Figure 1 shows the block diagram for an MIMO system with soft IDD, where a LDPC code is used as an FEC scheme and a PSK is used for modulation. At the transmitter, the bit information vector is encoded to produce codeword. Then codewords are interleaved and modulated succes- sively before they are mapped to the transmitted antennas, where is the number of bits per transmitted symbol. The interleaved bits

are divided and modulated to transmitted symbol vector, where represents the kth bit of the mth symbol which is independently chosen from a complex con- stellation O.

Suppose the received symbol vector is represented as, and a flat Rayleigh fading channel model is chosen as the MIMO channel, i.e., , where denotes the channel fading coefficient between the jth transmitted antenna and the ith received antenna. The elements of the channel matrix are modeled by independent and complex-valued Gaussian random variables with zero mean and unit variance. Then,

where is an vector whose elements are independent zero-mean complex Gaussian random variables with variance per dimension.

Figure 1 shows the soft IDD procedure. The SBI of the MIMO detector, is first calculated and subtracted by the a priori information of the detector, i.e.,; then the extrinsic information is passed through the de-interleaver, and its de-interleaved version is utilized by the channel decoder as the a priori information. The channel decoder estimates the infor- mation sequence and generates its soft output, by using soft iterative de- coding algorithm such as the min-sum-product algorithm. Subsequently, the interleaved version of extrinsic information, is fed back to the detector as the a priori information.

The PIC-MMSE MIMO detector is performed as follows. First, with the a priori information, from the channel decoder (at the first iteration it is set to zero), the expected value of the ith transmitted symbol, is calculated as follows [8]:

where is a constellation symbol from, and is set to be −1 and 1 according to the kth bit of is 0 and 1, respectively. At the same time, the variance of the ith transmitted symbol, is calculated by:

The second step is PIC on the received symbol vector, , along with MMSE filtering. The PIC process on is performed using the estimated expectation, in (2) as follows:

where is the interference canceled symbol for the ith layer, is the ith column vector of the channel matrix, and denotes the residual noise plus interference (NPI) term expressed by. At the same time, the MMSE filtering process is performed as follows:

where is a diagonal matrix with estimated in (3).

The third step is suppressing the NPI term in (4) using the MMSE filter in (5). Then, the filtered result for the th layer, i.e., the estimated value of the th symbol, can be expressed as:

where denotes the ith row of and corresponds to the MMSE filter for the ith layer. The last step is to calculate SBI contained in. Let us denote SBI of the kth bit contained in the ith symbol, in terms of log-likelihood ratio (LLR) estimated on the observation of y over the channel H, as, then

where and denote constellation symbols with the kth bit of 0 and 1, respectively, and

It is clear in (7) that the search process to find the solution of needs complexity of for every layer.

It was reported that there were no performance degradation even if the a priori information from the channel decoder in (7) was neglected, for the systems using binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) [9]. In this paper, we use this fact, and assume that the a priori infor- mation, can be neglected, then (7) can be simplified as follows.

With this simplification, the argument of operations in (9) is now. In other words, we need to find a symbol, with the minimum distance from for the kth bit. For this purpose, we map to a target unit range where there is only one constellation symbol to estimate the distance from. We utilize the symbol mapping technique using the symmetric characteristics of PSK symbol constellation for a given bit [7].

First, we consider that the constellation diagram can be divided into unit regions for -ary PSK modulation scheme, using the bisectors between the constellation symbols. Subsequently, is recursively mapped to a specific region in the first quadrant to, where, is the phase of the mapped symbol used for the kth bit of b. Then, at the ith layer, for the corresponding bit of is the same as the distance between and a constellation symbol located in the same unit region as. The mapped symbol, and the corresponding constellation symbol, are determined by the symmetric characteristics of the symbol constellation, based on Euclidean geometry for a given bit.

With the above property, (9) can be generalized as

where is a scaling parameter for the kth bit of the ith layer, which is a function of by reflecting the sign change introduced by the symbol mapping, and are the unique symbols nearest to, i.e., the mapped version of the detected symbol, with a phase of.

In addition, and are determined prior to the system implementation, and they are constant values independent of. For PSK with the Gray- mapping, and can be found as shown in Table 1 and Table 2. In 2, values can be recursively found as follows

where. In this way, the search process to find the minima can be eliminated. Thus, the complexity to estimate SBI is reduced to

linear-order for each independent layer without degrading the performance from the Max-approach.

Soft symbol value estimations specified in (2) and (3) require times of multiplications, which exponentially increases by. We reduce the com- plexities of computing (2) and (3), by finding the symbol values of. We modify the idea reported in [9] for QAM.

By denoting as the probability of the kth bit of the ith layer to be 0, the following equation applies:

where, then,

With this information, we simplify the expressions of and for PSK schemes, so that they can be used in real-number operations; First, in the case of BPSK, and can be derived as follows:

Then, we apply the same rule for the soft IDD scheme, and denote the iteration index inside the PIC-MMSE detector as, then (12) can be repre- sented by:

Equation (16) is then applied to QPSK, whereby QPSK can be decomposed into two independent BPSK. Then, expressions of the real and imaginary parts of the soft symbol values, and are formulated independently for QPSK with amplitude of and, respectively, as follows:

and accordingly, the of QPSK can be simplified according to the following equation:

The expressions for the real and imaginary part of for 8-PSK can be derived as follows:

where and. and for the 16-PSK can be derived with the same principle, and formulized as follows

where, , and . With the same principle, the in (2) for the higher-order PSK can be expanded.