BER Performance of the RAKE Receiver in a Realistic Mobile Radio Channel Syed Aun Abbas, NORTEL (Northern Telecom), 3500 Carling Avenue, Nepean, Ontario, Canada, Email: sabbas@nortel.com

Asrar U. Sheikh, WlSR Centre, Department of Electronic Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, Email: sheikh@encserver.en.polyu.edu.hk

Abstract: The performance of an ideal RAKE receiver, which exploits timelfrequency diversity, is no different than the per- formance of an L branch space diversity receiver as shown in [3]. Therefore, the expressions for average probability of error for space diversity systems can be used as a starting point for RAKE performance. This paper extends this analy- sis to incorporate the effects of random path arrival on RAKE BER performance assuming the path arrival may be modeled as a modified (doubly stochastic) Poisson process as was suggested in [4] and [5]. The analysis results in a new measure for diversity which is the average number of paths for a given length of the power delay profile. As average num- ber of paths (the Poisson parameter) is not necessarily an integer number, the notion of order of diversity, which is gen- erally described in integer numbers, can be extended to frac- tional diversity for realistic channels.

I. Introduction This paper addresses the BER performance of frequency

selective (time dispersive) signals received over fading channels using RAKE receivers with random path arrival character.

The wideband frequency selective signals posses the poten- tial to exploit the inherent time dispersive multipath character of the mobile radio channel. The frequency selective reception helps obtain the time/frequency diversity from the channel. The RAKE receiver has been suggested for above mentioned purpose. The RAKE receiver is used in IS95 based systems, [l]. One issue related to its usage for operation in a realistic channel is the ran- dom nature of the path arrival. Because paths arrive randomly at the receiver, efficient strategies are needed to maximize captured energy from the channel. In the classical RAKE receiver, a large number of fingers are available. Multipath combining is, how- ever performed by enabling only those fingers corresponding to delays at which useful multipath component is observed, [2]. In another approach, the position of the fingers on the delay line may be optimally selected by an adaptive algorithm, [l]. In this implementation, fingers are not necessarily separated by the fixed duration as in the classical RAKE receiver [2], [3], but are deter- mined by a search algorithm, [l]. This search algorithm makes the finger assignment as per following rule for IS95 base station receiver: 1) assign one finger-pair at the global maximum of mul- tipath intensity profiles, and 2) assign the second finger-pair at the local maximum of multipath intensity profile which was of greatest value, which was within 6dB of their global maximum, and for which two fingers were at least more than a chip duration apart.

The effect of the random path arrivals on RAKE perfor- mance has apparently not been investigated to date. An analysis which takes into account the random path arrivals is helpful in quantifying the performance degradation due to random path

arrivals and may serve as the theoretical benchmark for compari- son of multiple RAKE implementations designed to overcome this obstacle.

11. Performance Analysis The following expressions for average probability of error

for binary orthogonal signals for space diversity systems' can be used as a starting point for RAKE [3],

L - 1

k = O k where

< is the average SNR er branch and it is assumed that branch mean square values E( ak) are equal. When these are unequal we have [3]

P

Here, L

?'k - Yi i = 1

(3)

(4)

i # k

In the case of RAKE, the average probabilities of bit error given in (1) and (3) are functions of the number of diversity branches or resolvable paths. This probability may now be considered as the average probability of bit error, conditioned on the number of resolvable paths, L. Average bit error probability which is now conditioned on L is given by

CO

P , = P,(L = N ) P ( N ) . ( 5 ) N = 1

In this case P(N) is given by the Poisson distribution [6], as fol- lows;

Here, N is the number of paths and h is the average number of paths per total duratiodlength of RAKE receiver. Turin et. al. [4]

1 .with maximal ratio combining

0-7803-4320-4/98/$5.00 0 1998 IEEE 2037 VTC '98

suggested a doubly stochastic Poisson process for the statistical modeling of path arrivals where h is also a random variable.

The average probability of error for BPSK signals is now given by the following expression,

m N N - 1

N = 1 k = O

The average probability of error as a function of average signal to noise ratio per bit as given in (3) is plotted in Figure 1. The curves are drawn for the case of one two and four resolvable paths'. The average probability of error, as given in (7), as a function of average signal to noise ratio per bit is plotted in Fig- ure 2. These curves are drawn again for the three values of the average number of resolvable paths per given length of the RAKE, h , i.e., one two and four. These integer h values are deliberately chosen so that the average number of paths case may be compared to that of a known number of paths. Increasing the h results in improved performance2, however, the gain in perfor- mance is not as large as in the deterministic case. Thus, when performance is compared for the same value of number of paths and the parameter h , a degradation is experienced for the ran- dom path arrival case. Figure 3 - Figure 6 are plotted to help understand the nature and quantify the performance degradation for these cases. The solid lines in these figures represent the ran- dom number of paths case whereas the dashed line represent the known number of paths case. Figure 3 compares the case of only one path and h =I . It shows that h =1 case is always better than L=l for the range of average SNR per bit values chosen for that plot. Figure 4 and Figure 5 compare the cases of h =2 and h =4 respectively. In Figure 4, random number of paths case is slightly better up to an average SNR per bit of about 14 dB after which a very noticeable performance degradation is experienced com- pared to known numbers of paths case. In Figure 5 , however, increasing the value of h results in no advantage before the crossover point at about 10 dB after which a very noticeable per- formance degradation is experienced again when compared to the known number of paths case. Figure 3 through Figure 5 are plot- ted together in Figure 6 to make these differences in relative behavior more obvious.

Figure 7 and Figure 8 illustrate how performance is varied for L and h for an average SNR per bit of 10 and 20 dB. From Figure 6, it may be observed that at high average SNR per bit the difference in BER performance is higher than at the lower values. This is shown in an enlarged fashion in Figure 7 and Figure 8. Note the crossover point at h =3 and h =I .5 in Figure 7 and Fig- ure 8 respectively. Figure 7 and Figure 8 are plotted together in Figure 9 to help appreciate the relative magnitude of performance differences.

1.The number of resolvable paths is the same as space diversity branches. The term paths will be used for brevity in this paper when referring to resolvable paths. 2.The use of performance improvement here refers to the improvement in performance of a time/frequency diversity case with non-diversity (which thus becomes narrowband), fading signal reception scenario.

The P, expression in (7) was obtained for the case of classi- cal Poisson multipath arrival. If the modified Poisson process, i.e the A - K model, is assumed then we assume h to be a random variable. Thus, the expression for P, in (7) is to be regarded as a conditional one; conditioned on h . Turin [4] and Suzuki [SI have suggested that random variations in h may be modeled by a two state discrete Markov chain assuming A = 1 . This condition on P, may be removed by averaging the P, on the pdf of h . The pdf of h may be written as follows:

P ( h ) = P16(h-h1)+P,6(h-h,) . (8) Here PI is the state probability that h is in state h = h, and the P, is the state probability that h is in state h = h, . The expres- sion for P, may now be found as follows; -

P, = I P , P ( h ) d h , (9) 0

m

P, = j(P,Pi6(h - h,) + P,P,&h - h, ) )dh . (10) 0

Using (7) for P, in (10) and applying the sampling property of the impulse function, we obtain;

NN- 1

N = l % = O I . .\

N = 1 h = O

The effect of variability in the h is illustrated in Figure 10. The solid lines correspond to the known h case, i.e., when h =1 and h =4, respectively. These values are chosen to allow for suffi- cient spacing between the two curves to illustrate the effect of randomness in h clearly. For reference, the top and bottom dot- ted lines are plotted for the L=l and L=4 cases. Assuming that h varies randomly between the two values of h =1 and h =4 with equal probabilities, i.e, PI=Pr( h =1)=0.5 and P,=Pr( h =4)=0.5, the performance is now given by the dashed line in Figure 10. This figure also shows the results for the case where the probabil- ity that h assumes the value 1 or 4 is not equal; instead, it is given by P,=Pr( h =1)=0.2, and P,=Pr( h =4)=0.8. It can be noted that the two curves provide performance lying between the pure h =I and h =4 cases.

111. Practical Considerations Up to this point, the performance was compared on one-to-

one basis for a known number of paths versus a given average number of paths where the actual number is a random variable. It helped understanding of the general behavior of performance variation. A case of more practical interest is described below.

Given a maximum allowable transmitted power3 in a chan- nel and sensitivity of the receiver used, the length of the power delay profile can be measured or simulated. Furthermore, based on the targeted system capacity and other design factors, the opti- mal choice of transmitted pulse/symbol duration may be made.

0-7803-4320-4/98/$5.00 0 1998 EEE 2038 VTC '98

With the help of above information one can determine the maxi- mum order of multipath diversity available from the channel. Suppose this order of diversity is four. It is justified to ask, if measurements are made, what range of h values will be obtained. Intuitively, this value would not be greater than four because the constraints of the real equipment preclude the possi- bility of receiving more than four resolvable multipath symbols. As the arrival process over the duration of the average power delay profile is assumed to be approximately Poisson one can therefore use the Poisson distribution to find the probability of a given number of paths arriving over the duration of the average power delay profile. Thus, with a certain measure of confidence one can determine the upper bound on the value of h . For exam- ple, assuming that it is known that the probability of receiving at most 4 paths is 90%, one can solve the following equation to find the value of h which any h measurement in that channel will not exceed. If P( . ) represents the probability of the event in braces, the probability of at most four paths per duration of the average power delay profile will be given as follows,

P ( N 5 4 ) = P ( N = O ) + P ( N = l ) + P ( N = 2 ) + P ( N = 3) + P ( N = 4) . (12)

As P ( N ) is given by Poisson distribution. Therefore (12) can be written as;

The probability of at most four paths, given the probability of the number of paths given by Poisson distribution, is plotted in Fig- ure 11 as a function of h . The value of h corresponding to a probability of 0.9 (i.e. 90%) is 2.4325.

Thus, it is more realistic to compare the four finger RAKE performance with the h=2.4325 case instead of h=4. Three curves have been plotted in Figure 12. The solid curve corre- sponds to the fourth order diversity case; the dash-dot line repre- sents Rayleigh or Nakagami (m=l) fading with no diversity. The h =2.4325 case has been shown by a dashed line. To quantify the deterioration in performance for a BER of it may be noted that the theoretical 14 dB savings (approximate) due to the use of ideal four finger RAKE receiver compared with a simple matched filter receiver will be reduced to about 9.5 dB when the same RAKE receiver operates over a realistic channel with ran- dom path arrival at a rate of h =2.4325. This is a performance loss of about 4.5 dB due to randomness.

The order of diversity is commonly expressed as an integer value in relation with the number of diversity branches. In a real- istic channel, the notion of fractional diversity may be introduced relating to the fact that whatever the number of diversity branches, the diversity gain now depends on the average number

3.The maximum allowable transmitted power is typically specified in a related standard specification. Its choice is primarily dependent on the regulatory EM1 requirements in air as well as the emission regulations for equipment.

number of paths, h , which may assume real values and hence the name fractional diversity.

IV. Conclusions The analytical results reveal that when compared with the

one resolvable path per finger RAKE receiver performance to the case when path arrive with a Poisson path arrival, the RAKE per- formance deteriorates with the random path arrival. When quan- tifed for a BER of it was found that about 14 dB savings due to the use of ideal four finger RAKE receiver over a simple matched filter receiver was reduced to about 9.5 dB when the same RAKE receiver operates over a realistic channel with ran- dom path arrival at a rate of 2.4325, a performance loss of about 4.5 dB's. The analytical results presented in this paper may be used to analyze the implementation performance of different implementation schemes.

References

[ l ] Stewart, Kenneth, A., et. al. "Wide band channel measure- ments at 900 MHz", in Proceedings 45nd IEEE Veh. Tech. Con- ference, July 26-29, 1995, Chicago, Illinois, pp. 236-240 [2] Turin George L., "Introduction to Spread-Spectrum Antimul- tipath Techniques and their Application to Urban Digital Radio.", Proceedings IEEE, vol: 68, No. 3 March 1980, pp. 328-353 [3] Proakis, John G., "Digital Communications", McGraw Hill Book Co., 3rd edition, 1995, pp. 772-806 [4] Turin G. L., et. al. "A Statistical Model for Urban Multi- path Propagation", IEEE Transactions on Vehicular Technology, vol. VT-21, no. 1, February 1972, pp. 1-9 [5 ] Suzuki, H. "A Statistical Model for Urban Radio Propaga- tion", IEEE Transactions on Communications, vol. COM-25, no. 7, July 1977 [6] Yuanqing, Li, "A Theoretical Formulation for the Distribu- tion Density of Multipath Delay Spread in a Land Mobile Radio Environment", IEEE Transactions on Vehicular technology, vol. VT-43, No. 2, May 1994, pp. 379-388, in particular Appendix B

1 0" 1

t 10-6

" \ \_ 1 1

0 5 10 15 20 25 30 35 40 Average SNR per Bit(dE)

Figure 1 BER performance for L=l (solid) L=2 (dashed) and L=4 (dash-dot). Performance improves as L increases.

0-7803-4320-4/98/$5.00 0 1998 IEEE 2039 VTC '98

0 5 10 15 20 25 30 35 40 10-71

Average SNR per Bit(dB)

Figure 2 BER performance for h =I (solid) h =2 (dashed) and h =4 (dash-dot). Performance improves as h increases.

I 0 5 10 15 20 25 30 35 40

Average SNR per W d B )

Figure 3 BER performance Comparison for L=l (dashed) and h=l (solid).

1 0 5 10 15 20 25 30 35 40

Average SNR per Bit(dB)

Figure 5 BER performance Comparison for L=4 (dashed) and h =4 (solid).

0 5 10 15 20 25 30 35 40 Average SNR per Bit(dB)

Figure 6 BER performance Comparison for L=l (solid) and h =I (solid-circle), L=2 (dashed) and h =2 (dashed-circles), L=4 (dash-dot) and h =4 (dash-dot-circles).

100 J

Figure 4 BER performance Comparison for L=2 (dashed) and h =2 (solid).

t 1 I

10-1 1 2 3 4 5 6 7 8 9 10 Number Of Resobable Paths (dashed)/Average Number of Paths (solid)

Figure 7 BER performance comparison; BER as a function of number of resolvable paths, L, (dashed) and average number of paths, h , (solid) for an average SNR per bit of 10 dB.

0-7803-4320-4/98/$5.00 0 1998 EEE 2040 VTC '98

Figure 8 BER performance comparison; BER as a function of L and h for an average SNR per bit of 100 dB.

0.95 l l

2 0.9

i t a 0.85 \ 1

0.95 - 2 0.9-

a 0.85 -

= 0.8-

?

2 go.,, -

0.7-

0.65 -

0 0.5 1 1.5 2 2.5 3 3.5 4 Average Number of Paths

0.6

Figure 1 1 Probabilitv of at most four paths as a function of h ., A probability of 0.9 corresponds to h =2.4325.

lodO 5 10 15 20 25 30 35 40 Average SNR per Bit(dB)

Figure 9 BER performance comparison; BER as a function of L and h for an average SNR per bit of 10 dB and 100 dB together.

0 5 10 15 20 25 30 35 40 Average SNR per Bit (dB)

10-7

Figure 10 BER performance comparison when h is random: dotted lines are L=l and L=4 cases, solid lines are for h =1 h =4. dashed lines when h varies between 1 and 4 with eaual orobabilitv and dash-dot line when h =1 occurs with probability 0.2 and h =4 occurs with probability 0.8.

Figure 12 BER performance for a practical case; L=4 (solid), L=l (dash-dot) and h =2.4325 (dashed).

0-7803-4320-4/98/$5.00 0 1998 IEEE 204 1 VTC '98