Efficient Time/Frequency Permutation of MIMO-OFDM Systems through Independent and Correlated Nakagami Fading Channels

Space-Time Frequency (STF) codes for MIMO-OFDM over block-fading channel can achieve rate M t and full-diversity M t M r M b L which is the product of the number of transmit antennas M t , receive antennas M r , fading blocks M b and channel taps L . In this article, time permutation is proposed to provide independent block-fading over Jake’s Doppler power spectrum channel. Moreover, we show the performance variations of STF code as channel delay spread changes. Therefore, we introduce a frequency/time permutation technique in order to remove the frequency correlation among sub-carriers, which subsequently increases the coding gain and achieves maximum diversity. Finally, the symbol error rate (SER) performance of the proposed time/frequency permuted STF codes over independent and correlated MIMO antenna branches under Nakagami fading channel is simulated. We show that the proposed systems provide better performance and more robust to large values of antennas correlation coefficients in comparison with the un-interleaved one.


Introduction
Achieving high data rate, full diversity gain and higher network capacity becomes the major requirements of wireless system providers.MIMO-OFDM system is one of the most attractive techniques to provide these capabilities.
Recently, some attention has been devoted to design STF codes for MIMO-OFDM system with M t transmit antennas, M r receive antennas, and N-OFDM tones through L multi-path fading channel.There are several papers, which discussed the code structure to provide full diversity gain and high data rate.In [1], W. Su et al. proposed the design of full diversity space frequency block code (SFBC) with rate-1 for any number of transmit antennas and arbitrary power delay profiles.The rate-M t full diversity SFBC was proposed in [2] for any arbitrary number of transmit antennas.However, because a zeropadding matrix has to be used when N is not an integer multiple of M t L, the symbol transmission rate M t cannot be always guaranteed.
In [3], better diversity gains through block-fading channels can be obtained, that was done by spreading the coding across multiple fading blocks.In [4], they studied the error performance results of STF codes in MIMO-OFDM systems for a variety of system configurations and channel conditions.The maximum diversity is the product of time diversity, frequency diversity and space diversity as shown in [5].Recently in [6], W. Zhang et al. proposed a systematic design of high-rate STF codes for MIMO frequency-selective block-fading channels.By spreading the algebraic coded symbols across different OFDM sub-channels, transmit antennas and fading blocks, the proposed STF codes can achieve a rate-M t and a full diversity of M t M r M b L, where M b is the number of independent fading blocks in the code-words.To achieve the full-diversity performance of STF code, maximumlikelihood (ML) decoding must be employed.In order to decrease the large complexity of ML decoding, sphere decoder can be considered to achieve near-ML performance [7,8].For block-fading channels, the performance of STF-coded OFDM is much better than SF coding as demonstrated in [9].
In MIMO-OFDM systems, the DFT operation introduces correlation into the channel frequency response at different sub-carriers [10,11], making its performance var-ies as the delays between paths vary.
The outline of the paper is as follows.Section 2 describes the channel statistics and system model.The suggested time/frequency permutations of high rate STF codes structure proposed in [6] for independent and correlated Nakagami fading are introduced in Section 3. In Section 4, we provide simulation results for the performance of the proposed scheme.Finally, some conclusions are made in Section 5.

Channel Statistics and System Models
Before investigating permutation schemes for MIMO-OFDM systems equipped with M t transmit antennas, M r receive antennas in mobile radio channels, we briefly describe the channel statistics, emphasizing the separation property of mobile wireless channels, which is crucial for simplifying our time/frequency permutation.In this section we also briefly describe a MIMO-OFDM system.

Statistics of Mobile Radio Channels
The channels between each pair of transmit and receive antennas are assumed to have L independent delay paths and the same power delay profile.The channel impulse response between m t th transmit antenna and m r th receive antenna can be modeled as where τ l is the delay of the l th path, and is complex amplitude of the l th path between m t th transmit antenna and m r th receive antenna.'s are modeled as a complex random fading signals with Nakagarni-m distributed fading amplitudes and uniform phases.Nakagami m-distribution fading model [12] is one of the most versatile, in the sense that it has greater flexibility and accuracy in matching some experimental data than Rayleigh, log-normal, or Rician distributions.The Rayleigh distribution is a special case when the fading parameter m=1.It can approximate Rice distribution for m＞1.Moreover, it is assumed that all path gains between any pair of transmit and receive antennas follow the same power profile, i.e., The MIMO channel is assumed to be spatially correlated for any (m t , m r ), where m t =1,…M t , m r =1,…M r , and independent for any l where, l=0,…L-1.Let denotes the spatial correlation coefficient between and defined as The spatial correlation coefficient observed at the receiver has also been extensively studied in the literature and is given as Given Equations ( 3) and ( 4), the symmetrical correlation matrices at transmitter and the receiver can be defined respectively as

R
(5) and, The spatial correlation matrix R of the MIMO radio channel is the Kronecker product of the spatial correlation matrix at the transmitter and the receiver and is given by [13] where  denotes the Kronecker product.The correlation function of the frequency response for different times and frequencies is Assume Jake's Doppler power spectrum [14], therefore the correlation of the l th path is given by where represents the power of l th path, f D is the Doppler frequency, and J O (x) is the zero order Bessel function of the first kind.Substitute Equation (9) in Equation ( 8), then Equation ( 8) can be rewritten as where Φ t (Δt) is the time domain correlation function and Φ f (Δf) is the frequency domain correlation function.
From Equation ( 10), the time-frequency domain channel correlation function of H mt,mr (t,f) can be separated as the product of the spatial correlation coefficient, the time domain channel correlation, and the frequency domain channel correlation, which are dependent on the antenna separation, the Doppler frequency, and multi-path delay spread respectively.For an OFDM system with block length T and tone spacing (sub-channel spacing) Δf=1/T, the correlation function for different blocks and tones can be written as

MIMO-OFDM System Model
Consider a STF-coded MIMO-OFDM system with M t transmit antennas, M r receive antennas and N sub-carriers operating over a frequency-selective multi-path fading channel.The MIMO-OFDM system with code permutations considered in this paper is shown in Figure 1.
The source S generates N s =N M t M b information symbols from the discrete alphabet A, which are quadrature amplitude modulation (QAM) normalized into the unit power.Using a mapping f: S→C, an information symbol vector S∈A Ns is parsed into blocks and mapped onto a STF codeword to be transmitted over the M t transmit antennas and M b OFDM blocks.Each STF codeword C can be expressed as a N ×M b M t matrix.
where the N×M t matrix ensures that the average SNR at each receive antenna is independent on the number of transmit antennas.

Time/Frequency Permuted STF Codes
STF coding proposed in [6] can achieve rate of M t and full diversity for any number of transmit antennas and any arbitrary channel power delay profiles.It was constructed by applying the layering concept along with algebraic code components, which was introduced in the design of threaded algebraic space-time (TAST) code [15].The STF code structure spreads the algebraic code components in adjacent sub-carriers and adjacent time slots that suffer from high correlation introduced by DFT operation and time correlation respectively.In this section, time/frequency permuted STF code structure is introduced into STF code structure of [6] in order to remove the effect of channel correlation among the code components and achieve better diversity order.
2) Generate algebraic code sub-block q n X by apply- ing a fully-diverse unitary transformations into each information vector to generate N q threads by , and is the first principal unitary matrix of the following matrix where .
where and 4) Re-arrange the elements of 16) , and The STF coding applies the same coding strategy to ) is spread ove ti ency dim erefore, the STF code structure is not optimum in spreading the code components of each thread on adjacent sub-carriers that suffer from high correlation introduced by DFT operation.However, if the power delay profile of the channel is available at the transmitter side, further improvement can be achieved by developing an interleaving strategy (can reduce the correlation between adjacent sub-carriers) which explicitly considers the power delay profile.In addition, since the STF code structure maintains its diversity gain from sending the OFDM blocks through independent fading blocks, we shall introduce time permutation to achieve independent fading blocks through MIMO channels that suffer from high correlation introduced by Doppler power spectrum.r space, me and frequ ensions.Th

Time/Frequency Permutation Schemes
he assumption of independent fading at the branches is to the autocorrelatio (20) Obviously, the sources of channel correlation are ca T acceptable if the antennas are spaced sufficiently apart with respect to the radio frequency (RF) carrier wavelength.In this case, used by the time domain channel correlation, and the frequency domain channel correlation.Our objective is to find the separation parameters k and n for MIMO-OFDM system which produce zero time and frequency correlations then permute the algebraic code components of B j (j=1,…J) at zero time frequency correlation to maximize the diversity gain.

 
The zeros of the Bessel functions (Equation ( 21)) play a dominant role in our applications.The Bessel functions have infinite number of zeros.The maxima and minima of J 0 steadily decrease in absolute value as k increases.
The first five zeros of J 0 are 2.4048, 5.5201, 8.6537, 11.7915, and 14.9309.The interval between the last two is 3.1394, which is already close to π.The larger roots are approximately     22)) can be easily found via low-complexity computer search.However, closed-form solutions for specific cases are reported in [1].
Based on the knowledge of ch of B j' .her perm 3) Furt utation should be done to break the rest of channel frequency correlation by permuting each pair of rows ) , ( . By performing the ab gure 2, th ove steps as shown in Fi e code components ) ) of each t pendent fading blocks which subsequently achieve maximum diversity gain.
Examples of STF codes and pe hread of code m are affected by inde rmuted STF codes for M d tim

Simulation Results
In this section, we simulated the proposed permutation atrix B j t =2, L=2 are shown Figures 3 and 4. For M b =1, STF codes will be, in fact, the SF codes of [16].The rate-2 SF code structure and the suggested time/frequency permutation (antenna 1 is shown only) are shown in Figure 3.
The rate-2 STF code structure and the suggeste e/frequency permutation for M b =2 are shown in Figure 4.   cheme and compared with the non-permuted STF codes s for different power delay profiles of the channel.We present average symbol-error rate (SER) curves as functions of the average SNR.Then we illustrate the performance of the proposed permutation for SF codes through correlated Nakagami fading channels.To investigate the performance of the proposed time/frequency permutation of STF codes over frequency-selective fading channels, we perform the simulation experiments and compare with the STF codes [6] for MIMO-OFDM systems.In the simulation, we use a 2×2 system with 128 OFDM tones and 4QAM transmission scheme, thus the spectral efficiency is 4 bit/s/Hz, ignoring the cyclic prefix.The bandwidth of OFDM system is 1 MHz and the length of the cyclic prefix is 32, i.It is to be noted that m = 0.5 represents the worst fa It can be observed from these figures that the SER performance of STF codes [6] varied as the delay spread of the channel changed.The SER performance of STF codes is further improved as delay spread of the channel increased.Such an improvement is attributed to the large coding gain induced by multi-path fading channels with a larger delay spread.The performance of the STF code degraded significantly from the 20μs case to the 8μs case, whereas the performance of the STF code using time/ frequency permutation was almost the same for the two delay profiles.g situation that can be represented by Nakagami distribution.This case can be countered in bad urban mobile radio.When m=1, we obtain Rayleigh fading channel.Finally, m=2 represents the best considered situation in which the fading is less than that of Rayleigh.

Delay Spreads
We can see that the T/FP-STF codes have better SER performance than the non-permuted STF codes.For τ=8μs case, there is an improvement of about 3.2 dB for SF codes and an improvement of about 1.8 dB for the STF codes at a SER of 10 -4 when m=1. Therefore; the proposed interleaving method offering higher code gains making it more robust to small delay spread.This confirms that by careful interleaver design, the performance of the STF codes can be significantly improved.
T performance of the proposed scheme with STF codes for different path delay of the two-ray model.A simple two-ray, equal-power delay profile, with a delay τ microseconds between the two rays is assumed.Simulation is carried out for two cases: 1) 8μ sec (optimum permutation N c =8) and 2) 20μ sec (optimum permutation N c =16).For Doppler spread f D =200Hz the optimum time separation is 14 OFDM symbols to ensure independent fading blocks, therefore the interleaved STF code is spanned over 56 OFDM symbols. Figures

5, 6 and 7 de
From Table 1, it is clear that the SNR decreases with the increase of m.The performance of the interleaved codes is not sensitive to the variation in the channel time delay spread.In all of cases considered, the required SNR of the time/frequency interleaved codes is lower than that needed for the un-interleaved one to achieve the same SER.MIMO system with closely spaced antenna elements is considered here.Our aim is to analyze the influence of the Nakagami-m fading parameter and the effect of antenna correlation on the SER performance of the rate-2 SF code, and the proposed T/FP-SF code depicted in Figure 3.
Figure 8 shows the SER degradation as the correlation coefficients between the transmitting antenna branches ρ vary from 0 up to 0.8.Similar correlation is assumed between receiving antenna branches.Simulation is carried out for two cases: From these figures, it is clear that the systems under consideration appreciably dominate the systems considered in [6].In this paper, the limitation for achieving full-diversity of STF-coded OFDM is introduced.The limitation arises due to the fact that the algebraic code components are spread in adjacent sub-carriers that suffer from high correlation introduced by DFT operation.Assuming that the power delay profile of the channel is available at the transmitter, we proposed an efficient time-frequency interleaving scheme to further improve the performance.Based on simulation results, we can draw the following conclusions.

Conclusions
First, the proposed time/frequency permutations STF codes offer considerable performance improvement over previously reported results.Second, the applied interleaving scheme can have a significant effect on the overall performance of the STF code through correlated and independent Nakagami fading channels.

cZ
Figure 1.MIMO-OFDM system with code permutation to combat channel correlation.
of N s transmitted information symbols S=[S 1 ,S 2 ,...S NMtMb ] T are parsed into J(J=N/K) equal size sub-blocks.Each sub-block S j ∈AK MtMb (j=1,2…, J) is respectively encoded into an STF code matrix Bj of size K×M t M b through the following steps:

3 )
Applying the layering concept to construct the encoder sub-matrices

2 .
v is the number of the root.To break correlation of the channel, verify independent fading block and realize high-rate full-diversity STC of [6], the M b -OFDM blocks of STC matrix B j (j=1,…J) should be transmitted at time differrequired to break the memory of the channel.The optimum sub-carriers separation factor N c (see For large coherence time or equivalently low Doppler spread of the fading, high inter-Eq annel separations factors N independent fading bl 2) Apply fr uency permutation into each pair of code m uation (

c
and K c , time/frequency permuted STF code can be introduced using the following steps: 1) Distribute the STC blocks over ocks by permuting the u -OFDM blocks of STC ma- trix B j (j=1,…J) with hose blocks at time uK c , and B j' , where b

Figure 2 .
Figure 2. The suggested time/frequency permutation of STF codes.

. 1 .
e., 32μs.Hence the duration of one OFDM symbol (cyclic prefix excluded) is T=128μs.A two-ray Nakagami fading channel statis-din Performance Comparison for Different he first set of experiments is conducted to compare the pict the improvement in SER pe tics model is considered with the equal gain, Doppler spread f D =200Hz, and fading depth m = 0.5, 1 and 2.
rformance offered by the proposed time/frequency permutations through independent Nakagami fading channel with different m.The values of the fading depth considered are m = 0.5, 1, and 2 respectively.
1) Transmitter correlated Nakagami MIMO fading channel case: , and , and 2) Doubly correlated Nakagami MIMO fading channel case: , and .The values of the fading depth considered are m=0.5, 1, and 2 respectively.It is clear that the SER increases with the increase of correlation coefficient ρ.At ρ=0, the received signals are independent and the codes practically achieves full diversity reception gain.It is clear that the probability of error decreases with the increase of m, which is with the decrease of the severity of fading.