Orthogonal Constant-Amplitude Sequence Families for System Parameter Identification in Spectrally Compact OFDM

Abstract

In rectangularly-pulsed orthogonal frequency division multiplexing (OFDM) systems, constant-amplitude (CA) sequences are desirable to construct preamble/pilot waveforms to facilitate system parameter identification (SPI). Orthogonal CA sequences are generally preferred in various SPI applications like random-access channel identification. However, the number of conventional orthogonal CA sequences (e.g., Zadoff–Chu sequences) that can be adopted in cellular communication without causing sequence identification ambiguity is insufficient. Such insufficiency causes heavy performance degradation for SPI requiring a large number of identification sequences. Moreover, rectangularly-pulsed OFDM preamble/pilot waveforms carrying conventional CA sequences suffer from large power spectral sidelobes and thus exhibit low spectral compactness. This paper is thus motivated to develop several order-I CA sequence families which contain more orthogonal CA sequences while endowing the corresponding OFDM preamble/pilot waveforms with fast-decaying spectral sidelobes. Since more orthogonal sequences are provided, the developed order-I CA sequence families can enhance the performance characteristics in SPI requiring a large number of identification sequences over multipath channels exhibiting short-delay channel profiles, while composing spectrally compact OFDM preamble/pilot waveforms.

Appendix

(A) Proof of Lemma 1: Consider two M-tuples $\textbf{1}_{m}$ and $\textbf{x}=[x_{m};m\in \mathcal {Z}_{M}]$ where all entries $x_{m}$ are integers greater than one and $\textbf{1}_{m}$ contains one at the m-th entry and $M-1$ zeros elsewhere. With (11), $f(\textbf{x}^{t}+k_{m} \textbf{1}_{m}^{t})-f(\textbf{x}^{t})$ is given by

$$\begin{aligned}&f(\textbf{x}^{t}+k_{m}\textbf{1}_{m}^{t})-f(\textbf{x}^{t}) \\&\quad =\frac{(x_{m}+k_{m})\prod _{i\ne m}x_{i}-1}{(x_{m}+k_{m}-1)\prod _{i\ne m}(x_{i}-1)}-\frac{\prod _{i}x_{i}-1}{ \prod _{i}(x_{i}-1)} \\&\quad =\frac{1}{\prod _{i\ne m}(x_{i}-1)}\left[ \frac{(x_{m}+k_{m})\prod _{i\ne m}x_{i}-1}{x_{m}+k_{m}-1}-\frac{\prod _{i}x_{i}-1}{ x_{m}-1}\right] \\&\quad =\frac{k_{m}(1-\prod _{i\ne m}x_{i})}{(x_{m}+k_{m}-1)\prod _{i}(x_{i}-1)} \end{aligned}$$

for $m\in \mathcal {Z}_{M}$ and it is negative when $k_{m}$ is a positive integer. Thus, $f(\textbf{x}^{t}+k_{m}\textbf{1}_{m}^{t})<f(\textbf{x}^{t})$ if the integer $k_{m}$ is positive and obviously $f(\textbf{x}^{t}+k_{m} \textbf{1}_{m}^{t})=f(\textbf{x}^{t})$ if $k_{m}=0$.

Next, define another M-tuple $\textbf{k}=\textbf{b}-\textbf{a}$ and express $\textbf{b}$ in terms of $\textbf{a}$ and $\textbf{k}$ as

$$\begin{aligned} \textbf{b}=\textbf{a}+\textbf{k}=\textbf{a}+\sum _{m=0}^{M-1}k_{m} \textbf{1}_{m} \end{aligned}$$

where all integer-valued entries $k_{m}$ in $\textbf{k}=[k_{m};m\in \mathcal { Z}_{M}]$ are nonnegative and all integer-valued entries $a_{m}$ and $b_{m}$ in $\textbf{a}=[a_{m};m\in \mathcal {Z}_{M}]$ and $\textbf{b}=[b_{m};m\in \mathcal {Z}_{M}]$ are greater than one. With $f(\textbf{x}^{t}+k_{m}\textbf{1 }_{m}^{t})<f(\textbf{x}^{t})$ for a positive $k_{m}$, we have

$$\begin{aligned} f(\textbf{b}^{t})\le f(\textbf{a}^{t}+\sum _{m=0}^{M-2}k_{m}\textbf{ 1}_{m}^{t})\le \cdots \le f(\textbf{a}^{t}+k_{0}\textbf{1}_{0}^{t})\le f( \textbf{a}^{t}). \end{aligned}$$

Thus, $f(\textbf{a}^{t})\ge f(\textbf{b}^{t})$ if $1<a_{n}\le b_{n}$ for all $n\in \mathcal {Z}_{M}$, and $f(\textbf{a}^{t})>f(\textbf{b}^{t})$ if $1<a_{n}<b_{n}$ for some $n\in \mathcal {Z}_{M}$ and $1<a_{m}\le b_{m}$ for all $m\in \mathcal {Z}_{M}-\{n\}$. This completes the proof.

(B) Proof of Lemma 2: With (11), $f([P_{a},P_{d}])\times f([P_{b},P_{c}])-f([P_{a},P_{c}])\times f([P_{b},P_{d}])$ is given by

$$\begin{aligned}&\frac{(P_{a}P_{d}-1)(P_{b}P_{c}-1)-(P_{a}P_{c}-1)(P_{b}P_{d}-1)}{ (P_{a}-1)(P_{b}-1)(P_{c}-1)(P_{d}-1)} \nonumber \\&\quad =\frac{P_{a}P_{c}+P_{b}P_{d}-P_{a}P_{d}-P_{b}P_{c}}{ (P_{a}-1)(P_{b}-1)(P_{c}-1)(P_{d}-1)} \nonumber \\&\quad =\frac{(P_{b}-P_{a})(P_{d}-P_{c})}{(P_{a}-1)(P_{b}-1)(P_{c}-1)(P_{d}-1)}. \end{aligned}$$

(17)

Similarly, $f([P_{a},P_{c}])\times f([P_{b},P_{d}])-f([P_{a},P_{b}])\times f([P_{c},P_{d}])$ is given by

$$\begin{aligned} \frac{(P_{d}-P_{a})(P_{c}-P_{b})}{(P_{a}-1)(P_{b}-1)(P_{c}-1)(P_{d}-1)}. \end{aligned}$$

(18)

When $1<P_{a}\le P_{b}\le P_{c}\le P_{d}$, (17) and (18) are both nonnegative. This completes the proof.

(C) Proof of Lemma 3: With (11), $f([P_{a},P_{b},P_{c}])-f([P_{a},P_{d}])\times f([P_{b},P_{c}])$ is given by

$$\begin{aligned}&\frac{P_{a}P_{b}P_{c}-1}{(P_{a}-1)(P_{b}-1)(P_{c}-1)} \nonumber \\&\qquad -\frac{(P_{a}P_{d}-1)(P_{b}P_{c}-1)}{(P_{a}-1)(P_{b}-1)(P_{c}-1)(P_{d}-1)} \nonumber \\&\quad =\frac{P_{a}P_{d}+P_{b}P_{c}-P_{a}P_{b}P_{c}-P_{d}}{ (P_{a}-1)(P_{b}-1)(P_{c}-1)(P_{d}-1)} \nonumber \\&\quad =\frac{(P_{a}-1)(P_{d}-P_{b}P_{c})}{(P_{a}-1)(P_{b}-1)(P_{c}-1)(P_{d}-1)} \end{aligned}$$

(19)

which is nonnegative when $P_{b}P_{c}\le P_{d}$ and $1<P_{a}\le P_{b}\le P_{c}\le P_{d}$. This completes the proof.

(D) Gosper’s Hack Algorithm: Gosper’s Hack algorithm in [37, 38] can assist in finding all possible factor sets $\{A_{m};m\in \mathcal {Z}_{\Omega (N)-\kappa }\}$ which satisfy $\prod _{m=0}^{\Omega (N)-\kappa -1}A_{m}=\prod _{m=0}^{\Omega (N)-1}P_{m}$ and are all characterized by an admissible pattern $\varvec{\omega }=[\omega _{m};m\in \mathcal {Z}_{\Omega (N)-\kappa }]$ with $\omega _{m}=\Omega (A_{m})$. To find all possible factor sets $\{A_{m};m\in \mathcal {Z}_{\Omega (N)-\kappa }\}$, we aim to (i) first find all possible partitions of $\{P_{m};m\in \mathcal {Z}_{\Omega (N)}\}$ into $\Omega (N)-\kappa$ prime factor subsets $\{P_{m}^{(n)};m\in \mathcal {Z}_{\omega _{n}}\}$ for $n\in \mathcal {Z}_{\Omega (N)-\kappa }$, where $P_{0}^{(n)}\le P_{1}^{(n)}\le \cdots \le P_{\omega _{n}-1}^{(n)}$, with the aid of Gosper’s Hack algorithm and (ii) then compose all possible factor sets by computing $A_{n}=\prod _{m=0}^{\omega _{n}-1}P_{m}^{(n)}$ accordingly. To describe step (i), we define $\widetilde{ \varvec{\omega }}=[\widetilde{\omega }_{n};n\in \mathcal {Z}_{\Omega (N)-\kappa }]$ with $\widetilde{\omega }_{n}\triangleq \sum _{m=n}^{\Omega (N)-\kappa -1}\omega _{m}$ and $\textbf{b}^{(n)}\triangleq [b_{m}^{(n)};m\in \mathcal {Z}_{\widetilde{\omega }_{n}}]$ as a binary codeword with length $\widetilde{\omega }_{n}$ and Hamming weight $\omega _{n}$.² For a given $\varvec{\omega }$, there are a total of $\prod _{n\in \mathcal {Z}_{\Omega (N)-\kappa }}\left( \begin{array}{c} \widetilde{\omega }_{n}\\ {\omega _{n}}\end{array}\right)$ possible binary codeword sets for $\{ \textbf{b}^{(n)};n\in \mathcal {Z}_{\Omega (N)-\kappa }\}$ and they can be exclusively obtained by Gosper’s Hack algorithm in Fig. 3 [38, Algorithm 3.1 ]. To obtain a partition of $\{P_{m};m\in \mathcal {Z}_{\Omega (N)}\}$ for each given $\{\textbf{b}^{(n)};n\in \mathcal {Z}_{\Omega (N)-\kappa }\}$, a binary codeword set $\{\widetilde{\textbf{b}}^{(n)};n\in \mathcal {Z}_{\Omega (N)-\kappa }\}$ is converted from $\{\textbf{b} ^{(n)};n\in \mathcal {Z}_{\Omega (N)-\kappa }\}$ by the proposed codeword conversion algorithm in Fig. 4, in a way that each codeword $\widetilde{ \textbf{b}}^{(n)}\triangleq [\widetilde{b}_{m}^{(n)};m\in \mathcal {Z}_{\Omega (N)}]$ contains $\Omega (N)$ entries and the same Hamming weight as $\textbf{b}^{(n)}$. Notably, there are a total of $\Omega (N)$ ones in $\{ \widetilde{\textbf{b}}^{(n)};n\in \mathcal {Z}_{\Omega (N)-\kappa }\}$. From $\{\widetilde{\textbf{b}}^{(n)};n\in \mathcal {Z}_{\Omega (N)-\kappa }\}$, a partition of $\{P_{m};m\in \mathcal {Z}_{\Omega (N)}\}$ into $\Omega (N)-\kappa$ prime factor subsets $\{P_{\widetilde{m}}^{(n)};\widetilde{m} \in \mathcal {Z}_{\omega _{n}}\}$ can be thus specified by

$$\begin{aligned} P_{\varepsilon _{m}^{(n)}}^{(n)}=P_{m}\quad \text {if}\quad \widetilde{b}_{m}^{(n)}=1 \end{aligned}$$

(20)

for $n\in \mathcal {Z}_{\Omega (N)-\kappa }$ and $m\in \Omega (N)$, where $\varepsilon _{m}^{(n)}=\sum _{m^{\prime }=0}^{m}\widetilde{b}_{m^{\prime }}^{(n)}-1$. Accordingly, all possible partitions of $\{P_{m};m\in \mathcal {Z}_{\Omega (N)}\}$ and thereby all possible factor sets for $\{A_{m};m\in \mathcal {Z}_{\Omega (N)-\kappa }\}$ can be found in steps (i) and (ii) from $\prod _{n\in \mathcal {Z}_{\Omega (N)-\kappa }}\left( \begin{array}{c}{\widetilde{\omega }_{n}}\\ {\omega _{n}}\end{array}\right)$ possible codeword sets for $\{\textbf{b}^{(n)};n\in \mathcal {Z}_{\Omega (N)-\kappa }\}$.

Fig. 3

Gosper’s Hack algorithm

Fig. 4

Codeword conversion algorithm

Consider the example with $\Omega (N)=6$, $\kappa =3$, and a given pattern $\varvec{\omega }=[3,2,1]^{t}$. Such $\varvec{\omega }$ determines $\widetilde{ \varvec{\omega }}= [6,3,1]^{t}$ uniquely and thus fixes the lengths 6, 3, 1 and Hamming weights 3, 2, 1 of the binary codeword set $\{\textbf{b}^{(0)}, \textbf{b}^{(1)},\textbf{b}^{(2)}\}$ accordingly. From Gosper’s Hack algorithm, there are $\left( \begin{array}{c} 6\\ 3\end{array}\right) \left( \begin{array}{c} 3\\ 2 \end{array}\right) \left( \begin{array}{c} 1 \\ 1\end{array}\right) =60$ possible codeword sets meeting such length and weight distributions. For example, $\textbf{b}^{(0)}=[0,1,0, 1,1,0]^{t}$, $\textbf{b}^{(1)}=[0,1,1]^{t}$ and $\textbf{b}^{(2)}=[1]$ form one possible codeword set. From the codeword conversion algorithm, the corresponding codeword set $\{\widetilde{\textbf{b} }^{(n)};n\in \mathcal {Z}_{\Omega (N)-\kappa }\}$ is obtained as $\widetilde{ \textbf{b}}^{(0)}=[0,1,0,1,1,0]^{t}$, $\widetilde{\textbf{b}} ^{(1)}=[0,0,1,0,0,1]^{t}$ and $\widetilde{\textbf{b}} ^{(2)}=[1,0,0,0,0,0]^{t}$. In turns, such $\{\widetilde{\textbf{b}} ^{(n)};n\in \mathcal {Z}_{\Omega (N)-\kappa }\}$ determines a partition of $\{P_{m};m\in \mathcal {Z}_{\Omega (N)}\}$ into $\{P_{m}^{(0)};m\in \mathcal {Z} _{\omega _{0}}\}=\{P_{1},P_{3},P_{4}\}$, $\{P_{m}^{(1)};m\in \mathcal {Z} _{\omega _{1}}\}=\{P_{2},P_{5}\}$, and $\{P_{m}^{(2)};m\in \mathcal {Z} _{\omega _{2}}\}=\{P_{0}\}$. The corresponding $\{A_{m};m\in \mathcal {Z} _{\Omega (N)-\kappa }\}$ becomes $\{P_{1}P_{3}P_{4},P_{2}P_{5},P_{0}\}$. All 60 possible partitions can be thus obtained from 60 codeword sets $\{ \textbf{b}^{(0)},$ $\textbf{b}^{(1)},\textbf{b}^{(2)}\}$ exclusively obtained by Gosper’s Hack algorithm.