# Energy Efficient GNSS Signal Acquisition Using Singular Value Decomposition (SVD)
## 1 Introduction
It roughly takes 1 chip = 1 microsecond to travel the length of 300 m, and it takes 1 epoch = 1023 bits of PRN code 1 ms to travel 300 km (see Figure 1). The receiving power level at the Earth's surface of r(t) is extremely weak, well below the noise floor. The minimum received power on the ground, defined at the output of an ideally-matched right-hand circularly-polarized 0 dBi user receiving antenna when the satellite elevation angle is higher than 10 degrees, is \(-\)157 dBW, considering 50%/50% E1B/E1C power-sharing [8]. Galileo satellites transmit the E1 (L1) signal on the centered frequency 1575.42 MHz, the same as GPS and, with a reference bandwidth of 24.5520 MHz, the E1 signal contains pilot and data channels, and both use composite-binary offset carrier (CBOC) modulation(see Figure 2), which is multiplexed BOC(1,1) and BOC(6,1). Figure 1: PRN code, chips, and epoch. Code A has 1023 MHz chipping rate, the data channel has a navigation message with 250 bps rate. The pilot channel is called E1-C, and the data channel is called E1-B. This kind of modulation allows GPS and Galileo signals to occupy the same frequency while avoiding mutual interference, making building receivers that use both GPS and Galileo simpler because GPS and Galileo use the same frequency. A distinction is made between signals containing navigation data (the data channels) and signals carrying no data (pilot channels) [9]: the signals of the data and pilot channels are shifted by 90 degrees in phase, which allows for their separation in the receivers. Galileo allows the receiver to estimate the ionospheric delay error. This error is due to the delay that the navigation signals suffer when they travel through the ionosphere. This delay makes the distance from the satellite to the user, as measured by the receiver, appear longer than it actually is and, if not corrected, would lead to large positioning errors. Fortunately, this delay is proportional to the frequency of the signal, with lower frequency signals experiencing a longer delay than higher frequency signals. Therefore, by combining measurements to the same satellite at two different frequencies, it is possible to produce another measurement where the ionospheric delay error has been canceled out. This cancellation becomes more effective as the separation between the two frequencies increases. This is the reason why Galileo services are generally realized using pairs of signals [9]. Basic GPS receiver architecture is shown in Figure 3. The satellite signal binary phase-shift keyed (BPKS) signal arrives at the antenna with some radio frequency (RF) plus noise. The front-end purpose of the receiver is to filter, amplify, and down-convert the incoming signal from analog to digital (A-D) to an intermediate frequency (IF) or lower frequency that is easy to process and sample in the receiver baseband. It is important to know that the RF front-end contains analog components that generate thermal noise and in the majority of satellite-receiver design the noise comes not from the satellites or any external source, but from the receiver itself [1]. After the front end, there is the baseband section of the receiver. The IF to baseband mixer acts to remove the carrier from the signal, leaving the original binary sequence that was created at the satellite and the 50-bps data, but also noise. Figure 3: Basic GPS receiver architecture. Figure 2: Spreading code, subcarrier, carrier, and signal as a result of the BOC modulation principle. At the correlator, the receiver takes a replica of the PRN code and multiplies it by the received signal, then integrates. When the correlators are aligned with the incoming signal a correlation peak is observed, and a hit is declared if the integrated value crosses a predetermined threshold. Moreover, the baseband block is repeated once per each channel so that each channel can acquire a different satellite. Therefore, a standard receiver has more than one channel. One aspect to notice is that until the correlation peak is found, there are two unknowns.