Threshold-based frame error rate analysis of MIMO systems over quasistatic fading channels I. Chatzigeorgiou, I.J. Wassell and R. Carrasco Proper selection of a signal-to-noise ratio threshold largely determines the tightness of an approximation to the frame error rate of a system over a quasistatic fading channel. It is demonstrated that the expression for the optimal threshold value, which has been established for single- input single-output (SISO) channels, remains unchanged for the general case of multiple-input multiple-output (MIMO) channels. Introduction: In various practical systems, such as fixed wireless access networks, the communication channel experiences extremely slow fading conditions that can be characterised by the quasistatic fading model. In quasistatic fading, the instantaneous signal-to-noise ratio (SNR) at the receiver, denoted as g, remains constant for the duration of a frame but changes independently from frame to frame. El Gamal and Hammons [1] demonstrated that the average error probability of an iteratively decoded scheme over a SISO quasistatic fading channel can be accurately approximated by ~ P Q e ð  g; g w Þ¼ 1  e g w =  g ð1Þ where  g ¼ E½g is the average SNR and g w is a SNR threshold, based on which the error probability of the iterative decoder is either one or zero depending on whether the instantaneous SNR value is less than or greater than g w , respectively. For systems employing turbo codes, g w coincides with the convergence threshold of the iterative decoder [1, 2]. Motivated by the work of El Gamal and Hammons [1], Rodrigues et al. [2] and Bouzekri and Miller [3], we demonstrated in [4] that not only iterative but also non-iterative and even uncoded schemes over SISO quasistatic fading channels can also be characterised by a threshold based on which their frame error rate (FER) can be accurately approximated by ~ P Q e ð  g; g W Þ. In particular, we showed that the SNR threshold is given by g w ¼ ð 1 0 P G d ðgÞ g 2 dg 0 @ 1 A 1 ð2Þ where P d G ( g) is the probability of successful frame detection in additive white Gaussian noise (AWGN). In this Letter we investigate whether the expression for the SNR threshold still holds in the general case of MIMO quasistatic fading channels. Preliminaries: We consider a MIMO channel having N T inputs and N R outputs. The transmitter uses space– time block coding [5], while the receiver coherently combines the N ¼ N T N R independent fading paths. If g now corresponds to the instantaneous SNR at the output of the combiner, its probability distribution is given by [2, 6] p  g R ðgÞ¼ g N1 e g=ð  g R =NT Þ ð  g R =N T Þ N ðN  1Þ! ð3Þ where  g R is the average SNR per receive antenna. The approximated FER of the system for MIMO quasistatic fading channels can then be obtained from [2] ~ P Q e ð  g R ; g w Þ¼ ð g w 0 p  g R ðgÞdg ¼ 1  e g w NT =  g R X N1 k¼0 ðg w N T =  g R Þ k k ! ð4Þ It is important to note that ~ P Q e ð  g R ; g w Þ is accurate for low to moderate values of N; for large N, the MIMO channel effectively collapses into an AWGN channel, for which the framework for threshold-based FER analysis does not apply. In the following Section, we derive an exact expression for the optimal SNR threshold g w for the case when (4) can be used to closely approximate the FER of a MIMO system. SNR threshold evaluation: If an appropriate value for the SNR threshold is selected, we expect the approximated FER of a MIMO system on a quasistatic fading channel to closely represent the exact FER, denoted as P Q e ð  g R Þ, for a wide range of  g R values. Ideally, ~ P Q e ð  g R ; g w Þ should be identical to P Q e ð  g R Þ so that P Q e ð  g R Þ ~ P Q e ð  g R ; g w Þ¼ 0 ð5Þ We set l ¼ N T =  g R and express P Q e ð  g R Þ and ~ P Q e ð  g R ; g w Þ as functions of l, that is P Q e ðlÞ and ~ P Q e ðl; g w Þ, respectively. The change of variable will not have an effect on (5). Consequently, the area under the graph of P Q e ðlÞ should be equal to the area under ~ P Q e ð  g; g w Þ, for l [ ½0 ... L when L ! 1. We can thus write lim L!1 ð L 0 P Q e ðlÞdl  ð L 0 ~ P Q e ðl; g w Þdl 8 < : 9 = ; ¼ 0 ð6Þ Taking into account that P Q e ðlÞ can be computed by integrating the FER in AWGN, represented by P G e ðlÞ, over the distribution p l ( g) given in (3) for l ¼ N T =  g R [6], we expand the first integral in (6) into ð L 0 P Q e ðlÞdl ¼ ð L 0 ð 1 0 P G e ðgÞp l ðgÞdgdl ¼ ð L 0 ð 1 0 p l ðgÞdgdl  ð L 0 ð 1 0 P G e ðgÞp l ðgÞdgdl ð7Þ Note that we have expressed the frame error probability in terms of the probability of successful frame detection in AWGN, that is P e G (l) ¼ 1 2 P d G (l). Substituting p l ( g) into (7) gives ð L 0 P Q e ðlÞdl ¼ ð L 0 ð 1 0 l N g N1 ðN  1Þ! e lg dgdl  ð 1 0 P G d ðgÞ g(N  1)! ð L 0 ðlgÞ N e lg dldg ð8Þ Careful inspection of the function in the double integral reveals that it describes a Poisson distribution; the integral of this probability distri- bution type from zero to infinity is equal to one. Therefore ð L 0 ð 1 0 l N g N1 ðN  1Þ! e lg dgdl ¼ L ð9Þ Furthermore, if we define V n ðxÞ ; x n þ nx n1 þ nðn  1Þx n2 þ ... þ n! ð10Þ where x is a real number and n is a positive integer, we can determine that ð L 0 ðlgÞ N e lg dl ¼ N ! g  e Lg g V N ðLgÞ ð11Þ based on tables in [7]. Consequently, the first integral in (6) assumes the form ð L 0 P Q e ðlÞdl ¼ L  N ð 1 0 P G d ðgÞ g 2 dg þ ð 1 0 P G d ðgÞV N ðLgÞ g 2 ðN  1Þ! e Lg dg ð12Þ ELECTRONICS LETTERS 12th February 2009 Vol. 45 No. 4