Detection Methods

Precoding in OFDM systems exploits the frequency diversity the channel provides. However, the precoding also results in a MIMO system for the subcarriers in the same coding block. One of the advantage of OFDM systems is that the equalization can be conducted by a single-tap

FEQ. As we will see, the single-tap FEQ cannot be used in precoded systems since the diversity gain will be reduced back to one [44]. To explore the diversity the precoded system has, we then have to use more sophisticated detection methods such as the SIC and ML methods. The SIC is a simple MIMO detection method. Due to its lower computational complexity, it is frequently considered in real-world implementation. The ML method is the optimum detection method; however, its computational complexity grows exponentially along with the QAM size and the system dimension. In many cases, the computational complexity becomes prohibitively high. To solve the problem, many near-optimum detectors have been proposed. Among these detectors, SD is considered as one of the most efficient ML algorithms. It has been proved that for an L × L MIMO system, the diversity gain with the ML detector is L [44]. Thus, to obtain the full diversity gain for precoded OFDM systems, we then have to use the ML detector. In many real-world applications, the channel length (i.e., L) is generally large. Even with the SD algorithm, the computational complexity is still too high. In this section, we propose a new method to solve the problem. The proposed method, combining the merits of SIC and SD, can have a similar performance as that of SD. However, the required computational complexity can be significantly reduced.

Zero Forcing FEQ (ZF-FEQ)

When detecting a signal component, the ZF-FEQ completely removes the interference from other components. Let the receive signal vector ˜r_i in (5.15) be multiplied by a ZF-FEQ matrix, denoted as S. The output signal, denoted as ˜u⁰_i, is then passed through a hard-decision device and a decided signal vector, denoted as ˜u_i, is obtained. The equalized signal vector can be expressed as

˜

u⁰_i = S˜r_i

= S˜y_S,i+ S˜y_I,i+ S˜v_i

= S ˜G_Sd˜_i+ S ˜G_Id˜_i+ S˜v_i, (5.35)

where ˜y_S,i, ˜y_I,i, and ˜v_iare those defined in (5.17) and (5.18). To have a ZF-FEQ result, we have S as

S = ˜G⁻¹_S . (5.36)

As a result, we have ˜u⁰_i as

u˜⁰_i = ˜di+ ˜G⁻¹_S G˜Id˜i+ ˜G⁻¹_S v˜i. (5.37)

From (5.19), we can see that

G˜⁻¹_S = U^HGˆ⁻¹_S U (5.38)

Note that ˆG⁻¹_S is just the single-tap FEQ used in conventional OFDM systems. The ZF-FEQ in the precoded OFDM system is exactly the same as that in the uncoded system. Thus, the required computational complexity is low. However, as mentioned, the diversity gain for the ZF detection in MIMO systems is one [44]. As a result, no performance improvement can be obtained with precoding.

Successive Interference Cancelation (SIC)

The SIC is a nonlinear detection method. The main idea is to estimate and detect each signal component of the transmitted symbol sequentially. Each detected component is then removed from the received signal before the estimation of the next component [63]. To apply the SIC method, we first partition the receive OFDM symbol, ˜r_i in (5.15), into p_M sub-symbols with size p = M/pM. Also, ˜vi in (5.16) and ˜uiare also partitioned accordingly. Let

where

where ˆGS,k is a p × p diagonal matrix. Then, with the property of the permutation matrix, ˜GS

can be also represented as

G˜_S=

where the kth component is given by

G˜_S,k= U^H_p Gˆ_S,kU_p, for k = 0, · · · , p_M − 1. (5.43) As we can see, all ˜G_S,k, P_k, U_p, and ˆG_S,kare p×p matrices. The corresponding kth component of ISI matrix can be represented as

G˜_I,k = U^H_p Gˆ_I,kU_p, for k = 0, · · · , p_M − 1. (5.44)

As a result, we have the receive kth sub-symbol as

˜r_i,k = ˜GS,kd˜i,k + ˜GI,kd˜i,k+ ˜vi,k, for k = 0, · · · , pM − 1, (5.45) u˜i,k, ˜ri,k and ˜vi,k are those defined in (5.39), and ˜GS,k, ˜GI,k in (5.43) and (5.44), respectively.

As we can see, (5.45) is a MIMO system representation.

From [42], we see that the probability density function of the receive signal conditioned on the transmit signal is defined as the likelihood function. For our problem, the likelihood function is then p{˜r_i,k|˜d_i,k}. The criterion to choose ˜d_i,k that maximizes p{˜r_i,k|˜d_i,k} is called the ML criterion. It is simple to prove that the decision rule can be reduced to find the ˜d_ithat is closest in distance to the received signal vector ˜r_i,k. Therefore, the ML detection criterion for our precoded OFDM systems can be reduced to

˜

u_i,k = arg min

d˜i,k∈Ψp

k ˜r_i,k− ˜G_S,kd˜_i,k k², (5.46)

where Ψ_p is a set including all possible ˜d_i. Using the QR decomposition, we can decompose G˜_S,kinto ˜G_S,k = ˜Q_kR˜_k, where ˜R_kis an upper-triangular matrix given by

and ˜Qkis a unitary matrix. Then, we have the ML detection as

˜

where ˜r⁰_i,k = ˜Q^H_k˜r_i,k. The SIC tries to implement the ML detection and conducts signal detec-tion starting from the last data symbol. From (5.48), we see that the last symbol can be detected by

where Ψd is the set for all possible transmit ˜di,k(p − 1). Now, if the detection is correct, i.e., ˆ

u_i,k(p − 1) = ˜d_i,k(p − 1), we can subtract its interference from the received signal and this will

enhance the probability of correct detection of ˜d_i,k(p − 2). This process can be repeated until all the symbols are detected. For (m + 1)th symbol, we then have the detection as

ˆ SIC method cannot achieve the ML performance since detection errors can occur in any stage.

Also, an detection error in a certain stage will increase the probability of detection error in later stages. This is called error propagation.

Maximum Likelihood Sequential Estimation

The ML detector is an optimal detector and it needs an exhaustive search over the entire set of Ψ_p [54]. If the QAM constellation size is R and the size of the OFDM sub-symbol ˜d_i,k is p, the computational complexity for the ML detector is O(R^p). The complexity of the ML detec-tion can become extremely high for a high constelladetec-tion moduladetec-tion size R and large symbol size p. Many suboptimum methods have been developed to reduce the required computational complexity. These methods can have near-ML performance but the required computational complexity is much lower. Among them, the most well known is the SD method. In this disser-tation, we use the SD-based method for the implementation of the ML detector.

Sphere Decoding (SD)

From (5.48), we see that the ML detection can be conducted as u˜i,k = arg min Note that ˜R_k is an upper-triangular matrix. The main idea of the SD method is to search a subset of Ψp such that

k ˜r⁰_i,k − ˜R_kd˜_i,k k²< r²_SD, (5.52)

where r_SDis the radius of the searching sphere [54]. The search starts with the last symbol of ˜r⁰_i,k and forms a tree structure excluding unlikely candidates located out of the sphere. Considering the pth component ˜r⁰_i,k, we have

|˜r_i,k⁰ (p − 1) − ˜r_k(p − 1, p − 1) ˜di,k(p − 1)|² < r²_SD. (5.53) We then choose all possible ˜di,k(p−1)⁰s such that |˜r⁰_i,k(p−1)−˜r_k(p−1, p−1) ˜di,k(p−1)|² < r²_SD as the candidates for the pth component of ˜r_i,k. Now, consider the (p − 1)th and pth components of ˜r⁰_i,k in (5.52). For each candidate of ˜di,k(p − 1), we then choose all ˜di,k(p − 2)’s such that a similar manner. For a general expanding form of (5.52), we have

˜ can-didate (˜di,k). Since the tree has many paths satisfying (5.55), we then have a list of candidates.

Finally, we can find the one minimizing (5.51) as the detection output.

The efficiency of the SD method greatly depends on the choice of the radius rSD. The complexity will be high if r_SD is large. This is because more candidates will be included in the sphere of (5.55). If rSD is small, the optimum solution may not be included in the sphere.

In [55], a proper radius is suggested as:

r²_SD = C|det( ˜G_S,k)|^1p (5.56) where C is a constant, and det( ˜G_S,k) the determinant of ˜G_S,k. The matrix ˜G_S,k is defined in (5.46). It has been shown that the choice can have a good compromise between performance and computational complexity [56].

Ordering for SIC and SD

For the SIC method, detection is conducted in a backward fashion. As mentioned, the SIC method described in (5.50) has an error propagation problem. The diagonal element of ˜R_k, i.e, ˜r_k(m, m), 0 ≤ m ≤ p − 1, determines the SINR of the mth signal component. If ˜r_i,k can be ordered before the QR decomposition such that ˜r_k(m, m) (after QR decomposition) has an ascending order, the error propagation effect can be reduced. However, the optimum ordering resulting an ascending order of ˜r_k(m, m) has not been found yet. Some suboptimum ordering methods have been proposed in the literature [56], [64], [61], [62]. For the SD method, a proper ordering also gives better result. This is because for the determination of the candidates of the mth ˜d_i,k, the number of components involved in (5.55) is m − 1. When the tree is expanded in early stages, m is small. The distance calculation in (5.55) is not reliable. If an proper ordering is conducted, the SINR can be enlarged and the radius of the sphere can be reduced. As a result, the number of candidates can be reduced too. As we see in (5.56), the computational complexity of the SD method is related to the number of candidates. A proper ordering can then reduce the computational complexity of the SD method.

From (5.48), we see that the equivalent MIMO system obtained from the precoded OFDM system has a special structure. The existing ordering algorithms may not be proper for this application. For example, the scheme in [56] uses the column norms of the channel matrix for ordering. However, from (5.19), we see that the column norms are all the same and the method in [56] cannot be applied. Here, we propose an simple ordering scheme for the precoded OFDM system.

Recall that the ML detection for precoded OFDM systems can be expressed as u˜i,k = arg min

d˜i,k∈Ψp

k ˜ri,k− ˜GS,kd˜i,k k² (5.57) where

˜r_i,k = ˜G_S,kd˜_i,k + ˜G_I,kd˜_i,k+ ˜v_i,k, for k = 0, · · · , p_M − 1. (5.58) Here, ˜u_i,k, ˜r_i,k and ˜v_i,k are those defined in (5.39), and ˜G_S,k, ˜G_I,k in (5.43) and (5.44),

re-spectively. As defined, ˆG_S,k in (5.41) is a diagonal matrix. We propose ordering the diagonal elements of ˆG_S,ksuch that the elements have an ascending order. That is

g˜S,k = [˜gS,k(0), · · · , ˜gS,k(p − 1)]^T = Okdiag rewrite the detection problem as

u˜_i,k = arg min

is a unitary matrix and R_k is an upper-triangular matrix with the form

Rk =

The operations are the same as those in (5.48), (5.51) except that Q_kand R_kare used to replace Q˜_kand ˜R_k, respectively.

˜

u^o_i,k = arg min

d˜^o_i,k∈Ψp

k Q^H_k˜r^o_i,k − R_kd˜^o_i,k k²

= arg min

d˜^o_i,k∈Ψp

k ˜r^0o_i,k− R_kd˜^o_i,k k², (5.63)

where ˜r^0o_i,k = Q^H_k˜r^o_i,k. Note that ˜u^o_i,k, ˜r^o_i,k, ˜d^o_i,k, ˜r^0o_i,k are the ordered version of ˜u_i,k, ˜r_i,k, ˜d_i,k,

˜r⁰_i,k. From (5.60), we know that ˆG^o_S,k is a matrix whose diagonal elements are in an ascending order of those of ˆG_S,k. From simulations, we found that ther_k(m, m) tends to be equal or larger than ˜r_k(m, m) when m is close to q. The result is similar to the method in [56]. However, the theoretical proof will be difficult.

Hybrid SD-SIC (SDSIC)

As described, the SD method can efficiently implement the ML detector. However, when the dimension of the MIMO system is high, the computational complexity is still high. The com-putational complexity of the SIC is much lower, but it suffers from the error propagation effect.

For our precoded OFDM system, the equivalent MIMO system is of dimension L × L where L is the length of the time-domain channel response. For OFDM systems, the CP size indicates the maximum channel length. For wideband systems, the delay spread of the channel is usually large. For example, the CP size for the OFDM symbol defined in IEEE802.11a/g systems is 16.

Thus, the equivalent MIMO system in precoded OFDM systems will be of dimension 16 × 16.

The computational complexity of the SD algorithm for such system will be very high.

We now propose a new detection method to solve the problem. The proposed method com-bines the merits of SD and SIC methods and its performance can approach to that of the SD method. We call it the SDSIC method. The main idea comes from the fact that for SIC, the decision errors at its early stages is more damaging. In other words, if the detection is erroneous in early stages, it is likely to be erroneous in later stages. To solve the problem, we can use the SD method to obtain decisions in early stages. The proposed SDSIC method can be described

as follows. Let ˜u_i,k be divided into two parts, ˜u_D,i,kand ˜u_C,i,k, where

Note that the parameter p_kdetermines the dimension of the MIMO system that the SD will work on. The large the p_k, the higher the computational complexity the SD will require. To find the decision for ˜u_D,i,k, we modify the SD method in (5.55) as the one minimizing (5.65) as the detection result.

With the detected ˜u_D,i,k, we can subtract its interference to the system and then use the SIC method to detect the remaining vector, ˜u_C,i,k. The SIC method can be described as

ˆ