Semi - Blind signal detection for mimo and mimo - Ofdm systems

th is 18 ( 18L = ) 56 Fig. 3.5. The case where the channel length is 20 ( 20L = ) 3.4.3 Comparison To illustrate the impact of the channel length and the CP length on the proposed algorithm, the performance for 14,16,18,20L = cases are shown in Fig. 3.6. It is obvious that the performance is only slightly degraded when the channel length increases from 14 ( L D ). It demonstrates that the channel length and CP length have insignificant effect on the proposed algorithm. It also verifies that the proposed algorithm is applicable irrespective of whether the channel length is shorter than, equal to or longer than the CP length. As a result, using the proposed algorithm for MIMO-OFDM systems, the CP can be shortened to improve the bandwidth efficiency with slight performance degradation. 57 Fig. 3.6. Proposed algorithm for L=14, 16, 18, 20 ( 16D = ) 3.4.4 Data length effect In the proposed algorithm, second-order statistics of the received signal vector is utilized to design the equalizer. In practice, it is computed from finite number of received signal vectors (3-59) and data length may affect the performance of the algorithm. This is different from the existing algorithms in which the statistics of the received signals is not used. For comparison, the TEQ [Al-Dhahir, 2001] and the MMSE algorithms mentioned before are also implemented. Fig. 3.7 shows the performance of various algorithms under consideration when the number of blocks sN varies from 100 to 1 500. In this simulation, the SNR is fixed at 20 dB and 20L = . As expected, there is little variation in the performance of the TEQ and MMSE algorithms while the proposed algorithm perfoms better when more blocks are used. It ourperforms the TEQ and MMSE 58 algorithms when the number of blocks is roughly greater than 1230 and 1450, respectively. It indicates that the proposed algorithm can outperform the existing ones provided that a sufficient number of data blocks are sent between two consecutive transmissions of two pilot blocks. This is consistent with all SOS-based algorithms. In fact, the proposed algorithm is a semi-blind algorithm as sN received data blocks are utilized to compute SOS for the design of the equalizer. Fig. 3.7. Data length effect on the proposed algorithm ( 20L = and 20SNR dB= ) 3.5 Summary A time domain semi-blind signal detection algorithm for MIMO-OFDM systems with short CP has been proposed in this chapter. A new system model has been introduced in which the i th received OFDM symbol is left shifted by J samples. Based on some 59 structural properties of the new system model, an equalizer has been designed to cancel most of the ISI using the SOS of the received signals before signal detection. It has been shown that the channel length information is not needed and only 2P columns of the channel matrix need to be estimated with a minimum of 4P pilots for identifiability. In addition, it has been demonstrated that the proposed algorithm is applicable to general MIMO-OFDM systems irrespective of whether the CP length is longer than, equal to or shorter than the channel length. Simulation results have shown that the proposed algorithm outperforms the existing ones in all cases. 60 Chapter 4 Two-Step Semi-Blind Signal Detection for MIMO-OFDM Systems without Cyclic Prefix 4.1 Introduction As pointed out in Chapter 1 and Chapter 3, in conventional MIMO-OFDM systems operating over frequency selective fading channels, signals can be easily detected using a set of parallel one-tap linear equalizers. A cyclic prefix (CP) is generally inserted at the beginning of each OFDM symbol [van Zelst and Schenk, 2004]. The length of the CP is chosen longer than the channel length to eliminate the inter-carrier interference (ICI) and inter-symbol interference (ISI). For example, in the wireless local area network (IEEE 802.11a) standard, the length of CP is 25% of an OFDM symbol duration, resulting in a significant loss in bandwidth efficiency. It is apparent that if the CP is shortened or removed, substantial gain in bandwidth efficiency can be achieved. A MIMO-OFDM system with short CP and without CP (MIMO-OFDM- WCP) is therefore desirable. In the previous chapter, semi-blind signal detection for MIMO-OFDM systems with short CP has been discussed. In this chapter, the focus is on the semi-blind signal detection for MIMO-OFDM systems without CP (MIMO- OFDM-WCP). 61 As the CP is removed, ICI and ISI are introduced into the received signals. Their presence destroys the orthogonal property of the subcarriers, making signal detection very difficult, if not impossible. So far there are only a few published works on this type of systems. In [Yue and Fan, 2004], a linear smoothing signal detection algorithm is proposed based on second-order statistics (SOS) of the received signals. It requires precise estimation of the channel length, which is difficult to achieve in practice. The algorithms in [Wang and Zhou, 2004; Huang and Bi, 2003] apply decision-feedback equalization (DFE) method in which successive decision of the previous OFDM symbol is utilized. These algorithms do have error propagation problem which will limit their performance. In [Toeltsch and Molisch, 2000], a two- step method (ISI cancellation and then ICI cancellation) based on successive decision of the previous OFDM symbol is proposed. This method is rather complex and also has the error propagation problem. Note that all the above algorithms are only applicable in single transmit antenna systems, and their extension to multiple transmit antenna systems is by no means straightforward. In this chapter, semi-blind signal detection for MIMO-OFDM-WCP systems is considered and a two-step algorithm is proposed. By modeling the system using the shifting method which is introduced for MIMO-OFDM systems with short CP (MIMO-OFDM-SCP) in the previous chapter, some new special structural properties are derived. With these properties, it turns out that certain second-order statistical matrices of the shifted received OFDM symbols are similar to that of single carrier MIMO systems. It follows that the blind SOS-based zero-forcing equalization method proposed for single carrier MIMO system [Zhu, Ding and Cao, 1999] can be utilized here to cancel all the ICI and ISI in the first step. Then the signals are detected in the presence of multi-antenna interference (MAI) with the aid of only one pilot OFDM symbol, given that the number of transmit antennas is smaller than the number of subcarriers in the pilot OFDM symbol. In the proposed algorithm, the number of pilot OFDM symbols is less than that required in the conventional signal detection algorithm [van Zelst and Schenk, 2004] for MIMO-OFDM systems with long CP 62 (MIMO-OFDM-LCP), in which the minimum number of pilot OFDM symbols required is equal to the number of transmit antennas. In addition, precise channel length estimation is unnecessary and higher bandwidth efficiency is achieved as the CP is removed. Simulation results indicate that the proposed algorithm achieves comparable performance to that of the conventional signal detection algorithm [van Zelst and Schenk, 2004] for MIMO-OFDM-LCP systems, and it is robust against channel length overestimation. The rest of the chapter is organized as follows. In Section 4.2, the MIMO-OFDM- WCP system model is introduced. The two-step semi-blind signal detection algorithm is presented in Section 4.3 and in Section 4.4, the performance of the proposed algorithm is demonstrated by simulation. Finally, a summary of this chapter is given in Section 4.5. 4.2 System Model Consider a MIMO-OFDM-WCP system with P transmit antennas and M receive antennas, which is illustrated in Fig. 4.1. RX M RX 1 TX P TX 2 TX 1 Data stream Spatial Demux IDFT IDFT IDFT Receiver Frequency selective fading channels Fig. 4.1 Block diagram of MIMO-OFDM-WCP system 63 Since spatial multiplexing of MIMO is considered in this thesis, the data is demultiplexed into P parallel independent bit streams. Each bit stream is grouped into blocks, transformed into OFDM symbols by IDFT (Inverse Discrete Fourier Transform) and transmitted through one transmit antenna. Denote the i th block signal from the p th transmit antenna before IDFT as , , , ,[0] [1] [ 1] T i p i p i p i ps s s N⎡ ⎤= −⎣ ⎦s G G G G" , {1,2, , }p P∈ " , (4-1) where superscript T stands for the transposition operator and N is the number of subcarriers in one OFDM symbol. Without loss of generality, , [ ]i ps n G , {0,1, , 1}n N∈ −" , are assumed to be statistically independent and white with zero mean and unit variance. After IDFT, the transmitted OFDM symbol is generated as , ,i p N i p=s F sG , {1,2, , }p P∈ " , (4-2) in which , , , ,[0] [1] [ 1] T i p i p i p i ps s s N⎡ ⎤= −⎣ ⎦s " , (4-3) and NF is the N N× IDFT matrix with the ( 1, 1)n k+ + th entry being 2 / /j nk Ne Nπ , , {0,1, , 1}n k N∈ −" . It is obvious that the DFT matrix is *NF and *N N N=F F I where superscript * stands for conjugate transpose and aI denotes the a a× identity matrix. In conventional MIMO-OFDM systems, a cyclic prefix is generally added at the beginning of each OFDM symbol to eliminate ICI and ISI. Here, no cyclic prefix is added. The OFDM symbols from all transmit antennas are simultaneously transmitted into frequency selective fading channels. Denote the frequency selective fading channel between the p th transmit antenna and the m th receive antenna be ( )pmh l , which is generally modeled as a pmL th-order FIR filter. Assume the system is frequency and time synchronized with the aid of pilot symbols transmitted at the beginning of the data packet. The i th received OFDM symbol at the m th receive antenna is therefore written as , , , 1 0 [ ] ( ) [ ] [ ] P L i m pm i p i m p l y n h l s n l w n = = = − +∑∑ , 0,1, , 1n N= −" , 1,2, ,m M= " , (4-4) 64 where L represents the maximum channel length, 1 ,1max ( )p P m M pmL L≤ ≤ ≤ ≤= ; ( )pmh l is zero-padded for pmL l L< ≤ ; , [ ]i mw n is independently and identically distributed (i.i.d.) white Gaussian noise at the m th receive antenna and is uncorrelated with the transmitted signals; , 1,[ ] [ ]i p i ps n s n N−= + for 0N n− ≤ < , and , 1,[ ] [ ]i p i ps n s n N+= − for N n≤ . Here, the maximum channel length L is generally assumed to be less than the number of subcarriers N . This assumption is consistent with the conventional MIMO-OFDM-LCP systems where the maximum channel length is assumed to be shorter than the CP length. A practical example is IEEE 802.11a WLAN standard where the CP length is equal to 25% of N . By defining ,1 ,2 ,[ ] [ ] [ ] [ ] T i i i i Mn y n y n y n= ⎡ ⎤⎣ ⎦y " , (4-5) 1 2( ) ( ) ( ) ( ) T p p p pMl h l h l h l⎡ ⎤= ⎣ ⎦h " , (4-6) ,1 ,2 ,[ ] [ ] [ ] [ ] T i i i i Mn w n w n w n= ⎡ ⎤⎣ ⎦w " , (4-7) (3-6) can be expressed in vector form as , 1 0 [ ] ( ) [ ] [ ] P L i p i p i p l n l s n l n = = = − +∑∑y h w , 0,1, , 1n N= −" . (4-8) At the receiver, signal detection is traditionally performed based on the i th received OFDM symbol with N -sample signals as (0) [0] [1] [ 1] TT T T i i i i N⎡ ⎤= −⎣ ⎦y y y y" . (4-9) Similar to the previous chapter, here the i th received OFDM symbol shifted by k samples is collected and modeled as ( ) ( ) ( )k k k i i i= +y Hx w , 0, 1, 2,k = ± ± " , (4-10) where ( ) [ ] [ 1] [ 1 ] Tk T T T i i i ik k N k⎡ ⎤= − − + − −⎣ ⎦y y y y" , (4-11) ( ) ( ) ( ) ( ) ,1 ,2 ,( ) ( ) ( ) Tk k T k T k T i i i i P⎡ ⎤= ⎣ ⎦x x x x" , (4-12) ( ) , , , ,[ ] [0] [ 1 ] Tk i p i p i p i ps L k s s N k⎡ ⎤= − − − −⎣ ⎦x " " , {1,2, , }p P∈ " , (4-13) 65 [ ]1 2 P=H H H H" , (4-14) ( ) ( 1) (0) ( ) ( 1) (0) ( ) ( 1) (0) p p p p p p p p p p L L L L L L −⎡ ⎤⎢ ⎥−⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦ h h h 0 0 0 h h h 0 H 0 0 h h h " " " % # % % % % % # " " , {1,2, , }p P∈ " , (4-15) ( ) [ ] [ 1] [ 1 ] Tk T T T i i i ik k N k⎡ ⎤= − − + − −⎣ ⎦w w w w" . (4-16) In (4-11), when 0N n− ≤ < , [ ]i ny represents the signal in the ( 1)i − th received OFDM symbol and is equal to 1[ ]i N n− +y . Similarly, when N n≤ , [ ]i ny represents the signal in the ( 1)i + th received OFDM symbol and is equal to 1[ ]i n N+ −y . Note that the received OFDM symbol in the previous chapter contains short CP while the system under consideration does not, hence their structural properties are different. The ( )MN N L P× + matrix H in (3-18) is the so-called channel convolution matrix, and is generally assumed to have full column rank after removing all-zero columns. This is a reasonable assumption as the number of receive antennas, M , can be chosen to satisfy ( ) /M L N P N≥ + , such that the matrix H has more rows than columns. Therefore, in practical channels, it is most likely to be of full column rank (otherwise this can still be achieved by simple artificial loading of the matrix) and has the property [Golub and Loan, 1996] * * # ( )( ) N L P+=H HH H A , (4-17) in which ( )N L P+A is an ( ) ( )N L P N L P+ × + identity matrix with zero rows corresponding to the all-zero columns of H and superscript # stands for pseudo- inverse. In the shifted received signal model (4-10), the transmitted signal vector ( ), k i px (3-15) contains the ( 1)i − th OFDM symbol signals ( , ,[ ] [ 1]i p i ps L k s⎡ ⎤− − −⎣ ⎦" ) and the i th OFDM symbol signals ( , ,[0] [ 1 ]i p i ps s N k⎡ ⎤− −⎣ ⎦" ) when 0 k N L≤ ≤ − . It can be rewritten in terms of the signals before IDFT ,i ps G as ( ) ( ) , 2 , k k i p N i p=x F cG , 0 k N L≤ ≤ − , {1,2, , }p P∈ " , (4-18) 66 where , ( 1), , TT T i p i p i p−⎡ ⎤= ⎣ ⎦c s s G G G , (4-19) and ( ) 2 ( 1: ) (1: ) Nk N N N L k N N k − − +⎡ ⎤= ⎢ ⎥−⎣ ⎦ F 0 F 0 F . (4-20) In (3-22), ( : )N a bF denotes a submatrix composed of the a th to the b th row of NF . Since each column (row) of the IDFT matrix NF is orthogonal to all other columns (rows), it is apparent that the matrix ( )2 k NF in (3-22) satisfies (0) (0)* 0 2 2N N N L+= =F F J I , (4-21) ( ) (0)* 2 2 k k N N =F F J , 0 k N L≤ ≤ − , (4-22) (0) ( )* 2 2 k k N N −=F F J , 0 k N L≤ ≤ − , (4-23) where kJ denotes a ( ) ( )N L N L+ × + matrix with zero entries except along the lower k th subdiagonal, in which the entries are one, and k−J is equal to *( )kJ . Here the structural properties of (4-18) and (4-21)-(4-23) are essential for the signal detection algorithm to be derived in the following section. 4.3 Two-Step Semi-blind Signal Detection From the shifted received signal model (4-10), it is obvious that ( )kiy includes ( )N L+ path signals from each transmit antenna. ISI, ICI and MAI simultaneously exist in the shifted i th received OFDM symbol ( )kiy . In order to detect the transmitted signals from ( )kiy , a two-step semi-blind signal detection algorithm is proposed and illustrated in Fig. 4.2. 67 Received Signals RX M Blind ICI and ISI cancellation Signal detection only in the presence of MAI RX 1 RX 2 DFT DFT DFT Output Data Fig. 4.2 Block diagram of the receiver structure To simplify the algorithm derivation, zero noise is first assumed. The effect of noise on the algorithm is then examined. In the absence of noise, ( )kiy can be expressed as ( ) ( )k k i i=y Hx , 0, 1, 2,k = ± ± " . (4-24) 4.3.1 Blind ICI and ISI cancellation Now, consider the following second-order statistical matrices of the shifted i th received OFDM symbol ( )kiy , { } { }( ) (0)* ( ) (0)* *( ) K Ky i i i iK E E= =R y y Hx x H , (4-25) { } { }(0) ( )* (0) ( )* *( ) K Ky i i i iK E E− = =R y y Hx x H , (4-26) and assume 0 K N L≤ ≤ − . It follows from the signals defined in (4-12) and the property (4-18) that 68 { } { } { } ( ) * (0)* 2 ,1 ,1 2 ( ) (0)* * * ( ) * (0)* 2 , , 2 ( ) K N i i N K y i i K N i P i P N E K E E ⎡ ⎤⎢ ⎥= = ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ F c c F 0 0 R Hx x H H 0 0 H 0 0 F c c F G G % G G , (4-27) { } { } { } (0) * ( )* 2 ,1 ,1 2 (0) ( )* * * (0) * ( )* 2 , , 2 ( ) K N i i N K y i i K N i P i P N E K E E ⎡ ⎤⎢ ⎥− = = ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ F c c F 0 0 R Hx x H H 0 0 H 0 0 F c c F G G % G G . (4-28) As the signals before IDFT are assumed to be statistically independent and white with zero mean and unit variance, { }*, , 2i p i p NE =c c IG G , {1,2, , }p P∈ " . It follows that { }( ) (0)* * ( ) (0)* *2 2( ) ( )K Ky i i P N NK E= = ⊗R Hx x H H I F F H , (4-29) { }(0) ( )* * (0) ( )* *2 2( ) ( )K Ky i i P N NK E− = = ⊗R Hx x H H I F F H , (4-30) where ⊗ is the Kronecker product of matrices. Substituting (4-21)-(4-23) into (4-29) and (4-30), { }( ) (0)* * *( ) ( )K Ky i i PK E= = ⊗R Hx x H H I J H , (4-31) { }(0) ( )* * *( ) ( )K Ky i i PK E −− = = ⊗R Hx x H H I J H , (4-32) { }(0) (0)* * * *(0) ( )y i i P N LE += = ⊗ =R Hx x H H I I H HH . (4-33) It is found that the second-order statistical matrices (4-31)-(4-33) are similar to that of single carrier MIMO system [Zhu, Ding and Cao, 1999]. As a result, the SOS-based zero-forcing equalization method for single-carrier MIMO system [Zhu, Ding and Cao, 1999] can be readily applied here. An equalizer is thus constructed as 1K K K+= −G U U , (4-34) where # #( ) (0) ( ) (0)K y y y yK K= −U R R R R . (4-35) Using the property of the channel matrix H in (3-28) and applying (4-31)-(4-33), the matrix KU becomes 69 # # * * # * * # * * # ( ) (0) ( ) (0) ( ) ( ) ( ) ( ) ( ) ( ) K y y y y K K P P K K P K K − − = − = ⊗ ⊗ = ⊗ U R R R R H I J H HH H I J H HH H I J J H HH . (4-36) Substituting (4-36) into (4-34), the equalizer is equivalent to 1 ( 1) ( 1) * * #( ( )) ( ) K K K K K K K P + − − + + = − = ⊗ − G U U H I J J J J H HH . (4-37) It is observed that the matrix KJ satisfies [Zhu, Ding and Cao, 1999] ( ) ( ) ( ) N L K N L K KK K K N L K K K + − + − ×− × + − × ⎡ ⎤= ⎢ ⎥⎣ ⎦ I 0 J J 0 0 , (4-38) ( 1) ( 1) ( 1) 1 ( 1) ( 1) ( 1) 1 ( 1) 1 ( 1) 1 1 N L K N L K N L K N L K K K K K K N L K K K N L K K K K + − − × + − − + − − × + − − × − − + + × + − − × × + − − × × ⎡ ⎤⎢ ⎥− = ⎢ ⎥⎢ ⎥⎣ ⎦ 0 0 0 J J J J 0 0 0 0 0 , (4-39) where a b×0 is an a b× zero matrix. The matrix ( 1) ( 1)K K K K− − + +−J J J J in (4-39) is zero except the ( , )N L K N L K+ − + − entry which is one. From this observation, it follows that 1 * * # ( 1) 1 ( 1)( ) ( ) ( ) K K K MN N L K MN K MN N L K P MN KN L K N L K + × + − − × × + − − × = − ⎡ ⎤= + − + −⎣ ⎦ G U U 0 H 0 0 H 0 H HH" . (4-40) In (4-40), ( )p aH denotes the a th column of the matrix pH , 1,2, ,p P= " . Applying the equalizer KG into the shifted i th received OFDM symbol ( )k iy (4-24), we have ( ) ( ) ( ) , 1 * * # ( ) ( 1) 1 ( 1) ( ) ( ) ( ) ( ) k k k i K K i K K i k MN N L K MN K MN N L K P MN K iN L K N L K + × + − − × × + − − × = = − ⎡ ⎤= + − + −⎣ ⎦ o G y U U Hx 0 H 0 0 H 0 H HH Hx" , 0, 1, 2,k = ± ± " . (4-41) Using (4-12), (3-15) and (3-28), the equalizer output ( ), k i Ko becomes ( ) ( ) ( ) , , k K k i K part i part=o H x , 0, 1, 2,k = ± ± " , (4-42) where [ ]( ) 1 2( ) ( ) ( )Kpart PN L K N L K N L K= + − + − + −H H H H" . (4-43) ( ) , ,1 ,2 ,[ 1 ] [ 1 ] [ 1 ] Tk i part i i i Ps N k K s N k K s N k K= − − − − − − − − −⎡ ⎤⎣ ⎦x " . (4-44) 70 It is apparent that only one path signal from each transmit antenna is retained in the equalizer output. It means that both ICI and ISI are cancelled by the equalizer. 4.3.2 Signal detection in the presence of MAI Although the equalizer has cancelled ICI and ISI, MAI still exists in the equalizer output ( ), k i Ko . To proceed with signal detection, we notice that knowledge of the matrix ( )K partH in (4-43) is required. For better performance, pilots will be utilized to estimate ( )K partH . As ( )KpartH is an MN P× matrix, its estimation requires at least P sets of equation (4-42). Choosing the P sets of equation (4-42) and writing them in matrix form as ( )K pilot part pilot=O H X , (4-45) where (0) (1) ( ) , , , P pilot i K i K i K⎡ ⎤= ⎣ ⎦O o o o" , (0) (1) ( ), , ,Ppilot i part i part i part⎡ ⎤= ⎣ ⎦X x x x" , (4-46) an estimation of ( )KpartH can be easily obtained based on the least-squares criteria as ( ) * * #ˆ ( )Kpart pilot pilot pilot pilot=H O X X X . (4-47) In (4-46), the P P× matrix pilotX is the so-called pilot matrix and includes P samples from each transmit antenna. In order to obtain a unique estimation of ( )KpartH , it must be selected to have full rank. With knowledge of the matrix ( )KpartH , a number of algorithms [van Zelst and Schenk, 2004; Li and Cao, 2005; Alias, Samingan, Chen and Hanzo, 2003; Thoen, Deneire, Van der Perre, Engels and De Man, 2003; Giangaspero, Agarossi, Paltenghi, Okamura, Okada and Komaki, 2002; Yan, Sun and Lei, 2004; Park and Kang, 2004; Letaief, Choi, Ahn and Chen, 2003] can be applied to detect the transmitted signal ( ) , k i partx as the equalizer output (4-42) only contains MAI. Here, the least-squares detection method is selected for its simplicity and the transmitted signal is detected as ( ) ( ) , , ˆ k k i part LS i K=x G o , 0, 1, 2,k = ± ± " . (4-48) 71 where LSG is an one-tap linear equalizer given by *( ) * ( ) 1 ( )( )K K KLS part part part −=G H H H . It is obvious that , [ 1 ]i ps N k K− − − , {1,2, , }p P∈ " , is the signal after IDFT (see (4-2) and (4-44)). The signals , [ ]i ps n G can be recovered by performing DFT to the signals , [ ]i ps n , {0,1, , 1}n N∈ −" , which can be obtained from (4-48) by setting the parameter 1, 2, ,k N K N K K= − − − − −" . Apparently, only P pilot samples from each transmit antenna are utilized in the pilot matrix pilotX (4-46) for signal detection. When P is less than the number of subcarriers N in one OFDM symbol (which is usually the real situation), one pilot OFDM symbol is sufficient. The number of pilot OFDM symbols required here is less than that required in the conventional signal detection algorithm [van Zelst and Schenk, 2004] for MIMO-OFDM-LCP system, which needs at least P pilot OFDM symbols and requires CP to eliminate ICI and ISI. The pilot OFDM symbol for channel-state estimation can be inserted as part of the header in a packet for slow fading channels or can be inserted at regular intervals within a packet for fast fading channels. 4.3.3 Effect of channel noise Up to now, the algorithm is derived under the zero-noise assumption. When the additive white channel noise with variance 2σ is presented, the second-order statistical matrices of the shifted i th received OFDM symbol becomes { } { } { } { } ( ) (0)* ( ) (0)* * ( ) (0)* ( ) (0)* * 2 ( ) ( ) K K K y i i i i i i K K i i M K E E E E σ = = + = + ⊗ R y y Hx x H w w Hx x H I J , (4-49) { } { } { } { } (0) ( )* (0) ( )* * (0) ( )* (0) ( )* * 2 ( ) ( ) K K K y i i i i i i K K i i M K E E E E σ − − = = + = + ⊗ R y y Hx x H w w Hx x H I J , (4-50) { } { } { }(0) (0)* (0) (0)* * (0) (0)* * 2 (0)y i i i i i i MN E E E σ = = + = + R y y Hx x H w w HH I , (4-51) 72 where J is a N N× matrix with the same structure as the matrix J . From (4-51), it is well-known that 2σ is the least eigenvalue of the matrix (0)yR [Shen and Ding, 2000]. After estimating the noise variance 2σ , its effect can be subtracted from the second-order statistical matrices in (4-49) - (4-51). In general, error will exist in the estimation of 2σ and therefore it is not recommended to directly subtract the noise contribution (which often results in poorer performance). In the simulation in Section 4.4, the noise contribution is not subtracted from the second-order statistical matrices in the first step (blind ICI and ISI cancellation). For the second step (signal detection in the presence of MAI), pilots are utilized to estimate the matrix ( )KpartH and MMSE (Minimum Mean Square Error) criteria can be exploited to take the noise into account, instead of Least-Squares criteria. Simulation results in the next section will show that the effect of noise on the system performance. 4.3.4 Implementation In practice, the second-order statistical matrices ( (0)yR , ( )y KR and ( )y K−R ) can only be computed from some finite number of the received OFDM symbols. We use the most commonly used method to approximate them as ( ) (0)* 1 1( ) sN K y i i is K N = ≈ ∑R y y , (0) ( )* 1 1( ) sN K y i i is K N = − ≈ ∑R y y , (0) (0)* 1 1(0) sN y i i isN = ≈ ∑R y y , (4-52) where sN is the number of OFDM symbols used. We also assume that an upper bound of the maximum channel length, that is, a number uppL such that uppL L≥ , is known or estimated. The implementation of the proposed algorithm is summarized as follows. Algorithm: Two-step semi-blind signal detection for MIMO-OFDM systems without cyclic prefix Step 1): Choose the parameter K such that 0 uppK N L≤ ≤ − . Compute the second-order statistical matrices ( (0)yR , ( )y KR and ( )y K−R ). Step 2): Form the equalization matrix KG defined in (4-34). 73 Step 3): Perform equalization as ( ) ( ), k k i K K i=o G y , 1, 2, ,k N K N K K= − − − − −" . Step 4): Collect the equalizer outputs as pilotO . Estimate the matrix ( )K partH using (4-47). Step 5): Detect the signals from the equalizer outputs as *( ) ( ) * ( ) 1 ( ) ( ) , , ˆ ( )k K K K ki part part part part i K −=x H H H o , 1, 2, ,k N K N K K= − − − − −" . Step 6): Recover , [ ]i ps n G , 1,2, ,p P= " , by performing DFT to the signals , [ ]i ps n , {0,1, , 1}n N∈ −" , which is obtained from ( ),ˆ ki partx . 4.4 Simulation Results The performance of the proposed semi-blind signal detection algorithm has been investigated through computer simulations. In the following examples, a MIMO- OFDM-WCP system with 2P = transmit antennas and 3M = receive antennas ( 2 3× system) was considered. The number of subcarriers in one OFDM symbol was set as 64N = . The performance measure, BER (bit error rate), was computed by averaging the results over 100 Monte Carlo realizations. In each run: (1) a data packet with 200 random OFDM symbols was transmitted from each antenna through the wireless channel with random Gaussian noise; (2) all transmitted signals were modulated by QPSK scheme; (3) the frequency selective fading channel responses were randomly generated with Rayleigh probability distribution. Signal-noise-ratio ( SNR ) was defined as { } 2 , 1 1 0 2 , 1 ( ) [ ] [ ] M P L pm i p m p l M i m m E h l s n l SNR E w n = = = = ⎧ ⎫⎪ ⎪−⎨ ⎬⎪ ⎪⎩ ⎭= ∑ ∑∑ ∑ , (4-53) and the second-order statistical matrices were computed from 200 OFDM symbols. 4.4.1 Effect of SNR For comparison, the conventional signal detection algorithm [van Zelst and Schenk, 74 2004] was also implemented for a MIMO-OFDM-LCP system. In this system, a CP with length of 25% of one OFDM symbol, that is 25% 16N× = , was inserted at the beginning of each OFDM symbol to eliminate ICI and ISI. The signals were detected by a set of parallel one-tap least-squares linear equalizers on each subcarrier after DFT. The number of pilot OFDM symbols utilized was equal to 2P = , while only one pilot OFDM symbol was utilized in the proposed algorithm. The parameter K in the proposed algorithm was chosen as / 2N . Two cases ( 6L = and 8L = ) were considered. Results are shown in Fig. 4.3 and Fig. 4.4, respectively. It is obvious that the proposed algorithm performs slightly worse than the conventional algorithm [van Zelst and Schenk, 2004] for MIMO- OFDM-LCP systems when SNR is low, while its performance is better than that of the conventional algorithm for MIMO-OFDM-LCP systems when SNR is high. It demonstrates that the proposed algorithm for MIMO-OFDM-WCP systems can achieve comparable performance to that of the conventional signal detection algorithm for MIMO-OFDM-LCP systems, but with higher bandwidth efficiency as CP is removed. 75 Fig. 4.3 BER versus SNR when the channel length is 6 ( 6L = ) 76 Fig. 4.4 BER versus SNR when the channel length is 8 ( 8L = ) 4.4.2 Effect of the parameter K In the proposed algorithm, the parameter K must be chosen such that 0 K N L≤ ≤ − . Here, it was varied from 0 to N L− to test its effect on the proposed algorithm. Results are shown in Fig. 4.5 and Fig. 4.6 for the cases where 6L = and 8L = , respectively. In these simulations, 20SNR dB= . There is little variation in their performance when K varies and it indicates that the proposed algorithm is not sensitive to the parameter K . 77 Fig. 4.5 Effect of K when the channel length is 6 ( 6L = ) 78 Fig. 4.6 Effect of K when the channel length is 8 ( 8L = ) 4.4.3 Effect of channel length overestimation The maximum channel length L in the proposed algorithm determines the range of the parameter K , that is [0, ]K N L∈ − . The error in the estimation of L only affects the system performance via the value of K . When L is overestimated, smaller K may be chosen as K must lie in the range [0, ]N L− . From the results in Fig. 4.5 and 4.6, it is clear that the performance of the proposed algorithm is not affected by the value of K , which in turn implies that the proposed algorithm is robust against channel length overestimation. 79 4.5 Summary In this chapter, a two-step semi-blind signal detection algorithm for MIMO-OFDM- WCP systems has been proposed. The algorithm takes advantage of some structural properties of the shifted received OFDM symbols. An equalizer has been designed in the first step to cancel the ICI and ISI based on SOS of the received signals. Signal detection has been achieved in the second step from the equalizer output with the aid of one pilot OFDM symbol. Exact knowledge of the maximum channel length is unnecessary and higher bandwidth efficiency is achieved as CP is removed. Simulations have shown that the proposed algorithm achieves comparable performance to that of MIMO-OFDM-LCP systems and is also robust against channel length overestimation. 80 Chapter 5 Conclusions and Suggestions for Future Research In this thesis, semi-blind signal detection for MIMO and MIMO-OFDM systems have been considered. In order to achieve high system capacity or high transmission rate as well as good QOS, three algorithms have been proposed for MIMO, MIMO-OFDM with short CP, and MIMO-OFDM without CP, respectively. Chapter 2 has proposed a semi-blind Rake-based multi-user detection technique for quasi-synchronous MIMO systems, consisting of multi-user single-path signal separation, time delay estimation, and multi-path combining. Multi-path signals have been exploited to achieve time diversity, therefore improving the performance. A simple time delay estimation method, which exploits the structural property of the channel matrix, has been proposed. It has been shown that, with the estimated time delays, the choice of combining weights resulted in a straight forward multi-user detection. Furthermore, knowledge of the channel length and the time delays are not required, which renders the technique more practical. Simulation results have demonstrated that the proposed technique achieves good performance and is robust against over-estimation of the maximum channel length and the maximum time delay. In Chapter 3, a time domain semi-blind signal detection algorithm for MIMO- OFDM systems with short CP has been proposed. A new system model has been 81 introduced in which the i th received OFDM symbol is left shifted by J samples. Based on some structural properties of the new system model, an equalizer has been designed to cancel most of the ISI using the SOS of the received signals before signal detection. It has been shown that the channel length information is not needed and only 2P columns of the channel matrix need to be estimated with a minimum of 4P pilots for identifiability. In addition, it has been demonstrated that the proposed algorithm is applicable to general MIMO-OFDM systems irrespective of whether the CP length is longer than, equal to or shorter than the channel length. Simulation results have shown that the proposed algorithm outperforms the existing ones in all cases. In Chapter 4, a two-step semi-blind signal detection algorithm for MIMO-OFDM systems without CP has been proposed. The algorithm takes advantage of some structural properties of the shifted received OFDM symbols. An equalizer has been designed in the first step to cancel the ICI and ISI based on SOS of the received signals. Signal detection has been achieved in the second step from the equalizer output with the aid of one pilot OFDM symbol. Exact knowledge of the maximum channel length is unnecessary and higher bandwidth efficiency is achieved as CP is removed. Simulations have shown that the proposed algorithm achieves comparable performance to that of MIMO-OFDM systems with long CP and is also robust against channel length overestimation. Suggestions for Future Research • In MIMO systems, time diversity is achieved by multi-path combining for performance enhancement. 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Shaodan Ma and TungSang Ng, “Two-step signal detection for MIMO-OFDM systems without cyclic prefix”, submitted to IEEE Transactions on Wireless Communications 5. Shaodan Ma and TungSang Ng, “Semi-blind time domain equalization for MIMO-OFDM systems”, submitted to IEEE Transactions on Vehicular Technology 6. Shaodan Ma and TungSang Ng, “Timing estimation for quasi-synchronous SDMA systems”, Proceeding of IEEE International Conference on Communications systems (ICCS2004), pp. 410-413, 2004, Singapore 90 7. Yonghong Zeng, TungSang Ng and Shaodan Ma, “Blind MIMO channel estimation with an upper bound for channel orders”, Proceeding of IEEE International Conference on Communications (ICC2005), pp. 1996-2000, 2005, Korea 8. Yonghong Zeng, Shaodan Ma and TungSang Ng, “Semi-blind channel identification and symbol estimation for asynchronous MIMO systems”, Proceeding of IEEE International Symposium on Signal Processing and Its Applications (ISSPA2005), pp. 435-438, 2005, Australia 9. Shaodan Ma, Yonghong Zeng and TungSang Ng, “Signal detection with time delay estimation for quasi-synchronous MIMO systems on multipath channels”, Proceeding of IEEE Global Telecommunications Conference (GlobeCom2005), pp. 2322-2326, 2005, U. S. A 10. Yonghong Zeng, A. Rahim Leyman, Shaodan Ma and TungSang Ng, “Optimal pilot and fast algorithm for MIMO-OFDM channel estimation”, Proceeding of IEEE International Conference on Information, Communications and Signal Processing (ICICS2005), Thailand 11. Shaodan Ma, Ngai Wong and TungSang Ng, “Time domain equalization for OFDM systems”, to appear, Proceeding of IEEE International Symposium on Circuits and Systems (ISCAS 2006), Greece 12. Shaodan Ma, Ngai Wong and TungSang Ng, “Signal detection for MIMO- OFDM systems with time offsets”, accepted for publication, GlobeCom2006