2010 Second International Conference on Computational Intelligence, Communication Systems and Networks
Performance of Dynamic MIMO Systems in Presence of Nakagami Fading Channel Ravi kumar*, Rajiv Saxena, Abhay Sah, Saroj Gupta, Tapesh Chandgothia Department of Electronics & Communication, Jaypee University of Engineering & Technology, Raghogarh, Guna-473226 (M.P.), India *E-mail:-
[email protected] Abstract—This paper presents a new way to realize the dynamic model of MIMO channel in presence of m-Nakagami fading. The dynamic model is compared with the static model of MIMO channel and the performance analysis of SISO, SIMO, MISO and MIMO systems is evaluated in presence of mNakagami distribution. All the results are plotted between signal to noise ratio (SNR) and bit error rate (BER) which are well validated and in excellent agreement.
Figure 1. MIMO Model. The MIMO model realisation uses space time block codes and m-Nakagami distribution which is described below.
Keywords- Nakagami-m fading; MIMO.
I.
A. MIMO systems utilizing space time block coding In wired media, communication was limited to time codes only whereas in wireless communication there was a need of a block coding technique which could account for both space as well as time. This has lead to Space Time Code which is not only helpful in encoding &decoding of data but also helpful in multiplexing & demultiplexing, modulation & demodulation and equalization. A MIMO system [5] having MT transmit antennas and MR receive antenna is shown where ti denotes transmitted symbols, i=1,2,3…. MT and rj denotes received symbols, j=1,2,3….. MR. Channel transfer function from the ith transmitted antenna to the jth receive antenna is denoted by hij. The transmitted power from each antenna is P/ MT where P is the transmitted power of a SISO system. The channel introduces Additive White Gaussian Noise(AWGN) nj to each receive antenna, j=1,2…. MR where AWGN components are assumed independently and uniformly distributed. Therefore the MIMO system can be expressed in a matrix form as given below:
INTRODUCTION
The Signal Fading can drastically affect the performance of terrestrial communication systems. Fading caused by multipath propagation can degrade the bit-error-rate (BER) performance of a digital communication system resulting data loss or dropped calls in a cellular system. The Nakagami-m distribution has gained widespread application in the modeling of physical fading radio channels. The primary justification of the use of Nakagami-m fading model is its good fit to empirical fading data. It is versatile and through its parameter m, we can model signal fading conditions that range from severe to moderate, to light fading or no fading. This paper highlights the performance of MIMO systems when the mobile station is stationary or moving with a Doppler shift. In section II, a literature survey to design a static model using space time block coding in presence of m-Nakagami fading is presented. In section III, our proposed model of dynamic MIMO system realization using Doppler shift in presence of m-Nakagami fading is described. In section IV, simulated results for SISO, SIMO, MISO and MIMO are presented where the performance is compared between static m-Nakagami channel samples and dynamic m-Nakagami channel samples. Finally, the paper is concluded in section V. II.
MT
rj = ∑ h j ,i .ti + n j
(1)
i =1
⎡ r1 ⎤ ⎡ h1,1 ⎢r ⎥ ⎢h ⎢ 2 ⎥ ⎢ 2,1 ⇒⎢ . ⎥=⎢ . ⎢ ⎥ ⎢ ⎢ . ⎥ ⎢ . ⎢ rM ⎥ ⎢ h ⎣ R ⎦ ⎣ M R ,1
STATIC MODEL
In static model the mobile station is assumed to be stationary. At the transmitter side, the digital input data stream is first modulated using M-ary technique and is space time coded. The coded signal is transmitted where the channel is assumed to be m-Nakagami fading channel and the noise added is AWGN. At the receiver side, reverse space time coding is performed and the signal is demodulated. The block diagram is depicted in Fig-1.
h1,2
. .
h2,2
. .
. .
. . . .
hM R ,2 . .
h1,M T ⎤ ⎡ t1 ⎤ ⎡ n1 ⎤ ⎥ h2,M T ⎥ ⎢⎢ t2 ⎥⎥ ⎢⎢ n2 ⎥⎥ . ⎥⎢ . ⎥+⎢ . ⎥ ⎥⎢ ⎥ ⎢ ⎥ . ⎥⎢ . ⎥ ⎢ . ⎥ ⎥ hM R ,MT ⎦ ⎢⎣tM T ⎥⎦ ⎢⎣ nM R ⎥⎦
This can be generalized as R = H.T + N where R is the received signals rj matrix of order M R × 1 , T is the transmitted signals ti matrix of order M T ×1 , N is a noise components nj
978-0-7695-4158-7/10 $26.00 © 2010 IEEE DOI 10.1109/CICSyN.2010.47
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cases. Furthermore, it can be used to model fading conditions that are less or more severe than those modeled by the Rayleigh distribution. Nakagami-m distribution [4] has one more free parameter, therefore it allows for more flexibility. It moreover contains both the Rayleigh distribution (m = 1) and the uniform distribution on the unit circle (m Æ ∞) as special (extreme) cases. The probability distribution function (pdf) and cumulative distribution function (pdf) expressions of mNakagami distribution and their corresponding graphs are shown below:
matrix of order M R ×1 and H is the channel cofficients hi,j matrix. According to diversity offered in systems with several transmitting antennas, the STBCs are the simplest type of Spatial temporal codes that can exploit it. In 1998, Alamouti [1] designed a simple transmission diversity technique for 2×2 MIMO systems. This method provides transmission rate equal to SISO systems. The process of encoding and decoding has been performed using block of transmission symbols that provide full diversity and require simple linear equations at both transmitter and receiver side. In this Alamouti space-time encoder [2]-[3], each group of m information bits is first modulated (M-ary modulation scheme is used), where m = log2M. Then, the encoder takes a block of two modulated symbols x1 and x2 in each encoding operation and maps them to the transmit antennas according to a code matrix given by:
⎡x X =⎢ 1 ⎣ x2
− x2* ⎤ ⎥ x1* ⎦
Probability distribution function 2m m m 2 p(x) = x 2 m −1 e x p ( − x ) ω Γ ( m )ω m Where
(2)
x ≥ 0, w ≥ 0 , m ≥
(6)
1 2
w = E ( x 2 ) is an instantaneous power. m=
Hence, the information data bits are first modulated and mapped into their corresponding constellation point. In the Alamouti scheme [1] during the first time instance, the symbol x1 and x2 are transmitted by the first and the second antenna element, respectively. During the second time instance t2, the negative of the conjugate of second symbol i.e. –x2* is send to the first antenna while conjugate of the first constellation point i.e. x1*. It is clear that the encoding is done in both the space and time domains. Let us denote the transmit sequence from antennas one and two by x1 and x2, respectively.
x1 = ⎡⎣ x1
− x2* ⎤⎦
(3a)
x 2 = ⎡⎣ x2
x1* ⎤⎦
(3b)
E ( x2 ) is a shape factor. var( x 2 )
Cumulative distribution function x 2m m ⎛ m ⎞ p (t ) = t 2 m −1 exp ⎜ − t 2 ⎟dt m ∫ Γ ( m )ω 0 ⎝ ω ⎠
In the Alamouti scheme[3], it is shown that the transmit sequences from the two transmit antennas are orthogonal, since the inner product of the sequences x1 and x2 is zero, i.e. (4) 1 2 * *
x .x = x1 x2 − x2 x1 = 0
(7)
Figure 2. PDF of Nakagami Fading Channel
Now at the receiver receive antennas are used. The fading channel coefficients from the first and second transmit antennas to the receive antenna at time t are denoted by h1 (t) and h2 (t) respectively. Assuming that the fading coefficients are constant across two consecutive symbol transmission periods at the receive antenna, the received signals denoted by r1 and r2 for time t and t + T respectively, can be expressed as:
r1 = h1 x1 + h2 x2 + n1 r2 = − h1 x2* + h2 x1* + n2
(5a) (5b)
B. The m-Nakagami Distribution The Nakagami distribution (m-distribution) is a generalized distribution which can model different fading environments. It has greater flexibility and accuracy in matching some experimental data than the Rayleigh, lognormal or Rice distributions [4]. It also has the advantage of including the Rayleigh and the one-sided Gaussian distribution as special
Figure 3. CDF of Nakagami Fading Channel But in most of the practical situation the receivers are not stationary. Hence the static model cannot be used to design the
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channel when the receiver is mobile. There is a requirement of some modification in the channel modeling and Doppler shift should be considered. Therefore a new approach has been proposed to design the channel when the receiver is mobile which is discussed in the next section. III.
Algorithm for BER v/s SNR plot of dynamic MIMO system in presence of m- Nakagami Fading Channel is shown below. Generate random binary sequence of data
DYNAMIC SYSTEM DESIGN
Modulate the data using M-ary PSK
In this modeling m-Nakagami variate is generated by using level crossing rate and average fade duration which incorporates Doppler shift fd. Level Crossing Rate (LCR) [4] is defined as the number of times per unit duration that the envelope of a fading channel crosses a given value in the positive direction. Level Crossing Rate is given by:
NR =
2π f d mρ 2 Γ ( m)
(
)
m −1 2
Group them into pair of two symbols Code it as the Alamouti Space Time code
(8)
exp(− mρ 2 )
Multiply the symbols with the dynamic Nakagami Channel
Average Fade Duration (AFD) [4] corresponds to the average length of time the envelope remains under the threshold value once it crosses it in the positive direction. Closed form expression for AFD is:
TR =
2 m Ψ ( m, ρ )
Add white Gaussian noise Perform decoding, demodulation and count the bit errors
(9)
2π f d ρ 2 m−1 exp(− mρ 2 )
Where ρ is normalized signal strength given as ρ = R/Rrms, г(.) is gamma function and ψ(m,ρ) is (9a) ρ
Repeat for multiple values of Eb/No and plot the simulation Figure 5. Algorithm for BER v/s SNR plot of dynamic MIMO system in presence of Nakagami Fading Channel
Ψ ( m, ρ ) = ∫ z 2 m −1 exp(−mz 2 )dz 0
Now we generate the cdf of m-Nakagami distribution by using the LCR & AFD as illustrated below:
p [ r ≤ R ] = N R * TR
IV.
(10)
Using this cdf, we generate the pdf of m-Nakagami variate and analyze the performance of SISO, SIMO, MISO and MIMO systems as stated in static model. The algorithm for the realization of dynamic MIMO system in presence of Nakagami Fading Channel is shown below.
SIMULATED RESULTS
In this simulation, four cases SISO, SIMO, MISO and MIMO are evaluated. Each case is analyzed for random channel, static and dynamic Nakagami channel. The assumptions taken are for MISO MT =1, MR=2, for SIMO MT = 2, MR =1 and for MIMO MT =2, MR =2. The channel experience by each transmit antenna is independent from the channel experienced by other transmit antennas. The channel experienced between each transmit to the receive antenna is randomly varying in time. However, the channel is assumed to remain constant over two time slots. On the receiver antenna, the noise has the Gaussian probability density function with µ=0 and σ2 = No/2. The expression is given below.
Generate Level Crossing Rate (LCR) of Nakagami Fading Channel for a given Doppler shift Generate Average Fade Duration (AFD) of Nakagami Fading Channel for a given Doppler shift
p (n) =
Obtain cdf from the multiplication of LCR and AFD
1 2πσ 2
− ( n − μ )2
e
2σ 2
(11)
The range of SNR is 0dB to 20 dB. The obtained plots are given below.
Simulate dynamic Nakagami Fading Channel from the obtained cdf Figure 4. Algorithm for the realisation of dynamic MIMO system in presence of Nakagami Fading Channel
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V.
Figure 6. SISO (1x1)
CONCLUSION
The various models of MIMO systems are realised using BER performance with respect to different SNR values for random channel, static m-Nakagami channel and dynamic mNakagami channel. The keying technique used is m-ary PSK and the space time code given by Alamouti [1] is used to transmit the data. It is found that MIMO achieves the performance of SISO while improving the diversity and maintaining the spatial multiplexing. The results reveal that the BER of MIMO system in presence of m-Nakagami fading is comparable in both static and dynamic model while there is a significant increase in the BER for SIMO and MISO systems. MIMO systems show best performance among all both for lower and higher SNR values irrespective of the fact that the receiver is stationary or mobile. REFERENCES [1]
Alamouti, “A Simple Transmit Diversity Technique for Wireless Communications”, IEEE Sel Comm, 1998.
[2]
Anatonis D. Valkanas & Alexender D. Poularikas. “Introduction to Space Time Coding”,2004.
[3]
Gesbert et al., “From Theory to Practice: An Overview of MIMO Space-Time Coded Wireless Systems”, IEEE Sel Comm, 2003.
[4]
Dhiraj Dilip Patil, “On The Simulations of Correlated Nakagami-m Fading Channels Using Sum-Of-Sinusoids Method”, 2006.
[5]
A.J Paulraj, Gore, Nabar and Bolcskei, “An Overview of MIMO Communications – A Key to Gigabit Wireless”, IEEE Trans Comm, 2003.
Figure 7. MISO (2x1)
[6] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [7] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [8] Telatar, “Capacity of multi-antenna Gaussian channels,” Bell Labs Technical Memorandum, June 1995.
Figure 8. SIMO (1x2)
[9] E. Biglieri , R. Calderbank , A. Constantinides , A. Goldsmith , A. Paulraj , H. V. Poor, MIMO wireless communications, Cambridge, January 2007. [10] A. Dogandzic and A. Nehorai, "Space-time fading channel estimation and symbol detection in unknown spatially correlated noise," IEEE Trans. Signal Process., vol. 50, no. 3, Mar. 2002. [11] Z. Liu and G. B. Giannakis, "Space-time block coded multiple access through frequency-selective fading channels," IEEE Trans. Commun., vol. 49, no. 6, pp. 1033-1044, Jun. 2001. [12] B. M. Hochwald and T. L. Marzetta, "Unitary space-time modulationfor multiple-antenna communications in Rayleigh flat fading," IEEE Trans. Inf. Theory, vol. 46, no. 2, Mar. 2000.
Figure 9. MIMO (2x2)
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