LOW-COMPLEXITY OFDM CHANNEL PREDICTOR DESIGN WITH VARYING SPEEDRECEIVERS IN TIME-VARYING WIRELESS CHANNELS

Edwin Christopher, Gejie Liu, Xianbin Wang, Jagath Samarabandu

Department of Electrical and Computer EngineeringThe University of Western Ontario, London, Ontario, Canada N6A 5B9

ABSTRACT

Signals transmitted through a wireless channel undergo dis-tortion due to multi-path and Doppler shift effects which canbe recovered at the receiver through channel state informa-tion (CSI). The variation of wireless channels causes outdatedCSI to be used for optimization techniques. Channel predic-tion allows the system to adapt modulation methods to an es-timated future CSI. But this becomes complex when the re-ceiver moves with a varying speed since it directly affects thechannel parameters. This paper proposes a channel predictionscheme in OFDM systems with speed-varying receivers. Thepredictor first tracks such speed variations and if significantchange occurs, then the channel parameters are re-calculatedand updated for future channel prediction. The theoreticalarguments and simulations show that the proposed systemcan obtain good performance in multi-path channel condi-tions with low computational complexity.

1. INTRODUCTION

Orthogonal Frequency Division Multiplexing (OFDM) is wellknown for its high data rate communication with the maxi-mum usage of the available spectrum and robustness to thefrequency-selective fading. However, OFDM performance isseverely affected by Doppler spread which causes the loss oforthogonality among subcarriers and resultant Inter CarrierInterference (ICI). A widely explored approach for obtainingchannel state information (CSI) in OFDM systems is trainingdata-based channel estimation, but channel estimation has thedrawback of potentially yielding outdated channel state infor-mation, which can prevent the application of promising tech-niques requiring instantaneous channel measurements.

Taking into account the feedback delays, an effective ap-proach in adaptive systems is to predict the channel responsesand feed those information back to the transmitter. Most chan-nel prediction techniques assume that there is no channel vari-ation in the defined time duration [1, 2]. However, the varia-tion of the channel over the time in rapidly time varying mo-bile environments cannot be ignored. Although the channelprediction techniques can compensate for the performanceloss related to the feedback delays of CSI, those techniquescannot perfectly compensate for the performance degradationresulting from the rapid channel variation over the duration.

Therefore, prediction of channel responses for time vary-ing channel through tracking the changes of parameters hasbecome a more popular research area for the past two decades.In [3], Lee suggested different pilot arrangements for differentreceiver speed to increase the prediction accuracy and Hei-dari [4] came up with a channel tracking algorithm basedon the prediction error threshold. In this paper, we proposea low-complexity channel predictor design to track Dopplershift changes and update the channel parameters when thereis a significant change in Doppler shift. The predictor subsys-tem employs the following property: In single-carrier system,the received signal envelope will have relatively deep fading(nulls) around every half-wavelength distance [5]. In multi-carriers system like OFDM, we take advantage of this afterconsidering the interference among the subcarriers which willbe discussed in details later. Therefore, the first step in theproposed system is to track the speed varying of receiversthrough nulls calculation for the past fixed period. When thereis significant speed change the channel parameters are esti-mated and updated and then used for channel prediction.

The rest of this paper is organized as follows: In section2, the channel model the relation between nulls and speed ofreceiver is introduced. Section 3 describes the proposed chan-nel predictor. Sections 4 and 5 show the simulation results ofthe proposed system and conclusions achieved from the re-sults respectively.

2. CHANNEL MODEL AND NULLS ANALYSIS

Consider an OFDM system with 𝐾 subcarriers. While theOFDM channel is wideband and, therefore, frequency selec-tive, the bandwidth of each subcarrier is usually much smallerthan the coherence bandwidth of the channel, and, thus, thesignal associated with each subcarrier is flat fading [5]. De-note the flat fading complex channel gain at the 𝑛th sym-bol block and 𝑘th subcarrier (with center frequency 𝑓𝑘) as𝐻(𝑛, 𝑘). We express 𝐻(𝑛, 𝑘) as the 𝑛th sample of complexfading channel. From [6] we know the complex basebandrepresentation of the time-varying wireless channel is givenas

ℎ𝑐(𝑡, 𝜂) =

𝐿−1∑

𝑝=0

𝛾𝑝(𝑡)𝛿(𝜂 − 𝜂𝑝(𝑡)) (1)

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�����������������������������������

where 𝜂𝑝(𝑡) is the delay, 𝛾𝑝(𝑡) is the complex amplitude ofthe 𝑝th multi-path tap, and 𝐿 is the number of propagationpaths. Assuming a far-field discrete scatterer model, 𝛾𝑝(𝑡)can be further decomposed as

𝛾𝑝(𝑡) =

𝑀𝑝−1∑

𝑟=0

𝑎𝑟,𝑝𝑒𝑗2𝜋𝜈𝑟,𝑝(𝑡)𝑡 (2)

where 𝑀𝑝 is the number of rays contributing to the 𝑝th path,and 𝑎𝑟,𝑝 and 𝜈𝑟,𝑝(𝑡) are the complex amplitude and Dopplerfrequency, respectively, for the 𝑟th ray in the 𝑝th path. Notethat the random phase from the complex exponentials havebeen incorporated into 𝑎𝑟,𝑝. Also note that the time delays andDoppler frequencies are all dependent on time. However, wecan assume that the time delay 𝜂𝑝(𝑡) and Doppler frequency𝜈𝑟,𝑝(𝑡) parameters vary slowly when compared to the OFDMsymbol time, and can be considered constant within the es-timation and prediction time horizons. Combining (1) and(2) and taking its Fourier transform, we get the frequency re-sponse of the time-varying channel as

𝐻𝑐(𝑡, 𝑓) =𝐿−1∑

𝑝=0

𝑀𝑝−1∑

𝑟=0

𝑎𝑟,𝑝𝑒𝑗2𝜋𝜈𝑟,𝑝𝑡𝑒−𝑗2𝜋𝜂𝑝𝑓 (3)

Assuming that the OFDM system with symbol period 𝑇𝑠𝑦𝑚and subcarrier spacing Δ𝑓 has proper cyclic extension andsample timing, the sampled channel frequency response at the𝑘th subcarrier of the 𝑛th OFDM block can be expressed as

𝐻(𝑛, 𝑘) =𝐿−1∑

𝑝=0

𝑀𝑝−1∑

𝑟=0

𝑎𝑟,𝑝𝑒𝑗2𝜋(𝑓𝑟,𝑝𝑛−𝜏𝑝𝑘) (4)

where 𝑓𝑟,𝑝 = 𝜈𝑟,𝑝𝑇𝑠𝑦𝑚 is the normalized Doppler frequencyand 𝜏𝑝 = 𝜂𝑝Δ𝑝𝑓 . Based on the above analysis, we can seethat the multi-path fading effect on 𝑘th subcarrier is similar tothe effect on single carrier signal [5].

Now, for the number of nulls as an indicator of speed vary-ing, we need several relations below. If the speed of receiveris 𝑣 and carrier frequency is 𝑓𝑘 in 𝑘th subcarrier, maximumDoppler frequency is defined as 𝑓𝑑,𝑘 = 𝑓𝑘 ⋅ (𝑣/𝑐). where 𝑐 isthe speed of light. So we have the following

𝑇𝑘 ≈ 0.423

𝑓𝑑,𝑘=

0.423𝑐

𝑣 ⋅ 𝑓𝑘 (5)

𝑁𝑘 ≈ 𝑇𝑡𝑜𝑡,𝑘/𝑇𝑘 =𝑇𝑡𝑜𝑡,𝑘 ⋅ 𝑣 ⋅ 𝑓𝑘0.423 ⋅ 𝑐 (6)

where 𝑇𝑘 is coherence time, 𝑇𝑡𝑜𝑡,𝑘 is the total time consideredand 𝑁𝑘 is number of nulls in 𝑘th subcarrier. Because of therelation of 𝑁𝑘 ∝ 𝑣, it is obvious that number of nulls can beused to indicate the change of speed.

3. LOW-COMPLEXITY CHANNEL PREDICTIONSUBSYSTEM DESIGN

Fig. 1 exhibits the basic structure of the proposed channel pre-diction subsystem which consists mainly of two parts: nulls

detection and channel prediction.

Low PassFilter

ChannelEqualization

NullsDetection

ChannelPrediction

S/P DFTReceivedSignal

RecoveredSignal

Channel Predictor Subsystem

CarrierFrequencyX

Fig. 1. Prediction Subsystem for the proposed method.

3.1. Nulls Detection and Channel PredictionAs an indicator, number of nulls should be traced to the changeof receiver speed and feed it to channel predictor. The firststep here is to find a way to detect the number of nulls. Weemploy a method based on the first derivative and secondderivative of time-domain properties in subcarriers. As canbe seen in Fig. 2, the signal in subcarriers are influenced bymulti-path fading with many deep fading points (nulls). Thesenulls can be categorized into two cases: Case 1 and Case 2. Inorder to find these nulls, we use properties of first and secondderivatives as shown in Fig. 3.

1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32

x 104

0

0.05

0.1

0.15

0.2

0.25

Time (ms)

Filt

ered

Rec

eive

d S

igna

l Env

elop

e (d

B)

Case 1

Case 2

λ/2

Fig. 2. Received Multipath OFDM Signal in one subcarrier

The method can be generalized in this way, after receiving𝑝 previous OFDM symbols, we pick 𝐿 subcarriers and obtaintheir corresponding time-domain properties. For the 𝑘th sub-carrier of received signal in time-domain 𝑦𝑘(𝑡),

∙ For Case 1: Based on the first derivative 𝑦′𝑘(𝑡) and sec-

ond derivative 𝑦′′𝑘 (𝑡), if 𝑦

′𝑘(𝑡𝑛1−𝛿) > 0,𝑦

′𝑘(𝑡𝑛1+𝛿) < 0

and 𝑦′𝑘(𝑡𝑛1) = 0, 𝑡𝑛1 can be recognized as a null, where

𝛿 is a infinitesimally small positive number.

∙ For Case 2: In this case, the nulls are not easily iden-tified only through first derivative due to the noise ef-

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Stationary Reflection Points

Signal

Envelope(dB)

First

Derivative

Second

Derivative

Case: 2a Case: 2b

Local Minimum Points

Case: 1

Signal

Envelope(dB)

First

Derivative

Second

Derivative

Signal

Envelope(dB)

First

Derivative

Second

Derivative

time

time

time

time

time

time

time

time

time

Fig. 3. Identification of nulls through derivatives

fect. So we employ second derivative 𝑦′′𝑘 (𝑡) to help the

detection. As is shown in Fig.3, null 𝑡𝑛2 in the sec-ond derivative should meet one of the following con-ditions: 𝑎.𝑦

′′𝑘 (𝑡𝑛2 − 𝛿) < 0, 𝑦

′′𝑘 (𝑡𝑛2 + 𝛿) > 0 and

𝑦𝑘′′(𝑡𝑛2) = 0 or 𝑏. 𝑦

′′𝑘 (𝑡𝑛2 − 𝛿) > 0,𝑦

′′𝑘 (𝑡𝑛2 + 𝛿) < 0

and 𝑦𝑘′′(𝑡𝑛2) = 0.

When the speed of receiver varies, the parameters for chan-nel prediction should be performed accordingly. Therefore,we need to establish threshold value of nulls change to decidewhen the system parameters need to be updated. Assumingthat we have 𝑁1 nulls when received 𝑃1 OFDM symbols, af-ter receiving another 𝑃2 OFDM symbols, the nulls numberbecomes 𝑁2. The significant change can be identified onlyif ∣𝑁1 − 𝑁2∣ ≥ 𝑁𝑇ℎ where 𝑁𝑇ℎ is the threshold depend-ing on the speed of receivers. In order to define the 𝑁𝑇ℎ, thereceiver speed is divided into three categories: low speed (0-40 km/h), medium speed (40-75 km/h) and high speed (>75km/h). Table 1 shows the different 𝑁𝑇ℎ values used for dif-ferent receiver speed ranges based on the simulation. where

Table 1. Nulls Threshold for Different Speed RangeSpeed Range (km/h) Correlation Reference 𝑁𝑇ℎ

0 – 40 1.000 – 0.7377 640 – 75 0.7377 – 0.2298 5>75 0.2298 – -0.3999 4

T is the OFDM symbol duration and the value is 80𝜇𝑠 in ourcase, 𝑓𝑑,𝑘 is maximum Doppler shift which depends on sub-carrier carrier frequency 𝑓𝑘 and speed of receiver 𝑣 through𝑓𝑑,𝑘 = 𝑓𝑘 ⋅ (𝑣/𝑐), 𝑚 is the correlation index, which is chosenon the condition that correlation coefficients are monotonicand it is 30 in the simulations. On the other hand, if the num-ber change of nulls is less than the threshold, which demon-strates that the channel does not vary rapidly. In this case,we assume that the subsystem updates the parameters auto-

matically every 10000 OFDM symbols (0.1𝑠). Simulations insection 4 validate this assumption.

After obtaining the information of significant speed change,the system will re-calculate the channel parameters and usethem to predict the future channel responses. In this paper, weuse per subcarrier Kalman filter channel prediction [8] whichmake use of the autocorrelation of channel responses givenby,

𝑅𝑘(𝑚) = 𝐸{𝐻𝑘(𝑛)𝐻∗𝑘 (𝑛−𝑚)} (7)

= 𝐽0(2𝜋𝑓𝑑𝑚𝑇 )

where 𝐻𝑘(𝑛) is the channel gain for 𝑘th subcarrier, 𝑓𝑑 is themaximum Doppler frequency and 𝐽0(⋅) is the zeroth-orderBessel function of the first kind. Here, autocorrelation ofchannel responses is heavily depend on Doppler shift and theautocorrelation needs to be re-calculated when there is sig-nificant change in receiver speed or Doppler shift. The cal-culation of autocorrelation is the primary computation loadfor the whole system [8]. Traditionally, updating the CSI isrealized through calculating autocorrelation periodically forevery 𝑁 OFDM symbols [8]. In our system, the update ofparameters are done only when there is a significant changein speed, which is a good way to obtain low complexity andhigh efficiency.

4. SIMULATION RESULTS AND ANALYSISTo evaluate the performance of the proposed system, numer-ical simulations have been carried out. Table 2 summarizesthe OFDM system and channel parameters. It is assumed thatthe transmitter and the receiver are fully synchronized and thechannel introduces Rayleigh fading model [7].

Table 2. OFDM and Channel ParametersSampling Frequency 1MHzChannel Model Frequency Selective

Rayleigh Fading [7]Modulation QPSKNumber of subcarriers 64Cyclic prefix length 16Pilot Arrangement 1 Symbol/ OFDM FrameKalman filter order 3

The traditional assumption for channel prediction is thatthe speed is fixed for the receiver and the autocorrelation iscalculated periodically. Therefore, we generalized the pre-vious channel prediction methods into two types: 1) Type-1: the channel correlation is calculated every 3000 OFDMsymbols and 2) Type-2: the channel correlation is calculatedevery 4000 OFDM symbols. The simulation results are ob-tained considering a receiver moving with maximum acceler-ation (9𝑚/𝑠2) from 0𝑘𝑚/ℎ to 140𝑘𝑚/ℎ.

In Fig.4, we compared the proposed system with two typesof generalized systems in certain SNR level (20 dB). At lowspeed range (≤ 40𝑘𝑚/ℎ), the Mean Square Error(MSE) of

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0−20 20−40 40−60 60−80 80−100 100−120 120−140

10−3

10−2

Speed Range (km/h)

Mea

n S

qu

are

Err

or

(MS

E)

Type−1Type−2Proposed Method

Fig. 4. Comparison of MSE of the proposed system withType-1 and Type-2.

proposed system was similar to the traditional types. This isbecause at lower speed range, the channel is slowly varyingand there is not much need for the proposed system to updatesystem parameters. At high speed range (≥ 40𝑘𝑚/ℎ), theproposed system acquired much better performance in MSEthan the others for the reason that when the channel is rapidlyvarying, the traditional methods cannot keep up with the vari-ations.

Fig. 5 displays the comparison of computational com-plexity of the proposed method with traditional ones. Fun-damentally, the computational complexity is highly depen-dent on the number of autocorrelation. As mentioned earlier,the traditional methods update channel prediction parametersthrough autocorrelation periodically at fixed time intervals,but the update in our system takes place only when signifi-cant speed changes occur. Based on the simulation, it is evi-dent that the proposed system has lower computational com-plexity compared to other types. Furthermore, the numberof calculations increases with the speed, which demonstratesthat the proposed system can follow the rapid change of chan-nel at high speed range of receiver. Additionally, the systemperformance will be better if the acceleration of receiver isless than its maximum value since in this case, Kalman filterparameters will be valid for longer period.

5. CONCLUSION

In this paper, we proposed a low-complexity channel pre-dictor design for the vehicular application where the systemadapts itself for change of receiver speed. The proposed sys-tem is based on continuous nulls detection and updates sys-tem parameters when necessary. For specific OFDM case, thesubsystem traces the number of nulls change for several sub-carriers simultaneously to optimize performance which hassimple structure and provides a good tradeoff between per-

0−20 20−40 40−60 60−80 80−100 100−120 120−1400

10

20

30

40

50

60

Speed Range (km/h)

To

tal C

orr

elat

ion

Cal

cula

tio

n

Type−1Type−2Proposed Method

Fig. 5. Comparison of computational complexity for a re-ceiver accelerating with maximum acceleration of 9 𝑚/𝑠2.

formance and the computational complexity. From the sim-ulation results, it shows that the proposed system has lowerMSE than the traditional methods on channel prediction. Fur-thermore, our system provides good system performance andmoderate computational efficiency at different speed ranges.

6. REFERENCES

[1] S. Flahati, A. Svensson, T. Ekman, and M. Sternad, “Adaptivemodulation systems for predicted wireless channels,” IEEETrans. Comm., vol. 52, no. 2, pp. 307-316, Feb. 2004.

[2] T. Ekman, M. Sternad, and A. Ahln, “Unbiased power predic-tion on broadband channel,” Proc. IEEE Veh. Tech. Conf., vol.1 pp. 280-284, Sept. 2002.

[3] S. Lee, J.Y. Lee, and H.S. Lee, “Group-based Pilot DesignMethod in Mobile OFDMA Systems,” Intl. Conf. on Adv.Comm. Tech., ICACT, vol.1, pp. 183-186, Feb. 2008.

[4] A. Heidari, A.K. Khandani, and D. McAvoy, “Adaptive mod-elling and long-range prediction of mobile fading channels,”IET, Comm., vol. 4, Issue:1, pp.39-50, Jan. 2010.

[5] T.S. Rappaport et al. “Wireless communications: principlesand practice,” Prentice Hall PTR New Jersey, 1996.

[6] I.C. Wong and B.L. Evans, “Joint channel estimation and pre-diction for OFDM systems,” IEEE Global Telecomm. Conf.vol.4, pp.5, Dec. 2005.

[7] W.C. Jakes and D.C. Cox, Microwave mobile communications,Wiley-IEEE Press, 1994.

[8] Wei Chen and Ruifeng Zhang, “Kalman-filter channel estima-tor for OFDM systems in time and frequency-selective fadingenvironment,” IEEE Intl. Conf. on Acoustics, Speech, and Sig-nal Processing, ICASSP, vol.4, pp.377-380, May. 2004.

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