Paraunitary generation/correlation of QAM complementary sequence pairs

Abstract

A unique decomposition of arbitrary pairs of complementary sequences (including standard binary, polyphase and QAM sequences as well as non-standard sequences and kernels) based on paraunitary matrices is presented. This decomposition allows us to describe the internal structure of any sequence pair of length L using basic paraunitary matrices defined by an ordered set of L complex coefficients named the omega vector. When the omega vector is sparse, the canonic form is compact and leads to an efficient implementation of a generator/correlator. We show that sequences generated by the standard algorithm have the sparsest known omega vector (log₂ L non-zero elements out of L) and, thus, the most efficient generator/correlator. The equivalence of paraunitary matrices and Z transforms of complementary sequences allows us to apply the rich results from the theory of perfect reconstruction filter-banks to the field of sequence design. We introduce a new generator/correlator algorithm for sequences in standard and non-standard QAM constellations that is based on this equivalence. Both rectangular and hexagonal constellations are considered and the cardinality of the generated set of unique complementary sequences is either determined or estimated for a number of important cases. We show, in the case of the standard 16-QAM constellation, that the paraunitary algorithm generates the same number of sequences as the published algorithms based on generalized Boolean functions. In the case of 64-QAM, the proposed algorithm generates more sequences than known algorithms. We introduce an algorithm for generating 256-QAM sequences and derive a tight upper bound on the number of generated sequences.

Appendices

Appendix A

Here we prove that the decomposition algorithm shortens a pair of sequences by 1. We consider the transformation (23):

$$ \left[\begin{array}{l}A\prime \left({Z}^{-1}\right)\hfill \\ {}B\prime \left({Z}^{-1}\right)\hfill \end{array}\right]=\left[\begin{array}{ll}1\hfill & 0\hfill \\ {}0\hfill & Z\hfill \end{array}\right]\cdot \left[\begin{array}{ll}1\hfill & -\varOmega \hfill \\ {}{\varOmega}^{*}\hfill & 1\hfill \end{array}\right]\cdot \left[\begin{array}{l}A\left({Z}^{-1}\right)\hfill \\ {}B\left({Z}^{-1}\right)\hfill \end{array}\right] $$

(62)

where a and b are complementary sequences of length L represented by their Z transforms A(Z ⁻¹) and B(Z ⁻¹) and Ω = −b ^*(0)/a ^*(0). As the sequences are complementary: R _a (L−1) + R _b (L−1) = 0 so a ^*(0)·a(L−1) + a ^*(0)·a(L−1) = 0 and finally Ω = −b ^*(0)/a ^*(0) = a(L−1)/b(L−1)

We can write the transformed complementary pair as:

$$ \left[\begin{array}{l}A\prime \left({Z}^{-1}\right)\hfill \\ {}B\prime \left({Z}^{-1}\right)\hfill \end{array}\right]=\left[\begin{array}{l}A\left({Z}^{-1}\right)-B\left({Z}^{-1}\right)\cdot a\left(L-1\right)/b\left(L-1\right)\hfill \\ {}Z\cdot A\left({Z}^{-1}\right)\cdot {a}^{*}\left(L-1\right)/{b}^{*}\left(L-1\right)+Z\cdot B\left({Z}^{-1}\right)\hfill \end{array}\right] $$

(63)

We first determine A′(Z ⁻¹):

$$ \begin{array}{c}\hfill A\prime ={\displaystyle \sum_{n=0}^{L-1}a(n)\cdot {Z}^{-\mathrm{n}}-{\displaystyle \sum_{n=0}^{L-1}b(n)\cdot a\left(L-1\right)/b\left(L-1\right)\cdot {Z}^{-\mathrm{n}}}}\hfill \\ {}\hfill ={\displaystyle \sum_{n=0}^{L-1}\left(a(n)-b(n)\cdot a\left(L-1\right)/b\left(L-1\right)\right)\cdot {Z}^{-\mathrm{n}}={\displaystyle \sum_{n=0}^{L-2}\left(a(n)\cdot b(n)\cdot a\left(L-1\right)/b\left(L-1\right)\right)\cdot {Z}^{-\mathrm{n}}}}\hfill \end{array} $$

(64)

as (a(L−1)−b(N−1)·a(L−1)/b(L−1))·Z ^−L+1 = 0 so the length of the a′ sequence is L−1. In a similar way:

$$ B\prime ={\displaystyle \sum_{n=0}^{L-1}\left(a(n)\cdot {a}^{*}\left(L-1\right)/{b}^{*}\left(L-1\right)+b(n)\right)\cdot {Z}^{-\mathrm{n}+1}={\displaystyle \sum_{n=1}^{L-1}\left(a(n)\cdot {a}^{*}\left(L-1\right)/{b}^{*}\left(L-1\right)+b(n)\right)\cdot {Z}^{-\mathrm{n}+1}}} $$

(65)

(a(0)·a ^*(L−1)/b ^*(L−1) + b(0))·Z ⁻ⁿ = 0 because (a(0)·a ^*(L−1) + b(0)·b(L−1) = 0 from (9) and so

$$ B\prime ={\displaystyle \sum_{n=0}^{L-2}\left\{a\left(n+1\right)\cdot {a}^{*}\left(L-1\right)/{b}^{*}\left(L-1\right)+b\left(n+1\right)\right\}\cdot {Z}^{-n}} $$

(66)

which represents a sequence of length L−1.

Appendix B

Here we prove that the standard algorithm produces no overlapping of polynomials thus does not require any additions in the generating process. Let us consider the generating matrix:

$$ {\mathbf{\mathcal{M}}}_N\left({Z}^{-1}\right)={\boldsymbol{U}}_N\cdot {\boldsymbol{D}}^{P_N}\cdot {\boldsymbol{U}}_{N-1}\cdot {\boldsymbol{D}}^{P_{N-1}}\dots {\boldsymbol{U}}_1\cdot {\boldsymbol{D}}^{P_1}\cdot {\boldsymbol{U}}_0 $$

(67)

where U _m are wide sense unitary matrices of the form $\left[\begin{array}{ll}{C}_m\hfill & -{S}_m^{*}\hfill \\ {}{S}_m\hfill & {C}_m^{*}\hfill \end{array}\right],\boldsymbol{D}=\left[\begin{array}{ll}1\hfill & 0\hfill \\ {}0\hfill & {Z}^{-1}\hfill \end{array}\right]$ and [P ₁, P ₂, …, P _N ] are Standard delays. Its recursive form is:

$$ \begin{array}{l}{\mathbf{\mathcal{M}}}_0\left({Z}^{-1}\right)={\boldsymbol{U}}_0\hfill \\ {}{\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)={\boldsymbol{U}}_m\cdot {\boldsymbol{D}}^{P_m}\cdot {\mathbf{\mathcal{M}}}_{m-1}\left({Z}^{-1}\right)\hfill \end{array} $$

(68)

where ${\boldsymbol{U}}_m\cdot {\boldsymbol{D}}^{P_m}=\left[\begin{array}{ll}{C}_m\hfill & -{S}_m^{*}{Z}^{-{P}_m}\hfill \\ {}{S}_m\hfill & {C}_m^{*}{Z}^{-{P}_m}\hfill \end{array}\right]=\left[\begin{array}{ll}{C}_m\hfill & 0\hfill \\ {}{S}_m\hfill & 0\hfill \end{array}\right]+\left[\begin{array}{ll}0\hfill & -{S}_m^{*}\hfill \\ {}0\hfill & {C}_m^{*}\hfill \end{array}\right]\cdot {Z}^{-{P}_m}={\boldsymbol{\varTheta}}_m+{\boldsymbol{\varPhi}}_m\cdot {Z}^{-{P}_m}$ and ${\boldsymbol{\varTheta}}_m=\left[\begin{array}{ll}{C}_m\hfill & 0\hfill \\ {}{S}_m\hfill & 0\hfill \end{array}\right]\kern0.5em \mathrm{and}\kern0.5em {\boldsymbol{\varPhi}}_m=\left[\begin{array}{ll}0\hfill & -{S}_m^{*}\hfill \\ {}0\hfill & {C}_m^{*}\hfill \end{array}\right]$. Hence:

$$ {\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)={\displaystyle \prod_{k=m}^1\left({\boldsymbol{\varTheta}}_k+{\boldsymbol{\varPhi}}_k\cdot {Z}^{-{P}_k}\right)\cdot {\boldsymbol{U}}_0={\displaystyle \sum_{i=1}^{2^m}{\boldsymbol{\varPsi}}_{L_i}{Z}^{-{L}_i}}} $$

(69)

(69) is a product of m binomial elements. Each of its 2^m expansion terms is an m-element product where each element involves the first or the second binomial term. Here,

$$ {L}_i={\displaystyle \sum_{k=1}^m{\beta}_{ki}{P}_k} $$

where β _ki  = 1 when the second binomial term is the k-th product element of the i-th term in (65), otherwise β _ki  = 0, for i = 1, …, 2^m. In fact, L _i is a unique number with [β _1i , β _2i , …, β _mi ] being its binary representation in terms of the binary powers in [P ₁, P ₂, …, P _m ]. The next iteration

$$ {\mathbf{\mathcal{M}}}_{m+1}\left({Z}^{-1}\right)={\boldsymbol{\varTheta}}_{m+1}\cdot {\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)+{\boldsymbol{\varPhi}}_{m+1}\cdot {\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)\cdot {Z}^{-{P}_{m+1}} $$

(70)

always adds different powers of Z ⁻¹ to ${\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)$ as explained next. Note that

$$ {\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)\cdot {Z}^{-{P}_{m+1}}={\displaystyle \sum_{i=1}^{2^m}{\boldsymbol{\varPsi}}_{L_i}{Z}^{-{L}_i-{P}_{m+1}}} $$

(71)

does not share any powers of Z ⁻¹ with ${\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)$. Since P _m+1 is absent from [P ₁, P ₂, …, P _m ]. its corresponding binary expansion element of L _i is zero. On the other hand, the polynomial in (71) has exponents of the form L _i + P _m+1, which are always different from L _i . Hence, the addition in (70) results in a union of polynomial terms that corresponds to interleaving of sequences elements in the time domain. Consequently, we have established that the sum of the delayed and the original pair corresponds to interleaving of sequences, without additions.

Note also that at iteration m:

$$ {\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)=\left[\begin{array}{ll}{A}_m\left({Z}^{-1}\right)\hfill & -{\left\{{B}_m^R(Z)\right\}}^{*}\hfill \\ {}{B}_m\left({Z}^{-1}\right)\hfill & {\left\{{A}_m^R(Z)\right\}}^{*}\hfill \end{array}\right]. $$

It follows that

$$ {\boldsymbol{\varTheta}}_{m+1}\cdot {\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)=\left[\begin{array}{ll}{C}_m{A}_m\left({Z}^{-1}\right)\hfill & -{C}_m{\left\{{B}_m^R(Z)\right\}}^{*}\hfill \\ {}{S}_m{A}_m\left({Z}^{-1}\right)\hfill & -{S}_m{\left\{{B}_m^R(Z)\right\}}^{*}\hfill \end{array}\right]\kern0.5em \mathrm{and}\kern0.5em {\boldsymbol{\varPhi}}_{m+1}\cdot {\mathbf{\mathcal{M}}}_m\left({Z}^{-1}\right)=\left[\begin{array}{ll}-{S}_m^{*}{B}_m\left({Z}^{-1}\right)\hfill & -{S}_m^{*}{\left\{{A}_m^R(Z)\right\}}^{*}\hfill \\ {}{C}_m^{*}{B}_m\left({Z}^{-1}\right)\hfill & {C}_m^{*}{\left\{{A}_m^R(Z)\right\}}^{*}\hfill \end{array}\right] $$

also require no additions but only multiplications. In fact, in the case of polyphase sequences, each term requires a phase shift only, instead of a multiplication.

Appendix C

Here we determine the canonic form for the PU QAM sequences with one QAM matrix using the following steps:

• 1. First we represent the QAM-U matrix as a product of an omega matrix and a diagonal matrix:

  $$ \boldsymbol{U}=\left[\begin{array}{ll}C\hfill & -{S}^{*}\hfill \\ {}S\hfill & {C}^{*}\hfill \end{array}\right]=\left[\begin{array}{ll}1\hfill & -{S}^{*}/{C}^{*}\hfill \\ {}S/C\hfill & 1\hfill \end{array}\right]\left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]=\boldsymbol{\varOmega} \cdot \boldsymbol{C} $$

  (72)

• 2. Next we apply the quasi-commutativity property:

  $$ \boldsymbol{C}\cdot \boldsymbol{\varOmega} =\left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]\left[\begin{array}{ll}1\hfill & -{\varOmega}^{*}\hfill \\ {}\varOmega \hfill & 1\hfill \end{array}\right]=\left[\begin{array}{ll}1\hfill & -{\left(\varOmega \cdot {C}^{*}/C\right)}^{*}\hfill \\ {}\varOmega \cdot {C}^{*}/C\hfill & 1\hfill \end{array}\right]\left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]=\left[\begin{array}{ll}1\hfill & -{\varOmega}_1^{*}\hfill \\ {}{\varOmega}_1\hfill & 1\hfill \end{array}\right]\left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]={\boldsymbol{\varOmega}}_{\mathbf{1}}\cdot \boldsymbol{C} $$

  (73)

  (where ${\varOmega}_1$Ω₁ = $\varOmega \cdot$Ω⋅C ^*/C) in order to move the C matrix to the right side of a PU product past the omega matrix.

• 3. We use the fact that diagonal matrices are commutative to move the C matrix to the right past the D matrix:

  $$ \boldsymbol{C}\cdot \boldsymbol{D}=\boldsymbol{D}\cdot \boldsymbol{C}\kern0.5em \mathrm{or}\kern0.5em \left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]\cdot \left[\begin{array}{ll}1\hfill & 0\hfill \\ {}0\hfill & {Z}^{-1}\hfill \end{array}\right]=\left[\begin{array}{ll}1\hfill & 0\hfill \\ {}0\hfill & {Z}^{-1}\hfill \end{array}\right]\cdot \left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right] $$

  (74)

• 4. We repeat steps 2 and 3 until C appear at the right side of the PU product in (37). We can note that the:

  $$ {\varOmega}_K=S/C\kern0.5em \mathrm{and}\kern0.5em {\varOmega}_n={C}^{*}/C\cdot {w}_n $$

  (75)

Next, we give a detailed derivation of the canonic form:

$$ {\boldsymbol{X}}_{(QAM)}={\boldsymbol{G}}_{(QAM)}\left({Z}^{-1}\right)\cdot \left[\begin{array}{l}1\hfill \\ {}0\hfill \end{array}\right]=w\cdot {\boldsymbol{W}}_N\cdot {\boldsymbol{D}}^{2^{P_N}}\cdot {\boldsymbol{W}}_{N-1}\cdot {\boldsymbol{D}}^{2^{P_{N-1}}}\cdot \dots \cdot {\boldsymbol{U}}_{(K)}\cdot \dots \cdot {\boldsymbol{W}}_1\cdot {\boldsymbol{D}}^{2^{P_1}}\cdot {\boldsymbol{W}}_0\cdot \left[\begin{array}{l}1\hfill \\ {}0\hfill \end{array}\right] $$

(76)

Where ${\boldsymbol{U}}_{(K)}=\left[\begin{array}{ll}C\hfill & {w}_K\cdot S\hfill \\ {}-\left({w}_K\cdot S\right)\hfill & {C}^{*}\hfill \end{array}\right]$ and K indicates the position of the U _(K) matrix in the product. In fact, it replaces the W _K matrix at that position.

$$ \begin{array}{c}\hfill {\boldsymbol{X}}_{(QAM)}=w\cdot {\boldsymbol{W}}_N\cdot {\boldsymbol{D}}^{2^{P_N}}\cdot \dots \cdot \left[\begin{array}{ll}1\hfill & -{w}_K^{*}{S}^{*}/{C}^{*}\hfill \\ {}{w}_KS/C\hfill & 1\hfill \end{array}\right]\cdot \left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]\cdot {\boldsymbol{D}}^{2^{P_{K*1}}}\cdot {\boldsymbol{W}}_{K-1}\cdot \dots \cdot {\boldsymbol{D}}^{2^{P_1}}\cdot {\boldsymbol{W}}_0\cdot \left[\begin{array}{l}1\hfill \\ {}0\hfill \end{array}\right]=\hfill \\ {}\hfill {\boldsymbol{X}}_{(QAM)}=w\cdot {\boldsymbol{W}}_N\cdot {\boldsymbol{D}}^{2^{P_N}}\cdot \dots \cdot \left[\begin{array}{ll}1\hfill & -{w}_K^{*}{S}^{*}/{C}^{*}\hfill \\ {}{w}_KS/C\hfill & 1\hfill \end{array}\right]\cdot {\boldsymbol{D}}^{2^{P_{K*1}}}\cdot \left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]\cdot \left[\begin{array}{ll}1\hfill & -{w}_{K-1}^{*}\hfill \\ {}{w}_{K-1}\hfill & 1\hfill \end{array}\right]\cdot \dots \cdot {\boldsymbol{D}}^{2^{P_1}}\cdot {\boldsymbol{W}}_0\cdot \left[\begin{array}{l}1\hfill \\ {}0\hfill \end{array}\right]\hfill \\ {}\hfill {\boldsymbol{X}}_{(QAM)}=w\cdot {\boldsymbol{W}}_N\cdot {\boldsymbol{D}}^{2^{P_N}}\cdot \dots \cdot \left[\begin{array}{ll}1\hfill & -{w}_K^{*}{S}^{*}/{C}^{*}\hfill \\ {}{w}_KS/C\hfill & 1\hfill \end{array}\right]\cdot {\boldsymbol{D}}^{2^{P_{K*1}}}\cdot \left[\begin{array}{ll}1\hfill & -{\left({C}^{*}/C{w}_{K-1}\right)}^{*}\hfill \\ {}{C}^{*}/C{w}_{K-1}\hfill & 1\hfill \end{array}\right]\left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]\cdot \dots \cdot {\boldsymbol{D}}^{2^{P_1}}\cdot {\boldsymbol{W}}_0\cdot \left[\begin{array}{l}1\hfill \\ {}0\hfill \end{array}\right]\hfill \end{array} $$

with a = S/C a β = C ^*/C.

$$ \begin{array}{c}\hfill {\boldsymbol{X}}_{(QAM)}=w\cdot {\boldsymbol{W}}_N\cdot {\boldsymbol{D}}^{2^{P_N}}\cdot \dots \cdot \left[\begin{array}{ll}1\hfill & -{w}_K^{*}{\alpha}^{*}\hfill \\ {}{w}_K\alpha \hfill & 1\hfill \end{array}\right]\cdot {\boldsymbol{D}}^{2^{P_{K*1}}}\cdot \left[\begin{array}{ll}1\hfill & -{\left({w}_{K-1}\cdot \beta \right)}^{*}\hfill \\ {}{w}_{K-1}\cdot \beta \hfill & 1\hfill \end{array}\right]\cdot \left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]\cdot \dots \cdot {\boldsymbol{D}}^{2^{P_1}}\cdot {\boldsymbol{W}}_0\cdot \left[\begin{array}{l}1\hfill \\ {}0\hfill \end{array}\right]\hfill \\ {}\hfill {\boldsymbol{X}}_{(QAM)}=w{\boldsymbol{W}}_N{\boldsymbol{D}}^{2^{P_N}}\dots \left[\begin{array}{ll}1\hfill & -{w}_K^{*}{\alpha}^{*}\hfill \\ {}{w}_K\alpha \hfill & 1\hfill \end{array}\right]{\boldsymbol{D}}^{2^{P_{K*1}}}\left[\begin{array}{ll}1\hfill & -{\left({w}_{K-1}\cdot \beta \right)}^{*}\hfill \\ {}{w}_{K-1}\cdot \beta \hfill & 1\hfill \end{array}\right]\dots {\boldsymbol{D}}^{2^{P_1}}\left[\begin{array}{ll}1\hfill & -{\left({w}_0\cdot \beta \right)}^{*}\hfill \\ {}{w}_0\cdot \beta \hfill & 1\hfill \end{array}\right]\left[\begin{array}{ll}C\hfill & 0\hfill \\ {}0\hfill & {C}^{*}\hfill \end{array}\right]\left[\begin{array}{l}1\hfill \\ {}0\hfill \end{array}\right]\hfill \\ {}\hfill {\boldsymbol{X}}_{(QAM)}=w\cdot C\cdot {\boldsymbol{W}}_N\cdot {\boldsymbol{D}}^{2^{P_N}}\dots \left[\begin{array}{ll}1\hfill & -{w}_K^{*}{\alpha}^{*}\hfill \\ {}{w}_K\alpha \hfill & 1\hfill \end{array}\right]{\boldsymbol{D}}^{2^{P_{K*1}}}\left[\begin{array}{ll}1\hfill & -{\left({w}_{K-1}\beta \right)}^{*}\hfill \\ {}{w}_{K-1}\beta \hfill & 1\hfill \end{array}\right]\dots {\boldsymbol{D}}^{2^{P_1}}\left[\begin{array}{ll}1\hfill & -{\left({w}_0\beta \right)}^{*}\hfill \\ {}{w}_0\beta \hfill & 1\hfill \end{array}\right]\cdot \left[\begin{array}{l}1\hfill \\ {}0\hfill \end{array}\right]\hfill \end{array} $$

Thus, the non-zero elements of the omega vector are:

$$ {\boldsymbol{\Omega}}_{(QAM)}=\left[\underset{K}{\underbrace{\beta {w}_0,\kern1em \beta {w}_1,\kern1em \dots, \kern1em \beta {w}_{K-1}}},\alpha {w}_K,\underset{N-K}{\underbrace{w_{K+1},\dots {w}_{N-1},\kern1em {w}_N}}\right) $$

(77)

$$ {\boldsymbol{\Omega}}_{(QAM)}=\left[\underset{K}{\underbrace{w_0{C}^{*}/C,{w}_1{C}^{*}/C,\dots, {w}_{K-1}{C}^{*}/C}},{w}_KS/C,\underset{N-K}{\underbrace{w_{K+1},\dots {w}_{N-1},{w}_N}}\right) $$

(78)

The canonic form of the QAM complementary sequences is defined by:

• 1. Multiplicative constant: w·C

• 2. Standard delays: [P ₁, P ₂, …, P _N ]

• 3. Non-zero elements of the omega vector: Ω_(QAM ) = [Ω _S(0), Ω _S(1), Ω _S(2), …, Ω _S(N)] according (31).