# The Impact of Inter-Modulation Components on Interferometric GNSS-Reflectometry
## 2 Characteristics of the Multiplexed Signals  
Because of the limited availability of the spectrum allocated for navigation systems, the modernized GNSS signals are composed of several components multiplexed on the same carrier using the Coherent Adaptive Subcarrier Modulation (CASM), a multichannel modulation scheme also known as tricode hexophase modulation or interplex modulation [20; 21]. This technique was proposed to keep the power envelope of the transmitted signals constant, i.e., the total transmitted power does not vary over time. It is also important to allow the use of a high-power amplifier operating close to saturation with limited signal distortion. However, this technique yields a byproduct signal that is not useful in the navigation applications. Taking the GPS L1 band as an example, the modernized GPS satellites (GPS IIR-M and II-R) transmit three different navigation signals: the C/A, P and M code components. The CASM modulation places the P and the M codes in the in-phase component (I) and the C/A code in the quadrature component (Q) with the IM component. The transmitted L1 signal at baseband could be expressed as [22]  
\[s_{L1}(t)= \sqrt{2}\left(\sqrt{P_{PSP}}\left(t\right)-\sqrt{P_{M}}s_{M}\left( t\right)\right)+ \tag{1}\] \[j\sqrt{2}\left(\sqrt{P_{CA}}s_{CA}\left(t\right)+\sqrt{P_{IM}}s_ {IM}\left(t\right)\right)\]  
in which the subscripts denote the signal components of P, M, C/A and IM respectively, \(s_{*}(t)\) are the spreading code modulated baseband navigation signals, \(P_{*}\) are the powers of each signal component and the IM component is the product of the three useful components as \(s_{IM}(t)=s_{CA}(t)s_{p}(t)s_{M}(t)\). Since the IM component is not correlated with the other signal components, i.e., \(\langle s_{IM}(t)s_{X}(t)\rangle=0\), it can be neglected in general GNSS signal processing for navigation, as well as in the \(c\)GNSS-R and \(r\)GNSS-R approaches. In modernized GNSS systems, the Binary Phase-Shift Keying (BPSK) and Binary Offset Carrier (BOC) modulations are used in different navigation signals. The general expression for BPSK and BOC modulations could be written as BPSK\((n)\) and BOC\((n,m)\), in which \(n\) and \(m\) refer to the code chipping rate of \(n\times 1.023\) MHz and the subcarrier frequency of \(m\times 1.023\) MHz. Thus, the C/A code, P code and M code could be expressed as BPSK(1), BPSK(10) and BOC(10, 5) respectively, and the IM component as BOC(10, 10) [23]. Figure 2 summarizes the ACFs of the different signal components in the GPS modernized L1 band. In [22], the GPS L1 band signal from GPS Block IIR-M #6 (PRN 07) satellite was analysed using a high gain antenna and the relative power of each signal component was estimated. The main characteristics of these signal components are summarized in Table 1. In Figure 3, the ideal power  
Figure 2: Comparison between the auto-correlation functions (ACFs) for different signal components in GPS L1 band, i.e., the C/A code, P code, M code and Inter-Modulation (IM) component. spectral density (PSD) \(S_{\mathrm{L1}}(f)\) and the Woodward Ambiguity Function (WAF), i.e., the squared ACF of the fully composite GPS L1 signal \(\Lambda^{2}(\tau)\), are compared against those of its ranging components, i.e., the composite of C/A, P and M codes as in previous studies. As expected, the wide band and the split-spectrum properties of the IM component increase the spectral energy further away from the carrier frequency, therefore sharpening the main lobe and reducing the side lobes of the ACF. To further validate the existence of the IM component and its effect on the \(i\)GNSS-R waveform, the raw samples collected on 3 December 2015 during an airborne experiment over the Baltic Sea were processed. A new instrument, known as the Software PARIS Interferometric Receiver (SPIR) [24], was used in this campaign. Thanks to the flexibility of the SPIR instrument, the up-looking antenna array (with 8 elements) can be divided into two independent sub-arrays with four antenna elements each.