Información lexicográfica escolar y general

The properties of the GNSS signal enter the radar equation through the ambiguity function 𝜒.

This will now be defined through exploring the structure of the GNSS signals. A thorough

reference for GNSS receiver fundamentals and signal structures is provided by [Kaplan 2006]

and [Tsui 2005].

The signal model for an unperturbed direct signal as transmitted by a GNSS transmitter will be expressed as a carrier modulated by spreading code 𝑐(𝑡) and data 𝑑(𝑡),

𝑠(𝑡) = √𝑃 𝑑(𝑡) 𝑐(𝑡) exp(𝑗2𝜋𝑓𝑡) _(2.2)

It has transmission power 𝑃, and carrier frequency 𝑓. The modulating code 𝑐(𝑡) is a Pseudo-Random Noise (PRN) sequence 𝑐(𝑡) ∈ (−1,1), so that the modulation is Binary Phase Shift Keyed (BPSK). In Section 4.5 this will be extended to another class of GNSS modulation, although for much of the research in this thesis the discussion will relate to the GPS Coarse Acquisition (C/A) code, which uses BPSK.

The C/A code is a 𝑁_𝑐 = 1023 chip long sequence from the Gold code family [Gold 1967].

Each bit or ‘chip’ of the code is 𝑇_𝑐 = 1/1.023 μs long so has a chip rate of 1.023 Mcps. The term 𝑑(𝑡) is the navigation data signal, which is transmitted at a rate of 50 Hz. The

navigation data contains the information necessary for a navigation receiver to compute position, velocity and time solutions.

Following propagation of the signal to the receiver, the GNSS component, 𝑠(𝑡), is modified by the channel, becoming the signal as received, 𝑢(𝑡). This is buried in thermal additive thermal noise 𝑛(𝑡) and has remaining amplitude 𝐴 after attenuations from free-space path loss, antenna gains, atmospheric losses and other linear scaling. The signal additionally experiences a time delay,𝜏, and Doppler shift 𝑓_𝐷. This results in the GNSS signal component from equation (2.2) having the general form once received,

𝑢(𝑡) = 𝐴 𝑑(𝑡 − 𝜏) 𝑐(𝑡 − 𝜏) exp(𝑗2𝜋(𝑓 + 𝑓_𝐷)𝑡) + 𝑛(𝑡)

(2.3)

For the receiver to extract the signal back out of the noise, the signal is frequency translated, through multiplication by the local oscillator, exp(−𝑗2𝜋𝑓′𝑡 + 𝜙). Then it is cross-correlated with an internally generated replica, 𝑐(𝑡 − 𝑡^′). The cross-correlation result 𝑤 is taken for the replica with the general time misalignment of 𝑡′, carrier frequency 𝑓^′, and a carrier phase difference of 𝜙,

𝑤(𝑡^′, 𝑓^′) = 𝐴

𝑇_𝑐𝑜ℎ∫^{𝑇𝑐𝑜ℎ}[exp(−𝑗2𝜋𝑓^′𝑡 + 𝜙) 𝑐(𝑡 − 𝑡^′)

0

( 𝑑(𝑡 − 𝜏) 𝑐(𝑡 − 𝜏) exp(𝑗2𝜋(𝑓 + 𝑓_𝐷)𝑡) + 𝑛(𝑡)) ]𝑑𝑡

(2.4)

There are now two symbols for the time and frequency offsets: the propagation offsets, 𝜏, 𝑓_𝐷, and the offsets of the internal replica signal, 𝑡′, 𝑓^′. This integration retains the signal phase so is called the coherent integration over the time period, 𝑇_𝑐𝑜ℎ. The complex formulation is used here, whereas in a physical receiver the real and imaginary parts are conventionally called the I and Q channels. The data symbol changes at low rate, so if replica and signal are aligned and integrated over a time less than the data bit period, then 𝑑(𝑡) can be taken outside of the integral. Expanding out the terms, the correlation result is,

𝑤(𝑡^′, 𝑓^′) = 𝐴 𝑑

𝑇_𝑐𝑜ℎ∫^{𝑇𝑐𝑜ℎ}𝑐(𝑡 − 𝜏)𝑐(𝑡 − 𝑡^′) exp(𝑗2𝜋(𝑓 + 𝑓_𝐷 − 𝑓^′)𝑡 + 𝜙)

0

+𝑛(𝑡) exp(𝑗2𝜋𝑓^′𝑡) 𝑐(𝑡 − 𝑡^′)𝑑𝑡

(2.5)

It is helpful to interpret this as a matched filtering process. The thermal noise component post-correlation becomes the result of filtering the noise by the frequency response of the modulation code, 𝑐(𝑡). For convenience the noise component post-correlation will be redefined as the complex vector 𝑛_𝑤. The correlation result then becomes,

𝑤(𝑡^′, 𝑓^′) = 𝐴 𝑑

𝑇_𝑐𝑜ℎ∫^{𝑇𝑐𝑜ℎ}𝑐(𝑡 − 𝜏)𝑐(𝑡 − 𝑡^′) exp( 𝑗2𝜋(𝑓 + 𝑓_𝐷− 𝑓^′)𝑡 + 𝜙)

0

𝑑𝑡 + 𝑛_𝑤 ^(2.6)

As the modulation code 𝑐(𝑡) is a pseudo random sequence, the expectation of the product

〈𝑐(𝑡 − 𝜏)𝑐(𝑡 − 𝑡′)〉 can be considered to be time-invariant with minimal error. The

expectation is no longer a function of time and then can be removed from the integral. This leads to the result that the code and frequency can be held as independent variables.

𝑤(𝑡^′, 𝑓^′) = 𝐴 𝑑

The expectation 〈𝑐(𝑡 − 𝜏)𝑐(𝑡 − 𝑡′)〉 is the auto-correlation function (ACF) of the pseudo-random code. The form of the ACF can be investigated by setting the propagation delay, 𝜏 = 0, the expectation can be expressed in relation to the time delay of the replica 𝑡′. The GPS C/A code is well-approximated by the function Λ(𝑡′/𝑇_𝑐), which is defined,

〈𝑐(𝑡)𝑐(𝑡 − 𝑡′)〉 = Λ(𝑡′) = { −1/𝑁_𝑐, |𝑡′/𝑇_𝑐| ≥ 1

1 − |𝑡′/𝑇_𝑐|, |𝑡′/𝑇_𝑐| < 1 ^(2.8) where 𝑇_𝑐 is the time period of each PRN code chip. In practice the auto-correlation function for the set of Gold codes used in for GPS C/A codes are not ideal, exhibiting additional cross correlation amplitudes of {-1/1023, -65/1023, 63/1023}. Outside of the time-aligned interval the cross-correlation function is almost zero, except due to the odd number of chips it reduces down to −1/𝑁_𝑐. The true auto-correlation of GPS C/A code is calculated for GPS PRN 2 and shown in Figure 2.5.

Figure 2.5 Normalised auto-correlation function for GPS C/A PRN 1. Left: Full code auto-correlation.

Right: zoom to chip range -10 to +10.

Due to the relatively small errors from the cross correlations, the idealisation Λ is often used in GNSS-R. The idealised ACF is now substituted into the correlation result of equation (2.7). The signal propagation delay, 𝜏, causing the time offset of the ACF function,

𝑤(𝑡^′, 𝑓^′) = 𝐴 𝑑

This leaves the remaining integral of just the carrier remaining after the receiver’s frequency translation. Evaluating the integral, the complex correlation result becomes,

𝑤(𝑡^′, 𝑓^′) = 𝐴

𝑇_𝑐𝑜ℎΛ(𝜏 − 𝑡^′)sin(𝜋(𝑓 + 𝑓_𝐷 − 𝑓^′)𝑇_𝑐𝑜ℎ) 𝜋(𝑓 + 𝑓_𝐷− 𝑓^′)𝑇_𝑐𝑜ℎ

exp(𝑗𝜋(𝑓 + 𝑓_𝐷− 𝑓^′) + 𝜙) + 𝑛_𝑤

(2.10)

Taking terminology from the field of radar, this auto-correlation result is the ambiguity function (AF) of the signal. This can be separated out in to a factorized form,

𝜒(𝜏 − 𝑡′, 𝑓 + 𝑓_𝐷 − 𝑓^′) = Λ(𝜏 − 𝑡′)sin(𝜋(𝑓 + 𝑓_𝐷− 𝑓^′)𝑇_𝑐𝑜ℎ)

𝜋(𝑓 + 𝑓_𝐷 − 𝑓^′)𝑇_𝑐𝑜ℎ ^(2.11) The single GNSS signal component used in this analysis is equivalent to the direct signal or a point scattering source. The spread in delay and Doppler of the AF limits the radar resolution, as the signal power is spread over an area of the signal space. The magnitude of the AF, |𝑤|, is plotted in delay and Doppler dimensions in Figure 2.6. For this the integration time 𝑇_𝑐𝑜ℎ was chosen to be 1 ms – this being the GPS C/A code length. Other integration times could be used to increase frequency resolution, the null-to-null frequency width post-integration of 2/𝑇_𝑐𝑜ℎ Hz. To centre the ACF, the offsets of the signal component are chosen as zero, 𝜏 = 0 and (𝑓 + 𝑓_𝐷) = 0, the axes of the plot are then the receiver’s replica offsets, 𝑡′ and 𝑓^′.

Figure 2.6 The ambiguity function for the BPSK, GPS C/A code on L1, using a coherent integration time of 1ms

It can be seen that when cross-correlating the incoming signal with a locally generated replica, components with aligned delay and Doppler are selected and misaligned components are filtered out. A navigation receiver would use a discriminator based on this to track the delay and frequency of the direct signal. A GNSS-R receiver instead uses this to select part of the signal space, which in turn corresponds to selecting part of the Earth’s surface. The bandwidth (chip rate) and coherent integration periods result in an ambiguity function that additionally spreads the signal, so reducing the achievable surface resolution.