Treatment of the Climate Change and International Security Discourse in Asia: A Distant Look

Figure 3-7. Chip shapes and autocorrelation functions for different signals.

Figure 3-7 illustrates the chip shape and the autocorrelation function for signals of interest. In red are presented results for the GPS L1 C/A signal, in blue results for Galileo E5a and L5 quadrature -phase signal components and in black results for Galileo E1C signal. The abscissa is given in GPS L1 C/A chip unit it means 1.023 × 10⁻⁶ s. The Galileo E1C correlation function corresponds to the 𝐶𝐵𝑂𝐶(6,1, 1 11⁄ , −) autocorrelation function. The Galileo E1C signal is normalized to have the 𝐵𝑂𝐶(1,1) component amplitude equal to 1 in the chip domain.

3.2 Analog processing of GNSS receiver

When reaching the receiver antenna, the incoming signal sent by the satellite and generated by the payload components, including the satellite antenna, has gone through the propagation channel that was described in the previous chapter (free space, ionosphere, troposphere, potential obstruction, multipath, interference). First of all this signal passes through the receiver antenna and then through the analog section of the GNSS receiver. In this section, the signal is processed as an analog signal before being digitized (sampled in time and quantified in amplitude) by the analog to digital converter.

3.2 Analog processing of GNSS receiver

69

The analog section is made of the antenna, a Low Noise Amplifier (LNA), mixers, local oscillators and filters. This receiver portion is of primary importance because it pre-conditions the signal that is then processed digitally.

3.2.1 Antenna

The antenna is the first component of the receiver encountered by the incoming signal. As GNSS signals are right hand circularly polarized, GNSS antennas are also right hand circularly polarized. Desired characteristics of GNSS antennas are: the frequency selectivity, a high gain towards the satellites, multipath and interference rejection capabilities, low gain in directions where no satellite is located and a stable phase and group delays.

The antenna has an impact on the signal quality. As an example, a high-gain dish antenna with a large diameter is directive and allows to amplify a signal arriving from one direction while attenuating all other. An omnidirectional antenna will have less gain in a given direction but will receive several signals with a fair 𝐶 𝑁⁄ 0. Depending on the usage, different antennas can be selected. For the observation of tiny signal distortions, high-gain dish antennas are preferred. However, typical civil aviation users have antenna with a positive gain in the up direction and good rejection in the down direction.

3.2.2 RF front-end

After the antenna, the signal is passing through the RF front-end where it is amplified, down-converted and filtered. The down-conversion consists in reducing the signal carrier frequency in order to reach intermediate frequencies (IF) and filter the IF signal more selectively. The down-conversion is realized by multiplying the incoming signals by local sinusoidal waves generated by local oscillators. Several stages are usually necessary to translate the signal to IF or baseband.

The antenna and the summed effect of the different equivalent baseband filters and electronic components, part of the RF front-end, determine the so-called equivalent selective filter or pre-correlation filter of the GNSS receiver. In general, the last filter of the RF front-end (the most selective) is the one that will dominate the pre-correlation filter.

Figure 3-8 gives the chip shapes after applying a 6^th-order Butterworth pre-correlation filter of 24 MHz (in red) and 12 MHz (in blue) double-sided. It is noticeable that a delay visible on the chip affects filtered measurements. The delay is higher when the filter bandwidth is lower.

At the output of the RF front-end, the GNSS signal is dominated by the noise. The next step of the signal processing which is the quantization of the signal consists essentially of a noise quantization more than a GNSS signal quantization.

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Figure 3-8. Influence of the filter bandwidth on chip shapes.

3.2.3 Analog to digital converter (ADC)

The ADC is the last step of the GNSS receiver analog processing. The purpose of this device is to digitize the analog signal. The signal is quantized in amplitude and in time. The process of time quantization is called sampling. The sampling period 𝑇𝑠 is an important parameter because it is one of the parameters linked to the resolution with which the signal is observed. Another parameter linked to the resolution of the digitized signal is the number of bits used to quantize the amplitude of the signal. If only few quantization levels are available, the digitized signal will suffer from quantization losses. In general, to avoid these losses, the receiver can use a multi-bit ADC. In this case, an Automatic Gain Controller (AGC) is necessary to adapt the power of the received signal to the ADC quantization range and avoid signal distortions [Parkinson and Spilker, 2006].

3.2.4 Signal expression at the output of the analog section

Considering the ideal GNSS signal at the output of the satellite 𝑠𝑠(𝑡), the signal 𝑠𝑖𝑛𝑝𝑢𝑡(𝑡) at the receiver antenna input can be written as a function of the propagation medium impulse response 𝑔_{𝑝𝑟𝑜𝑝}(𝑡):

When the signal goes through the antenna section, it is convolved with the impulse response of the antenna 𝑔𝑎𝑛𝑡(𝑡). At the output of the antenna 𝑠(𝑡), the signal can be modeled as:

The signal at the output of the RF front-end 𝑠̃(𝑡) can be expressed as a function of the impulse response of the RF front-end ℎ𝑅𝐹(𝑡).

The last step is the A/D conversion which consists in a quantization of the signal. The signal 𝑠̃𝑛 at the output of the ADC can be modeled as:

𝑠_{𝑖𝑛𝑝𝑢𝑡}(𝑡) = 𝑔_{𝑝𝑟𝑜𝑝}(𝑡) ∗ 𝑠_𝑠(𝑡) (3-20)

𝑠(𝑡) = 𝑔_𝑎𝑛𝑡(𝑡) ∗ 𝑠_{𝑖𝑛𝑝𝑢𝑡}(𝑡) (3-21)

𝑠̃(𝑡) = ℎ_𝑅𝐹(𝑡) ∗ 𝑠(𝑡) (3-22)

3.2 Analog processing of GNSS receiver

71 where 𝑓_𝑠= 1 𝑇⁄ is the sampling frequency in hertz. _𝑠

Finally, the signal at the GNSS receiver analog section output can be written as a function of the signal at the satellite output:

Where 𝑔_𝑡𝑜𝑡(𝑡) = (ℎ_𝑅𝐹∗ 𝑔_𝑎𝑛𝑡∗ 𝑔_{𝑝𝑟𝑜𝑝})(𝑡) is the impulse response of the propagation medium, the antenna and the RF front-end equivalent filter. 𝑔𝑡𝑜𝑡(𝑡) is equivalent to a delay 𝜏_𝑐(𝑡) on the code and a delay 𝜏_𝜑(𝑡) on the carrier phase that are time-dependent.

Expressions of the two delays are function of parameters defined in 2.1.3 and can be expressed in seconds as:

and

The carrier phase delay caused by the propagation of the signal is usually given in radian and is function of the carrier frequency:

It is important to notice that the Doppler Effect, caused by the relative velocity between the satellite and the receiver and signals propagation effects, is included in the definition of the phase delay 𝜑𝑝(𝑡).

At a given time, the Doppler frequency 𝑓𝑑𝑜𝑝 is linked to the phase delay by:

It entails that 𝜑𝑝(𝑡) can be written as a function of the signal initial phase 𝜑_𝑝0:

As an example, considering the expression of the GPS L1 C/A signal 𝑠_{𝑠_𝐶 𝐴}⁄ (𝑡) at the satellite antenna output defined by equation (3-5):

72 where

- 𝑛̃_𝑠 is an additive perturbation that affects the GNSS signal. It is modeled as a filtered white Gaussian noise (thermal noise).

- 𝜑_𝑝(𝑡) is the phase delay of the signal induces by the propagation, the antenna and the RF front-end in radian.

- 𝑓_𝐼𝐹 is the intermediate frequency after the down-conversion in hertz.

- 𝑓𝑑𝑜𝑝(𝑡) is the Doppler frequency affecting the signal in hertz. This term is time-dependent.

- 𝑐̃(𝑡, 𝑓_𝑑𝑜𝑝) is the filtered PRN chips sequence affected by a Doppler 𝑓_𝑑𝑜𝑝 at time 𝑡.

- 𝑑̃(𝑡, 𝑓_𝑑𝑜𝑝) is the filtered data sequence affected by a Doppler 𝑓_𝑑𝑜𝑝 at time 𝑡.

The same concept can be applied to other signals.

To simplify formula without losing generality, the influence of the ADC is not taken into account in the expression of the signal at receiver digital section input. In the following, derivations are proposed considering a continuous signal.