an efficient ifft modulator for ofdm systems operating

AN EFFICIENT IFFT MODULATOR FOR OFDM SYSTEMS OPERATING ON NON-SAMPLE SPACED CHANNELS El Kefi Hlel, Fethi Tlili, Sofiane Cherif, Mohamed Siala {el-kefi.hlel, fethi.tlili, sofiane.cherif, mohamed.siala}@supcom.rnu.tn Ecole Sup´erieure des Communications de Tunis, 2083 cit´e El-Ghazala/Ariana, TUNISIA ABSTRACT In this paper, we propose a flexible Inverse Fast Fourier Transfrom (IFFT) modulator for Orthogonal Frequency Division Multiplexing (OFDM) systems operating in nonsample spaced time varying channles where the delays are randomly distributed over the length of the cyclic prefix. After application of the Shannon criterion to the OFDM transmitted signal, the choice of the modulator IFFT length is based to some known system informations such as the symbol duration and the used bandwidth. The evaluation of the proposed IFFT scheme is assessed through computer simulations in indoor environment. Test results are presented to show the IFFT length effect to the performance of the least square channel estimator.

Keywords: OFDM, IFFT, Non-sample spaced channel, Sampling duration, Least Square channel estimator.

1. INTRODUCTION In recent years, there has been a lot of interest in applying Orthogonal Frequency Division Multiplexing (OFDM) in wireless and mobile communication systems because of its various advantages in resolving the severe effects of frequency selective channels. OFDM is being commercially applied for wireless local area network (IEE802.11a and HIPELAN/2), terrestrial digital audio broadcasting (DABT), and terrestrial digital video broadcasting (DVB-T), and it is being considered for wireless broadband access systems by the IEEE802.16 work group [1, 2, 3]. In order to avoid Inter Symbol Interference (ISI), conventional OFDM systems first take the Inverse Fourier Transform (IFFT) of data symbols and then insert redundancy in the form of a Cyclic Prefix (CP) of length larger than the multipath channel delay spread. At the receiver the guard interval is discarded and a FFT on the received block of signal samples is performed [4]. A combination of IFFT and CP at the transmitter with the FFT at the receiver converts the frequency-selective channel to separate flat-fading subchannels.

In this paper, we analyse the effect of the IFFT block size on the performance of the Least Square channel estimator. Infact, the performance of this estimator is very sensitive to the chosen channel model. If the channel is sample spaced, i.e. the channel tap delays are multiples of the sampling duration, the estimator present a very good performance because the OFDM system is simply described by a set of parallel Gaussian channels. In the non-sample spaced channel context, which appears to be a more realistic model for a wireless OFDM system, the least square channel estimator present an error floor. To solve this issue, we propose in this paper a flexible IFFT modulator where the determination of the required IFFT length is based to Shannon criterion and exploits some basics known system informations such as the OFDM symbol duration and the used bandwidth. This paper is structured as follows: the baseband model of the OFDM system and the used channel model are briefly described in section II. The equivalent Input/Output system model for sample spaced and non-sample spaced channels are derived in section III. In Section IV, we assess the performance of our proposed IFFT scheme through computer simulations. Finally, Section V concludes the paper. 2. CONVENTIONNAL OFDM BASEBAND MODEL We consider an OFDM system which consists of N subcarriers. Each transmission subcarrier is modulated by a data symbol xk,i , where k ∈ {0, 1, ..., N − 1} represents the subcarrier number and i ∈ Z is the OFDM symbol index. The transmit symbols xk,i are supposed independent h i 2

and identically distributed with variance Es = E |xk,i | . The baseband model of our OFDM system is presented in figure 1.

Figure 1.

Baseband model of an OFDM system.

In the transmitter side, an Inverse Fast Fourier Transform (IFFT) is performed in each OFDM symbols with duration Tu , and subsequently a guard interval with duration ∆ in the form of a cyclic prefix having Lcp samples is inserted for every OFDM symbol to avoid intersymbol interference caused by multipath fading channels. As a result, the output baseband signal of the transmitter can be represented as [4] s (t) =

−1 X NX i

xk,i e

j2π Tku

t

.

(1)

The whole OFDM symbol duration including the guard interval is Ts = Tu + ∆. The OFDM subcarrier spacing is T1u . After that, the signal is transmitted over a multipath Rayleigh fading channel characterized by [5]: h (τ, t) =

yk,i = Hk,i xk,i + ωk,i ,

hk (t) δ (τ − τk ) ,

(5)

where Hk,i is the frequency response channel coefficients given by the following expression: Hk,i =

k=0

M X

In this case the input-output relation of the overall OFDM system can be modelled as [10]

L X

kl

hl,iN1 e−j2π N ,

(6)

l=0

and ωk,i is an additif white Gaussian noise. Here N1 = N + Lcp is the size of the transmitted OFDM symbol. According to (5), we can describe our system as a set of parallel Gaussian channels with correlated attenuations Hk,i . The description of our system is shown in figure 2.

(2)

k=1

where M is the paths number, τk is the time delay of the k th path and hk (t) is the corresponding complex attenuation, assumed to be Wide-Sense Stationnary (WSS) narrow-band complex Gaussian processes with the so-called Jakes’ power spectrum. We assume too that the different path gains are uncorrelated with respect to each other where the average energy of the total channel energy is normalized to one. At the receiver side, the guard interval is removed and a N -points FFT on the received block of signal samples is performed. We assume the cyclic prefix to yield perfect orthogonality and the channel varies considerably between OFDM symbols but negligibly within an OFDM symbol, so that the channel path complex gain hk,m can be considered constant during one OFDM symbol. Here, the quatitie hk,m is given by: hk,m = hk (tm ) ,

tm ∈ [mTs ,

(m + 1) Ts [

(3)

3. CONVENIENT OFDM MODULATOR REPRESENTATION BY IFFT In the transmitter side, the determination of the efficient IFFT size is related to the multipath channel models. we consider next two channel models: sample spaced and nonsample spaced channels. 3.1. Sample spaced channel For this channel, the different path time delay τk are multiple of the OFDM system sampling duration, i.e αk τk = , Tu

k = 0, 1, ...M − 1;

(4)

where αk is an integer. Then, the transmitted signal after N -points IFFT do not require any resampling before convolution with the multipath channel impulse response.

Figure 2. OFDM system, modelled as a set of parallel Gaussian channels

3.2. Non-sample spaced channel A more realistic channel is a non-sample spaced one, where the delays τk are randomly distributed over the length of the guard interval. In order to do the convolution operation without loss of any information, the transmitted signal at the output of the IFFT transform should be resampled to obtain the same sampling duration as the channel. We can do easily this operation by means of J-points IFFT, where J is an integer greater than N . The determination of the required IFFT size is calculated according to the Shannon criterion. The bandwidth of the transmitted signal s(t) is given by [6]: B=

2 1 + (N − 1) . Ts Tu

(7)

According to Shannon theorem, the sampling frequency fs of the transmitted signal should verifies the following condition: fs ≥ 2B.

(8)

For reason of simplification, we assume that fs is a multiple of the system OFDM subcarrier spacing, i.e fs =

N2 , Tu

where N2 is an integer greater than N .

(9)

After sampling s(t) with respect to fs , we obtain the following samples:

0.2

0.18

sm,i =

NX 2 −1

xk,i e

j2π mk N 2

0.16

,

(10)

0.14

k=0

J≥

Tu (2 + 2N) + 1. Ts

(11)

4. PERFORMANCE EVALUATION BY COMPUTER SIMULATION 4.1. System Parameters OFDM system parameters used in the simulation are indicated in Table 1. Some of this parameters are based on the IEEE 802.11a specifications [7]. Table 1. Simulation Parameters Parameters Specifications FFT size 64 Guard interval 16 Guard type Cyclic extension Signal canstellation 16-QAM Tu 3.2 µs Ts 4 µs Bandwidth 20 MHz ∆f 0.3125 MHz

0.08

0.06

0.04

0.02

0

50

100

150

200

250

300

350

400

τk(ns)

Figure 3. Power delay profile evolution of the channel Model used for the simulations

Figure 4 compares the least square channel estimate ˆ with the real channel state for 3 values of IFFT size: H N , 2N and 3N . We can see that, for IFFT length equal to N or 2N , the input/output relation given by (5) is not respected for all the flat-fading sub-channels. The same condition is respected for IFFT length that verifies (11). Figure 5 shows the evolution of the N M SEH versus Eb /N0 ratio for four values of the IFFT size: N , 2N , 3N and 4N . The quantitie N M SEH is the LS channel estimate Normalized Mean Square Error, which is defined by: 2 NP −1 ˆ Hk,i − Hk,i . (13) N M SEH = k=0N −1 P 2 |Hk,i | k=0

4.2. Channel Model One multi-path fading channel model is used in the simulations. This non-sample spaced channel corresponds to a typical office environment, where the delays are not multiples of the sampling system duration. Figure 3 shows the power delay profile of this channel model [8, 9]. 4.3. Simulation Results The evaluation of the proposed IFFT size for OFDM modulator is shown according to the performance of the least square channel estimator where the transmitted symbols xk,i are supposed known by the receiver. The LS channel estimation is done in the frequency domain on sub-channel by sub-channel basis and given by: k = 0, 1, ..., N − 1;

0.1

0

If the IFFT size respect the condition (11), the inputoutput relation of the overall OFDM system can be modelled as a set of parallel Gaussian channels and verifies (5).

ˆ k,i = yk,i , H xk,i

0.12

P(τk)

where m = 0, 1, ..., N2 − 1 and xk,i = 0 for k ≥ N . The last expression indicates that the sampling operation is similar to an N2 - points IFFT. After substitution of (9) in (8) and replacing B by (7), the size of the IFFT verifies the following condition:

(12)

The evolution of the N M SEH versus the modulator IFFT length for four values of Eb /N0 ratio: 10, 15, 20 and 25 dB, is shown in figure 6. The performance of the channel estimator get worse when the IFFT block size is greater than 3 N because this size is greater than the theorical limit and in this case the noise power increase.

5. CONCLUSION In this paper, we have presented an efficient IFFT modulator for OFDM systems operating in non-sample spaced channels. Basing to the channel model, this modulator is caracterized by the IFFT size. After application to the shannon criterion for sampling the transmitted signal, the determination of the required IFFT length is related to some known system parameters such as the OFDM symbol duration and the used bandwidth. Simulation results demonstrate that the proposed IFFT based modulator gives a substantial improvement in least square channel estimator performance.

(a)

0

10

Eb/N0 E /N b 0 Eb/N0 E /N

2.2 HLS H

2

b

0

= = = =

10 15 20 25

dB dB dB dB

1.8

1.6

−1

10

|H|, |HLS|

1.4

NMSEH

1.2

1

0.8

0.6

−2

10

0.4

0.2

0

10

20

30

40

50

60

70

f/∆ f

(b) −3

2.4

10

H

LS

H

2.2

1

2

3

4

5

(p): Length of FFT = pN

2

1.8

Figure 6. N M SEH versus IFFT length for 4 values Eb /N0 ratio: 10, 15, 20 and 25 dB

|H|, |HLS|

1.6

1.4

1.2

1

6. REFERENCES

0.8

0.6

0.4

0

10

20

30

40

50

60

[1] O. Edfors, M. Sandell, J. van de Beek, S. K. Wilson and P.O. Borjesson, OFDM channel estimation by singular value decomposition, IEEE Trans. on Commun., vol. 46, no. 7, pp. 931 939, July 1998.

70

f/∆ f

(c) 2.4 HLS H

2.2

[2] V.P.Gil Jimnez, M.J.F.Garcia, and A.G.Armada Channel Estimation for Bit-loading in OFDM-based WLAN, in IEEE ISSPIT-2002, Marrakesh, December 2002, pp. 581 585.

2

1.8

|H|, |HLS|

1.6

[3] D. Schafhuber, G. Matz and F. Hlawatsch, Adaptive Prediction of Time-Varying Channels for Coded OFDM Systems, in IEEE ICASSP-2002, Orlando, May 2002, pp. 2549 2552.

1.4

1.2

1

0.8

[4] E.K. Hlel, S. Cherif, F. Tlili and M. Siala, Improved estimation of time varying and frequency selective channel for OFDM systems, IEEE ICECS 2005, Tunis, Tunisia.

0.6

0.4

0

10

20

30

40

50

60

70

f/∆ f

Figure 4. |H| and |HLS | versus frequency: (a)IFFT Length = N , (b)IFFT Length = 2N , (c)IFFT Length = 3N

[6] J.C.Bic, D.Duponteil and J.C.Imbeaux, Elments de communications numriques, Dunod, 1986.

0

10

IFFT IFFT IFFT IFFT

size size size size

: : : :

2N 3N 4N 5N

[7] IEEE, IEEE802.11a-1999 part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications, IEEE 1999.

−1

[8] J. Medbo, Radio Wave Propagation Characteristics at 5 GHz with Modeling Suggestions for HIPERLAN/2, Ericsson Radio Systems, January 1998.

NMSEH

10

[9] J. Medbo, P. Schramm Channel Models for HIPERLAN/2 in different indoor scenarios, Ericsson Radio Systems AB, March 1998.

−2

10

−3

10

[5] Y. Zhao and A. Huang, A novel channel estimation method for OFDM mobile communication systems based on pilot signals and transform domain processing, in Proc. of 1997 IEEE 47th Vehicular Technology Conference, Vol. 3, May 1997, pp. 2089 2093. 1996.

4

6

8

10

12

14

16

18

20

E /N (dB) b

0

Figure 5. N M SEH versus Eb /N0 for 4 values of IFFT length : N , 2N , 3N and 4N

[10] E.K. Hlel, S. Cherif, F. Tlili and M. Siala, Channel parameters estimation for OFDM systems, IEEE-EURASIP ISCCSP 2006, Marrakech, Morocco, March 2006.