BS-ACEBOC: a generalized low-complexity dual-frequency constant-envelope multiplexing modulation for GNSS

Abstract

In order to satisfy the need for combining signals in a constant-envelope modulation, we propose a new dual-frequency constant-envelope modulation (DCEM), called asymmetric constant-envelope binary offset carrier multiplexing modulation with bipolar subcarrier (BS-ACEBOC). It can combine up to four signal components from two separate sidebands with low complexity and an arbitrary power ratio between signal components. We briefly introduce the design principles and address the construction method of the BS-ACEBOC. The high flexibility of the proposed method is proven, and several special cases for practical applications are presented. The complexity comparison for the signal generation shows that the computational requirements of BS-ACEBOC are significantly lower than that of other DCEMs. Finally, the performance of BS-ACEBOC is analyzed. With low complexity and high flexibility, BS-ACEBOC is a promising solution for signal design of global navigation satellite systems including Beidou Phase III.

Appendix: The PSD of BS-ACEBOC([1, β 2, 1, β 2])

Under the assumption that the signal is stationary in wide sense and that the PRN codes have ideal correlation characteristics, the products of signal components can also be regarded as spreading signals. The autocorrelation function of BS-ACEBOC can be shown as

$${\mathcal{R}}_{{\text{BS-ACEBOC} }} (\tau ) = \left\{ \begin{aligned} {\mathcal{R}}_{{{\text{UI}},\tilde{\gamma }_{sc,\text{U}} }} (\tau ) + \beta^2 {\mathcal{R}}_{{{\text{UQ}},\tilde{\gamma }_{sc,\text{U}} }} (\tau ) + {\mathcal{R}}_{{{\text{LI}},\tilde{\gamma }_{sc,\text{L}} }} (\tau ) + \beta^4 {\mathcal{R}}_{{{\text{LQ}},\tilde{\gamma }_{sc,\text{L}} }} (\tau ) + \\ \beta^{4} {\bar{\mathcal{R}}}_{{{\text{UI}},\tilde{\gamma }_{sc,\text{U}} }} (\tau ) + \beta^2 {\bar{\mathcal{R}}}_{{{\text{UQ}},\tilde{\gamma }_{sc,\text{U}} }} (\tau ) + \beta^{4} {\bar{\mathcal{R}}}_{{{\text{LI}},\tilde{\gamma }_{sc,\text{L}} }} (\tau ) + \beta^2 {\bar{\mathcal{R}}}_{{{\text{LQ}},\tilde{\gamma }_{sc,\text{L}} }} (\tau ) \\ \end{aligned} \right.$$

(40)

where ${\mathcal{R}}_{{m,\tilde{\gamma }_{sc,n} }} (\tau )$ and ${\bar{\mathcal{R}}}_{{p,\tilde{\gamma }_{sc,q} }} (\tau )$ stand for the autocorrelation function (ACF) of $b_{m} (t) \cdot \tilde{\gamma }_{sc,n} (t)$ and $\bar{b}_{p} (t) \cdot \tilde{\gamma }_{sc,q} (t)$, respectively, with m, p ∊ {UI, UQ, LI, LQ} and n, q ∊ {U, I}.

Since all the signal components are BPSK-R(n) modulated with the same code rate, the ACF of BS-ACEBOC $([1,\beta^{2} ,1,\beta^{2} ])$ can be simplified as

$${\mathcal{R}}_{{\text{BS - ACEBOC}}} (\tau ) = 2\left( {1 + 2\beta^2 + \beta^{4} } \right)\left[ {{\mathcal{R}}_{{{\text{UI}},\tilde{\gamma }_{{sc,{\text{U}}}} }} (\tau ) + {\mathcal{R}}_{{{\text{LI}},\tilde{\gamma }_{{sc,{\text{L}}}} }} (\tau )} \right]$$

(41)

and the corresponding PSD can be obtained via the Fourier transforms (FT) of ${\mathcal{R}}_{{\text{BS - ACEBOC}}} (\tau )$, which is

$$G_{{\text{BS - ACEBOC}}} (f) = 2\left( {1 + 2\beta^2 + \beta^{4} } \right)\left[ {G_{{{\text{UI}},\tilde{\gamma }_{{sc,{\text{U}}}} }} (f) + G_{{{\text{LI}},\tilde{\gamma }_{{sc,{\text{L}}}} }} (f)} \right]$$

(42)

where $G_{{{\text{UI}},\tilde{\gamma }_{{sc,{\text{U}}}} }} (f)$ and $G_{{{\text{LI}},\tilde{\gamma }_{{sc,{\text{L}}}} }} (f)$ are the FTs of ${\mathcal{R}}_{{{\text{UI}},\tilde{\gamma }_{{sc,{\text{U}}}} }} (\tau )$ and ${\mathcal{R}}_{{{\text{LI}},\tilde{\gamma }_{{sc,{\text{L}}}} }} (\tau )$, respectively.

Note that the upper sideband complex subcarrier piecewise can be expressed as

$$\tilde{\gamma }_{{sc,{\text{U}}}} (t) = \sum\limits_{k = 0}^{\varPhi - 1} {( - 1)^{k} \mu_{{T_{s} /2}} \left( {t - k\frac{{T_{s} }}{2}} \right)}$$

(43)

where

$${\mu _{{T_s}/2}}\left( t \right) = \left\{ {\begin{array}{ll}{1 + {\mathop{\rm j}\nolimits} }&{\left[ {0,\frac{{{T_s}}}{4}} \right)}\\ { - 1 + {\mathop{\rm j}\nolimits} }&{\left[ {\frac{{{T_s}}}{4},\frac{{{T_s}}}{2}} \right)} \end{array}} \right.$$

(44)

The FT of $\tilde{\gamma }_{{sc,{\text{U}}}} (t)$ is

$$\tilde{\varGamma }_{{sc,{\text{U}}}} (f) = \frac{{\text{e}^{{ - \text{j} \pi f\frac{{T_{s} }}{2}}} }}{{2\text{j} \pi f}}\left\{ {2\cos \left( {\pi f\frac{{T_{s} }}{2}} \right) - 2\sin \left( {\pi f\frac{{T_{s} }}{2}} \right) - 2} \right\}\sum\limits_{k = 0}^{\varPhi - 1} {( - 1)^{k} \text{e}^{{ - \text{j} \pi fkT_{s} }} }$$

(45)

Thus, $G_{{{\text{UI}},\tilde{\gamma }_{{sc,{\text{U}}}} }} (f)$ can be expressed as

$$G_{{s,\tilde{\gamma }_{{sc,{\text{U}}}} }} (f) = f_{c} \left| {\tilde{\varGamma }_{{sc,{\text{U}}}} (f)} \right|^{2} = \frac{{f_{c} }}{{\pi^{2} f^{2} }}\left\{ {\cos \left( {\pi f\frac{{T_{s} }}{2}} \right) - \sin \left( {\pi f\frac{{T_{s} }}{2}} \right) - 1} \right\}^{2} \left\| {\sum\limits_{k = 0}^{\varPhi - 1} {( - 1)^{k} \text{e}^{{ - \text{j} \pi fkT_{s} }} } } \right\|^{2}$$

(46)

where $\varPhi = 2f_{s} /f_{c}$. It is easy to recognize that the value of $G_{{s,\tilde{\gamma }_{{sc,{\text{U}}}} }} (f)$ will be different depending on whether $\varPhi$ is even or odd. If $\varPhi$ is even, $G_{{{\text{UI}},\tilde{\gamma }_{{sc,{\text{U}}}} }} (f)$ can be further simplified as

$$G_{{{\text{UI}},\tilde{\gamma }_{{sc,{\text{U}}}} }} (f) = \frac{{f_{c} }}{{\pi^{2} f^{2} }}\left\{ {\cos \left( {\pi f\frac{{T_{s} }}{2}} \right) - \sin \left( {\pi f\frac{{T_{s} }}{2}} \right) - 1} \right\}^{2} \frac{{\sin^{2} \left( {\frac{\pi f}{{f_{c} }}} \right)}}{{\cos^{2} \left( {\frac{\pi f}{{2f_{s} }}} \right)}}$$

(47)

while if $\varPhi$ is odd, $G_{{\text{UI}},\tilde{\gamma }_{\text{sc,U}} } (f)$ is

$$G_{{{\text{UI}},\tilde{\gamma }_{{sc,{\text{U}}}} }} (f) = \frac{{f_{c} }}{{\pi^{2} f^{2} }}\left\{ {\cos \left( {\pi f\frac{{T_{s} }}{2}} \right) - \sin \left( {\pi f\frac{{T_{s} }}{2}} \right) - 1} \right\}^{2} \frac{{\cos^{2} \left( {\frac{\pi f}{{f_{c} }}} \right)}}{{\cos^{2} \left( {\frac{\pi f}{{2f_{s} }}} \right)}}$$

(48)

In a similar way, it can be shown that $G_{{{\text{LI}},\tilde{\gamma }_{sc,\text{L}} }} (f)$ can be expressed as

$$G_{{{\text{LI}},\tilde{\gamma }_{{sc,{\text{L}}}} }} (f) = \left| {\tilde{\gamma }_{{sc,{\text{L}}}} (f)} \right|^{2} = \left\{ \begin{aligned} \frac{{f_{c} }}{{\pi^{2} f^{2} }}\left[ {\cos \left( {\frac{\pi f}{{2f_{s} }}} \right){ + }\sin \left( {\frac{\pi f}{{2f_{s} }}} \right) - 1} \right]^{2} \frac{{\sin^{2} \left( {\frac{\pi f}{{f_{c} }}} \right)}}{{\cos^{2} \left( {\frac{\pi f}{{2f_{s} }}} \right)}}\quad \varPhi \,{\text{is}}\,{\text{even}} \hfill \\ \frac{{f_{c} }}{{\pi^{2} f^{2} }}\left[ {\cos \left( {\frac{\pi f}{{2f_{s} }}} \right){ + }\sin \left( {\frac{\pi f}{{2f_{s} }}} \right) - 1} \right]^{2} \frac{{\cos^{2} \left( {\frac{\pi f}{{f_{c} }}} \right)}}{{\cos^{2} \left( {\frac{\pi f}{{2f_{s} }}} \right)}}\quad \varPhi \,{\text{is}}\,{\text{odd}} \hfill \\ \end{aligned} \right.$$

(49)

Substituting (47), (48) and (49) into (42), we derive the PSD of BS-ACEBOC $([1,\beta^{2} ,1,\beta^{2} ])$, which can be normalized as

$$G_{{\text{BS - ACEBOC}}} (f) = \left\{ \begin{aligned} \frac{{f_{c} }}{{\pi^{2} f^{2} }}\left[ {1 - \cos \left( {\frac{\pi f}{{2f_{s} }}} \right)} \right]\frac{{\sin^{2} \left( {\frac{\pi f}{{f_{c} }}} \right)}}{{\cos^{2} \left( {\frac{\pi f}{{2f_{s} }}} \right)}}\quad \varPhi \,{\text{is}}\,{\text{even}} \hfill \\ \frac{{f_{c} }}{{\pi^{2} f^{2} }}\left[ {1 - \cos \left( {\frac{\pi f}{{2f_{s} }}} \right)} \right]\frac{{\cos^{2} \left( {\frac{\pi f}{{f_{c} }}} \right)}}{{\cos^{2} \left( {\frac{\pi f}{{2f_{s} }}} \right)}}\quad \varPhi \,{\text{is}}\,{\text{odd}} \hfill \\ \end{aligned} \right.$$

(50)