Showing the decoding radius of SIC is larger than that of ZF

April 27, 2015 Uncategorized No comments

Zero forcing (ZF) and successive interference cancellation (SIC) are among the most popular sub-optimal approaches to perform a MIMO detection.  Ref. [1] serves as a great analytic perspective to look at the decoding radius of ZF, SIC and their lattice reduction counter parts. However, the statement about the size comparison of SIC and ZF decoding radius is still away from ‘nice and clear’, i.e.,

(pp. 2799, para 2 on the left column)”Since \hat{b}_i only needs to be orthogonal to {b}_1, \ldots,b_{i-1}, we must have \theta_i\leq\phi_i and hence d_{i,ZF}\leq d_{i,SIC}“.

In this post, we try to establish a proof about “d_{i,ZF}\leq d_{i,SIC}“.

First of all, necessary notations are given, as those in Ref. [1]. Let d_{i,ZF} be the Euclidean distance from point 0 to the i-th facet of the decision regions of ZF and SIC, respectively. Then the distance spectrum of them are given by:

d_{i,ZF}=\frac{1}{2}||\mathbf{b}_i||\sin \theta_i\\ d_{i,SIC}=\frac{1}{2}||\mathbf{b}_i||\sin \phi_i

where \theta_i  denotes the angle between \mathbf{b}_i and the space span by all the rest columns of matrix \mathbf{B}, and \phi_i is the angle between \mathbf{b}_i and the space span by the columns before i. We dubs the two notations as

\theta_i=angle(\mathbf{b}_i,span(\mathbf{B_{[-i]}))

\phi_i=angle(\mathbf{b}_i,span(\mathbf{B_{[1\sim i-1]}))

In order to show that \sin \theta_i \leq \sin \phi_i, it is indeed showing

\frac{||Proj(\mathbf{b}_i,span^\perp (\mathbf{B}_{[-i]}))||}{||\mathbf{b}_i||}\leq\frac{||Proj(\mathbf{b}_i,span^\perp (\mathbf{B}_{[1\sim i-1]}))||}{||\mathbf{b}_i||}

where the “Proj” operator means taking the projection of vectors onto their orthogonal complement. Let the normalized Gram-Schmidt basis of \mathbf{B} note as \hat{\mathbf{B}}=[\hat{\mathbf{b}}_1,...,\hat{\mathbf{b}}_n], then we have

{||Proj(\mathbf{b}_i,span^\perp (\mathbf{B}_{[-i]}))||}= ||\hat{\mathbf{B}}_{i}\hat{\mathbf{B}}_{i}^{\mathrm{T}}\mathbf{b}_i||

{||Proj(\mathbf{b}_i,span^\perp (\mathbf{B}_{[1\sim i-1]}))||}= ||\hat{\mathbf{B}}_{[i\sim n]}\hat{\mathbf{B}}_{[i\sim n]}^{\mathrm{T}}\mathbf{b}_i||

where ||\hat{\mathbf{B}}_{i}\hat{\mathbf{B}}_{i}^{\mathrm{T}}\mathbf{b}_i||=||\hat{\mathbf{b}}_{i}\hat{\mathbf{b}}_{i}^{\mathrm{T}}\mathbf{b}_i|| and ||\hat{\mathbf{B}}_{[i\sim n]}\hat{\mathbf{B}}_{[i\sim n]}^{\mathrm{T}}\mathbf{b}_i||= ||\hat{\mathbf{b}}_{i}\hat{\mathbf{b}}_{i}^{\mathrm{T}}\mathbf{b}_i+\hat{\mathbf{b}}_{i+1}\hat{\mathbf{b}}_{i+1}^{\mathrm{T}}\mathbf{b}_i+\ldots+\hat{\mathbf{b}}_{n}\hat{\mathbf{b}}_{n}^{\mathrm{T}}\mathbf{b}_i||, among which the vectors are pair-wise orthogonal. Thus ||\hat{\mathbf{B}}_{i}\hat{\mathbf{B}}_{i}^{\mathrm{T}}\mathbf{b}_i||\leq||\hat{\mathbf{B}}_{[i\sim n]}\hat{\mathbf{B}}_{[i\sim n]}^{\mathrm{T}}\mathbf{b}_i||, and we arrive in the conclusion that

\sin \theta_i\leq \sin \phi_i

So the decoding radius of SIC is larger.

 

References:

[1] C. Ling, “On the proximity factors of lattice reduction aided decoding”, IEEE Transactions on signal processing, Vol.59, No.6, pp. 2795–2808, 2011.

[2] Y.H. Gan, C. Ling and W.H. Mow, “Complex lattice reduction algorithm for low-complexity full-diversity MIMO detection”, IEEE Transactions on signal processing, Vol.57, No.7, pp. 2701–2710, 2009.

 

 

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