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Table 1 Computational complexity of the proposed QL-QR algorithm

From: Low-complexity QL-QR decomposition- based beamforming design for two-way MIMO relay networks

Step

Operations

FLOPS

Case: (2,2,2)×6

1

V 1,V 2

\(2\times K\left (40{N_{i}^{3}}-24{N^{2}_{i}}+17N_{i}\right)\)

1560

2

Q L,1 L 1,Q L,2 L 2

\(2\times 16K\left ({N^{2}_{T}}N_{i}-N_{T}{N^{2}_{i}}+\frac {1}{3}{N^{3}_{i}}\right)\)

4864

3

Q R,1 R 1,Q R,2 R 2

\(2\times 16K\left ({N^{2}_{T}}N_{i}-N_{T}{N^{2}_{i}}+\frac {1}{3}{N^{3}_{i}}\right)\)

4864

4

\(\mathbf {H}^{\mathrm {T}}_{1,1}\mathbf {F}_{1}\mathbf {H}_{1,2}\)

\(8{N^{2}_{T}}N_{i}+4N_{T}{N^{2}_{i}}+2N_{T}N_{i}\)

696

5

\(\mathbf {H}^{\mathrm {T}}_{2,1}\mathbf {F}_{2}\mathbf {H}_{2,2}\)

\(8{N^{2}_{T}}N_{i}+4N_{T}{N^{2}_{i}}+2N_{T}N_{i}\)

696

6

\(\mathbf {\widehat {C}}_{1}\)

\(2K\left (32{N^{2}_{T}}N_{i}+8N_{T}N_{i} +2{N^{2}_{T}}-4N_{i}+3N_{T}\right)\)

14,856

7

\(\left (\mathbf {\Xi }^{\mathrm {H}}_{i}\mathbf {\Xi }_{i}\right)^{-1}\)

\(K\left (\frac {14}{3}{N^{3}_{T}}-2{N^{2}_{T}}+N_{T}\right)\)

2826

8

\(\text {det}\mathbf {B}^{2}_{1}\)

\(4K\left ({N^{3}_{T}}+{N^{2}_{T}}+2N_{T}\right)\)

3168

Total

  

33,530