A Study of Channel Estimation for OFDM Systems and System Capacity for MIMO - OFDM Systems

an random vectors Xc and Xs, given by 152 1 2[ ] T c c cX X X= , 1 2[ ]Ts s sX X X= (6-19) with zero mean and covariance matrices Kcc, Kss and cross-covariance matix Kcs. That is, 2 1 1 2 2 1 2 2 cc cc cc K σ ρ σ σ ρ σ σ σ ⎡ ⎤= ⎢ ⎥⎣ ⎦ 2 1 1 2 2 1 2 2 ss ss ss K σ ρ σ σ ρ σ σ σ ⎡ ⎤= ⎢ ⎥⎣ ⎦ 1 2 1 2 0 0 cs cs sc K ρ σ σ ρ σ σ ⎡ ⎤= ⎢ ⎥⎣ ⎦ (6-20) meanwhile satisfying ,cc ss cs scρ ρ ρ ρ= = − (6-21) , where ( )T⋅ denotes conjugate, ρcc, ρss, ρcs and ρsc are the correlation coefficients of (Xc1, Xc2), (Xs1, Xs2), (Xc1, Xs2), (Xs1, Xc2) , respectively. And 21σ is the variance of Xc1, 22σ is the variance of Xc2, we have that the joint PDF of the bivariate r1 and r2, is given by 1 2 2 2 , 02 2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 1( , ) exp 4 (1 ) 2(1 ) 1 cc cs r r cc cs cc cs cc cs xyx yf x y I ρ ρ σ σ ρ ρ ρ ρ σ σ ρ ρ σ σ ⎡ ⎤⎡ ⎤ +⎛ ⎞− ⎢ ⎥= + ⋅⎢ ⎥⎜ ⎟− − − − − −⎢ ⎥⎝ ⎠⎣ ⎦ ⎣ ⎦ , for x, y ≥ 0. (6-22) where 2 21 1 1c sr X X= + , 2 22 2 2c sr X X= + , 0 ( )I ⋅ is the modified zero order Bessel function of the first kind. Let Xc1 = HRe(i, k1), Xc2 = HRe(i, k2), Xs1 = HIm(i, k1), and Xs2 = HIm(i, k2), we have 2 2 1 2 0.5σ σ= = , 1 2 1 2 0 2 ( )cos L l cc ss l l k k N π τρ ρ σ− = −⎡ ⎤= = ⎢ ⎥⎣ ⎦∑ , and 1 2 1 2 0 2 ( )sin L l cs sc l l k k N π τρ ρ σ− = −⎡ ⎤= − = ⎢ ⎥⎣ ⎦∑ . Since the two 2×1 Gaussian random vectors, [HRe(i, k1) HRe(i, k2)]T and [HIm(i, k1) HIm(i, k2)]T, satisfy the conditions (6-20) and (6-21), the joint 153 PDF of |H(i, k1)|2 and |H(i, k2)|2 is given by 2 2 1 2 0| ( , )| ,| ( , )| 21( , ) exp , for , 0. 1 1 1H i k H i k x yf x y I xy x yγγ γ γ ⎡ ⎤⎡ ⎤+= − ⋅ ≥⎢ ⎥⎢ ⎥− − −⎢ ⎥⎣ ⎦ ⎣ ⎦ (6-23) where 1 1 2 2 1 2 1 2 1 2 1 0 2 0 2 ( )cos ( ) L L l l l l l l k k N πγ σ σ τ τ− − = = −⎡ ⎤= −⎢ ⎥⎣ ⎦∑∑ . (6-24) Therefore, the correlation between the capacity of subcarrier k1, 1k C and the capacity of subcarrier k2, 2k C , is given by ( ) 2 21 2 1 2, , 2 2 | ( , )| ,| ( , )|0 0 log (1 ) log (1 ) ( , )i k i k H i k H i kE C C SNR x SNR y f x y dxdy+∞ +∞= + ⋅ ⋅ + ⋅ ⋅∫ ∫ . (6-25) where SNR is the system signal to noise ratio, defined by (6-4). 2 2 1 2| ( , )| ,| ( , )| ( , ) H i k H i k f x y is the joint PDF of |H(i, k1)|2 and |H(i, k2)|2, expressed by (6-23). Therefore, the variance of OFDM system capacity 2Cσ is finally obtained by substituting above formulas (6-14), (6-18) and (6-25) into (6-16). 6.3.2 OFDM system capacity over Ricean fading channels This subsection provides the mean value and variance of OFDM system capacity over Ricean fading channels. First, the correlation coefficient between two arbitrary channel frequency responses is presented for further derivation of the variance of OFDM system capacity. Second, the mean and variance of OFDM system capacity are derived. 6.3.2.1 The correlation coefficient between two channel frequency responses It is assumed that the paths h(i, τl) (l = 0,1,...,L-1) are statistically independent and each path h(i, τl) is given by 154 ,( , )l l i lh i Aτ β= + (6-26) where Al is the LOS for the l-th path, ßi,l is the scattering signal following circulant complex Guassian distribution with zero mean and 2lσ ′ variance. Therefore, the envelope of each path is Ricean distributed, expressed by 2 2 02 2 2 | | 2 | |2( ) exp l ll l l l x A x Axf x Iσ σ σ ⎛ ⎞ ⎛ ⎞+= −⎜ ⎟ ⎜ ⎟′ ′ ′⎝ ⎠ ⎝ ⎠ (6-27) where 0 ( )I ⋅ is the 0-th modified Bessel function of the first kind. The channel is normalized so that 1 2 2 0 1 L h l l σ σ− = = =∑ , where 2 2 2| |l l lAσ σ ′= + is the total power of the l-th path, composed of the LOS signal and Non-LOS signal power. The Ricean K-factor for the l-th path is 2 2| | /l l lK A σ ′= and the Ricean K-factor for the multipath Ricean fading channel is defined by 1 1 2 2 0 0 | | / L L l l l l K A σ− − = = ′= ∑ ∑ (6-28) Then, the frequency response of the channel H(i, k) can be expressed by N 1 2 / 0 1 1 2 / 2 / , 0 0 ( , ) FFT ( ( , )) ( , ) ( , ) ( , ) l l l L j k N l l L L j k N j k N l i l l l H i k h i n h i e A e e a i k b i k π τ π τ π τ τ β − − = − −− − = = = = = + = + ∑ ∑ ∑ (6-29) where a(i, k) is the LOS term of H(i, k) and b(i, k) is the Non-LOS term of H(i, k). And it is easy to verify that the variance of b(i, k) is 1 2 0 L l l σ− = ′∑ and 2 21 | ( , ) | | |l k l a i k A N =∑ ∑ . Thus, the correlation coefficient between two arbitrary frequency responses, γk 1,k 2, is derived as 155 ( ) ( )( )1 2 1 2 1 2 1* 2 , 1 2 2 2 0 1 2 / 2 / 2 , , 0 1 1 2 ( ) /2 2 0 0 ( , ) ( , ) ( , ) ( , ) / / / l l l L k k l l L j k N j k N i l i l l l l l L L j k k N l l l l E H i k E H i k H i k E H i k E e e e π τ π τ π τ γ σ β β σ σ σ − = −− = − −− − = = ′= − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎛ ⎞ ′= ⋅⎜ ⎟⎝ ⎠ ′ ′= ∑ ∑ ∑ ∑ ∑ ∑ (6-30) 6.3.2.2 The mean value of OFDM system capacity The capacity of the k-th subcarrier within an OFDM symbol is also given by equation (6-13). From (6-29), since the envelope of the channel frequency response |H(i, k)| follows Ricean distribution, the PDF of |H(i, k)|, | ( , )| ( )H i kf x , is given by 2 2 | ( , )| 01 1 1 2 2 2 0 0 0 2 | ( , ) | 2 | ( , ) |( ) exp , 0.H i k L L L l l l l l l x x a i k x a i kf x I x σ σ σ− − − = = = ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟+⎜ ⎟ ⎜ ⎟= − ≥⎜ ⎟ ⎜ ⎟′ ′ ′⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑ ∑ (6-31) Therefore, the mean value of OFDM system capacity over Ricean fading channels, Cμ ′ , is given by 2 2 1 2 2 | ( , )|0 0 log (1 | ( , ) | ) 1 log (1 ) ( ) C N H i k k E SNR H i k SNR x f x dx N μ − +∞ = ′ ⎡ ⎤= +⎣ ⎦ = + ⋅∑∫ (6-32) 6.3.2.3 The variance of OFDM system capacity The variance of OFDM system capacity for Ricean fading channels is also given by equation (6-16). Note that Cμ ′ , instead of Cμ , should be used in (6-16). The first term in (6-16) is further derived as ( )1 1 22 2, 2 | ( , )|2 2 0 0 0 1 1( ) log (1 ) ( ) N N i k H i k k k E C SNR x f x dx N N − − +∞ = = = + ⋅∑ ∑∫ (6-33) 156 where f|H(i, k)| is given by formula (6-31). Next, we further derive the second term of formula (6-16). Since |H(i, k1)| and |H(i, k2)| are two Ricean random variables with correlation coefficient γk 1,k 2 given by (6-30), the joint PDF of |H(i, k1)| and |H(i, k2)| is given by (Appendix C) 1 2 1 2 | ( , )|,| ( , )| 1 2 21 2 2 1,2 0 2 2 1 2 1,2 1 2 1,2 1 2 1 2 2 1,2 0 2 2 1 2 1 2 1,2 1,2 2 1,2 2( , ) (1 ) | ( , ) | | ( , ) | 2 | ( , ) ( , ) | cos( ) exp (1 ) 2 cos( ) exp (1 ) H i k H i k L l l L l l x xf x x r a i k a i k r a i k a i k w r x x x x r w r π σ θ θ σ α − = − = = ⋅⎛ ⎞′− ⎜ ⎟⎝ ⎠ ⎡ ⎤⎢ ⎥+ − − − +⎢ ⎥− ⋅⎢ ⎥′−⎢ ⎥⎣ ⎦ + − −− ′− ∑ ∑ 1 2 1,2 1 1,2 2 1 2 2 1,2 0 1 1 2 0 2 210 2 ( ) 2 1,2 20 ( ) 1 1,2 1 2 (1 ) | | ( , ) | | ( , ) | | ( , ) | | ( , ) | | L l l j j j L j w jl l j w r x a i k e I x a i k e e d x r a i k e e x r a i k e θ π θ α θ α θ σ α σ − = − − − − + = − ⎛ ⎞⋅⎜ ⎟′⎜ ⎟−⎜ ⎟⎜ ⎟⎡ ⎤ ⎜ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⋅ +⎜ ⎟⎢ ⎥ −⎜ ⎟⎢ ⎥⎣ ⎦ ⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ∑ ∫ ∑ (6-34) where r1,2 is the envelope of γk 1,k 2, w1,2 is the angle of γk 1,k 2, θ1 is the angle of a(i, k1), θ2 is the angle of a(i, k2),. So the correlation between the capacity of subcarrier k1, 1k C and the capacity of subcarrier k2, 2k C , is expressed by ( )1 2 1 22 2, , 2 2 | ( , )|,| ( , )|0 0 log (1 ) log (1 ) ( , )i k i k H i k H i kE C C SNR x SNR y f x y dxdy+∞ +∞= + ⋅ ⋅ + ⋅ ⋅∫ ∫ (6-35) where the function 1 2| ( , )|,| ( , )| ( )H i k H i kf ⋅ is given by equation (6-34). Then the variance of OFDM system capacity 2Cσ for Ricean fading channels is finally obtained by substituting above formulas (6-32), (6-33) and (6-35) into (6-16). 157 6.4 Numerical and Simulation Results Both numerical method and computer simulation have been employed to investigate the OFDM system capacity. For computer simulation, the number of the subcarriers of the OFDM system, N, is 512. Both Rayleigh fading channel and Ricean fading channel are constructed for simulation, respectively. For the multipath Rayleigh fading channel, Jakes model [65] is applied to construct a Rayleigh fading channel for each path. The decaying factor β in equation (6-6) is 0.15. For the multipath Ricean fading channel, the scattering signal in formula (6-26) adopts Jakes model and the decaying factor β is also 0.15. The line of sight signal Al in equation (6-26) only exists in the first path, that is, Al = 0, for l = 1, 2,…,L-1. In addition, the bandwidth of OFDM system is 20 MHz. The Doppler shift is 100 Hz. A. OFDM system capacity over the Rayleigh fading channel Fig 6.1 depicts The PDF of the capacity at a subcarrier, , ( ) i kC f x , for SNR = 0 dB, 5 dB, 10 dB, and 20 dB, respectively. The PDF of Ci,k can be derived from (6-13) and it is given by , ln 2 2 1 2( ) exp i k x x Cf x SNR SNR ⎛ ⎞⋅ −= ⎜ ⎟⎝ ⎠ . (6-36) Observe that the PDF for SNR = 0 dB decreases monotonically and the PDFs for SNR = 10 dB and SNR = 20 dB have their maximum values of 0.282 when x = 3.3 and 0.257 when x = 6.6, respectively. The values of PDFs at x = 0 are nonzero and the PDF value for 158 x = 0 is decreased when SNR is increased. Fig 6.2 depicts the joint PDF of of |H(i, k1)|2 and |H(i, k2)|2, 2 2 1 2| ( , )| ,| ( , )| ( , ) H i k H i k f x y ,for the coefficient of equation (6-23), γ = 0.61. Observe that the joint PDF decrease rapidly from 0.3 to nearly zero when increasing x and y from 0 to 3. Fig 6.3 depicts the coefficient of equation (6-24), γ, versus different subcarrier gap between k1 and k2. The CP length is 64, the number of paths is 8, and the delay of each path is uniformly distributed over the CP length. Observe that the coefficient γ varies periodically from 0 to 0.28 and the period is 64. Since the coefficient γ is composed of a few cosine waves and the minimum frequency among the cosine waves is ( ) , 0,1,..., 1 min / 8 / 512i ji j L Nτ τ= − − = , the frequency of γ is equal to the minimum frequency among the cosine waves. That is, the period of γ is 512/8 = 64, in unit of sample point. Fig 6.4 depicts the variance of OFDM system capacity for the number of channel paths L = 2, 4, and 8, by computer simulation and numerical method. The CP length is 64 and the delay of each path is uniformly distributed over the CP length. The variance of OFDM system capacity by numerical method is calculated by equation (6-16). Observe that the numerical results marginally overlap with the computer simulation and hence are well verified by that of computer simulation. The variance of OFDM system capacity increases with the increase of SNR linearly from 0 dB to 15 dB at a fixed number of channel paths. However, the capacity variance only marginally increases from 20 dB to 25 dB, especially when L is large. Observe that the variance of OFDM system capacity decreases when increasing the number of channel paths L, at a fixed SNR. 159 Fig 6.5 depicts the variance of OFDM system capacity versus the CP of an OFDM symbol in unit of sample point. The number of resolvable paths L is 4. Observe that the variance for SNR = 10 dB increases from 0.54 to 1.69 when increasing the CP from 16 to 128. However, further increase of CP only marginally increases the variance of OFDM system capacity. Similar results are found for the other two cases of SNR = 15 dB and SNR = 20 dB, respectively. Fig 6.6 shows the variance of OFDM system capacity versus the number of subcarriers of one OFDM symbol. The variance is evaluated by computer simulation. The CP length is 64 and the number of resolvable paths for the channel, L = 4. The delay of each path is uniformly distributed over the CP length. Observe that the variance of OFDM system capacity does not vary significantly for the number of subcarriers of one OFDM symbol N = 256, 512, 1024, and 2048. B. OFDM system capacity over the Ricean fading channel Fig 6.7 depicts the variance of OFDM system capacity over the Ricean fading channel, for the number of resolvable paths L = 2, 4, 8, respectively. The CP length is 64 and the Ricean factor K in (6-28) is 0 dB. The variance of OFDM system capacity by numerical method is calculated by equation (6-16). Observe that the numerical results are well verified by computer simulation results. The variance increases for the increase of SNR and decreases with the increase of the number of resolvable paths L. For instance, for SNR = 15 dB, the variance with L = 2 is 1.46 and the variance with L = 8 is 0.9. 160 C. The comparison of OFDM system capacity between the Rayleigh fading channel and the Ricean fading channel Fig 6.8 shows the mean value of OFDM system capacity for Rayleigh fading channel and Ricean fading channel, calculated by equation (6-14) and (6-32), respectively. The Ricean factor K defined in (6-28) is set to be 0 dB, 10 dB, and ∞, respectively. Observe that the OFDM system capacity increases nearly linearly with the increase of SNR, for both Rayleigh fading channel and Ricean fading channel. The OFDM system capacity with Ricean fading channel is larger than that of Rayleigh fading channel, especially at large Ricean factor K. The system capacity with Ricean fading channel increases with the increase of K until K goes to infinite. Fig 6.9 depicts the variance of OFDM system capacity for Rayleigh fading channel and Ricean fading channel, by numerical method and computer simulation. The numerical results are calculated by equation (6-16). Observe that the numerical results are well verified by simulation results. The capacity variance with Rayleigh fading channel is larger than that of Ricean fading channel. The capacity variance with Ricean fading channel decreases with the increase of Ricean factor K. In addition, the variance with Ricean fading channel increases nearly linearly at low SNR, and increases to an asymptotic level at high SNR. Further increase of SNR only marginally increases the capacity variance. For instance, for the variance curve with Ricean factor K = 10 dB, the variance increases almost linearly from 0.078 to 0.33 when SNR increases from 0 dB to 10 dB. However, for 161 further increasing SNR the capacity variance approaches an asymptotic level of 0.40. 6.5 Conclusion The variance of OFDM system capacity under the Rayleigh fading channel and the Ricean fading channel with finite paths have been derived and thoroughly investigated. The numerical and simulation results reveal the follows. First, the variance of OFDM system capacity increase almost linearly with the increase of SNR and it decreases with the increase of the multipath number of the channel, for both Rayleigh fading channels and Ricean fading channels. Second, the capacity variance with Rayleigh fading channels increases when increasing the CP length at a fixed multipath number. However, the variance increases to a asymptotical level at a certain value of CP and further increase of CP only marginally increases the system capacity variance. Third, the capacity mean value with Ricean fading channel is larger than that of Rayleigh fading channel, especially at large Ricean factor. The capacity variance with Rayleigh fading channel is larger than that of Ricean fading channel. In addition, the joint probability density function of two arbitrary correlated Ricean random variables has been presented in an integral form. 162 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 The capacity of a subcarrier, x P ro ba bi lit y de ns ity SNR = 0dB SNR = 5 dB SNR = 10dB SNR = 20dB Fig 6.1: The PDF of the capacity at a certain subcarrier, , ( ) i kC f x in (6-36), for SNR = 0 dB, 5 dB, 10 dB, and 20 dB, respectively. 0 1 2 3 0 1 2 3 0 0.1 0.2 0.3 Fig 6.2: The joint PDF of of 21| ( , ) |H i k and 22| ( , ) |H i k , 2 2 1 2| ( , )| ,| ( , )| ( , ) H i k H i k f x y ,for the coefficient of equation (6-23), γ = 0.61. 163 0 100 200 300 400 500 600 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |k1-k2| co ef fic ie nt r Fig 6.3: The coefficient of equation (6-24), γ, versus different subcarrier gap between 1k and 2k . 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 SNR (dB) Th e va ria nc e of O FD M s ys te m c ap ac ity L = 2, simulation L = 4, simulation L = 8, simulation L = 2, numerical method L = 4, numerical method L = 8, numerical method Fig 6.4: The variance of OFDM system capacity for the number of channel paths L = 2, 4, and 8, over the Rayleigh fading channel. 164 50 100 150 200 0.5 1 1.5 2 2.5 3 The CP length of an OFDM symbol in unit of sample point Th e va ria nc e of O FD M s ys te m c ap ac ity SNR = 10 dB, simulation SNR = 15 dB, simulation SNR = 20 dB, simulation SNR = 10 dB, numerical method SNR = 15 dB, numerical method SNR = 20 dB, numerical method Fig 6.5 The variance of OFDM system capacity versus the CP of an OFDM symbol in unit of sample point, over the Rayleigh fading channel. 0 500 1000 1500 2000 2500 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 The number of subcarriers of an OFDM symbol, N Th e va ria nc e of O FD M s ys te m c ap ac ity SNR = 10 dB SNR = 15 dB SNR = 20 dB Fig 6.6: The variance of OFDM system capacity versus the number of subcarriers of one OFDM symbol, for Rayleigh fading channels. 165 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 SNR (dB) Th e va ria nc e of O FD M s ys te m c ap ac ity L = 2, Simulation L = 4, Simulation L = 8, Simulation L = 2, Numerical method L = 4, Numerical method L = 8, Numerical method Fig 6.7: The variance of OFDM system capacity over Ricean fading channels for L = 2, 4, 8, respectively. 0 5 10 15 20 25 0 1 2 3 4 5 6 7 8 9 SNR (dB) Th e m ea n va lu e of O FD M s ys te m c ap ac ity (b it/ s/ H z) Ricean channel, K = 0 dB Ricean channel, K = 10 dB Ricean channel, K = Rayleigh channel Fig 6.8: The mean value of OFDM system capacity for Rayleigh fading channel and Ricean fading channel, by numerical method. 166 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 SNR (dB) Th e va ria nc e of O FD M s ys te m c ap ac ity Ricean channel, K = 0 dB, Simulation Ricean channel, K = 10 dB, Simulation Ricean channel, K = 20 dB, Simulation Rayleigh channel, Simulation Ricean channel, K = 0 dB, Numerical method Ricean channel, K = 10 dB, Numerical method Ricean channel, K = 20 dB, Numerical method Rayleigh channel, Numerical method Fig 6.9: The variance of OFDM system capacity for Rayleigh fading channel and Ricean fading channel, by computer simulation and numerical method. 167 Chapter 7: Conclusions and future works 7.1 Conclusions The thesis addresses the problems of channel estimation and system capacity for OFDM-based systems. The major conclusions are summarized as follows. 1. A fast LMMSE channel estimation method has been proposed and thoroughly investigated for OFDM systems over slow fading channels. Our proposed method can marginally achieve the same performance with the convention method, in terms of NMSE and BER. The use of the improved MST algorithm and Kumar’s fast algorithm in the calculation of channel autocorrelation matrix has been proposed, so that the computation complexity can be reduced significantly. 2. The MSE analysis for conventional LMMSE channel estimation method has been presented. And we also provide the MSE for the proposed fast LMMSE channel estimation method, for both the matched SNR and unmatched SNR. 3. To eliminate the effect of ICI due to Doppler shift in fast fading channels, we propose a new pilot pattern and corresponding channel estimation method and data detection for OFDM systems. The proposed pilot pattern consists of two classical pilot patterns, which are the comb-type pilot pattern and the grouped pilot pattern. Computer simulation shows that the proposed channel estimation and data detection based on the proposed pilot pattern can eliminate ICI effect effectively. Compared with the 168 algorithm of [29], the proposed algorithm achieves almost the same BER performance while reducing the number of pilots significantly. 4. The MSE analysis for the MST algorithm based on comb-type pilot pattern has been presented, considering ICI effect. MSE analysis of channel estimation based on the grouped and equi-spaced pilot pattern has also been provided. 5. The closed-loop MIMO-OFDM capacity with imperfect feedback has been derived. A system capacity indicator, namely, the feedback SNR, that reflects the gain of closed-loop capacity over that of open-loop capacity, has been proposed. The lower thresholds of the feedback SNR has been provided and investigated by numerical method. Numerical results show that the lower threshold of feedback SNR is proportional to the number of antennas and also proportional to the MIMO-OFDM system SNR. 6. The variance and mean value of OFDM system capacity over Rayleigh fading channels and Ricean fading channels have been derived and thoroughly investigated. The system capacity variances over Rayleigh and Ricean fading channels have been evaluated by computer simulation and verified by numerical method. The results show that the variance of OFDM system capacity increase almost linearly with the increase of SNR and it decreases with the increase of the multipath number of the channel, for both Rayleigh fading channels and Ricean fading channels. The capacity variance with Rayleigh fading channel is larger than that of Ricean fading channel provided that the SNR, the channel delay, and the non-LOS power decaying factor are the same. 169 7. The joint probability density function of two arbitrary correlated Ricean random variables has been presented in an integral form. 7.2 Future works Two possible topics for future researches are given by the followings. 1. The proposed channel estimation methods can be further extended to MIMO-OFDM systems. Since the pilot design for MIMO-OFDM systems should satisfy that the pilots for different transmitter antennas should be orthogonal between each other, the channel estimation will be more complicated, considering the orthogonal pilot design. 2. The research on MIMO-OFDM system capacity with imperfect feedback is still not immature, since the proof for the lower thresholds of feedback SNR has not been provided. We will further consider the derivation of lower thresholds of feedback SNR so that the results in Chapter 5 could be verified. 170 APPENDIX A: The derivation of the rank of channel frequency autocorrelation matrix RHH in Chapter 3 In this appendix, we will prove that the rank of RHH is equal to L and the rank of 2wσ+HHR Ι is equal to N. We can obtain from (3-7) and (3-9) that 1 0 1 1 2 0 0 (0, )exp{ 2 / } exp{ 2 / }exp{ 2 / } N k HH n N L l l n l R n j nk N j n N j nk N λ π σ πτ π − = − − = = = − = − ∑ ∑∑ 1 2 0 0, 0, , , . L l l for k for k N for k N for k α α σ α α− = ∉⎧ ∉⎧⎪= =⎨ ⎨∈ ∈⎩⎪⎩ ∑ (A-1) where { | 0,1,..., 1}l l Lα τ= = − , lτ is the delay of the l-th path, L is the number of resolvable paths. Thus, the number of non-zero eigenvalues of HHR is equal to L. Denote the eigenvalues of the matrix 2wσ+HHR Ι by , 0,1,..., 1k k Nμ = − . We can obtain that 0 1 1 2 N 2 2 2 0 1 1 [ ] [FFT ( (0,0) (0,1) (0, 1))] [ ]. N HH w HH HH w w N w R R R N μ μ μ σ λ σ λ σ λ σ − − = + − = + + + " " " (A-2) Therefore the number of nonzero eigenvalues of the matrix 2wσ+HHR Ι is N and the rank of the matrix 2wσ+HHR Ι is N. 171 APPENDIX B: The derivation of equation (3-20) in Chapter 3 In this appendix, we will show the derivation of (3-20). Since the matrix kSNR β+ p pH H R I is circulant, the inverse matrix k 1 SNR β −⎛ ⎞+⎜ ⎟⎝ ⎠p pH HR I  can be obtained by Kumar’s fast algorithm [22]. Denote the first row of kSNR β+ p pH H R I byCand we have k[ (0, 0) (0,1) (0, 1)].p p p p p pH H H H H H pR R R NSNR β= + −C   " (B-1) Kumar’s fast algorithm can be summarized as follows. Step 1: Compute pN points FFT of the vectorCand we obtain 0 1 1[ ] FFT ( ).p pN Nd d d −= =D C" (B-2) Step 2: E can be obtained from (B-2) 0 1 1[1/ 1/ 1/ ].pNd d d −=E " (B-3) Step 3: Denote the first row of the matrix k 1 SNR β −⎛ ⎞+⎜ ⎟⎝ ⎠p pH HR I  by F and F can be given by computing pN points IFFT of the vector E pN IFFT ( ).=F E (B-4) The above three steps can be combined as p p 1 N NIFFT ( [ {FFT ( )}] )diag −= ⋅F 1 C (B-5) where 1[1 1 1] pN×=1 " and { }diag i denotes diagonalization operation. The matrix k 1 SNR β −⎛ ⎞+⎜ ⎟⎝ ⎠p pH HR I  can be acquired from the 1 by Np vector F by circle shift. Denote the 172 first row of the matrix k 1 SNR β −⎛ ⎞+⎜ ⎟⎝ ⎠p p p pH H H HR R I   by B , the first column of the matrix k 1 SNR β −⎛ ⎞+⎜ ⎟⎝ ⎠p pH HR I  by G . It follows that 1 0 ( ) ( ) (( ) mod ), 0,1,..., 1. pN p p i B j A i G i j N j N − = = − = −∑   (B-6) where ( )B i , ( )A i and ( )G i are the i-th elements of the vector B , A and G , respectively. A is the first row of the matrix p pH H R . Since H=G F and *( ) ( )pG i G N i= − , where *( )i denote conjugate, ( )Hi denotes Hermitian transpose, equation (B-6) can be equivalently written as 1 0 ( ) ( ) (( ) mod ), 0,1,..., 1. pN p p i B j A i F j i N j N − = = − = −∑  (B-7) Or equivalently, = ⊗B A F (B-8) where⊗ denotes circulant convolution, ( )F i is the i-th entry of the vector F . Using the property of DFT, (B-8) can be written as 1IFFT {FFT [ [ {FFT ( )}] }. p p pN N N diag − = ⊗ = ⋅ ⋅ B A F A 1 F   (B-9) Using equation (3-17), (B-1) and (B-5), equation (B-9) can be further written as k k k ( 1)(0) (1)IFFT . (0) (1) ( 1) p MST pMST MST N MST MST MST p p p p P NP P P P P N N SNR N SNR N SNR β β β ⎡ ⎤⎢ ⎥−⎢ ⎥= ⎢ ⎥+ + − +⎢ ⎥⎣ ⎦ B " (B-10) 173 APPENDIX C: The derivation of the joint PDF of two arbitrary correlated Ricean random variables In this appendix, we derive the joint PDF of two arbitrary correlated Ricean random variables. Consider two circulant complex Gaussian random variables X1, X2 with mean mX1, mX2 and variance σ2 1 , σ2 2 , respectively. The complex random variables X1, X2 can be further written as X1 = A1 + jB1, X2 = A2 + jB2 (C-1) The covariances between Ai and Bj are as follows. Cov (A1, B1) = Cov (A2, B2) = 0 Cov (A1, B2) = - Cov (A2, B1) = u2σ1σ2/2 Cov (A1, A2) = Cov (B1, B2) = u1σ1σ2/2 (C-2) The correlation coefficient between x1 and x2 is given by ( ) ( )*1 1 2 2 1 2 1 2 X Xj E X m X m e u juωρ σ σ ⎡ ⎤− −⎣ ⎦= = + (C-3) where (·)* denotes conjugate. Then, the joint PDF for A1, A2, B1, and B2 is expressed as [72] 174 1 2 1 2, , , 1 2 1 2 2 2 2 2 2 1 2 1 2 2 2 1 1 1 2 2 2 2 2 2 2 1 1 1 2 2 1 1 1 2 2 1 2 1 1 1 2 22 2 2 2 2 2 1 1 2 2 2 2 2 1 1 1( , , , ) (1 ) ( ) / ( ) / ( ) / 1exp 2 ( )( )(1 ) 2 ( )( ) ( ) / 2 ( )( ) 2 ( )( ) A A B B a a b a a b b b a b a b f a a b b u u a m a m b m u a m a mu u u b m b m b m u a m b m u a m b m π σ σ σ σ σ σ = ⋅− − − + − + − − ⋅ − −⎡− − + − −+ − − + − − − − − 1 2/σ σ ⎧ ⎫⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎪⎪⎨ ⎨ ⎬⎬⎤⎪ ⎪ ⎪⎪⎢ ⎥⎪ ⎪ ⎪⎪⎢ ⎥⎪ ⎪ ⎪⎪⎢ ⎥⎪ ⎪ ⎪⎪⎢ ⎥⎪ ⎪ ⎪⎪⎣ ⎦⎩ ⎭⎩ ⎭ (C-4) where ma1, ma2, mb1, and mb2 are the mean values of A1, A2, B1, and B2, respectively. Let 2 2 1 1 1R A B= + , 2 22 2 2R A B= + , ( )11 1 1tan /A B−Φ = , and ( )12 2 2tan /A B−Φ = , (C-5) we have the Jacobian determinant of (A1, A2, B1, B2) with respect to (R1, R2, Φ1, Φ2) is R1R2. Thus, the joint PDF of the phases and amplitudes, 1 2 1 2, , , 1 2 1 2 ( , , , )R Rf r r ϕ ϕΦ Φ , is given by 1 2 1 2 1 2 1 2 , , , 1 2 1 2 1 2 , , , 1 1 2 1 2 2 1 2 1 2 1 1 2 1 2 2 1 2 1 2 2 2 2 2 1 1 2 2 1 2 2 1 1 2 1 2 2 2 2 2 2 1 2 1 ( , , , ) ( ( , , , ), ( , , , ), ( , , , ), ( , , , )) 2/ / cos( ) exp (1 ) 1 exp R R A A B B X X X X f r r r r f a r r a r r b r r b r r m m m m r r r ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ρσ σ θ θ ωσ σ π ρ σ σ ρ Φ Φ = ⎡ ⎤+ − − −⎢ ⎥⎢ ⎥= ⋅ −− −⎢ ⎥⎢ ⎥⎣ ⎦ ⋅ −     2 2 2 2 1 2 2 1 2 2 1 1 2 2 2 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 2 2 1 1 2 1 2 1 2 / / 2 cos( ) / 1 / cos( ) / cos( )2exp 1 / cos( ) / cos( ) X X X X r r r m r m r r m r m σ σ ρ ϕ ϕ ω σ σ ρ σ ϕ θ σ ϕ θ ρ ρ σ σ ϕ θ ω ρ σ σ ϕ θ ω ⎡ ⎤+ − − −⎢ ⎥−⎣ ⎦ ⎡ ⎤⎛ ⎞− + −⋅ ⎢ ⎥⎜ ⎟− − − − − − + −⎢ ⎥⎝ ⎠⎣ ⎦     (C-6) where 1Xm and 2Xm are the envelopes of mX1 and mX2, θ1 and θ2 are the angles of mX1, mX2, respectively. Therefore, the joint PDF of two arbitrary correlated Ricean random variables R1, R2 is give by 175 1 2 1 2 1 2 2 2 , 1 2 , , , 1 2 1 2 1 20 0 2 2 2 2 1 2 1 1 2 2 1 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 1 2 2 1 2 2 1 1 2 2 1 2 ( , ) ( , , , ) / / 2 / cos( )exp (1 ) 1 / / 2 cos( ) /exp 1 2exp 1 R R R R X X X X X f r r f r r d r r m m m m r r r r m π π ϕ ϕ ϕ ϕ σ σ ρ σ σ θ θ ω π ρ σ σ ρ σ σ ρ ϕ ϕ ω σ σ ρ ρ Φ Φ= ⎡ ⎤+ − − −= ⋅ −⎢ ⎥− −⎣ ⎦ ⎡ ⎤+ − − −⋅ −⎢ ⎥−⎣ ⎦ ⋅ − ∫ ∫      2 21 1 1 1 2 2 2 2 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 2 2 2 2 1 2 1 1 2 2 1 2 1 2 2 1 2 2 2 2 1 2 2 2 1 1 / cos( ) / cos( ) / cos( ) / cos( ) 2 / / 2 / cos( )exp (1 ) 1 /exp X X X X X X X r m r d r m r m r r m m m m r σ ϕ θ σ ϕ θ ϕ ϕρ σ σ ϕ θ ω ρ σ σ ϕ θ ω σ σ ρ σ σ θ θ ω π ρ σ σ ρ σ ⎡ ⎤⎛ ⎞− + −⎢ ⎥⎜ ⎟− − − − − + −⎢ ⎥⎝ ⎠⎣ ⎦ ⎡ ⎤+ − − −= ⋅ −⎢ ⎥− −⎣ ⎦ ⋅ −        1 2 1 2 2 22 2 2 1 2 1 2 20 ( ) ( ) ( )2 2 0 1 1 1 2 2 2 2 2 1 2 1 1 1 22 / 2 cos( ) / 1 2 | / / / / | 1 j j j j X X X X r r r I r m e r m e r m e r m e d π θ α θ α ω θ ω θ σ ρ α ω σ σ ρ σ σ ρ σ σ ρ σ σ αρ − − − − − ⎡ ⎤+ − −⎢ ⎥−⎣ ⎦ ⎡ ⎤⋅ + − −⎢ ⎥−⎣ ⎦ ∫     (C-7) 176 Appendix D: List of Abbreviations AWGN additive white Gaussian noise BER bit error rate BLAST Bell Laboratories Layered Space Time CFO carrier frequency offset CP cyclic prefix CSI channel state information DAB digital audio broadcasting DFT Discrete Fourier Transform DOA direction of arrival DSL digital subscriber line DVB digital video broadcasting DVB-T digital video broadcasting terrestrial FFT Fast Fourier Transform GSM Global System for Mobile Communications ICI inter-carrier interference IFFT inverse Fast Fourier Transform IMT-Advanced International Mobile Telecommunication Advanced ISI inter-symbol interference ITU International Telecommunication Union 177 KL Karhunen-Loeve LMMSE linear minimum mean square error LOS line of sight signal LS least square LST Layered Space Time LSTC Layered Space Time Code LTE 3GPP Long Term Evolution MIMO multiple-input multiple-output MIMO-OFDM multiple-input multiple-output orthogonal frequency division multiplexing MISO multiple input single output ML maximum likelihood MMSE minimum mean square error MST most significant taps NLOS non line of sight NMSE normalized mean square errors OFDM Orthogonal Frequency Division Multiplexing OFDMA Orthogonal Frequency Division Multiple Access PSD power spectrum density RV random variable SIC successive interference cancellation 178 SINR signal to interference and noise ratio SNR signal to noise ratio STBC Space Time Block Code STC Space Time Code STTC Space Time Trellis Code SVD singular value decomposition ULA uniform linear array UMTS Universal Mobile Telecommunications System UWB Ultra-Wideband VA Viterbi algorithm V-BLAST Vertical Bell Laboratories Layered Space Time WLAN Wireless Local Area Network WiMAX World Interoperability for Microwave Access ZF zero forcing 3G 3rd Generations 3GPP 3rd Generation Partnership Project 179 REFERENCES [1] S. 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Lam, “Capacity of OFDM systems over time and frequency selective fading channels”, submitted to IEEE Transactions on Vehicular Technology. [4] W. Zhou, W. H. Lam, “MIMO-OFDM system Capacity with imperfect feedback channel”, submitted to IET Communications. Conference papers: [1] W. Zhou and W. H. Lam, “A novel method of Doppler shift estimation for OFDM systems”, IEEE Military Communication Conference (MILCOM 2008), pp. 1-7, San Deigo, USA, November 2008. [2] W. Zhou and W. H. Lam, “Channel Estimation and Data Detection for OFDM systems over Fast Fading Channels”, IEEE International Symposium on Personal, Indoor and Mobile Communications (PIMRC 2009), pp. 3109-3113, Tokyo, Japan, September 2009.