Time-Frequency Interferometry for OFDMChang-Jun Ahn†, Satoshi Takahashi, and Kazuhisa Haeiwa

Hiroshima City University3-4-1 Ozuka-Higashi, Asa-Minami, Hiroshima, 731-3194 Japan

E-mail: junny@hiroshima-cu.ac.jp†

Abstract— In OFDM systems, the pilot signal averaging chan-nel estimation is generally used to identify the channel state in-formation (CSI). In this case, large pilot symbols are required forobtaining an accurate CSI. As a result, the total transmission rateis degraded due to large number of pilot symbols transmission.To reduce this problem, in this paper, we propose time-frequencyinterferometry (TFI) for OFDM to achieve an accurate CSI. TFI-OFDM can multiplex the same impulse responses in twice on thetime domain without overlapping to each other. By averagingof these impulse responses, we can obtain the accurate channelimpulse response.

keywords: OFDM, time-frequency interferometry, CSI.

I. INTRODUCTION

High data rate and high quality multimedia services aredemanded in a fourth generation mobile communication, sinceapplication services are increasing. To meet this demand,orthogonal frequency division multiplexing (OFDM) is at-tractive and widely studied in recent years [1], [2]. Sincethe signals are transmitted in parallel by using many sub-carriers that are mutually orthogonal and the correspondingspectrum is shaped like rectangle, OFDM can achieve highfrequency efficiency and high data rate. Moreover, OFDMhas been chosen for several broadband WLAN standards likeIEEE802.11a, IEEE802.11g and European HIPERLAN/2, andterrestrial digital audio broadcasting (DAB) and digital videobroadcasting (DVB) was also proposed for broadband wirelessmultiple access systems such as IEEE802.16 wireless MANstandard and interactive DVB-T [3]-[5].

In OFDM systems, the pilot signal averaging channelestimation is generally used to identify the channel stateinformation (CSI) [6]. In this case, large pilot symbols arerequired to obtain an accurate CSI. As a result, the totaltransmission rate is degraded due to transmission of largepilot symbols. To reduce this problem, in this paper, wepropose time-frequency interferometry (TFI) for OFDM toachieve an accurate CSI. This paper is organized as follows.Configuration of the proposed TFI-OFDM system is describedin Section II. In Section III, we show the computer simulationresults. Finally, the conclusion is given in Section IV.

II. TFI-OFDM SYSTEM

This section describes the proposed TFI-OFDM system,which employs time division multiplexing (TDM) transmis-sion for multiple users. The proposed system is illustrated inFig. 1.

A. Channel Model

We assume that a propagation channel consists of L discretepaths with different time delays. The impulse response h(τ, t)is represented as

h(τ, t) =L−1∑l=0

hl(t)δ(τ − τl), (1)

where hl and τl are the complex channel gain and thetime delay of the lth propagation path, respectively, and∑L−1

l=0 E|h2l | = 1, where E| · | denotes the ensemble average

operation. The channel transfer function H(f, t) is the Fouriertransform of h(τ, t) and is given by

H(f, t) =∫ ∞

0

h(τ, t) exp(−j2πfτ)dτ

=L−1∑l=0

hl(t) exp(−j2πfτl). (2)

In any radio transmission, the channel spectral response isnot flat. When L > 1, H(f, t) is no longer constant overthe signal bandwidth. This channel is called the frequency-selective fading channel, and in this paper, we consider it forthe purpose of evaluating TFI-OFDM system.

B. TFI-OFDM Transmitter

The transmitter block diagram of TFI-OFDM system isshown in Fig. 1(a). Firstly, the coded binary information datasequence is modulated, and Np pilot symbols are appended atthe beginning of the sequence. The TFI-OFDM transmit signalcan be expressed in its equivalent baseband representation as

s(t) =Np+Nd−1∑

i=0

g(t − iT ) ·{√

2S

Nc

Nc−1∑k=0

u(k, i)

· exp [j2π(t − iT )k/Ts]}

, (3)

where Nd and Np are the number of data and pilot symbols,Nc is the number of carriers, Ts is the effective symbollength, S is the average transmitting power, T is the OFDMsymbol length, respectively. The frequency separation betweenadjacent orthogonal subcarriers is 1/Ts and can be expressed,by using the kth subcarrier of the ith modulated symbol d(k, i)with |d(k, i)| = 1 for Np ≤ i ≤ Np + Nd − 1, as

u(k, i) = cPN (k) · d(k, i), (4)

where cPN is a long pseudo-noise (PN) sequence as a scram-bling code to reduce the peak average power ratio (PAPR).

Proceedings of Asia-Pacific Conference on Communications 2007

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Data/FEC Interleaver

Pilot generation

Mapper

S/P +GIQuadmod.

U/C

FEC/Data Deinterleaver P/S Demod. - Pilot signal FFT -GIA/D

A/D

Quadmod.

D/C

IFFT

D/A

D/A

Timewindows

I

I

Q

Q

(a) Transmitter

(b) Receiver...

.

....

..

..

..

..

IFFTScramblingDemux

Phase offsetting

..

Descrabling

Averaging /FFT ..

..

..

I

Q

I

Q

..

I

Q

I

Q

h (k)

Fig. 1. The proposed TFI-OFDM system.

The guard interval Tg is inserted in order to eliminate theinter-carrier interference (ICI) due to the frequency selectivefading, and hence, we have

T = Ts + Tg. (5)

In OFDM systems, Tg is generally considered as Ts/4 or Ts/5.Thus, we assume Tg = Ts/4 in this paper. In Eq. (3), g(t) isthe transmission pulse given by

g(t) ={

1 for −Tg ≤ t ≤ Ts

0 otherwise .(6)

For 0 ≤ i ≤ Np − 1, the transmitted pilot signal of kthsubcarrier is given by

d(k, i) = exp(−j2πk/Ts) + exp(−j4πkTg/Ts) (7)

where Np is the number of pilot symbols. Due to the super-position of Eq. (7), the transmission power of pilot signals is1/2 for 0 ≤ i ≤ Np − 1.

C. Receiver Structure

The receiver structure is illustrated in Fig. 1(b). By applyingthe FFT operation, the received signal r(t) is resolved intoNc subcarriers. The received signal r(t) in the equivalentbaseband representation can be expressed as

r(t) =∫ ∞

−∞h(τ, t)s(t − τ)dτ + n(t), (8)

where n(t) is additive white Gaussian noise (AWGN) with asingle sided power spectral density of N0. The kth subcarrier

r(k, i) is given by

r(k, i) =1Ts

∫ iT+Ts

iT

r(t) exp [−j2π(t − iT )k/Ts] dt

=√

2S

Nc

Nc−1∑e=0

u(e, i) · 1Ts

∫ Ts

0

exp[j2π

·(e − k)t/Ts] ·{∫ ∞

−∞h(τ, t + iT )g(t − τ)

· exp(−2πeτ/Ts)dτ}

dt + n(k, i), (9)

where n(k, i) is AWGN noise with zero-mean and a varianceof 2N0/Ts. Assuming that the maximum τl is shorter than theguard interval Tg , the integral with respect to τ becomes, fromEq. (6),∫ ∞

−∞h(τ, t + iT )g(t − τ) exp(−j2πeτ/Ts)dτ

=∫ Ts

0h(τ, t + iT ) exp(−j2πeτ/Ts)dτ

= H(e/Ts, t + iT ). (10)

Assuming that εi(t) remains almost constant over the symbollength T ,

εi(t + iT ) ≈ εi(iT ) for 0 ≤ t ≤ T , (11)

and hence, we have

H(k/Ts, t + iT ) ≈ H(k/Ts, iT ), for 0 ≤ t ≤ T . (12)

As a result, Eq. (9) can be rewritten as

r(k, i) ≈ 1Ts

√2S

Nc

Nc−1∑e=0

u(e, i) ·∫ Ts

0

exp[j2π

·(e − k)t/Ts] ·{∫ ∞

−∞h(τ, t + iT )

154

·g(t − τ) exp(−2πeτ/Ts)dτ}

dt + n(k, i)

=√

2S

NcH(k/Ts, iT )u(k, i) + n(k, i). (13)

After scrambling, the output signal r(k, i) is given by

r(k, i) =c∗PN (k)

|cPN (k)|2{r(k, i)

},

=√

2S

NcH(k/Ts, iT )d(k, i) + n(k, i), (14)

where c∗P N (k)|cP N (k)|2 is the descrambling operation.

D. Proposed Channel Estimation scheme

Since Tg = Ts/4, TFI-OFDM can multiplex the sameimpulse responses in twice on the time domain. In this case,by averaging of these impulse responses, we can obtain anaccurate channel impulse response. After the pilot signalseparation, the pilot signal is converted to the time domainsignal r(t) again as

r(t) =Np−1∑i=0

√2P

Nc

Nc−1∑k=0

r(k, i) exp [j2π(t − iT )k/Ts]

=Np−1∑i=0

√2P

Nch(τ, t + iT )

Nc−1∑k=0

d(k, i)

· exp [j2π(t − iT )k/Ts] + n(t)

=Np−1∑i=0

√2P

Nc

L−1∑l=0

hl(t + iT )

· 1√2

{δ(τ − τl) + δ(τ − τl − 2Tg)

}+n(t), (15)

where P is the power of pilot signals. From Eq. (7),∑Nc−1k=0 d(k, i) exp [j2π(t − iT )k/Ts] shows two impulses

with time shift as δ(τ − 2Tg), and the output signals areequivalent to time domain multiplexed impulse responses.Therefore, the impulse response of kth subcarrier H(k) isobtained by

H(k) =1√

P/Nc

Nc−1∑e=0

1Ts

∫ Ts

0

{ L−1∑l=0

hl(t + iT )

{δ(τ − τl) + δ(τ − τl − 2Tg)

}

· exp(−2πeτ/Ts)dτ}

dt + η(k), (16)

where η(k) is AWGN component with E[|η(k)|]2 =E[| n(k,i)

2 |]2 = σ2

2 . Observing Eq. (16), we can see that theestimated CSI of TFI-OFDM can be obtained an improvedaccuracy compared with the conventional OFDM.

-GI FFT IFFTDescrambling

t

Time windows[ 0, Tg -1 ],

[ 2Tg, 3T -1g ]

... ...

Averaging

... ...

FFT ... CSI

t

t

Impulse response

Pilot IFFT +GIScrambling t

(a) Transmitter

(b) Receiver

S/P Tg0

TgTs/4 =t

without scrambling with scrambling

Phase offsetting

2Tg 3Tg

Fig. 2. The concept of TFI-OFDM system.

III. COMPUTER SIMULATED RESULTS

In this section, the performance of the proposed TFI-OFDMis compared with the conventional pilot signal averaging basedOFDM. Fig. 1 shows a simulation model of the proposed TFI-OFDM. On the transmitter, the pilot signals are assigned foreach transmitter using Eq. (7). In this case, TFI-OFDM canmultiplex the same impulse responses in the receive antennain twice on the time domain without overlapping to eachother as shown in Fig. 2(a). The data stream is encoded.Here, convolutional codes (rate R=1/2, constraint length K=7)with bit interleaving are used. These have been found to beefficient for transmission of an OFDM signal over a frequencyselective fading channel. The coded bits are QPSK modulated,and then the pilot signal and data signal are multiplexed withscrambling using PN code to reduce the PAPR. The OFDMtime signals are generated by an IFFT and transmitted to thefrequency selective and time variant radio channel after cyclicextensions have been inserted. The transmitted signals aresubject to broadband channel propagation. In the simulation,we assume that OFDM symbol period is 10µs, guard intervalis 2µs, and L = 15 path Rayleigh fadings have exponentialshapes and a path separation Tpath = 125nsec. The maximumDoppler frequencies are assumed to be 10Hz and 300Hz. In

TABLE I

SIMULATION PARAMETERS.

Data Modulation QPSKData detection Coherent

Symbol duration 10µsFrame size 21 symbols

(Np = 1, Nd = 20)FFT size 64

Number of carriers 64Guard interval 16 sample times

Fading 15 path Rayleigh fadingDoppler frequency 10, 300Hz

FEC Convolutional code(R=1/2, K=7)

155

10-5

10-4

10-3

10-2

10-1

100

0 5 10 15 20 25 30

Pilot avearging OFDM (Np=1,fd=10Hz)Pilot avearging OFDM (Np=1,fd=300Hz)Pilot avearging OFDM (Np=2,fd=10Hz)Pilot avearging OFDM (Np=2,fd=300Hz)Proposed TFI-OFDM (Np=1,fd=10Hz)Proposed TFI-OFDM (Np=1,fd=300Hz)

BE

R

Eb/No[dB]

Fig. 3. BER of the conventional pilot signal averaging based OFDM withNp = 1 and Np = 2, and TFI-OFDM at Doppler frequencies of 10Hz and300Hz.

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25

Pilot avearging OFDM (Np=1)Pilot avearging OFDM (Np=2)Proposed TFI-OFDM (Np=1)

Thr

ough

put [

Mbp

s]

Eb/No[dB]

Fig. 4. Throughput of the conventional pilot signal averaging based OFDMwith Np = 1 and Np = 2, and TFI-OFDM at Doppler frequency of 10Hz.

the receiver, the received signals are erased the guard intervaland S/P converted. The parallel sequences are passed to anFFT operator, which converts the signal back to the frequencydomain. After descrambling and IFFT, each impulse responsecan be estimated by extracting and averaging twice impulseresponses using the time windows with Eq. (16) as shown inFig. 2(b). The frequency domain data signal is detected anddemodulated using the estimated channel impulse response.After detection, bits are detected by the Viterbi soft decodingalgorithm. The packet consists of Np=1 pilot symbol andNd=20 data symbols. Table 1 shows the simulation parameters.

Fig. 3 shows the BER of the conventional pilot signal

averaging based OFDM with Np = 1 and Np = 2, andTFI-OFDM at Doppler frequencies of 10Hz and 300Hz. Theproposed scheme can estimate the accurate CSI comparedwith the conventional pilot signal averaging based OFDMwith Np = 1 by averaging these channel impulse responses.From the simulation results, it is shown that our proposedscheme achieves 2.8dB and 0.5dB gains compared with theconventional pilot signal averaging based OFDM with Np = 1and Np = 2 at Doppler frequencies of 10Hz and 300Hz,respectively.

Fig. 4 shows the throughput of the conventional pilotsignal averaging based OFDM with Np = 1 and Np = 2, andTFI-OFDM at Doppler frequency of 10Hz. Since the proposedscheme uses the reduced number of pilot symbols comparedwith the conventional pilot signal averaging based OFDMwith Np = 2, the total transmission rate is increased. As aresult, the proposed scheme achieves about 5% improvementto compare with the conventional pilot signal averaging basedOFDM with Np = 2 in high Eb/N0.

IV. CONCLUSION

In this paper, we have proposed TFI-OFDM systems toachieve the accurate CSI without increasing the number ofpilot symbols. TFI-OFDM can multiplex the same impulseresponses in twice on the time domain without overlappingto each other. By averaging of these impulse responses,we can obtain the accurate channel impulse response.From the simulation results, it is shown that our proposedscheme achieves 2.8dB and 0.5dB gains compared withthe conventional pilot signal averaging based OFDM withNp = 1 and Np = 2 at Doppler frequency of 10Hz.

REFERENCES

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[2] J. A. C. Bingham, “Multicarrier modulation for data transmission: an ideawhose time has come,” IEEE Comm. Mag., vol. 28, pp. 5-14, May, 1990.

[3] ETSI ETS 301 958, “Digital Video Broadcasting (DVB); interactionchannel for digital terrestrial television (RCT) incorporating multipleaccess OFDM,” ETSI, Tech. Rep., March 2002.

[4] “IEEE draft standard for local and metropolitan area network-part 16: Airinterface for fixed broadband wireless access systems - medium accesscontrol modifications and additional physical layer specifications for 2-11GHz,” IEEE LAN MAN Standards Committee, 2002.

[5] I. Koffman and V. Roman, “Broadband wireless access solutions basedon OFDM access in IEEE802.16,” IEEE Commun. Mag., vol. 40, pp.96-103, April 2002.

[6] C. Ahn, and I. Sasase, “The effects of modulation combination, targetBER, Doppler frequency, and adaptive interval on the performance ofadaptive OFDM in broadband mobile channel,”IEEE Trans. ConsumerElectronics, vol. 48, no.1, pp.167-174, Feb., 2002

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