Using tensor contractions to derive the structure of slice-wise multiplications of tensors with applications to space–time Khatri–Rao coding for MIMO-OFDM systems

EURASIP Journal on Advances in Signal Processing

Table 1 Link between the diagonalized tensor structures and their generalized unfoldings

Nonzero elements	Generalized unfoldings
\({\varvec{{D}}}_{(m,m)} = {\varvec{{a}}}_{(m)}\)	\({\varvec{{D}}} = {\varvec{{I}}}_M \diamond {\varvec{{a}}}^{\mathrm{T}}\)
\({\varvec{{{\mathcal {D}}}}}_{(m,n,n)} = {\varvec{{A}}}_{(m,n)}\)	\(\left[ {\varvec{{{\mathcal {{\varvec{{{\mathcal {D}}}}}}}}}} \right] _{([1,3],[2])} = {\varvec{{I}}}_N \diamond {{\varvec{{A}}}}\)
\({\varvec{{{\mathcal {D}}}}}_{(m,m,n)} = {\varvec{{A}}}_{(m,n)}\)	\(\left[ {\varvec{{{\mathcal {{\varvec{{{\mathcal {D}}}}}}}}}} \right] _{([3,2],[1])} = {\varvec{{I}}}_M \diamond {{\varvec{{A}}}}^{\mathrm{T}}\)
\({\varvec{{{\mathcal {D}}}}}_{(m,m,n,n)} = {\varvec{{A}}}_{(m,n)}\)	\(\left[ {\varvec{{{\mathcal {{\varvec{{{\mathcal {D}}}}}}}}}} \right] _{([1,3],[2,4])} = {\varvec{{I}}}_M \diamond \mathop {\mathrm{{vec}}}\left( {\varvec{{A}}}\right) ^{\mathrm{T}}\)
\({\varvec{{{\mathcal {D}}}}}_{(m,n,k,k)} = {\varvec{{A}}}_{(m,n,k)}\)	\(\left[ {\varvec{{{\mathcal {{\varvec{{{\mathcal {D}}}}}}}}}} \right] _{([1,2,4],[3])} = {\varvec{{I}}}_K \diamond \left[ {\varvec{{{\mathcal {{\varvec{{{\mathcal {A}}}}}}}}}} \right] _{([1,2],[3])}\)
\({\varvec{{{\mathcal {D}}}}}_{(m,m,n,k)} = {\varvec{{A}}}_{(m,n,k)}\)	\(\left[ {\varvec{{{\mathcal {{\varvec{{{\mathcal {D}}}}}}}}}} \right] _{([3,4,2],[1])} = {\varvec{{I}}}_M \diamond \left[ {\varvec{{{\mathcal {{\varvec{{{\mathcal {A}}}}}}}}}} \right] _{([2,3],[1])}\)