Exact Closed-Form Expressions for the Distribution, Level-Crossing ...
Exact Closed-Form Expressions for the Distribution, Level-Crossing Rate, and Average Duration of Fades of the Capacity of MIMO Channels Bjørn Olav Hogstad and Matthias P¨atzold Department of Information and Communication Technology Faculty of Engineering and Science, Agder University College Servicebox 509, NO-4898 Grimstad, Norway Email: {bjorn.o.hogstad, matthias.paetzold}@hia.no
Abstract— This paper deals with some important statistical properties of the multiple-input multiple-output (MIMO) channel capacity. We assume that all the subchannels are uncorrelated. In case of single-input multiple-output (SIMO) and multiple-input single-output (MISO) channels, exact closed-form expressions are derived for the probability density function (PDF), the cumulative distribution function (CDF), the level-crossing rate (LCR), and the average duration of fades (ADF) of the channel capacity. Furthermore, these exact closed-form expressions are extended to characterize the statistical properties of the maximum MIMO channel capacity. The correctness of the derived closed-form expressions is confirmed by simulations.
I. I NTRODUCTION Exact knowledge of the statistical properties of the MIMO channel capacity is an important issue in the development of future mobile communication systems. Some of these statistical properties are for examples the PDF, the CDF, the LCR, and the ADF of the channel capacity. In [1], exact closed-form expressions for the PDFs of the SIMO and MISO channel capacity are derived. Also from [1], the exact PDF of the maximum MIMO channel capacity can easily be obtained. The maximum MIMO channel capacity is reached in the unrealistic situation when each of the MT transmitted signals is received by the same set of MR antennas without interference. It can also be described as if each transmitted signal where received by a separate set of receive antennas, giving a total number MT · MR receiving antennas [2]. In our paper, we give a detailed proof of the exact closed-form expression for the PDF of the maximum MIMO channel capacity. Even though the maximum MIMO channel capacity is an upper bound for the capacity, our derived PDF shows the potential of the MIMO technology. From the PDF, we can easily obtain the corresponding CDF. For SIMO and MISO channels, where the power of all branches are different, expression for the CDF of the capacity can be found in [3]. In this paper, we present the CDF when all the branch powers are equal. It should be mentioned that the obtained CDF in [3] is not valid in this case. Even when the PDF and CDF are important statistical quantities of the MIMO channel capacity, they do not show the dynamic temporal behavior of the capacity in time-varying MIMO fading channels. To fulfill this gap, we will study the
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LCR and ADF of the MIMO channel capacity. To the best of the authors knowledge, the closed-form expressions for the LCR and ADF are completely new, except that a similar formula for the LCR of the SISO channel capacity can be found in [4]. Most of the LCR and ADF related research has been carried out for single-input single-output (SISO) mobile fading channels [5], [6], [7]. Therein, the LCR is defined as how often the signal envelope crosses a certain threshold from up to down (or from down to up) within one second. Analogously, the ADF is defined as the expected time length the signal envelope stays below a given threshold. The ADF is an important dynamic characteristic quantity of the channel. For example, the ADF enables one to get the statistics of burst errors occurring in mobile communication systems [8], and thus, can provide useful information for the design of error-correcting schemes. Furthermore, the LCR and ADF have found a variety of other applications such as the finite-state Markov modeling of fading channels [9], and the analysis of handoff algorithms [10]. In this paper, the LCR of the MIMO channel capacity is defined as the average number of up-crossings (or downcrossings) of the capacity through a fixed level within one second. From the LCR, one can easily calculate the ADF, which is defined as the expected value of the length of the time intervals in which the capacity is below a given level. A common approach for calculating the LCR of a white sense stationary random process is to employ Rice’s formula [11], [12], which expresses the LCR in terms of the joint PDF of the process and its derivative. Furthermore, in [13] Rice’s LCR formula is rewritten in terms of the joint characteristic function (CF) of the random process and its derivative. This latter formula is useful for many cases where the joint CF is simpler to derive than the associated joint PDF. Nevertheless, in this paper, we will consider the original formula proposed by Rice. Simulation results for the channel capacity’s PDF, LCR, and ADF by using the geometrical one-ring scattering model can be found in [14], [15], and for the two-ring scattering model in [16]. Furthermore, analytical approximations of the LCR and ADF of SIMO and MISO channel capacity are presented
455
in [14]. The remainder of this paper is organized as follows. In Section II, we will describe the Rayleigh process with crosscorrelated inphase and quadrature components. A detailed study of the statistical properties of the MIMO channel capacity is given in Section III. The analytical results obtained in Section III will be compared with simulation results in Section IV. Finally, the conclusions are drawn in Section V. II. T HE R AYLEIGH PROCESS Throughout the paper, we will denote the time-varying complex channel gain between a single transmit and a single receive antenna by h(t). The complex process h(t) can be expressed as I
Q
h(t) = h (t) + jh (t)
−∞
=
e −z ,
follows a Rayleigh distribution. The time derivative of ζ(t) ˙ will be denoted by ζ(t). Throughout the paper, we let the overdote of a process denote the time derivative. From [17], ˙ the joint PDF of ζ(t) and ζ(t) at the same time t, can be expressed as 2 2 2 ze −z −z˙ /(2β) , ˙ = z ≥ 0, |z| ˙ < ∞ (3) pζ ζ˙ (z, z) πβ where β is given by [17, eq. (21b)]. If the inphase and quadrature components hI (t) and hQ (t) are uncorrelated and we consider isotropic scattering, then β = 2(πσfmax )2 , where fmax is the maximum Doppler frequency. For the calculations of the statistical properties of the MIMO channel capacity (see Section III), we need the joint PDF of ζ 2 (t) and ζ˙ 2 (t). After applying the concept of transformation of random variables [18], we obtain pζ 2 ζ˙2 (z, z) ˙ =
(5)
−∞
=
1 . 1 − jω
(6)
For the calculation of the LCR of the capacity of MIMO channels, we need the joint CF of ζ 2 (t) and ζ˙ 2 (t), denoted by ˙ This function can be expressed as Φζ 2 ζ˙2 (ω, ω). ∞ ∞ ˙ Φζ 2 ζ˙2 (ω, ω) ˙ = pζ 2 ζ˙2 (z, z)e ˙ j(ωz+ω˙ z) dz dz˙ −∞
=
0
1 + 2βω 2 + j ω˙ . (1 + 2βω 2 )2 + ω˙ 2
(7)
In the next section, we will derive the capacity of MIMO channels making use of the results presented above. III. S TATISTICAL PROPERTIES OF THE CHANNEL CAPACITY We divide the capacity studies into three parts, namely, the SIMO/SISO channel capacity, the MISO channel capacity, and finally the MIMO channel capacity. A. The SIMO/SISO Channel Capacity In this subsection, we consider a SIMO channel with MR receive antennas. Note that we consider a SISO channel when MR = 1. The time-varying complex channel gains will be denoted by hm (t) (m = 1, 2 . . . , MR ). We assume that the complex channel gains are independent. The SIMO channel capacity CSIMO (t) is defined as [19] CSIMO (t) = log2 (1 + ρhH (t)h(t))
(8)
where h(t) = [h1 (t), . . . , hMR (t)]T is the MR × 1 complex channel gain vector. The transpose and the complex conjugate transpose operators are denoted by (·)T and (·)H , respectively. Finally, the quantity ρ is the signal-to-noise ratio (SNR). Alternatively, the SIMO channel capacity CSIMO (t) in (8) can be expressed in component form as MR 2 ζm (t) CSIMO (t) = log2 1 + ρ
z ≥ 0, |z| ˙ < ∞. (4)
From (4), we can easily see that we cannot express the joint ˙ as a product of two marginal PDFs, denoted PDF pζ 2 ζ˙2 (z, z) by pζ 2 (z) and pζ˙2 (z). ˙ Hence, the stochastic processes ζ 2 (t) and ζ˙ 2 (t) are not statistical independent. Nevertheless, we can
z ≥ 0.
The formula presented in (5) will be used in the next section together with the CF of pζ 2 (z) to derive the PDF of the capacity of MIMO channels. The CF of pζ 2 (z) is denoted by Φζ 2 (ω). This function is defined as the Fourier transform of pζ 2 (z), i.e., ∞ pζ 2 (z)e jωz dz Φζ 2 (ω) =
(1)
where the real and imaginary components of h(t) are denoted by hI (t) and hQ (t), respectively. In general, it is usually assumed that hI (t) and hQ (t) are uncorrelated real Gaussian noise processes with zero means and identical variances σh2 I = σh2 Q = σ 2 [5]. In this paper, hI (t) and hQ (t) are allowed to be correlated. The correlation properties are described by the correlation matrix in [17, eq. (17)]. In the following, we assume that h(t) is a zero-mean complex Gaussian process with unit variance. Consequently, the absolute value of h(t) (also denoted as envelope) (2) ζ(t) = |h(t)| = [hI (t)]2 + [hQ (t)]2
√ √ 1 p ˙ ( z, z/(2 ˙ z)) 4z ζ ζ 2 1 = √ e −z−z˙ /(8βz) , 2 2πβz
obtain the PDF pζ 2 (z) as follows ∞ pζ 2 ζ˙2 (z, z) ˙ dz˙ pζ 2 (z) =
(9)
m=1 2 (t) = |hm (t)|2 . For simplicity, we denote where ζm
Λ(t) =
MR m=1
456
2 ζm (t) .
(10)
2 Since we have assumed that hm (t), and hence ζm (t) are independent stochastic processes, the CF of Λ(t) can be expressed as
ΦΛ (ω)
M R
=
Φζ 2 (ω)
m=1
1 . (1 − jω)MR
=
(11)
The PDF of Λ(t) can be obtained by taking the inverse Fourier transform of ΦΛ (ω), i.e., ∞ 1 pΛ (z) = ΦΛ (ω)e −jωz dω 2π −∞ 1 z MR −1 e −z , = z≥0 (12) Γ(MR ) where Γ(·) is the Gamma function [20, eq. (8.310)]. By applying a transformation from Λ(t) to CSIMO (t), we can find the PDF of CSIMO (t). Hence, the PDF of CSIMO (t), denoted by pC, SIMO (r), is given by [1] r ln 2 2r (2r − 1)MR −1 e −(2 −1)/ρ , r ≥ 0. pC, SIMO (r) = M R Γ(MR )ρ (13) Furthermore, the CDF of CSIMO (t), denoted by FC, SIMO (r), can be expressed as
FC, SIMO (r)
=
r
1 − ρ1−MR e −(2 −1)/ρ (2r − 1)MR −1 × M R −1 ρk . (14) Γ(MR − k)(2r − 1)k k=0
To find the LCR and ADF of the SIMO channel capacity CSIMO (t), we continue as follows. In the Appendix, it is shown ˙ that the joint PDF of Λ(t) and Λ(t) is given by 2
˙ = pΛΛ˙ (z, z)
z MR −1 e −z−z˙ /(8βz) √ , 2Γ(MR ) 2πβz
z
It should be mentioned that the same formula also can be found in [4, Eq. (22)], if we let β = 2(πfmax )2 . Finally, by means of [21], the ADF TC, SIMO (r) of the SIMO channel capacity is obtained as FC, SIMO (r) . (20) TC, SIMO (r) = NC, SIMO (r) All the results above can easily be applied to characterize the MISO channel capacity. This is the topic of the next subsection. B. The MISO Channel Capacity In this subsection, we consider a MISO channel with MT transmit antennas. The MISO channel capacity CC, MISO (t) is defined as ρ H CMISO (t) = log2 1 + h (t)h(t) (21) MT where h(t) = [h1 (t), . . . , hMT (t)]T is the MT × 1 complex channel gain vector. Again, we assume that the complex channel gains are independent. After substituting ρ byρ/MT in Section III.A, we obtain the PDF, CDF, LCR, and ADF of the MISO channel capacity as follows. From (13), the PDF pC, MISO (r) can be expressed as pC, MISO (r) =
z ≥ 0, |z| ˙ < ∞. (15)
From (9) and (10), it follows that CSIMO (t) is a function of Λ(t). Thus, by applying the concept of transformation of random variables [18], we obtain 2z ln 2 2 (z, z) ˙ = pΛΛ˙ ((2z − 1)/ρ, 2z z˙ ln 2/ρ) pC C,SIMO ˙ ρ 22z−3/2 (ln 2)2 (2z − 1)MR −3/2 √ = Γ(MR )ρMR +1/2 πβ × e −(2
It should be noted that NC, SIMO (r) is proportional to fmax , as can easily be shown by substituting β by the expression in [17, eq. (21b)]. Thus, the normalization of NC, SIMO (r) onto fmax removes the influence of the vehicle speed. Let us consider the LCR of the SISO channel capacity. When MR = 1, then (18) reduces to 2β(2r − 1) −(2r −1)/ρ e . (19) NC, SISO (r) = πρ
−1)/ρ−(2z z˙ ln 2)2 /(8βρ(2z −1))
. (16)
The LCR NC, SIMO (r) of the SIMO channel capacity CSIMO (t) is defined as ∞ zp ˙ C C,SIMO (r, z) ˙ dz, ˙ r ≥ 0. (17) NC, SIMO (r) = ˙ 0
After substituting (16) in (17) and carrying out some lengthy algebraic computations, we finally find the result r 2ρβ(2r − 1) r √ (2 −1)MR −1 e −(2 −1)/ρ . (18) NC, SIMO (r) = M R Γ(MR )ρ π
r (MT )MT ln 2 r 2 · (2r − 1)MT −1 e −MT (2 −1)/ρ , M T Γ(MT )ρ r ≥ 0. (22)
Similarly, using the expression in (14), we obtain the CDF FC,MISO of CMISO (t) as ρ 1−MT r FC, MISO (r) = 1 − e −MT (2 −1)/ρ (2r − 1)MT −1 × MT M T −1 ρk . (23) Γ(MT − k)(MT )k (2r − 1)k k=0
The LCR NC,MISO (r) of CMISO (t) follows easily from (18), i.e., (MT )MT −1/2 2ρβ(2r − 1) √ NC, MISO (r) = Γ(MT )ρMT π r × (2r − 1)MT −1 e−MT (2 −1)/ρ . (24) Note that when MT = 1 in (24), we obtain the formula presented in (19). Finally, the ADF TC, MISO (r) of the MISO channel capacity is obtained as TC, MISO (r) =
457
FC, MISO (r) . NC, MISO (r)
(25)
is nearly independent of the number of transmit antennas. By In the artificial case of no interference between the received comparing Figs. 1 and 2, we observe that to gain high capacity, signals, the MIMO channel capacity can be written as [2], [22] it is more important to have a high number of receive antennas rather than a high number of transmit antennas. In Figs. 4–6, ρ H h (t)h(t) (26) we have presented the normalized LCR of the capacity. In the CMIMO (t) = MT log2 1 + MT SIMO and MISO cases, the spread of the LCR of the capacity T where h(t) = [h1 (t), . . . , hMR (t)] is the MR × 1 complex decreases with increasing number of antennas. Figure 4 shows channel gain vector. In general, the formula in (26) represents that the maximum LCR is nearly independent of MR for an upper bound of the MIMO channel capacity. By applying MR ≥ 2. Analogously, Fig. 5 shows that the maximum LCR a transformation of CSIMO (t) to CMIMO (t) and substituting ρ by of CMISO (t) is nearly independent of MT for MT ≥ 2. Again, there is an excellent correspondence between the theory and ρ/MT , we obtain the PDF pC, MIMO (r) of CMIMO (t) as the simulation results. Figure 7 presents the normalized ADF 1 pC, MIMO (r) = pC, SIMO (r/MT ) of the SIMO channel capacity. This figure shows that the MT mean value for the length of the time intervals in which the ln 2(MT )MR 2r/MT (2r/MT − 1)MR −1 capacity CSIMO (t) is below a given length r, is decreasing with = Γ(MR )MT ρMR the number of receive antennas. Furthermore, Fig. 8 presents r/MT −1)/ρ × e −MT (2 , r ≥ 0. (27) the ADF of the MISO channel capacity. Around the mean value of CMISO (t), the ADF of CMISO (t) is nearly independent Analogously, the CDF FC, MIMO (r) of CMIMO (t) can be expressed of the number of transmit antennas. For levels r less than the as mean value of CMISO (t), the ADF decreases with the number of 1−MR
transmit antennas. The opposite occurs for levels r larger than r/M ρ T −1)/ρ e −MT (2 (2r/MT−1)MR−1× the mean value of CMISO (t). Finally, in Fig. 9 we can observe FC, MIMO (r)=1− MT the behaviour of the ADF of the MIMO channel capacity. M R −1 ρk . (28) Similarly to the SIMO case, the ADF of CMIMO (t) decreases k (2r/MT − 1)k Γ(M − k)(M ) with the number of antennas. R T k=0 C. The MIMO Channel Capacity
Finally, the LCR NC, MIMO (r) and ADF TC, MIMO (r) of the MIMO channel capacity are given by 1
NC, MIMO (r) = NC, SIMO (r/MT ) (MT )MR −1/2 2ρβ(2r/MT − 1) √ = Γ(MR )ρMR π and TC, MIMO (r) =
FC, MIMO (r) NC, MIMO (r)
r/MT
(1 × 9)
0.8
−1)/ρ
pC, SIMO (r)
× (2r/MT − 1)MR −1 e −MT (2
Theory Simulation
(29)
(1 × 7) (1 × 5)
0.6
(1 × 3) (1 × 2)
0.4 (1 × 1)
(30)
0.2
respectively. In the next section, we compare our analytical expressions with simulation results.
0 0
2
4
6
8
10
Level, r Fig. 1. The PDF of the (1 × MR ) SIMO channel capacity.
IV. S IMULATION RESULTS In the following, we present analytical and simulation results of the statistical properties of the capacity for various MIMO channels. To generate mutually uncorrelated Rayleigh fading waveforms, we have used the sum-of-sinusoids principle. For the computation of the model parameters, we have used the generalized method of exact Doppler spread (GMEDS1 ) [23]. In the applied Rayleigh fading channel simulator, the following parameters have been used. The numbers of sinusoids were N1 = 35 and N2 = 36. The maximum Doppler frequency fmax was 91 Hz. For the simulations of the capacity of MIMO channels, the SNR was set to 17 dB. Firstly, we consider the PDF of the capacity for various numbers of receive and transmit antennas in Figs. 1–3. In all cases, there is an excellent fitting between the analytical and the simulation results. Figure 2 shows that the expected value of CMISO (t)
1
Theory Simulation
(9 × 1)
0.8
pC, MISO (r)
(7 × 1) (5 × 1)
0.6
(3 × 1)
0.4
(2 × 1) (1 × 1)
0.2
0 0
2
4
6
8
Level, r Fig. 2. The PDF of the (MT × 1) MISO channel capacity.
458
10
3
0.2
10
Theory Simulation
(2 × 2)
Theory Simulation
(1 × 1) (1 × 2) (1 × 3) (1 × 5) (1 × 7) (1 × 9)
2
10
TC, SIMO (r) · fmax
0.15
pC, MIMO (r)
(4 × 4) (6 × 6)
0.1
0.05
1
10
0
10
−1
10
−2
10 0 0
10
20
30
40
0
2
4
6
50
Level, r
3
10
Theory Simulation
(1 × 1)
1
(1 × 2) (1 × 3)(1 × 5)(1 × 7)
(1 × 1) (2 × 1) (3 × 1) (5 × 1) (7 × 1) (9 × 1)
10
TC, MISO (r) · fmax
NC, SIMO (r)/fmax
Theory Simulation
2
(1 × 9)
0.8 0.6 0.4
1
10
0
10
−1
10
0.2
−2
0 0
2
4
6
8
10
10
0
2
4
6
8
10
Level, r
Level, r
Fig. 8. The normalized ADF of (MT × 1) the MISO channel capacity.
Fig. 4. The normalized LCR of the (1 × MR ) SIMO channel capacity.
2
1.2
10
Theory Simulation
1
Theory Simulation
1
10
0.8
(1 × 1)
TC, MIMO (r) · fmax
NC, MISO (r)/fmax
10
Fig. 7. The normalized ADF of the (1 × MR ) SIMO channel capacity.
Fig. 3. The PDF of the (MT × MR ) MIMO channel capacity. 1.2
8
Level, r
(2 × 1)
0.6 (3 × 1) (5 × 1) (7 × 1) (9 × 1)
0.4
(4 × 4)
(2 × 2)
(6 × 6)
0
10
−1
10
−2
10
0.2
−3
0 0
2
4
6
8
10
10
0
10
20
30
40
50
60
Level, r
Level, r
Fig. 9. The normalized ADF of the (MT × MR ) MIMO channel capacity.
Fig. 5. The normalized LCR of the (MT × 1) MISO channel capacity.
V. C ONCLUSIONS 1.2 (2 × 2)
(4 × 4)
NC, MIMO (r)/fmax
1
(6 × 6)
Theory Simulation
0.8 0.6 0.4 0.2 0 0
5
10
15
20
25
30
35
40
Level, r Fig. 6. The normalized LCR of the (MT × MR ) MIMO channel capacity.
In this paper, we have studied the statistical properties of the capacity for various MIMO channels. Exact closed-form solutions for the PDF, CDF, LCR, and ADF of the capacity have been derived. The analytical expressions are valid for any numbers of transmit and receive antennas. Our analytical developments help us to understand and predict the capacity gain expected from the MIMO technique, in terms of the channel dimension and parameters. Simulation results show an excellent correspondence between theory and simulation.
45
A PPENDIX In general, if the joint CF of two random variables exists, there must be a one-to-one correspondence between the joint PDF
459
and the joint CF. In this paper, we can easily find the joint CF ˙ Hence, the formula in (15) is correct if the joint of pΛΛ˙ (z, z). ˙ is equal to MR times Φζ 2 ζ˙2 (ω, ω). ˙ The CF CF of pΛΛ˙ (z, z) ˙ denoted by ΦΛΛ˙ (ω, ω), ˙ is defined as of pΛ2 Λ˙ 2 (z, z), ∞ ∞ ˙ ˙ = pΛ2 Λ˙ 2 (z, z)e ˙ j(ωz+ω˙ z) dz dz˙ . (31) ΦΛΛ˙ (ω, ω) −∞
0
After some lengthy algebraic computations, we find ΦΛΛ˙ (ω, ω) ˙ = =
1 (1 + 2βω 2 − j ω) ˙ MR M R Φζ 2 ζ˙2 (ω, ω) ˙ .
(32)
m=1
[17] M. P¨atzold, U. Killat, and F. Laue, “An extended Suzuki model for land mobile satellite channels and its statistical properties,” IEEE Trans. Veh. Technol., vol. 47, no. 2, pp. 617–630, May 1998. [18] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes. New York: McGraw-Hill, 4th edition, 2002. [19] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. Telecommun. Related Technol., vol. 10, pp. 585–595, 1999. [20] I. S. Gradstein and I. M. Ryshik, Tables of Series, Products, and Integrals, vol. I and II. Frankfurt: Harri Deutsch, 5th edition, 1981. [21] W. C. Jakes, Ed., Microwave Mobile Communications. Piscataway, NJ: IEEE Press, 1993. [22] G. H. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, pp. 311–335, 1998. [23] M. P¨atzold and B. O. Hogstad, “Two new methods for the generation of multiple uncorrelated Rayleigh fading waveforms,” in Proc. 163th IEEE Semiannual Vehicular Technology Conference, VTC 2006-Spring, Melbourne, Australia, May 2006, vol. 6, pp. 2782–2786.
Hence, the proof of (15) is complete. R EFERENCES [1] H. Ge, K. D. Wong, M. Barton, and J. C. Liberti, “Statistical characterization of multiple-input multiple-output (MIMO) channel capacity,” in Proc. IEEE Wireless communications and networking conference, Mar. 2002, vol. 2, pp. 789–793. [2] B. Holter, “On the capacity of the MIMO channel - a tutorial introduction,” in Proc. IEEE Norwegian Symposium on Signal Processing, Trondheim, Norway, Oct. 2001, pp. 167–172. [3] P. J. Smith, L. M. Garth, and S. Loyka, “Exact capacity distributions for MIMO systems with small numbers of antennas,” IEEE Commun. Lett., vol. 7, no. 10, pp. 481–483, Oct. 2003. [4] S. Wang and A. Abdi, “On the second-order statistics of the instantaneous mutual information of time-varying fading channels,” in Proc. 6th IEEE Workshop on Signal Processing Advances in Wireless Commun., New York, NY, USA, 2005, pp. 405–409. [5] M. P¨atzold, Mobile Fading Channels. Chichester: John Wiley & Sons, 2002. [6] M. P¨atzold and F. Laue, “Level-crossing rate and average duration of fades of deterministic simulation models for Rice fading channels,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1121–1129, Jul. 1999. [7] N. Youssef, C.-X. Wang, and M. P¨atzold, “A study on the second order statistics of Nakagami-Hoyt mobile fading channels,” IEEE Trans. Veh. Technol., vol. 54, no. 4, pp. 1259–1265, 2005. [8] K. Ohtani, K. Daikoku, and H. Omori, “Burst error performance encountered in digital land mobile radio channel,” IEEE Trans. Veh. Technol., vol. 23, no. 1, pp. 156–160, 1981. [9] H. S. Wang and N. Moayeri, “Finite-state Markov channel – A useful model for radio communication channels,” IEEE Trans. Veh. Technol., vol. 44, pp. 163–171, Feb. 1995. [10] R. Vijayan and J. M. Holtzman, “Foundations for level crossing analysis of handoff algorithms,” in Proc. ICC, May 1993, pp. 935–939. [11] S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J., vol. 23, pp. 282–332, Jul. 1944. [12] S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J., vol. 24, pp. 46–156, Jan. 1945. [13] A. Abdi and M. Kaveh, “Level crossing rate in terms of the characteristic function: a new approach for calculating the fading rate in diversity systems,” IEEE Trans. Commun., vol. 50, no. 9, pp. 1397–1400, Sept. 2002. [14] A. Giorgetti, P. J. Smith, M. Shafi, and M. Chiana, “MIMO capacity, level crossing rates and fades: The impacts of spatial/temporal channel correlation,” Journal of Communications and Networks, vol. 5, no. 2, pp. 104–115, 2003. [15] B. O. Hogstad and M. P¨atzold, “Capacity studies of MIMO models based on the geometrical one-ring scattering model,” in Proc. 15th IEEE Int. Symp. on Personal, Indoor and Mobile Radio Communications, PIMRC 2004, Barcelona, Spain, Sept. 2004. [16] B. O. Hogstad and M. P¨atzold, “A study of the MIMO channel capacity when using the geometrical two-ring scattering model,” in Proc. 8th International Symposium on Wireless Personal Multimedia Communications, WPMC 2005, Aalborg, Denmark, Sept. 2005, pp. 1790–1794.
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Abstract— This paper deals with some important statistical properties of the multiple-input multiple-output (MIMO) channel capacity. We assume that all the subchannels are uncorrelated. In case of single-input multiple-output (SIMO) and multiple-input single-output (MISO) channels, exact closed-form expressions are derived for the probability density function (PDF), the cumulative distribution function (CDF), the level-crossing rate (LCR), and the average duration of fades (ADF) of the channel capacity. Furthermore, these exact closed-form expressions are extended to characterize the statistical properties of the maximum MIMO channel capacity. The correctness of the derived closed-form expressions is confirmed by simulations.
I. I NTRODUCTION Exact knowledge of the statistical properties of the MIMO channel capacity is an important issue in the development of future mobile communication systems. Some of these statistical properties are for examples the PDF, the CDF, the LCR, and the ADF of the channel capacity. In [1], exact closed-form expressions for the PDFs of the SIMO and MISO channel capacity are derived. Also from [1], the exact PDF of the maximum MIMO channel capacity can easily be obtained. The maximum MIMO channel capacity is reached in the unrealistic situation when each of the MT transmitted signals is received by the same set of MR antennas without interference. It can also be described as if each transmitted signal where received by a separate set of receive antennas, giving a total number MT · MR receiving antennas [2]. In our paper, we give a detailed proof of the exact closed-form expression for the PDF of the maximum MIMO channel capacity. Even though the maximum MIMO channel capacity is an upper bound for the capacity, our derived PDF shows the potential of the MIMO technology. From the PDF, we can easily obtain the corresponding CDF. For SIMO and MISO channels, where the power of all branches are different, expression for the CDF of the capacity can be found in [3]. In this paper, we present the CDF when all the branch powers are equal. It should be mentioned that the obtained CDF in [3] is not valid in this case. Even when the PDF and CDF are important statistical quantities of the MIMO channel capacity, they do not show the dynamic temporal behavior of the capacity in time-varying MIMO fading channels. To fulfill this gap, we will study the
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LCR and ADF of the MIMO channel capacity. To the best of the authors knowledge, the closed-form expressions for the LCR and ADF are completely new, except that a similar formula for the LCR of the SISO channel capacity can be found in [4]. Most of the LCR and ADF related research has been carried out for single-input single-output (SISO) mobile fading channels [5], [6], [7]. Therein, the LCR is defined as how often the signal envelope crosses a certain threshold from up to down (or from down to up) within one second. Analogously, the ADF is defined as the expected time length the signal envelope stays below a given threshold. The ADF is an important dynamic characteristic quantity of the channel. For example, the ADF enables one to get the statistics of burst errors occurring in mobile communication systems [8], and thus, can provide useful information for the design of error-correcting schemes. Furthermore, the LCR and ADF have found a variety of other applications such as the finite-state Markov modeling of fading channels [9], and the analysis of handoff algorithms [10]. In this paper, the LCR of the MIMO channel capacity is defined as the average number of up-crossings (or downcrossings) of the capacity through a fixed level within one second. From the LCR, one can easily calculate the ADF, which is defined as the expected value of the length of the time intervals in which the capacity is below a given level. A common approach for calculating the LCR of a white sense stationary random process is to employ Rice’s formula [11], [12], which expresses the LCR in terms of the joint PDF of the process and its derivative. Furthermore, in [13] Rice’s LCR formula is rewritten in terms of the joint characteristic function (CF) of the random process and its derivative. This latter formula is useful for many cases where the joint CF is simpler to derive than the associated joint PDF. Nevertheless, in this paper, we will consider the original formula proposed by Rice. Simulation results for the channel capacity’s PDF, LCR, and ADF by using the geometrical one-ring scattering model can be found in [14], [15], and for the two-ring scattering model in [16]. Furthermore, analytical approximations of the LCR and ADF of SIMO and MISO channel capacity are presented
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in [14]. The remainder of this paper is organized as follows. In Section II, we will describe the Rayleigh process with crosscorrelated inphase and quadrature components. A detailed study of the statistical properties of the MIMO channel capacity is given in Section III. The analytical results obtained in Section III will be compared with simulation results in Section IV. Finally, the conclusions are drawn in Section V. II. T HE R AYLEIGH PROCESS Throughout the paper, we will denote the time-varying complex channel gain between a single transmit and a single receive antenna by h(t). The complex process h(t) can be expressed as I
Q
h(t) = h (t) + jh (t)
−∞
=
e −z ,
follows a Rayleigh distribution. The time derivative of ζ(t) ˙ will be denoted by ζ(t). Throughout the paper, we let the overdote of a process denote the time derivative. From [17], ˙ the joint PDF of ζ(t) and ζ(t) at the same time t, can be expressed as 2 2 2 ze −z −z˙ /(2β) , ˙ = z ≥ 0, |z| ˙ < ∞ (3) pζ ζ˙ (z, z) πβ where β is given by [17, eq. (21b)]. If the inphase and quadrature components hI (t) and hQ (t) are uncorrelated and we consider isotropic scattering, then β = 2(πσfmax )2 , where fmax is the maximum Doppler frequency. For the calculations of the statistical properties of the MIMO channel capacity (see Section III), we need the joint PDF of ζ 2 (t) and ζ˙ 2 (t). After applying the concept of transformation of random variables [18], we obtain pζ 2 ζ˙2 (z, z) ˙ =
(5)
−∞
=
1 . 1 − jω
(6)
For the calculation of the LCR of the capacity of MIMO channels, we need the joint CF of ζ 2 (t) and ζ˙ 2 (t), denoted by ˙ This function can be expressed as Φζ 2 ζ˙2 (ω, ω). ∞ ∞ ˙ Φζ 2 ζ˙2 (ω, ω) ˙ = pζ 2 ζ˙2 (z, z)e ˙ j(ωz+ω˙ z) dz dz˙ −∞
=
0
1 + 2βω 2 + j ω˙ . (1 + 2βω 2 )2 + ω˙ 2
(7)
In the next section, we will derive the capacity of MIMO channels making use of the results presented above. III. S TATISTICAL PROPERTIES OF THE CHANNEL CAPACITY We divide the capacity studies into three parts, namely, the SIMO/SISO channel capacity, the MISO channel capacity, and finally the MIMO channel capacity. A. The SIMO/SISO Channel Capacity In this subsection, we consider a SIMO channel with MR receive antennas. Note that we consider a SISO channel when MR = 1. The time-varying complex channel gains will be denoted by hm (t) (m = 1, 2 . . . , MR ). We assume that the complex channel gains are independent. The SIMO channel capacity CSIMO (t) is defined as [19] CSIMO (t) = log2 (1 + ρhH (t)h(t))
(8)
where h(t) = [h1 (t), . . . , hMR (t)]T is the MR × 1 complex channel gain vector. The transpose and the complex conjugate transpose operators are denoted by (·)T and (·)H , respectively. Finally, the quantity ρ is the signal-to-noise ratio (SNR). Alternatively, the SIMO channel capacity CSIMO (t) in (8) can be expressed in component form as MR 2 ζm (t) CSIMO (t) = log2 1 + ρ
z ≥ 0, |z| ˙ < ∞. (4)
From (4), we can easily see that we cannot express the joint ˙ as a product of two marginal PDFs, denoted PDF pζ 2 ζ˙2 (z, z) by pζ 2 (z) and pζ˙2 (z). ˙ Hence, the stochastic processes ζ 2 (t) and ζ˙ 2 (t) are not statistical independent. Nevertheless, we can
z ≥ 0.
The formula presented in (5) will be used in the next section together with the CF of pζ 2 (z) to derive the PDF of the capacity of MIMO channels. The CF of pζ 2 (z) is denoted by Φζ 2 (ω). This function is defined as the Fourier transform of pζ 2 (z), i.e., ∞ pζ 2 (z)e jωz dz Φζ 2 (ω) =
(1)
where the real and imaginary components of h(t) are denoted by hI (t) and hQ (t), respectively. In general, it is usually assumed that hI (t) and hQ (t) are uncorrelated real Gaussian noise processes with zero means and identical variances σh2 I = σh2 Q = σ 2 [5]. In this paper, hI (t) and hQ (t) are allowed to be correlated. The correlation properties are described by the correlation matrix in [17, eq. (17)]. In the following, we assume that h(t) is a zero-mean complex Gaussian process with unit variance. Consequently, the absolute value of h(t) (also denoted as envelope) (2) ζ(t) = |h(t)| = [hI (t)]2 + [hQ (t)]2
√ √ 1 p ˙ ( z, z/(2 ˙ z)) 4z ζ ζ 2 1 = √ e −z−z˙ /(8βz) , 2 2πβz
obtain the PDF pζ 2 (z) as follows ∞ pζ 2 ζ˙2 (z, z) ˙ dz˙ pζ 2 (z) =
(9)
m=1 2 (t) = |hm (t)|2 . For simplicity, we denote where ζm
Λ(t) =
MR m=1
456
2 ζm (t) .
(10)
2 Since we have assumed that hm (t), and hence ζm (t) are independent stochastic processes, the CF of Λ(t) can be expressed as
ΦΛ (ω)
M R
=
Φζ 2 (ω)
m=1
1 . (1 − jω)MR
=
(11)
The PDF of Λ(t) can be obtained by taking the inverse Fourier transform of ΦΛ (ω), i.e., ∞ 1 pΛ (z) = ΦΛ (ω)e −jωz dω 2π −∞ 1 z MR −1 e −z , = z≥0 (12) Γ(MR ) where Γ(·) is the Gamma function [20, eq. (8.310)]. By applying a transformation from Λ(t) to CSIMO (t), we can find the PDF of CSIMO (t). Hence, the PDF of CSIMO (t), denoted by pC, SIMO (r), is given by [1] r ln 2 2r (2r − 1)MR −1 e −(2 −1)/ρ , r ≥ 0. pC, SIMO (r) = M R Γ(MR )ρ (13) Furthermore, the CDF of CSIMO (t), denoted by FC, SIMO (r), can be expressed as
FC, SIMO (r)
=
r
1 − ρ1−MR e −(2 −1)/ρ (2r − 1)MR −1 × M R −1 ρk . (14) Γ(MR − k)(2r − 1)k k=0
To find the LCR and ADF of the SIMO channel capacity CSIMO (t), we continue as follows. In the Appendix, it is shown ˙ that the joint PDF of Λ(t) and Λ(t) is given by 2
˙ = pΛΛ˙ (z, z)
z MR −1 e −z−z˙ /(8βz) √ , 2Γ(MR ) 2πβz
z
It should be mentioned that the same formula also can be found in [4, Eq. (22)], if we let β = 2(πfmax )2 . Finally, by means of [21], the ADF TC, SIMO (r) of the SIMO channel capacity is obtained as FC, SIMO (r) . (20) TC, SIMO (r) = NC, SIMO (r) All the results above can easily be applied to characterize the MISO channel capacity. This is the topic of the next subsection. B. The MISO Channel Capacity In this subsection, we consider a MISO channel with MT transmit antennas. The MISO channel capacity CC, MISO (t) is defined as ρ H CMISO (t) = log2 1 + h (t)h(t) (21) MT where h(t) = [h1 (t), . . . , hMT (t)]T is the MT × 1 complex channel gain vector. Again, we assume that the complex channel gains are independent. After substituting ρ byρ/MT in Section III.A, we obtain the PDF, CDF, LCR, and ADF of the MISO channel capacity as follows. From (13), the PDF pC, MISO (r) can be expressed as pC, MISO (r) =
z ≥ 0, |z| ˙ < ∞. (15)
From (9) and (10), it follows that CSIMO (t) is a function of Λ(t). Thus, by applying the concept of transformation of random variables [18], we obtain 2z ln 2 2 (z, z) ˙ = pΛΛ˙ ((2z − 1)/ρ, 2z z˙ ln 2/ρ) pC C,SIMO ˙ ρ 22z−3/2 (ln 2)2 (2z − 1)MR −3/2 √ = Γ(MR )ρMR +1/2 πβ × e −(2
It should be noted that NC, SIMO (r) is proportional to fmax , as can easily be shown by substituting β by the expression in [17, eq. (21b)]. Thus, the normalization of NC, SIMO (r) onto fmax removes the influence of the vehicle speed. Let us consider the LCR of the SISO channel capacity. When MR = 1, then (18) reduces to 2β(2r − 1) −(2r −1)/ρ e . (19) NC, SISO (r) = πρ
−1)/ρ−(2z z˙ ln 2)2 /(8βρ(2z −1))
. (16)
The LCR NC, SIMO (r) of the SIMO channel capacity CSIMO (t) is defined as ∞ zp ˙ C C,SIMO (r, z) ˙ dz, ˙ r ≥ 0. (17) NC, SIMO (r) = ˙ 0
After substituting (16) in (17) and carrying out some lengthy algebraic computations, we finally find the result r 2ρβ(2r − 1) r √ (2 −1)MR −1 e −(2 −1)/ρ . (18) NC, SIMO (r) = M R Γ(MR )ρ π
r (MT )MT ln 2 r 2 · (2r − 1)MT −1 e −MT (2 −1)/ρ , M T Γ(MT )ρ r ≥ 0. (22)
Similarly, using the expression in (14), we obtain the CDF FC,MISO of CMISO (t) as ρ 1−MT r FC, MISO (r) = 1 − e −MT (2 −1)/ρ (2r − 1)MT −1 × MT M T −1 ρk . (23) Γ(MT − k)(MT )k (2r − 1)k k=0
The LCR NC,MISO (r) of CMISO (t) follows easily from (18), i.e., (MT )MT −1/2 2ρβ(2r − 1) √ NC, MISO (r) = Γ(MT )ρMT π r × (2r − 1)MT −1 e−MT (2 −1)/ρ . (24) Note that when MT = 1 in (24), we obtain the formula presented in (19). Finally, the ADF TC, MISO (r) of the MISO channel capacity is obtained as TC, MISO (r) =
457
FC, MISO (r) . NC, MISO (r)
(25)
is nearly independent of the number of transmit antennas. By In the artificial case of no interference between the received comparing Figs. 1 and 2, we observe that to gain high capacity, signals, the MIMO channel capacity can be written as [2], [22] it is more important to have a high number of receive antennas rather than a high number of transmit antennas. In Figs. 4–6, ρ H h (t)h(t) (26) we have presented the normalized LCR of the capacity. In the CMIMO (t) = MT log2 1 + MT SIMO and MISO cases, the spread of the LCR of the capacity T where h(t) = [h1 (t), . . . , hMR (t)] is the MR × 1 complex decreases with increasing number of antennas. Figure 4 shows channel gain vector. In general, the formula in (26) represents that the maximum LCR is nearly independent of MR for an upper bound of the MIMO channel capacity. By applying MR ≥ 2. Analogously, Fig. 5 shows that the maximum LCR a transformation of CSIMO (t) to CMIMO (t) and substituting ρ by of CMISO (t) is nearly independent of MT for MT ≥ 2. Again, there is an excellent correspondence between the theory and ρ/MT , we obtain the PDF pC, MIMO (r) of CMIMO (t) as the simulation results. Figure 7 presents the normalized ADF 1 pC, MIMO (r) = pC, SIMO (r/MT ) of the SIMO channel capacity. This figure shows that the MT mean value for the length of the time intervals in which the ln 2(MT )MR 2r/MT (2r/MT − 1)MR −1 capacity CSIMO (t) is below a given length r, is decreasing with = Γ(MR )MT ρMR the number of receive antennas. Furthermore, Fig. 8 presents r/MT −1)/ρ × e −MT (2 , r ≥ 0. (27) the ADF of the MISO channel capacity. Around the mean value of CMISO (t), the ADF of CMISO (t) is nearly independent Analogously, the CDF FC, MIMO (r) of CMIMO (t) can be expressed of the number of transmit antennas. For levels r less than the as mean value of CMISO (t), the ADF decreases with the number of 1−MR
transmit antennas. The opposite occurs for levels r larger than r/M ρ T −1)/ρ e −MT (2 (2r/MT−1)MR−1× the mean value of CMISO (t). Finally, in Fig. 9 we can observe FC, MIMO (r)=1− MT the behaviour of the ADF of the MIMO channel capacity. M R −1 ρk . (28) Similarly to the SIMO case, the ADF of CMIMO (t) decreases k (2r/MT − 1)k Γ(M − k)(M ) with the number of antennas. R T k=0 C. The MIMO Channel Capacity
Finally, the LCR NC, MIMO (r) and ADF TC, MIMO (r) of the MIMO channel capacity are given by 1
NC, MIMO (r) = NC, SIMO (r/MT ) (MT )MR −1/2 2ρβ(2r/MT − 1) √ = Γ(MR )ρMR π and TC, MIMO (r) =
FC, MIMO (r) NC, MIMO (r)
r/MT
(1 × 9)
0.8
−1)/ρ
pC, SIMO (r)
× (2r/MT − 1)MR −1 e −MT (2
Theory Simulation
(29)
(1 × 7) (1 × 5)
0.6
(1 × 3) (1 × 2)
0.4 (1 × 1)
(30)
0.2
respectively. In the next section, we compare our analytical expressions with simulation results.
0 0
2
4
6
8
10
Level, r Fig. 1. The PDF of the (1 × MR ) SIMO channel capacity.
IV. S IMULATION RESULTS In the following, we present analytical and simulation results of the statistical properties of the capacity for various MIMO channels. To generate mutually uncorrelated Rayleigh fading waveforms, we have used the sum-of-sinusoids principle. For the computation of the model parameters, we have used the generalized method of exact Doppler spread (GMEDS1 ) [23]. In the applied Rayleigh fading channel simulator, the following parameters have been used. The numbers of sinusoids were N1 = 35 and N2 = 36. The maximum Doppler frequency fmax was 91 Hz. For the simulations of the capacity of MIMO channels, the SNR was set to 17 dB. Firstly, we consider the PDF of the capacity for various numbers of receive and transmit antennas in Figs. 1–3. In all cases, there is an excellent fitting between the analytical and the simulation results. Figure 2 shows that the expected value of CMISO (t)
1
Theory Simulation
(9 × 1)
0.8
pC, MISO (r)
(7 × 1) (5 × 1)
0.6
(3 × 1)
0.4
(2 × 1) (1 × 1)
0.2
0 0
2
4
6
8
Level, r Fig. 2. The PDF of the (MT × 1) MISO channel capacity.
458
10
3
0.2
10
Theory Simulation
(2 × 2)
Theory Simulation
(1 × 1) (1 × 2) (1 × 3) (1 × 5) (1 × 7) (1 × 9)
2
10
TC, SIMO (r) · fmax
0.15
pC, MIMO (r)
(4 × 4) (6 × 6)
0.1
0.05
1
10
0
10
−1
10
−2
10 0 0
10
20
30
40
0
2
4
6
50
Level, r
3
10
Theory Simulation
(1 × 1)
1
(1 × 2) (1 × 3)(1 × 5)(1 × 7)
(1 × 1) (2 × 1) (3 × 1) (5 × 1) (7 × 1) (9 × 1)
10
TC, MISO (r) · fmax
NC, SIMO (r)/fmax
Theory Simulation
2
(1 × 9)
0.8 0.6 0.4
1
10
0
10
−1
10
0.2
−2
0 0
2
4
6
8
10
10
0
2
4
6
8
10
Level, r
Level, r
Fig. 8. The normalized ADF of (MT × 1) the MISO channel capacity.
Fig. 4. The normalized LCR of the (1 × MR ) SIMO channel capacity.
2
1.2
10
Theory Simulation
1
Theory Simulation
1
10
0.8
(1 × 1)
TC, MIMO (r) · fmax
NC, MISO (r)/fmax
10
Fig. 7. The normalized ADF of the (1 × MR ) SIMO channel capacity.
Fig. 3. The PDF of the (MT × MR ) MIMO channel capacity. 1.2
8
Level, r
(2 × 1)
0.6 (3 × 1) (5 × 1) (7 × 1) (9 × 1)
0.4
(4 × 4)
(2 × 2)
(6 × 6)
0
10
−1
10
−2
10
0.2
−3
0 0
2
4
6
8
10
10
0
10
20
30
40
50
60
Level, r
Level, r
Fig. 9. The normalized ADF of the (MT × MR ) MIMO channel capacity.
Fig. 5. The normalized LCR of the (MT × 1) MISO channel capacity.
V. C ONCLUSIONS 1.2 (2 × 2)
(4 × 4)
NC, MIMO (r)/fmax
1
(6 × 6)
Theory Simulation
0.8 0.6 0.4 0.2 0 0
5
10
15
20
25
30
35
40
Level, r Fig. 6. The normalized LCR of the (MT × MR ) MIMO channel capacity.
In this paper, we have studied the statistical properties of the capacity for various MIMO channels. Exact closed-form solutions for the PDF, CDF, LCR, and ADF of the capacity have been derived. The analytical expressions are valid for any numbers of transmit and receive antennas. Our analytical developments help us to understand and predict the capacity gain expected from the MIMO technique, in terms of the channel dimension and parameters. Simulation results show an excellent correspondence between theory and simulation.
45
A PPENDIX In general, if the joint CF of two random variables exists, there must be a one-to-one correspondence between the joint PDF
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and the joint CF. In this paper, we can easily find the joint CF ˙ Hence, the formula in (15) is correct if the joint of pΛΛ˙ (z, z). ˙ is equal to MR times Φζ 2 ζ˙2 (ω, ω). ˙ The CF CF of pΛΛ˙ (z, z) ˙ denoted by ΦΛΛ˙ (ω, ω), ˙ is defined as of pΛ2 Λ˙ 2 (z, z), ∞ ∞ ˙ ˙ = pΛ2 Λ˙ 2 (z, z)e ˙ j(ωz+ω˙ z) dz dz˙ . (31) ΦΛΛ˙ (ω, ω) −∞
0
After some lengthy algebraic computations, we find ΦΛΛ˙ (ω, ω) ˙ = =
1 (1 + 2βω 2 − j ω) ˙ MR M R Φζ 2 ζ˙2 (ω, ω) ˙ .
(32)
m=1
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