system with linear-phase transmultiplexer

Seog Geun Kang and Eon Kyeong Joo

An orthogonal frequency division multiplexing system with a linear-phase transmultiplexer is presented. The proposed system has much lower sidelobe power and better subband isolation characteristics than the conventional system. In addition, the proposed system shows an improved error performance, especially at low SNR.

4 ho(n) $13'~) .--- 4 hl(n) signal ,,

'--+

Introduction: In an orthogonal frequency division multiplexing (OFDM) system, the inverse discrete Fourier transform (IDFT) is employed to modulate a block of parallelised complex signals at the transmitter. In addition, DFT is used to separate each sub- band signal from the received symbol. To maintain the orthogo- nality among subband signals and to meet the requirement of the Nyquist property, systems in general have employed rectangular filters as prototype filters [l]. As a result, each subband has a sig- nificant spectral overlap with its neighbouring subbands.

To deal with this problem, an OFDM system which is designed with a linear-phase transmultiplexer is presented in this Letter. According to the McClellan-Parks algorithm [2], a linear-phase paraunitary filter has been designed and employed as a prototype fdter. The synthesis and analysis filter banks of the transmulti- plexer have been designed using cosine modulation of the proto- type. As a result, the peak power of the first sidelobe of the presented prototype filter is lower than that of the rectangular fd- ter by - 15dB. Also, the presented OFDM system shows a reduced bit error rate (BER) especially at low signal-to-noise ratio (SNR), as compared to the conventional system in an additive white Gaussian noise (AWGN) environment.

W W W &-, (n) 4 hM-l (n) m-*

Design with transmultiplexer: A general multichannel filter bank is constructed using two types of filter; synthesis fdters, g,, and anal- ysis fdters, h,, 0 5 k 5 M - 1. Each filter is usually implemented with cosine modulation of a linear-phase lowpass prototype fdter, p(n), as follows:

P/s mapper -+x(n)

h k ( n ) 2p (n ) COS - (k + 0 5 ) n - - (; ' ( Y ) + @ k )

where N is the length of a filter and O,=(-l)W4. The structure of a transmultiplexer allows the composition mul-

titude of data at the transmitter and the decomposition of the data at the receiver [3]. Such a multiple-input and output structure and the operation of the fdter banks can be well adapted to the OFDM system. In Fig. 1, an OFDM system which is designed with a transmultiplexer is depicted, where S/P and P/S denote a serial to parallel and parallel to serial converter, respectively. JM represents an M-fold downsampler, and 1'M an upsampler.

Each parallelised signal is interpolated and fdtered independ- ently to produce a composite signal, y(n). The synthesis fdter bank, g(n)=k,(n), gl(n), ..., gM-,(n)], allocates each input signal to a different frequency band by selecting a set of M centre frequen- cies. Thus, the frequency response of a transmitted signal can be represented as

k=O

where Y(z) is the z-transform of y(n). Gk(z) and X,(z") represent the frequency responses of the Mh synthesis filter and interpolated signal, respectively.

A parallel structure of separation fdters in the receiver should ensure that each recovered signal depends only on its correspond- ing input. Let h(n)=[h,(n), h,(n), ..., hM.,(n)] be the impulse response of the analysis filter bank. The composite signal, ~ ( n ) , is fed into M analysis filters and produces i;(n)=[;,(n), G1(n), ..., i M-l(n)], where ;,(n)=h,(n)*y(n). Thus, the overall transfer func- tion is

Fk,(z) = Z-(~-~)G~(Z'/~W~)H~(Z~/~W')~-("-") (4) for 0 I k, i I M - 1. In the case of k # i in eqn. 4, F,,(z) may be thought of as interchannel interference YCI) which is the influence of the undesired input signal X;(z) on X,(z). IC1 can be eliminated when each output signal depends only on its corresponding input signal. Owing to the orthogonality between the synthesis and anal- ysis fdters in eqns. 1 and 2, Fkr(z) = 0 for k # i and the transfer characteristic of the presented OFDM system are

FTMUX = HTG = diag[Q,cl, ..., C M - ~ ] Z - ( " - ' ) (5) where c, = l /M Z k 1 G,(z~/~W)H,~(Z~/~W) is an arbitrary con- stant and diag[c,, c1, .-,cM-,] represents an M x M diagonal matrix. Thus, the recovered signal of the kth path can be represented as a scaled and delayed version of the input signal and is given as fol- lows:

normalised frequency

Fig. 2 Frequency characteristics of prototype filters - - _ _ rectangular filter

paraunitary filter

Table 1: Coefficients for prototype fdter

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Performance analysis: To evaluate the performance of the pro- posed system, computer simulations for an OFDM system with a 16-channel transmultiplexer have been performed. A square 16- QAM (quadrature amplitude modulation) constellation has been used as a signal mapper, and a linear-phase lowpass prototype fil- ter of length N = 32 has been designed using the McClellan-Parks algorithm [2]. The coefficients of the filter are shown in Table 1, where p(N - 1 - n) = p(n) for 0 5 n < Ni2. Frequency characteris- tics of the prototype and rectangular filter are shown in Fig. 2.

The peak power of the first sidelobe of the rectangular fiter is lower by - 13.ldB than that of the main lobe. In comparison, the difference in the paraunitary prototype filter is - 28.5dB. In addi- tion, the latter has a much lower normalised power in all sidelobes than does the rectangular filter. The influence of spectral overlap, therefore, can be much more severe in the rectangular filter bank than in the transmultiplexer. Thus, the paraunitary filter bank is expected to be more robust to ICI. The error performance of OFDM systems in an AWGN channel are depicted in Fig. 3.

I I I I b 1 2 3 4 5 -81

Eb/N,,dB

Fig. 3 Bit error rates of OFDM systems 0 conventional system 0 proposed system

When the signal-to-noise ratio (SNR) is > 5.5dB, the BERs of both systems are almost the same. The BER of the proposed sys- tem, however, is much lower than that of the conventional system for low E,IN,. Owing to ICI, the performance of the conventional system is believed to be seriously degraded at low values of SNR, where the effect of noise is relatively large.

Conclusions: Owing to the significant spectral overlap of the rec- tangular filter bank, the orthogonality of the conventional OFDM system can hardly be maintained in noisy channels. As compared to that, the paraunitary prototype filter shows a much lower sidelobe power and better subband isolation characteristics than the conventional rectangular fdter bank. In addition, the presented OFDM system shows improved error performance especially at low SNR as compared to the conventional system in an AWGN environment.

0 IEE 1998 Electronics Letters Online No: 19980911 Seog Geun Kang and Eon Kyeong Joo (School of Electronic and Electrical Engineering, Kyungpook National University, Taegu, 702-701, Korea) Corresponding author: Eon Kyeong Joo E-mail: ekjoo@ee.kyungpook.ac.kr

20 April 1998

References

1 RIZOS, A.D., PROAKIS, J.G., and NGUYEN, T.Q.: ‘Comparison of DFT and cosine modulated filter banks in multicarrier modulation’. Proc. IEEE Globecom’94, San Francisco, November 1994, Vol. 2, pp. 687-691

2 MCCLELLAN, J.H., PARKS, T.w., and RABINER, L.R.: ‘A computer program for designing optimum FIR linear phase digital filters’, IEEE Trans. Signal Process., 1973, SP-21, (6), pp. 506-526 VAIDYANATHAN, P.P.: ‘Multirate systems and filter banks’ (Prentice- Hall, Englewood Cliffs, New Jersey, 1993)

3

Reuse efficiency for non-uniform traffic distributions in CDMA systems

C.K. d’Avila and M.D. Yacoub

An exact and simple method to evaluate the frequency reuse efficiency for the reverse link of cellular CDMA systems over a non-uniform traffc distribution is presented. This method is based on a traffic redistribution and decomposition process and on standard curves with whch partial reuse effciencies are obtained and conveniently added to give the fmal parameter.

Introduction: The frequency reuse efficiency F indicates the capac- ity reduction of a multiple cell system as compared to a one-cell network, and can be estimated for a given cell j as

where N is the number of cells within the system and k, is the interference factor of cell i measured at cellj. The parameter k, can be calculated as a function of the propagation conditions and traf- fic density as

k = Ls C(Ts/Tt)yp(x,Y) d A (2)

where r, is the distance from the interfering mobile to its serving base station, r, is the distance from this mobile to the target (inter- fered) base station, y is the path loss exponent, p(xy) is the traffic density as a function of the positional variables (x,y), A, is the integration area of cell i, dA is the infinitesimal area, and C is a constant which includes the mobile transmitted power, the voice activity factor, and other factors.

The frequency reuse efficiency has a direct impact on both the capacity and the radio coverage prediction of a CDMA system. An accurate estimation of such a parameter, therefore, is of para- mount importance in cellular system design. On the other hand, this parameter varies substantially according to each particular propagation condition and also with the traffk distribution. Its precise estimation is rather intricate and it is usually attained by means of long and time consuming processes, such as complex numerical integration [l] or Monte Carlo simulation [2]. As a con- sequence, in practice, planning and design generally make use of ‘typical’ values (assumed 0.65 [3], for instance), obtained for the uniform traffic distribution case.

This Letter proposes an exact and simple method to compute F for any traffic distribution. The overall procedure comprises four steps: (i) traffic redistribution process (TRP), (ii) traffic decompo- sition process (TDP), (iii) partial evaluation process (PEP), and (iv) final evaluation process (FEP). These steps are detailed below.

Traffic redistribution process: For a given target cell j , the traffic of the cells in each tier surrounding cell j is redistributed all around this cell within the respective tier so that cellj becomes the central cell of the system. The only constraint for this redistribution is that the sum of the traffic of each tier must be unaltered. This step is obviously unnecessary if the target cell is the central cell of the original system.

Traffic decomposition process: Assume that, in an N-cell system, a non-uniform traffc distribution p(x,y) can be written as

N

P ( S , Y) = P d Z ( G Y) (3) 2=1

where pt is the mean traffic density of cell i and S,(x,y) = 1 for (x,y) c A, and S,(x,y) = 0, otherwise. We define pmin = min{pz}, i = 1, 2, ..., Nand N,,, as the number of cells with p, = pmin. For each

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