Because it is hard to calculate the counts of the computational operations of the proposed detection methods precisely, the first and last detection layers of the two methods are only considered and compared instead. If we can make sure that the counts of the computational operations of the proposed detection methods are less than that of Choi’s method in these two detection layers, we can claim that the pro-posed detection method has less computational complexity than Choi’s method.
However, the difference between the two successive detection methods in the last
layer is only Step P-3. It is easy to show that the proposed methods have the less computational complexity. For more serious analysis, the detection layers before the last detection layers of the two successive detection methods are considered instead.
By comparison of Choi’s method and the proposed methods shown in Figure 3.2 and Figure 4.1 respectively, the quantities of complexity reduction of the proposed methods in the first detection layer are listed in the descending order as: [16]
z Computational complexity of the pseudo inversion in Step P-1 is reduced from ( 3)
O N to O N q( ( +1) )3 .
z Computational complexity of updating the pseudo inversion in Step P-5 is reduced from O N N( ( −1) )2 to O q q( ( +1)q2).
z Computational complexity of the norm operation of the equalization vector in Step P-2 and Step P-6 is reduced from O N( 2) to O q(( +1) )2 .
z The number of norms of the updated equalization vectors in Step P-6 is reduced from N -1 to q.
z Taps of the equalization vectors in Step P-3 is reduced from N to (q+1).
The counts of the computational operations of the proposed and Choi’s successive detection methods in the first layered detection are shown in Table 4.1, clearly. In the following tables, the numbers of the complex divisions, multiplications, and additions are listed, respectively. Gaussian elimination is adopted for the matrix inversion [19].
It is assumed that the matrix inversions in the detection procedures all exist.
The quantities of the complexity reduction of the proposed methods in the detec-tion layer before the last one are listed in the descending order as
z Computational complexity of updating the pseudo inversion in Step P-5 is reduced from O N( 2 )2 to O(2(q+1)2 )2 or to O(2(q+1)).
z Computational complexity of the norm operation of the equalization vector in Step
2 + 2
z Taps of the equalization vectors in Step P-3 is reduced from N to (q+1).
The counts of the computational operations of the proposed and Choi’s successive detection methods in the detection layer before the last one are shown in Table 4.2.
Table 4.1 Complexity statistics of the proposed and Choi’s successive detection methods in the first detection layer
# of divisions # of multiplications # of additions Choi’s method [7] 2N2−4N+2 6N3−7N2+8N−2 6N3−11N2+9N−3
Table 4.2 Complexity statistics of the proposed and Choi’s successive detection methods in the detection layer before the last one
# of divisions # of multiplications # of additions
Choi’s method [7] 2 12N+4 10N−3
Table 4.3 Complexity statistics of the proposed and Choi’s successive detection methods in the first detection layer with N = 64, and q=8, 4
# of divisions # of multiplications # of additions
Choi’s method [7] 7938 1544702 1528381
q = 8 5056 156081 164838
Table 4.4 Complexity statistics of the proposed and Choi’s successive detection methods in the detection layer before the last one with N = 64, and q = 8, 4
# of divisions # of multiplications # of additions
Choi’s method [7] 2 772 637
q = 8 4 or 2 235 or 127 196 or 104 Proposed method
q = 4 4 or 2 159 or 99 136 or 84
q = 8 0% or 0% 69.56% or 83.55% 69.23% or 83.67%
Reduction %
q = 4 0% or 0% 79.40% or 87.18% 78.65% or 86.81%
The complexity statistics of the proposed and Choi’s successive detection methods in the targeted simulation environments are shown in Table 4.3 and Table 4.4. The percentages of complexity reductions of the proposed methods compared with Choi’s method are listed in the bottoms of the tables.
Table 4.3 shows that the reduction percentages of the multiplication and the addi-tion operaaddi-tions of the proposed methods is roughly up to 90% in the first layered de-tection even if q = 8. However, the reduction of the divisions is relatively small be-cause the number of divisions is proportional to O N( 2) not O N( 3). The computa-tional complexity of updating the matrix inversion of the proposed methods in the succeeding layered detection gradually approaches that of Choi’s method, because the dimensions of matrices for inversions in Choi’s method become small. Therefore, the benefits for utilizing the spare matrix structure of Jeon’s LS method decrease. In Table 4.4, the reduction percentages of the multiplication and the addition operations of the proposed methods drop to 70% in the detection layer before the last one. Unfortu-nately, additional divisions for the matrix inversions of the block matrices are required when un-demodulated signals gather together. In conclusion, it is supposed that the percentages of the complexity reductions excluding divisions of the proposed
detec-tion methods with q = 8 and q = 4 are at least 75% and 85%, respectively.
The computational complexities of the other mentioned methods are shown in Ta-ble 4.5, where I indicates the number of the iterations of the ICI cancellation method.
The ratios of the computational complexities of different data detection methods to that of Choi’s method in the first detection layer in the targeted simulation environ-ments are shown in percentage in Table 4.6. The percentages of complexity ratios of the proposed methods only include the computational operations in the first detection layer, because it is hard to evaluate the exact computational operations of the pro-posed methods.
Table 4.5 Complexity statistics of the other mentioned methods
# of divisions # of multiplications # of additions
LS detection N2−N N3 3N3−2N2−N Complexity ratio % Division Multiplication Addition
LS detection 50.79% 50.91% 50.92%
In Table 4.6, it is known that the ICI cancellation method with 3 detection itera-tions has the least computational complexity of all the compared detection methods.
For achieving whole detection, the numbers of the multiplication and addition opera-tions are less than 1% those of Choi’s method in the first layered detection. Jeon’s method of LS detection with a worse performance than that of the ICI cancellation method with 3 detection iterations, however, is more than 1.5 times the numbers of the multiplications and additions of the ICI cancellation method with 3 detection it-erations. The computational complexity of the proposed method with q = 4 is larger than those of Jeon’s method of LS detection and ICI cancellation method with 3 de-tection iterations. It is noted that the percentages of complexity ratios of the proposed methods shown in Table 4.6 only include the computational operations in the first de-tection layer.