The Cumulative Distribution Function for the Joint Fading and Two Path Shadowing Channel : Expression and Application I. Dey and G. G. Messier
S. Magierowski
Electrical and Computer Engineering University of Calgary Calgary, Canada Email:
[email protected],
[email protected]
Electrical Engineering and Computer Science York University Ontario, Canada Email:
[email protected]
Abstract—A new expression for the Cumulative Distribution Function (CDF) of the Joint Fading and Two-path Shadowing (JFTS) distribution is derived in this paper. The derived theoretical CDF expression is shown to agree with the experimental results. The CDF expression is used to derive an outage expression for non-diversity transmission which is then used to illustrate the performance of adaptive transmission over the JFTS link.
I. I NTRODUCTION Attempts to model the combined effect of small scale fading and shadowing have long been introduced in the literature [1]– [7] in order to characterize land mobile satellite (LMS) and macro-cellular channels. In an outdoor environment, macrocellular and LMS communication users are highly mobile and cover considerable distances which allows them to visit several scattering clusters. As a result, a range of main waves arrive at the mobile resulting from these clusters, where the strength of each can be drawn from the log-normal distribution. In an indoor wireless environment like an open office or laboratory, there are not enough large obstacles to reflect or refract the main waves contributed by the scattering clusters. Moreover, due to the inability of most WLAN standards in handling hand-offs, mobile WLAN users have to limit their movements within a small area traveling at most between one or two scattering clusters. An appropriate composite fading / shadowing channel model that characterizes the transition from local small scale fading to global shadowing statistics in a large office indoor wireless environment is proposed in [8] based on an indoor measurement campaign. A joint distribution called the Joint Fading and Two-path Shadowing (JFTS) distribution combining Rician fading and the two wave with diffuse power (TWDP) shadowing models is shown to fit the measurement data. However, the work in [8] is limited to the derivation of the expression for the probability density function (PDF) of the JFTS distribution. Statistical characterization of the received signal envelope in terms of its moments and cumulative distribution function (CDF) in faded and shadowed indoor wireless environments, is useful in the design of a mobile radio communication system and the analysis of its performance. The CDF expression is
of particular interest, as it can be directly applied to analyze the outage probability performance of communication systems. Outage probability is an appropriate performance indicator for adaptive modulation techniques, as these techniques attempt to maximize the average data rate under a bit error rate (BER) constraint or minimize outage probability under a delay constraint. Hence, the expression for CDF can be used to analyze the outage probability of optimal adaptive modulation techniques like optimal rate adaptation with constant transmit power (ORA), optimal simultaneous power and rate adaptation (OPRA), and sub-optimal policies like total channel inversion with fixed rate (CIFR) and truncated channel inversion with fixed rate (TIFR) [9]. CDF expressions can also be used to calculate higher order statistics like Level Crossing Rate (LCR) or Amount of Fading (AF), as is done in [10]. The primary contribution of this paper is two-fold. Firstly, a new closed-form expression for the CDF of signal envelope at the output of a JFTS channel is derived. Secondly, the derived expression for CDF is used to develop a new expression for outage probability. Finally, the analytical expressions for CDF and outage probability are used as a basis for numerical analysis of few adaptive modulation techniques, like OPRA and TIFR. The outage probability performances are plotted as functions of the parameters of the communication channel model. The analytical results for both CDF and outage probability are compared with the simulation results in order to verify the validity of the derived expressions. The rest of the paper is organized as below. Section II presents the PDF of the JFTS faded received signal envelope. The analytical derivation of the CDF of the JFTS faded received signal envelope is presented in Section III. Section IV derives the expression for outage probability. Numerical results for the CDF of the JFTS distribution and simulation results for the outage probability performance of two different adaptive transmission policies, OPRA and TIFR, are presented in Section V. Concluding remarks are provided in Section VI.
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II. PDF OF S IGNAL E NVELOPE In an indoor wireless LAN (WLAN) communication scenario representing an open concept office or laboratory layout, the PDF fZ (z) of the received signal envelope, Z(t), can be given by [8], 4 2K z (−K−Sh ) e bi D z ; Sh , Δ fZ (z) = P1 P2 P1 P2 i=1 m 2 rh 2 (2P1 −1) − 2P z r 2 π wh 2P1 2 h · cos(i − 1) e 7 |rh | h=1 (1)
where, D (c; d, e) = 1/2 exp (ed)I0 c 2d(1 − e) + exp (−ed)I0 c 2d(1 + e) , I0 (·) is the zeroth order modified Bessel function of the first kind and bi = ai I0 (1), where 751 3577 49 2989 , a2 = 17280 , a3 = 640 and a4 = 17280 for a1 = 17280 i = 1, 2, 3, 4 respectively. The parameter K is the Rician K-factor, Sh is the range of discrete shadowing values experienced by a user as it transits through different scattering clusters, Δ represents the transition from one scattering cluster to the next one, P1 is the mean-squared voltage of the diffused components and P2 is the mean-squared voltage of the shadowed components. The variable, m is the quadrature order (determining approximation accuracy) and the multiplier wh denotes the Gauss-Hermite quadrature weight factors which is tabulated in [11] and is given by, √ wh = (2m−1 m! π)/(m2 [Hm−1 (rh )]2 ) where Hm−1 (.) is the Hermite polynomial with roots rh for h = 1, 2, . . . , m. For our analysis, we have chosen m = 20, as is done for parameter estimation of composite gamma log-normal fading channels in [12]. Following extensive data fitting, several ranges of the JFTS model parameters have been recommended in [8] that are suitable for different scenarios within an indoor office environment. As for example, K will vary between 7 dB to 9 dB, Sh between 6 dB to 11 dB, and Δ between 0.3 to 0.58, if the user and the access point are in different rooms separated by one set of dry-wall or partition. In the next section Section III, we will be using the expression for the joint PDF in order to derive the expression for CDF of the JFTS distribution, which will subsequently be used to derive the expression for outage probability without diversity combining in presence of composite JFTS fading and shadowing in Section IV. III. CDF OF S IGNAL E NVELOPE The Cumulative Distribution Function (CDF) of the composite fading / shadowing process, Z(t), can be defined as FZ (z) = Pr{Z ≤ z} and the complimentary CDF (CCDF) of that process can be written as, F¯Z (z) = Pr{Z > z} = 1 − FZ (z). By definition, the expression for JFTS CDF can be determined as, z FZ (z) = fU (u)du (2) −∞
where, U is the JFTS distributed received signal envelope with PDF given by (1). Putting (1) in (2) and rearranging, we obtain m 4 bi
FZ (z) = A1
2
R
z
−∞
u exp (−A2 u2 )
i=1 h=1 · B3 I0 (2uB1 ) + B4 I0 (2uB2 ) du
(3)
where, exp (−K − Sh ) P1 P2 1 2P2 rh2 KSh 2 (1 − Δ cos(i − 1)π/7) P1 P2 KSh 2 (1 + Δ cos(i − 1)π/7) P1 P2 exp (Sh Δ cos(i − 1)π/7) exp (− Sh Δ cos(i − 1)π/7) 2 rh (2P1 − 1) wh exp |rh | 2P1
A1 = A2 = B1 = B2 = B3 = B4 = R =
Using the infinite series expansion of the modified Bessel +∞ /2)2k function of the first kind, I0 (f ) = k=0 (f(k!) 2 , (3) can be expressed as, FZ (z) = A1 ·
m 4 bi
i=1 z −∞
4 u
+∞ B3 B2k + B4 B2k 1 2 k!
R
h=1 2k+1
k=0
exp (−A2 u2 ) du
(4)
Using the closed form solution of the integral α exp −
δβ d and the incomplete gamma function given in [13], we can write (4) as, +∞ 2k B3 B2k 1 + B4 B2 4 k! i=1 h=1 k=0 2 Γ(k + 1, A2 z ) · 1 − 2Ak+1 2
FZ (z) = A1
m 4 bi
R
(5)
Thereby, we can arrive at the final expression for CDF of JFTS distribution as, 4 m bi
FZ (z) =
i=1
2
R e−K−Sh
h=1
+∞ k=0
K k Shk (k!)2 (P
· (1 − ΔMi )k + (1 + ΔMi )k e− ·
k
1 − 2
(P2 rh2 )k+1
k+1
1 P2 )
Sh ΔMi
z2 Γ k + 1, 2P2 rh2
eSh ΔMi
(6)
where, Mi = cos(i − 1)π/7. The validity of the derived CDF expression (6) for JFTS distribution will be evaluated in Section V, by comparing it with the CDF generated using the measured data reported in [8].
IV. O UTAGE P ROBABILITY
T2 : R1
where, fΓ (γ) is the PDF of the instantaneous SNR per symbol. Since data transmission is suspended when γ < γ0 , AM will suffer a probability of outage, γ0 fΓ (γ)dγ = FΓ (γ0 ) (8) Pout = Pr{γ ≤ γ0 } = 0
where FΓ (γ) is the CDF of γ. The instantaneous SNR per Es , where Z is the JFTS symbol is given by, γ = Z 2 N 0 distributed composite fading/shadowing amplitude, Es is the average symbol energy and N0 is the one-sided power spectral density of the additive white Gaussian noise (AWGN). Since, the average SNR per symbol is given by, γ¯ = E{γ} = γ ¯ 2 Es 2 E{Z } N0 , we can write γ = z E{Z 2 } . Using this transformation in terms of instantaneous and average SNR 2 per γE{Z } . symbol, the CDF of γ is simply FΓ (γ) = FZ γ ¯ Finally the expression for outage probability can be read γ0 E{Z 2 } using (6). Using ily obtained as, Pout = FZ γ ¯ 4 2 ˆ ˆ E{Z } = 8P1 P2 K ai Sh , we have, i=1
Pout
JFTS
=
R e−K−Sh
+∞
K k Shk
k+1 2 2 h=1 k=0 (k!) (P1 P2 )
S ΔM · e h i (1 − ΔMi )k + (1 + ΔMi )k − Sh ΔMi 1 − 2k (P2 rh2 )k+1 · e i=1
ˆ Sˆh γ0 4ai P1 K · Γ k + 1, rh2 γ
(9)
ˆ = 1 + K and Sˆh = 1 + Sh . where, K The outage probability of any AM technique, whether using optimal or sub-optimal adaptation, is given by (9). However, the main difference between optimal and sub-optimal AM, in terms of outage probability, is that for optimal adaptation, the optimal threshold SNR γ0 is always less than 0 dB. While
1
0.8 Measured Empirical
0.6
0.4
0.2
0 0
0.5
1
1.5
2
Received Signal Envelope, z
2.5
CDF, FZ (z) = Pr {Z ≤ z}
CDF, FZ (z) = Pr {Z ≤ z}
In this section, we will use the JFTS CDF to analyze the outage probability of a communication system in presence of JFTS composite fading / shadowing. Outage probability is an essential performance metric for adaptive modulation (AM) techniques, as these techniques seek to maximize spectral efficiency under a BER constraint over a fading, shadowing or composite fading / shadowing channel by minimizing the outage probability. Outage probability, Pout , is defined as the probability that the instantaneous signal-to-noise ratio (SNR) per symbol γ, falls below a specified threshold SNR per symbol γ0 . In case of any existing AM technique, when the receiver detects that the channel SNR cannot guarantee the target BER constraint and falls below the optimal cut-off SNR (γ0 ), the adaptive transmission system enters into “no transmission” mode. Hence, the optimal threshold γ0 must satisfy, +∞ 1 1 fΓ (γ)dγ = 1 − (7) γ0 γ γ0
4 m bi
T3 : R9
1
0.8 Measured Empirical
0.6
0.4
0.2
0 0
0.5
1
1.5
2
2.5
Received Signal Envelope, z
Fig. 1. Set 1 : Measured and Empirical CDFs for JFTS at fc = 2450 MHz.
for sub-optimal adaptation, the optimal cut-off SNR becomes larger than 0 dB for γ¯ > 5 dB [9]. As a result, for a fixed γ, sub-optimal techniques will have a higher outage probability than optimal schemes. In Section V, we will utilize the JFTS CDF in (9) to quantify OPRA and TIFR outage performance as described in [9] and determine how much the TIFR outage probability exceeds that of OPRA, assuming perfect channel state information (CSI) at the receiver. For our analysis, we will be considering continuous-rate AM with M -ary Quadrature Amplitude Modulation (M -QAM), where M denotes the number of signal constellation points. Adaptive continuous rate (ACR) refers to the technique in which the number of bits per symbol is not restricted to integer values. We assume that the transmission is done with a fixed symbol rate Ts and ideal Nyquist data pulses (sinc [t/Ts ]) are transmitted for each constellation. Since each of our M QAM constellations has Nyquist data pulses (B = 1/Ts ), the average Es /N0 equals the average SNR, Es /N0 = γ, where B (Hz) is the channel bandwidth. The outage probability of ACR M -QAM over JFTS fading / shadowing communication link will be given by (9) for both OPRA and TIFR policies V. N UMERICAL R ESULTS Cumulative Distribution Function (CDF): In this subsection, we will evaluate the validity of the expression (6) for CDF of JFTS distribution, by comparing it with CDF curves generated using the measured data reported in [8]. In [8], measurements were collected in a typical open office or laboratory WLAN scenario, where the transmitter was fixed at one location while the receiver was moved through several receiver routes. Several fixed transmit positions were selected and for each pair of a transmit position (T d) and receiver route (Rd), a separate measurement file is generated, where d is an index used to indicate different measurement sets. For the first set of figures in Fig. 1, we have used measurements collected for the T 2 : R1 and T 3 : R9 scenarios at fc = 2450 MHz, where fc is the center frequency of the measurement bandwidth. As reported in [8], the best fit JFTS parameters for these two scenarios are K = 8.19 dB, Sh = 11.51 dB, Δ = 0.323 and K = 6.21 dB, Sh = 4.32 dB, Δ = 0.48, respectively. We have used these same set of parameters for CDF curve fitting.
D = max |FZ (z) − FX (x)|
(10)
We defined H0 as the null hypothesis that the measured data X belongs to the derived analytical CDF of the JFTS distribution, as is done in [14]. The K-S test is used to compare D to a critical level Dmax = 0.04301 and a significance level α = 0.05. Any H0 for which, D < Dmax is accepted with a significance level of 1 − α . For all the measurement data sets considered in both Set 1 and Set 2, the null hypothesis is accepted with 95% significance, which establishes the fact that our derived expression for CDF offer a good fit to the CDFs of the measured data, with the distribution of the measurement errors being non-normal. Outage Probability: The first set of curves shown in, Fig. 3 are generated by plotting the transmission outage probability of ACR M -QAM with OPRA as a function of target BERs over the JFTS faded / shadowed communication channel with different K-factors but fixed Sh and Δ. The outage expressions are generated by using the JFTS CDF expression derived in Section III to analyze AM performance as described in [9]. The curves are plotted at a fixed γ of 10 dB. As the target BER increases, the probability of no transmission decreases. As the JFTS K parameter decreases, the power contributed by the diffused and scattered components exceeds the power contributed by the specular components. As a result the transmission outage probability increases for the same target BER. The second set of curves in Fig. 4 are generated by comparing the transmission outage probability of ACR M QAM and OPRA (optimal adaptation), with that of ACR T1 : R4
T1 : R7
0.8 Measured Empirical
0.6
0.4
0.2
0 0
0.5
1
1.5
2
Received Signal Envelope, z
2.5
K= 8 dB K= 5 dB K= 2 dB
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −9
−8
−7
−6
−5
−4
−3
−2
−1
Target BER (1.0eX)
Fig. 3. Transmission outage probability (Pout ) of ACR M -QAM with optimal adaptation techniques at different target bit error rates over JFTS faded / shadowed channel at a fixed average received carrier-to-noise ratio (γ) of 10 dB, where the curves are generated by varying the K-parameter of the JFTS distribution, with fixed Sh = 2 dB and fixed Δ = 0.3.
M -QAM and TIFR (sub-optimal adaptation) as a function of average SNR for two sets of JFTS parameters. As both K and Sh decrease, the channel condition deteriorates and consequently the transmission outage probability increases. However, at a particular target BER and average received SNR, the probability of outage for a sub-optimal adaptation policy is higher than that for optimal techniques confirming the inference made in Section IV. The final set of curves in this subsection (Fig. 5) are generated by plotting the transmission outage probability of ACR
0
10
−1
10
−2
10
Optimal Adaptation (Set 1) Truncated Channel Inversion (Set 1) Optimal Adaptation (Set 2) Truncated Channel Inversion (Set 2)
0.8 Measured Empirical
0.6
K= 10 dB
0.8
1
CDF, FZ (z) = Pr {Z ≤ z}
CDF, FZ (z) = Pr {Z ≤ z}
1
0.9
Transmission Outage Probability ( Pout )
1
Probability of Outage
For the second set of figures in Fig. 2, we have used data collected for the T 1 : R4 and T 1 : R7 scenarios at fc = 2450 MHz. In this case, as mentioned in [8], the best fit JFTS parameters are obtained as K = 6.48 dB, Sh = −6.08 dB, Δ = 0.155 and K = 5.92 dB, Sh = −8.28 dB, Δ = 0.1973, respectively. We have again used these same set of parameters for the second set of CDF curve fitting. In order to measure the goodness-of-fit between the CDF of the measured data, FX (x) and the derived analytical CDF, FZ (z), we used the Kolmogorov-Smirnov (K-S) test, which can be defined as,
−3
10 0.4
10
15
20
25
30
Average Received SNR (dB)
0.2
0 0
5
0.5
1
1.5
2
2.5
Received Signal Envelope, z
Fig. 2. Set 2 : Measured and Empirical CDFs for JFTS at fc = 2450 MHz.
Fig. 4. Probability of no transmission (Pout ) of ACR M -QAM over JFTS faded / shadowed communication channel as a function of average received SNR of (γ) for different optimal adaptation policies like OPRA and ORA in comparison with TIFR, where the curves are generated for two sets of JFTS parameters, Set 1 : K = 8 dB, Sh = 8.5 dB, Δ = 0.45 and Set 2 : K = 6.5 dB, Sh = −2.5 dB, Δ = 0.25.
1 Avg. received SNR = 10 dB Avg. received SNR = 15 dB Avg. received SNR = 20 dB
Transmission Outage Probability
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −9
−8
−7
−6
−5
−4
−3
−2
−1
Target BER (1.0eX)
Fig. 5. Transmission outage probability (Pout ) of ACR M -QAM with optimal adaptation techniques at different target bit error rates over JFTS faded / shadowed channel with parameters K = 5.5 dB, Sh = −7.5 dB, Δ = 0.15.
M -QAM with optimal adaptation techniques as a function of target bit error rates for four different sets of JFTS parameters. The curves are plotted at three sets of γ, 10 dB, 15 dB and 20 dB. Transmission outage probability decreases with the increase in target BER for all three SNRs. However, at a particular target BER, the outage probability at a higher γ is lower than at a lesser γ. This set of results can also be compared with the outage probability results obtained in [15] for ACR M -QAM over Rayleigh and Rician fading channels. For all the three choices of γ outage probability over JFTS fading / shadowing channel with JFTS parameters K = 5.5 dB, Sh = −7.5 dB, Δ = 0.15 is higher than over both Rayleigh or Rician fading channels. The lower performance over the JFTS channel relative to [15] can be attributed to low Sh factor, which represents a scenario where each scattering cluster contributes a very small range of discrete shadowing values, that are encountered repeatedly resulting in an increased severity in shadowing. VI. C ONCLUSION The main aim of this paper is to derive a new closed-form expression for the CDF of the JFTS distribution. The CDF expression is next used to find an expression for probability of outage for transmission without diversity. Results indicate that the analytical CDF expression is a good fit for CDFs generated with measured data. The transmission outage probability of some well-known adaptive transmission techniques are also simulated over the JFTS fading / shadowing communication link, assuming perfect CSI at the receiver, using the newly derived expression for outage probability. Simulation result show that outage probability increases with decreasing JFTS parameters K, Sh and increasing Δ. R EFERENCES [1] H. Suzuki, “A statistical model for urban radio propagation,” in IEEE Trans. Communication, vol. 25, pp. 673–680, Jul. 1977.
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