Binary offset carrier modulation^[1][2] (BOC modulation) was developed by John Betz in order to allow interoperability of satellite navigation systems. It is currently used in the US GPS system, IndianIRNSS system and inGalileo^[3] and is a square sub-carriermodulation, where a signal is multiplied by a rectangularsub-carrier of frequency${\displaystyle f_{\text{sc}}}$ equal to or greater than thechip rate. Following thissub-carriermultiplication, thespectrum of the signal is divided into two parts, therefore BOC modulation is also known as a split-spectrum modulation. Their major advantages are, that one can shape the spectrum to allow inter-system-compatibility and better theoretically achievable tracking capabilities, due to higher frequencies if downmixed to the complex baseband. On the other hand, a huge variety of different implementations or instantiations was set up, making it difficult to get the whole picture. Early (and sometimes recent) publications dealing with that topic usually do not include matched filters for pulse shaping as well as the concept of complex Gaussian noise - which is very often not treated correctly - to yield a mathematically consistent baseband description that, although complicated looking, models the physics correctly. I.e. if these standards are not treated correctly, theoretical results are not reliable. This is independent of the media and the peer-review and the person, who published it.

The main idea behind BOC modulation is to reduce the interference withBPSK-modulated signals, which have asinc function shaped spectrum. Therefore, BPSK-modulated signals such as C/AGPS codes have most of their spectral energy concentrated around thecarrier frequency, while BOC-modulated signals (used inGalileo system) have low energy around the carrier frequency and two main spectral lobes further away from the carrier (thus the name of split-spectrum).

BOC modulation has several variants: sine BOC (sinBOC), cosine BOC (cosBOC),^[4] alternative BOC (altBOC),multiplexed BOC (MBOC),^[5] double BOC (DBOC)^[4] etc. and some of them have been currently selected for GalileoGNSS signals.^[6]

A BOC waveform is typically denoted via BOC(m, n) or BOC${\displaystyle (f_{\text{sc}},f_{\text{c}})}$, where${\displaystyle f_{\text{sc}}}$ is the sub-carrier frequency,${\displaystyle f_{\text{c}}}$ is the chip frequency,${\displaystyle m=f_{\text{sc}}/f_{\text{ref}}}$,${\displaystyle n=f_{\text{c}}/f_{\text{ref}}}$, and${\displaystyle f_{\text{ref}}=1.023}$ Mcps is the reference chip frequency of C/AGPS signal.

A sine BOC(1, 1) modulation is similar toManchester code; that is, in the digital domain, a '+1' is encoded as a '+1 −1' sequence, and a '0' is encoded as a '−1 +1' sequence. For an arbitrary${\displaystyle N_{\text{BOC}}=2m/n}$ modulation order, in sine BOC(m,n) case, a '+1' is encoded as an alternating sequence of '+1 −1 +1 −1 +1 ...', having${\displaystyle N_{\text{BOC}}}$ elements, and a '0' (or '−1') is encoded as an alternating '−1 +1 ...' sequence, also having${\displaystyle N_{\text{BOC}}}$ elements.

BOC modulation is typically applied onCDMA signals, where each chip of thepseudorandom code is split into BOC sub-intervals, as explained above (i.e., there are${\displaystyle N_{\text{BOC}}}$ BOC intervals per chip).

Thepower spectral density of a BOC-modulated signal depends on the BOC modulation order${\displaystyle N_{\text{BOC}}=2\frac{f_{\text{sc}}}{f_{\text{c}}}=2\frac{m}{n}}$.^[4]

BOC-modulated signals, by difference with BPSK signals, create the so-called ambiguities in the correlation function. The BOC-modulated signals in GNSS can be processed either with a Full BOC receiver or via various unambiguous approaches.^[7][8]

• Multiplexed binary offset carrier

• Quantized radio modulation modes

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