሺ୩ሻۿሺ୩ሻିଵ
6. Prediction fornext iteration σ୩ାଵȁ୩ൌ σ୩ȁ୩ ۾ሺ୩ሻ ɐ
୴ ଶ۷
ൌ ൜ቂ۱୩ାଵെ ۱൫ ͳห൯ቃ ቂ۱୩ାଵെ ۱൫ ͳห൯ቃୌฬܚሺ୩ሻൠǡ
۱ሺ୩ାଵȁ୩ሻൌ ۱൫ห െ ͳ൯ ܕሺ୩ሻ , ۱ሺȁିଵሻൌ ۱
Steps 1 to 4 and step 6in Table I are the relevant
Kalman Filtersteps forour proposedstructure that can
be written comparing Eqs. (4), (5-b) with (A1) and (A2)
in Appendix.Step 5has been also written considering the proposed dynamic model in [20], as we explained earlier. In Table I,σ൫ห െ ͳ൯is the covariance matrix
of the prediction error, σȁିଵ is considered to be an identity matrix ۷ሺభାమሻൈሺభାమሻ. ɐܖ
۷ shows the
covariance matrix of the measurement noiseܖሺ୩ሻ,ɐ୴ଶ۷is
the covariance matrix of thestatenoise܄ሺ୩ሻ,and۱ is
the initial estimation of the channel state information۱ሺ୩ሻ. The term ۱isobtained from training sequenceasisexplainedin thenext section. Theinitial
distribution for the primitiveequation wasset toǡ ܕൌ ǡ ۾ൌ ͳͲିସ۷. According to the [20], in all the experiments performedin this paper, at aSNR of 5dB,
thediscount factorɁ wasset to 0.999. Thenas theSNR
increases, Ɂ is steadily decreased to 0.89 at 25dB.
Experimentally, the discount factor in this range was
found to produce the best performanceresults.
Therefore,ineach timeinstant , the relay receives
compositesignal ܚሺ୩ሻ, updatesitsestimationof the two
uplink channels, de/modulates and decodes data information from bothnodes,re-encodesand forwards
thedata toendnodes using the concept of PLNC.
As mentioned in the previous section, indownlink
phase, the received signal at destinationଵ can be
presented by Eq. (2). Equation (2) (that isin frequency
domain) canalso be writtenin timedomainas:
ܡሺ୩ሻൌ ቂ܆෩
୰ ሺ୩ሻቃቂܐ
୰
ሺ୩ሻቃ ܖ
୰ሺ୩ሻ, (6)
where܆෩୰ሺ୩ሻ, thenormalized IDFT of ܆୰ሺ୩ሻin Eq. (2),isa
ൈ ୰ circulant matrix formed with the modulated
block ܠ୰ሺ୩ሻൌ ൣ୰ሺ୩ሻሺͲሻǡ ୰ሺ୩ሻሺͳሻǡ ǥ ǡ ୰ሺ୩ሻሺ െ
ͳሻ൧(expressing the convolution process by a matrix product which makes the notation much easier).
ܖ୰ሺ୩ሻ̱घԧ൫ǡ ɐ୬౨
ଶ ۷൯ and ܄ ୰
ሺܓሻ
̱घԧሺǡ ɐ୴౨
ଶ۷ሻ are the
measurement and the state noise models, respectively, andૄ୰ሺ୩ሻ̱घԧሺܕ୰
ሺ୩ሻ
ǡ ۾୰ሺ୩ሻሻ.
Also thedynamic model fordownlink channel from the relay tonodeଵisexpressedas,
ܐ୰ሺ୩ሻൌ ܐ୰ሺ୩ିଵሻ ૄ୰ሺ୩ሻ ܄୰ሺ୩ሻ (7) Therefore, Eqs. (6), (7) make the statespaceequations
for the downlink phase equivalent with (A1), (A2) in
Appendix.Now, with the same procedure the relevant
KF-CE algorithm can be written for downlink channel
estimation.
As a result, we can see that using the KF-CE
algorithm channel tracking is performed with high accuracy,no phaseambiguity (the problem whichsome
methods [4] suffer from it), and stability. By this
method,it isenough tosenda training OFDM blockat the start as well as after every transmission, to
update primitiveestimationof channelsinformation,۱.
Thishasan important advantage that wedon’t have to send training pilots in each OFDM block to track the
channels variations. Comparing with other methods
which need some pilots (at least ሺଵ ͳሻ ሺଶ ͳሻ,
[4]), in each OFDM block, our proposed algorithm
needs much less training data, and so achieves much
more throughput. A special training sequence which can
be usedinoursystem will be proposedin thenext part.
3.2 ProposedTrainingSequence
Toavoid the problem stemming from overlapping of the channel impulse responses from two users
throughout training phase,at the first training time, both nodes ଵ and ଶ transmit the same training OFDM
block with uniform scaling onall thesub-carriers,say
ଶሺሻሺሻ ൌ ଵሺሻሺሻ ൌ , ൌ ͳǡʹǡ ǥ ǡ െ ͳ,
ȁȁଶൌ ͳ.
Andat thesecond training time they transmit
ଶሺଵሻሺሻ ൌ െଵሺଵሻሺሻ ൌ .
Assuming fading channel gains are constant over
two consecutive OFDM blocks (we need this assumptionjust for training phase),according to Eq. (1),
two consecutive OFDM block (after CP removal andN -point normalized DFT) received at the relay can be express,as:
ܚሺሻൌ ۶ሺሻ ۵ሺሻ ܖሺሻ (8) ܚሺଵሻൌ ۶ሺሻെ ۵ሺሻ ܖሺଵሻ (9)
Using (8) and (9), the least-square channel estimate, ۶ሺሻൎ ۶ሺሻ and ۵ሺሻൎ ۵ሺሻ, can be easily obtained
from the following equations:
۶ሺሻൌ ଵ
ଶୟ൫ܚ
ሺሻ ܚሺଵሻ൯ ൌ ۶ሺሻ ܖ
ୡ୦ (10)
۵ሺሻൌ ଵ
ଶୟ൫ܚ
ሺሻെ ܚሺଵሻ൯ ൌ ۵ሺሻ ܖ
ୡ୦ (11)
whereܖୡ୦ൌ ଵ ଶୟ൫ܖ
ሺሻേ ܖሺଵሻ൯̱घ ԧሺǡ
మ ଶ ۷ሻ.
Consequently, the relay isable toestimate the CSIs
from both users in the training phase, whichis used to
calculate the first estimationof ۱,and updatesit every
OFDM blocks. Between two training phase, the relay usesKF-CE algorithm to track the CSIs from both
users. According to this method for training, we useʹ כ
Iranian Journal of Electrical & Electronic Engineering, Vol. 12, No. 3, September 2016 191
modulatedsymbols per OFDM symbols compared
withכ ሺଵ ଶ ʹሻonesin pilot based training [4]. For example if you consider ൌ (although,
usually ܰ) andଵൌ ଶൌ ʹ, bandwidthefficiency
in our proposed method is three times more than [4], andif ଵൌ ଶൌ ͷ (as weassumedinoursimulations), here bandwidthefficiency issix times more than [4]. Note that the assumption of the constant channel between two consecutive OFDM blocksisareasonable assumption. However if it does not hold, the primary channel estimate can be done in other ways like what was proposedin [23].
4 Computer ResultsandDiscussions
This section represents the simulation results, with
the useof our proposedapproachof channel estimation and tracking algorithm in time-domain for an OF DM-PLNC system.
Channels from both୨’s to R are assumed to have
five taps, each presented by a symmetric complex Gaussian random process. The Doppler spectrum is
Jake’s and ɐ୴౨ଶ ൌ ɐ୴ଶൌ ͳ െ ሺሺʹɎୈୱሻሻଶ, where ୈ
denotes the Doppler frequency between moving transmitter and receiver, and ୱൌ
ଵ
౩ is the rate of sampling. For example, if the normalized considered
fading rate is ୈୱൌ ͲǤͲͳ (an example of fading rate
for fast fading channel), the noise variances of the
model are considered as ɐ୴౨ ଶ ൌ ɐ
୴
ଶൌ ͳǤͻʹ ൈ
ͳͲିଷǡaccording to [18], [20]. Weassumed that thereis no frequency offset between transmitters and channels are assumed constant during one OFDM symbol. We always fix ଵൌ ଶൌ ୰ (the average transmission
power ofଵ, ଶ and R) and define SNR as ଵȀɐ୬ଶ. Numberof subcarriers, following IEEE 802.11a, wasset
as ൌ Ͷ, and the lengthof CP isେൌ ͳsamples.
QPSKsymbol modulation isemployedand Re-training
symbols are sent repeatedly every ൌ ͳͲͲ OFDM
symbols in our simulations. No channel coding was
considered.
Monte Carlo simulations over ͳͲସ runs are
performed for averaging and MSE figureof merit was
used whichisdefinedas;
ൌ ଵ
ହൈଵరσ ฮܐመሺ୩ሻെ ܐሺ୩ሻฮ ଶ ଵర
୩ୀଵ (12)
where ܐመሺ୩ሻ is the ୲୦ estimation of ܐሺ୩ሻ, and 5 in denominatorisjust forscaling andrefer to thenumber of channel taps.
At first, the performance of proposed method is studied in the time domain for tracking the amplitude and phaseimpulseresponseof the channel. Magnitude and phase tracking of (the first tap of ) isshownin
Fig. 3,in the caseof ୈୱൌ ͲǤͲͳ andat averageSNR
15dB. Channel estimation only based on training
sequence (without KF-CE tracking) is also illustrated.
This figure shows that the time variations of channel tapsare traced with precision,evenin low instantaneous SNR.
Fig. 4 illustrates theaverage MSE of ܐ andat the relay. As tworeferences curves, this figurealsoshows
theaverage MSE of ܐandat therelay obtained with
the perfect knowledgeof data (܊ଵ܊ଶሻ,and theother obtained with channel estimationonly basedon training
sequence and without KF-CE tracking. Besides, the average MSE of therelevant downlink channel at node ܖଵ is also shownin this figure. Note that the channel
estimation at the relay (uplink channels estimation) is
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
100
Time (s)
Am
p
litu
d
e
r
s
po
n
s
e
(
|h
0
(t)|
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4 -2 0 2 4
Time (s)
Ph
a
s
e
r
e
s
p
o
n
s
e
(
a
rg
(h
0
(t)) )
actual channel tracking with KF-CE
channel estimation with training sequence without tracking
Fig.3 Time-domain Channel tracking fora typical Doppler faded channel (SNR=15dB)
done based on Eq. (1), estimating of ܐ and after receiving the superimposed signal ܚሺ୩ሻ. However, the estimation at the end nodes is done based on Eq. (2) whichisin fact,similar toa channel estimationin point to point communications. In the uplink phase the estimation error of both ܊ଵ ܊ଶ effects on the
channel estimation error, whilein the downlink phase, at nodeܖଵforexample,only theestimationerrorof ܊ଶis effective. For this reason, the average MSE of uplink
channelsestimation is more than thedownlink channel, asFig. 4 illustrates. It isseenin this figure the MSE of uplinkanddownlink channelsestimationarenearly the same under the perfect knowledgeof data consideration. Since data detection is carried in the frequency
domain, hence the accuracy of estimating the channel
frequency responses in the frequency of sub-carriers is alsoneeded to beevaluated.Fig.5showsamplitudeand
phase responses, for the actual channel and its estimation,at a given OFDM block time. Asit isseen,
channel transfer functionsarealsoestimatedaccurately, in bothamplitudeand phase.
The averageBER of ܊ଵǡ ܊ଶ at the relay,as well as, BER of ܊ଶat nodeଵ(end toendBER),arealsoshown inFig.6.
InFig. 7 weshow the average MSE of tracking for different receiver types as the channel dispersion increases. The signal-to-noise ratio of the channel is
considered to be 15 dB in this figure. For small
normalized Dopplerrate (< 0.001), the channel isalmost constant between tandem training symbols. However,
by increasing the Dopplerrate, the dispersivenatureof the rises and tracking the channel becomes more difficult. As you canseein this figure, theoverall MSE
of the downlink channel is lower than the uplink
channels. This is because only one channel in the downlinkdirectionisrequired toestimate, but therelay
estimates two uplink channels at the same time in the
uplink phase. As it is seen, all these figures predict a
very good performanceof the proposedalgorithm ina
time-varying frequency selective fading channel conditions.
5 Conclusion
In this paper, we proposeda time varying frequency fading channel tracking algorithm for OFDM-PLNC
system, where, two users exchange their information
Fig. 4 Theaverage MSE of the uplink channelsestimationat the relay anddownlink channel estimationat nodeܖ
5 10 15 20 25
10-5
10-4
10-3
10-2
10-1
100
101
102
SNR (dB)
M
ea
n
S
qu
a
re
E
rro
r
o
f
ch
an
n
e
l
tra
c
k
i
ng
UplinkKF-CEwithoutperfectknowledgeof data X1 & X2
UplinkKF-CEwithperfectknowledgeof data X1 & X2
UplinkCEwithtrainingsequencewithoutKF-CEtracking
DownlinkKF-CEatT1withoutperfectknowledgeof data X2
DownlinkKF-CEatT1withperfectknowledgeof data X2
0 10 20 30 40 50 60
0 5 10 15 20 25
Subcarrier Index
Am
p
li
tude
re
s
pons
e
s
(|
H
(
f
)|
)
0 10 20 30 40 50 60
-4 -2 0 2 4
Subcarrier Index
P
h
ase
re
s
pons
e
s
(a
rg
(
H
(
f
)))
trackingwithKF-CE
actual channel
channel estimationwithtraining
sequencewithouttracking
Fig. 5Frequency-domain Channel tracking fora typical Doppler faded channel (SNR=5dB, t=60(s))
Iranian Journal of Electrical & Electronic Engineering, Vol. 12, No. 3, September 2016 193
througha relay using the decode-and-forward relaying
scheme. A Kalman Filter Algorithm working in time domain is the base of this approach where the initial condition needed by KF, comes from a simple and
linear precoding algorithm implemented over every
OFDM symbols. Accuracy,no phaseambiguity,and stability of this methodin fast fading channels are its advantages, in comparison with other tracking
algorithms running in frequency domain. Furthermore, our method uses less training symbolsand gives much
more throughput in comparison withother approaches
that needs pilot insertionoverdifferent OFDM symbols ina frame. Thesimulationresults,illustratea very good
performance. In this paper, equations are written assuming thesame fading rate for uplink channels (ǡ ).
This could bean inspiring direction for future work to extendKF-CE algorithm to the GeneralizedKF-CE that
isable to track channels withdifferent fading ratesina
PLNC system.
Appendix
The Kalman filter is an efficient recursive
filter that estimates theinternal stateof a lineardynamic
system discretized in the time domain from a series of noisy measurements. A general model for such a
system isdepicted inFig. A1,anddescribed forܓ
by the following equations:
ܢܓൌ ۶ܓܠܓ ܞܓ (A1)
ܠܓାൌ ۴ܓܠܓ ۵ܓܟܓ (A2)
For the linear, finitedimensional,discrete-timesystem
of (A1) and (A2) defined forܓ ,suppose that ܞܓand ܟܓ are independent, zero mean, Gaussian White
processes with
۳ሾܞܓܞܓᇱሿ ൌ ܀ܓ઼ܓ۳ሾܟܓܟܓᇱሿ ൌ ۿܓ઼ܓ (A3)
Suppose further that the initial state ܠ is a Gaussian random variable with mean ܠത and covariance ۾, independent of ܞܓandܟܓ. Determine theestimates
ܠොܓȁܓିൌ ۳ሾܠܓȁ܈ܓିሿܠොܓȁܓൌ ۳ሾܠܓȁ܈ܓሿ (A4)
and the associated error covariance matrix σܓȁܓି and
σܓȁܓ, where܈ܓିൌ ሼܢǡ ܢǡ ǥ ǡ ܢܓିሽand
σܓȁܓିൌ ۳ሼሾܠܓെ ܠොܓȁܓିሿൣܠܓെ ܠොܓȁܓି൧ᇱȁ܈ܓିሽ (A5)
showshow good theestimateܠොܓȁܓିis.
The Kalman Filter comprises the system depicted in
Fig. A2 and is described for ܓ by the following
equations. The proof of theseequations can beseenin
[22].
ܠොܓȁܓିൌ ሾ۴ܓെ ۹ܓ۶ܓᇱሿܠොܓȁܓି ۹ܓܢܓ (A6)
With
ܠොȁିൌ ܠത (A7)
5 10 15 20 25
10-4 10-3 10-2 10-1 100
SNR(dB)
B
i
t
Er
ro
r
Ra
te
AverageBERofb1andb2 intheRelay withKF-CE AverageBERofb1andb2 intheRelay withperfectCSIs AverageBERofb1andb2 intheRelay withoutChannel tracking Estimationofb2atnoden1withKF-CE
Estimationofb2atnoden1withperfect CSI
Fig. 6 TheaverageBERof ܊ǡ ܊at therelay,as well as,
BERof ܊at nodeܖ(withand without channel state
information)
Fig. 7 Meansquarederror tracking performance (averagedover
all channels) vs.normalized Dopplerrate.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
10-4
10-3
10-2
10-1
100
101
SNR(dB)
Me
a
n
Squa
re
Er
ro
r
AverageMSEofuplinkchannelsh&g
MSEofdownlinkchannel hr
ݖ݇ ݒ݇
ݕ݇ ݔ݇ͳ
ݓ݇ ܩ ܼെͳ
݇ ܪ݇
Ԣ
ܨ݇
Fig. A1Basic signal model
The gain matrix ۹ܓ is determined from the error
covariance matrix by
۹ܓൌ ۴ܓσܓȁܓି۶ܓሾ۶ᇱܓσܓȁܓ۶ܓ ܀ܓሿି (A8)
Assuming the inverse exists, and the conditional error
covariance matrix is givenrecursively by
σܓାȁܓൌ ۴ܓሾσܓȁܓିെ σܓȁܓି۶ܓሺ۶ܓᇱσܓȁܓି۶ܓ ܀ܓሻି۶ܓᇱσܓȁܓିሿ۴ܓᇱ ۵ܓۿܓ۵ܓᇱ (A9)
Thisequationisinitialized by
σȁିൌ ۾ (A10)
Oneobtains ܠොܓȁܓandσܓȁܓas follows:
ܠොܓȁܓൌ ܠොܓȁܓି σܓȁܓି۶ܓሺ۶ܓᇱσ
ܓȁܓି۶ܓ ܀ܓሻିሺܢܓെ ۶ܓᇱܠොܓȁܓିሻ (A11)
σܓȁܓൌ σܓȁܓିെ σܓȁܓି۶ܓሺ۶ܓᇱσ
ܓȁܓି۶ܓ
܀ܓሻି۶
ܓᇱσܓȁܓି (A12)
References
[1] S. Zhang, S. C. ILLKM, and P. P. Lam, "Hot topic: Physical-layer network coding," in Proc.
12th Annual Int. Conf Mobile Computing and Networking (MobiCom '06). New York, NY. USA: ACM, pp. 358-365, 2006.
[2] R. Ahlswede, N. Cai, S.Y. R. Li, and R. W.
Yeung, "Network information flow," IEEE
Trans. Inf. Theory, Vol. 46, No. 4, pp.
1204-1216, 2000.
[3] J. Laneman, D. Tse, and G. Wornell,
"Cooperative diversity in wireless networks: efficient protocols and outage behavior", IEEE
Trans. Inf. Theory, Vol. 50, No. 12, pp. 3062–
3080, 2004.
[4] F. Gao, R. Zhang and Y.C. Liang, "Channel
estimation for OFDM modulated two-way relay
networks," IEEE Trans. Sig. Proc., Vol.57,No.
11, pp. 4443-4455, 2009.
[5] Rossetto, F., Zorzi, M. "On the design of practical asynchronous physical layer network
coding". IEEE 10th Workshop on SPAWC, pp.
469–473, 2009.
[6] H. Gacanin and F. Adachi, "Performance of physical layer network coding in a frequen
cy-selective fading channel," IEEE Int. Symp. on
Personal, Indoor and Mobile Radio Comm., pp.
928-932, 2009.
[7] Jie Huang; Zhaohui Wang; Shengli Zhou; Zhengdao Wang "Turboequalization for OFDM modulated physical layer network coding", Signal Processing Advances in Wireless
Communications (SPAWC), IEEE 12th
International Workshop, pp. 291 – 295, 2011.
[8] Desi Luo; Qiyue Yu; Weixiao Meng "Optimum power allocation for OFDM in physical-layer network coding over a flat frequency-selective
fading channel", Information, Communications
and Signal Processing (ICICS), 9th International
Conference, pp. 1 – 5, 2013.
[9] Tomas Sjodin: "A Channel Estimation Scheme
for Analog Network Coding basedon OFDM ina
MultipathFading Environment",Master’s thesis
in computing science, UMEA University, 2009.
[10] Min Zhou, Qimei Cuiand M. Valkana, "Energy Efficient Resource Allocation for OFDMA based
Two-Way Relay Channel with Physical-Layer Network Coding",EURASIP Journal on Wireless
Communication and Networking, Vol. 16,No.6,
pp. 3311-3321, 2012.
[11] Fei He, Sun Yin, Chen Xiang, Xiao Limin and
Zhou Shidong, "Optimal Power Allocation for
Two-Way Decode-and-Forward OFDM Relay
Network", in Proc. IEEE ICC’12, Ottawa,
Canada, pp. 4463-4467, 2012.
[12] R. C. Cannizzaro, P. Banelli, and G. Leus,
"Adaptive channel estimation for OFDM systems
with Doppler spread", IEEE Signal Processing
Advances in Wireless Communications, pp. 1–5,
2006.
[13] P. Banelli, R. C. Cannizzaro, and L. Rugini,
"Data-aided Kalman tracking for channel
estimation in Doppleraffected OFDM systems",
inIEEE ICASSP, 2007.
[14] Al-Naffouri, T.Y., "An EM-Based Forward
-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM", Signal
Processing, IEEE Transactions on, Vol.55, No.
7, pp.3924-3930, 2007.
[15] K. MuralidharandK.H. Li, "A Low-Complexity
Kalman Approach for Channel Estimation in
Doubly-Selective OFDM Systems",IEEE Signal
Process. Letters, Vol. 16, No. 7, pp. 632–635,
2009.
ܭ݇ ݖെͳ
ܨ݇െ ܭ݇ܪ݇Ԣ
ݖ݇ ݔො݇ͳȁ݇ ݔො݇ȁ݇െͳ
Fig.A2Structureof KalmanFilter
Iranian Journal of Electrical & Electronic Engineering, Vol. 12, No. 3, September 2016 195
[16] L. Ros, H. Hijazi, E.-P. Simon, "Complex AmplitudesTrackingLoopformultipathchannel estimation in OFDM systems under slow to moderatefading",SignalProcessing,2014.
[17] P. Popovski and H. Yomo, "Physical network coding in two-way wireless relay channels", Proc. ICC 2007, Glasgow, UK, pp. 707-712, 2007.
[18] A. Bello, "Characterization of randomly time
-variant linear channels", IEEE Trans. Commun.
Systems,Vol.11,No.4,pp.360-393,1963.
[19] H. Wang and P. Chang, "On verifying the first order Markovian assumption for a Rayleigh fading channel model", IEEE Trans. Vehicle.
Technol.,Vol.45,No.2,pp.353-357,1996.
[20] K. Huber and S. Haykin "Improved Bayesian MIMO Channel Tracking for Wireless Communications: Incorporating a Dynamical Channel", IEEE Trans.WirelessCommun.,Vol. 5,No.9,pp.2458-2466,2006.
[21] Barbieri,A.;Piemontese,A.;Colavolpe,G.,"On the ARMA approximation for fading channels describedby the Clarkemodel withapplications to Kalman-based receivers", IEEE Transactions
on WirelessCommunications,Vol.8, No. 2,pp.
535-540,2009.
[22] Brian D. O. Anderson, John B. Moore, "OPTIMAL FILTERING", Englewood Cliffs,
N.J.Prentice-Hall,1979.
[23] T.Sjodin,H.GacaninandF.Adachi "Two-slot channel estimation for analog network coding based on OFDMin afrequency-selectivefading channel", inProc.IEEE VTC,pp.1-5,2010.
S. Khosroazad received the B.S.
and the M.S. degrees in electrical
engineering from Ferdowsi
University of Mashhad, Mashhad,
Iran,in2006and2009,respectively. She is currently pursuing the PhD degreeintheelectricalandcomputer
engineering department at the
UniversityofBirjand,Birjand,Iran. In 2016 shehas receiveda visiting fellowshipfromUniversityofMaine,Maine,USA,andspent six monthsin WiSe-Net Laboratory workingwith Prof. Ali Abedi’sgroup.Herresearchinterestsincludecommunication
systems theory, Wireless Communications, Cooperative
Networks, Physical Layer network Coding and Cognitive
Networks.
N.NedareceivedtheB.S.degreein electricalEng.fromtheUniversityof
Tehran and the M.S. degree in
communication Eng. from Sharif
University of Technology (SUT),
both in Tehran, Iran, in 1990 and
1994 respectively. He received the
PhD degree in communication Eng.
from the University of Surrey
(CCSR),Guildford,UK,in2003.Heiscurrentlyanassistant professorofcommunicationengineeringwiththeDepartment ofElectricalandComputerEng.attheUniversityofBirjand, Iran. His research-interests include signal Processing for
communication systems, physical layer of
CDMA/MCCDMA/OFDMnetworksandsensornetworks.
H. Farrokhi received the B.S.E.E.
degree from theSharif University of
Technology (SUT), Tehran, Iran in
1989, the M.S. degree in electronics
engineering from the Iranian
UniversityofScienceandTechnology (IUST), Tehran,Iranin 1996and the PhDdegree(withdistinction)fromthe Universityof Regina, Canada in2005. Hewas a researcher withtheIranianTelecommunicationResearchCenter(ITRC) during 1990-1991. He is currentlyan associate professor of
communication engineering with the Department of
Engineering at the University of Birjand, Iran. His current
research interests include spread spectrum systems and
positioning, next generation wireless systems and cognitive radio.
