Estimation of Noise and Interfering Power for Transmissions over Rayleigh Fading Channels Lionel HUSSON, Armelle WAUTIER, Jacques ANTOINE and Jean-Claude DANY e-mail :
[email protected] SUPELEC, Service Radioélectricité et Electronique 3 rue Joliot-Curie, Plateau de Moulon F-91192 Gif sur Yvette, France
Abstract The topic of the paper is the evaluation by the receiver of the power of noise and co-channel interferences altering transmissions over Rayleigh fading channels. In our study we display results for TDMA mobile radio systems such as GSM or EDGE, but the approach is valuable for CDMA as well. We propose two methods to carry out the evaluation of the signal to noise ratio (SNR) or the carrier to interference ratio (CIR) at each received burst . The accuracy of these methods are derived in a closed form in the case of an additive noise. In the case of interferences, we display results of simulations for both methods when the number of interferers is between 1 and 6.
I) INTRODUCTION The supervision of the transmission quality is a key factor to implement the correction mechanisms (for instance : power control, handover, link adaptation) which make possible to provide the target services [1] [2] [3]. In this paper, we address to the estimation of the interfering power by the receiver, without resorting to any additive signal or signaling (i. e. pilot-aided estimation see for example [4]). We describe and compare two methods that are applicable after the decision of the transmitted symbols. The definition of the proposed estimators and the analytical evaluation of their performance when the transmissions are altered by additive noises are the matter of section II. Section III displays results of simulation when considering both the accuracy of channel estimation and the influence of the decision of the transmitted symbols. This section also deals with the influence of the structure of the interference (additive noise, one or several interferers). Finally, we draw the paper conclusions in section IV.
II) PROPOSED METHODS AND THEORETICAL ANALYSIS II-1) Notation and hypotheses We assume transmissions over slow time varying channels with slow frequency hopping, i. e. the channel response is constant during a burst but it varies from one burst to another. The nth received symbol of a burst is expressed by : N
rn = ¦ ci .sn-i + ηn
(1)
i=0
where { ck } are the channel coefficients, { sk } the emitted symbols and { ηk } the additive noise and interferences.
II-2) Regenerated Signal (RS) method a) principle. The nth received symbol of a burst is expressed by (1). Having the estimation of the channel of propagation and the decided symbols, the receiver builds a “noiseless regenerated signal” which is subtracted to the received signal to obtain an evaluation of the additive noise. N
N
N
i=0
i=0
i=0
^ = r - ¦ ^c .s^ = η + ¦ c .s - ¦ ^c .s^ η n n i n-i n i n-i i n-i
(2)
The power of perturbations is evaluated by the following computation : ^ 2 = E[ |η ^ |2] σ (3) η n
In the case of perfect knowledge of the channel and perfect decision of the symbols, the estimation of the power of perturbation is unbiased. It follows that ^ 2=σ 2 (4) σ η η The computation of the estimated noise power depends on the accuracy of the channel estimate and on the reliability of the symbols decision. We derive the impact
of these defaults on the estimation of the power of the additive noise. b) influence of the errors of estimation of the symbols. Let us assume that the channel is fixed and perfectly known, then ^ci = ci for any i between 0 and N. The estimated noise is expressed by : N
^ =η + η n n
¦
i=0
ci .(sn-i -s^n-i)
(5)
c) influence of the estimation of the channel. We assume that the symbols are perfectly detected, i. e. ^s i = si for any i. The estimated noise is expressed by : N
^ =η + η n n
¦
i=0
N
^ |2 = |η |2 + |η n n
+ 2 Real(ηn* .
N
¦¦
i=0 k=0 N
¦
i=0
+ 2 Real(ηn* .
ci . ck* . εn-i . εn-k*
ci . εn-i )
(6)
with εi = si -.s^i In the case of an AWGN channel with one path, the analytical expression of the estimated power can be derived (complete computation can be found in [5]). -γ ^ 2 = σ 2 [ 1 + 2 γ erfc( γ ) - 2 γ e ] (7) σ η η π where γ denotes the instantaneous signal to noise ratio The estimated noise power is always less than the effective power. In the case of a Rayleigh fading channel, the instantaneous SNR γ changes at each burst according to an exponential distribution. For an average SNR (denoted by Γ), the estimated noise power is expressed by the following expression (the complete computation, based on the average of expression (7) can be found in [5]). Γ ^ 2=σ 2[1+2Γ][1σ ] (8) η η 1+Γ
Average error of estimation of interfering power(dB)
0.5 0
N
¦¦
i=0 k=0 N
¦
i=0
sn-i . sn-k* . (ci -c^i). (ck -c^k)*
sn-i . (ci -c^i) )
(10)
Considering that the symbols are independent, and that the channel is estimated using a CAZAC(P,N) (Constant Amplitude Zero Autocorrelation) sequence of length P+N, the noise power can be expressed by : [5], [6] ^ 2 = σ 2 [ 1 + N+1 ] σ (11) η η P This result is available for AWGN and for any Rayleigh fading channel. The estimation of the interfering power is biased and the bias only depends on the length of the channel response and the CAZAC sequence. d) influence of both errors of estimation of the channel and of detection of the symbols. Monte Carlo simulations have been carried out for AWGN and Rayleigh fading channel. 0.5 Average error of estimation of interfering power (dB)
N
(9)
and
Then ^ |2 = |η |2 + |η n n
^ sn-i .(ci -ci)
0 -0.5 -1 -1.5 -2 -50
-0.5 -1
-30 -20 -10 0 -40 Average signal to noise ratio (dB)
10
20
Figure 2 : average error of estimation of the noise power for transmissions over a one-path Rayleigh fading channel. SR method used, channel estimation with a CAZAC (16, 0)
-1.5 -2 -50
-40 -30 -20 -10 0 Average signal to noise ratio (dB)
10
20
Figure 1 : average error of estimation of the noise power for transmissions over a one-path Rayleigh fading channel. SR method used, perfect knowledge of the channel
II-3) Identical Sequences (IS) method a) principle. We consider the received signal rn at time tn which depends on the channel response, the sequence of symbols {sn, sn-1, ... ,sn-N}.and the noise ηn. Its expression is given by (1).
N
¦
i=0
ci .sm-i + ηm
(12)
The difference between the signals corresponding to a matching sequence gives an estimate of the difference between two noises samples. N
dn,m = rn - rm =
¦
i=0
ci .(sn-i - sm-i ) + (ηn - ηm)
= (ηn - ηm) if the sequences are identical. The estimation of interfering power is given by : ^ 2 = 1 E[ |d |2 ] σ (14) η n,m 2 Having the decided symbols of the burst, the receiver establishes the greater number of pairs of identical decided sequences ( ^s n, ^s n-1, ..., ^s n-N) and ( ^s m, ^s m-1, ..., ^s m-N) the length of which is equal to the channel memory The mean number of pairs of such sequences in a burst of M symbols is M - N -1 - 2N+1 [5]. If the symbols are perfectly decided the estimation is unbiased, thus ^ 2=σ 2 σ (15) η η This estimation is independent of the channel estimation and only depends on the reliability of the decided symbols.
b) influence of the decision of the symbols. It is possible to derive analytically the impact of the errors of decision of the symbols on the estimation of the noise power for a AWGN channel with one path. In this case the received signal is expressed by : rn = c0 .sn+ ηn (16) and dn,m = c0 .εn,m + (ηn - ηm)
0 -0.5 -1 -1.5 -2
(13)
(17)
with εn,m= sn - sm It follows that : |dn,m|2=|ηn-ηm|2 + |c0|2.|εn,m|2+2.Real((ηn-ηm)*.εn,m.c0)(18) The estimated interfering power is given by : ^ 2 = 1 E[ |d |2 ] = σ η n,m 2 e-2γ e-γ ση2[1+2γ.erfc γ-γ(erfc γ)2-2(1-erfc γ) γ ] (19) π π In the case of a Rayleigh fading channel, the average estimated noise power for an average SNR Γ,is given by [5] : Γ Γ ^ 2=σ 2[1- 1 +Γ- 2 (1+2Γ) arctan ] (20) σ η η 1+Γ 1+Γ π π
-30
-20 -10 0 10 Average signal to noise ratio (dB)
20
Figure 3 : average error of estimation of the noise power for transmissions over a one-path Rayleigh fading channel. SI method used, perfect knowledge of the channel
III) SIMULATIONS In this part, we compare the performance of the considered method. In a first time, we analyse the sensibility to the accuracy of the channel estimation when the interfering power is an additive noise. Then we display simulation results in the case of several interferers.
III-1) AWGN CHANNELS Figures 4 to 7 display the error of estimation of noise power in presence of errors in symbols decision for both RS and IS method in the case of perfect knowledge of the channel and in the case of its estimation resorting to a CAZAC(16,0). Error of estimation of noise power (dB)
rm =
0.5 Average error of estimation of interfering power (dB)
At another time tm, the received signal rm depends on the channel response, the sequence of symbols {sm, sm-1, ... ,sm-N}..and the noise ηm:
1 0.5 0 -0.5 -1 -1.5 -2 -50
-40
-30 -20 -10 0 Signal to noise ratio (dB)
10
20
Figure 4 : mean value and dispersion (interval of 80% of values) of the error of estimation of noise power for transmissions over an AWGN channel. RS method used, perfect knowledge of the channel
0
-0.5 -1
-1.5 -2 -50
-40
-30 -20 -10 0 Signal to noise ratio (dB)
10
20
Error of estimation of noise power (dB)
Figure 5 : mean value and dispersion (interval of 80% of values) of the error of estimation of noise power for transmissions over an AWGN channel. RS method used, channel estimation with a CAZAC(16,0) 0.5
Error of estimation of noise power (dB)
0.5
0.5 0 -0.5 -1 -1.5 -2 -2.5 -30
-20
-10 0 Signal to noise ratio (dB)
10
20
Figure 7 : mean value and dispersion (interval of 80% of values) of the error of estimation of noise power for transmissions over an AWGN channel. IS method used, channel estimation with a CAZAC(16,0) The comparison of figures 4 and 6 shows that the dispersion of the estimation values is lower for the RS method than the IS method in the case of perfect knowledge of the channel. However, the RS method is sensitive to the accuracy of the channel estimation (compare figures 4 and 5) while the IS method is designed to be independent of the channel estimation (see figures 6 and 7). In the case of inaccurate estimation of the channel IS method outperforms RS method.
0
-0.5 -1
-1.5 -2
III-2) RAYLEIGH FADING CHANNELS AND INTERFERENCES
-2.5 -30
-20
-10 0 Signal to noise ratio (dB)
10
20
Figure 6 : mean value and dispersion (interval of 80% of values) of the error of estimation of noise power for transmissions over an AWGN channel. IS method used, perfect knowledge of the channel In these simulations, 3200 bursts of 200 bits have been transmitted through an AWGN channel. In the plots, the plain curves represent the mean value of the error of estimation and the dashed curves stand for the limits of the interval including 80 % of the estimated values.
Average error of estimation of interfering power (dB)
Error of estimation of noise power (dB)
1
0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
noise,1,2,3,6
-0.9 -1 -30
-25
-20
-15
-10
-5
0
5
10
15
20
Average signal to interference ratio (dB)
Figure 8 : average error of estimation of interfering power for transmissions over a flat Rayleigh fading channel. RS method used, perfectly known channel. The parameter specifies the number of interferers or the additive noise.
Average error of estimation of interfering power (dB)
Figures 8 and 9 display the error of estimation of the average interfering power for transmission over a flat Rayleigh fading channel and in presence of errors in symbols decision. The number of interferers (1,2 3 or 6) is displayed as a parameter on the plots. The parameter “noise” indicates that the interference is noise only. 0 -0.5 -1 -1.5
noise
-2 -2.5
6
-3
3
-3.5 -4
2
1
-4.5 -5 -30
-25
-20
-15
-10
-5
0
5
10
15
20
Average signal to interference ratio (dB)
Figure 9 : average error of estimation of interfering power for transmissions over a flat Rayleigh fading channel. IS method used, perfectly known channel. The parameter specifies the number of interferers or the additive noise. Figure 8 shows that the behavior of the RS method is independent of the properties of the interferences while the IS is very sensitive on the characteristics of the interferences. Figure 9 illustrates the dependence of IS method on the number of interferers; the performance becomes closer to the case of an additive noise as the number of interferers increases. For the IS method, the major advantage is its independence to the channel estimation but it leads to considerable error of estimation for negative values of the CIR in the case of a little number of interferers. In this situation, the matching sequences are built on symbols corresponding to the interfering signal instead of the desired signal. This effect is expected to be decreased for multipath channels. Notice that for both methods when the CIR exceeds 0 dB, the average error of estimation is less than 1 dB in any case.
IV) CONCLUSIONS In this paper, we presented two methods of estimation of interfering power. The RS method is based on the signal regeneration and the IS method on the search of matching sequences. The performance of the estimators have been studied analytically in the case of additive noises. Simulations allowed to evaluate the performance in the case of interferences. For CIR over 0 dB both methods perform very well and lead to average error of estimation less than 1 dB, whatever the characteristics of the interferences. The results presented on the paper are considering transmissions over flat Rayleigh fading channel in order to establish performances independent of the process of equalization, nevertheless the proposed methods can be used on fading channels with any number of paths. The major advantage of the RS method is its independency on the characteristics of the interferences; however it is sensitive to the accuracy of the channel estimation. As to the IS method, the greater advantage is its robustness on channel estimation but it can be inaccurate in the case of transmissions altered by only few interferers and with poor CIR. In CDMA systems where the same sequences are used periodically and have good inter-correlation properties, this method should be advantageous.
REFERENCES [1] J. K. Cavers, “Variable-rate transmission for Rayleigh fading channels”, IEEE Trans. Commun., vol. 20, pp. 1522, Feb. 1972 [2] A. J. Goldsmith and S. G. Chua, “Adaptive coded modulation for fading channels”, IEEE Trans. Commun., vol. 46, pp. 595-601, May 1998 [3] J. Zander, “Performance of optimum transmitter power control in cellular radio systems”, IEEE Trans. Vehic. Technol., vol. 41, pp. 57-62, Feb. 1992. [4] S. Kozono, “Co-channel interference measurement method for mobile communication”, IEEE Trans. Vehic. Technol., vol. 36, pp. 7-13, Feb. 1987. [5] L. Husson, “Evaluation par le récepteur de la qualité du signal reçu dans les systèmes de radiocommunication avec les mobiles et amélioration des performances par l’égalisation conditionnelle”, Ph. D; dissertation, Univ. Paris XI, France, Jan. 1998 [6] A. Wautier, “Influence de l’estimation du canal sur les performances d’un égaliseur dans le cadre des radiocommunications avec les mobiles”, Ph. D; dissertation, Univ. Paris XI, France, Dec. 1992.
