Can you list all the properties of the carrier-smoothing filter?

Q: Can you list all the properties of the carrier-smoothing filter?

A: Carrier-smoothing filters, also known as Hatch filters, are commonly used to reduce (“smooth”) the noise and multipath errors in pseudorange measurements by exploiting the high-precision relative distance information from carrier phase measurements. However, carrier-smoothing filters operate on more than just noise and multipath, and this article summarizes the response of such filters to all relevant inputs.

Q: Can you list all the properties of the carrier-smoothing filter?

A: Carrier-smoothing filters, also known as Hatch filters, are commonly used to reduce (“smooth”) the noise and multipath errors in pseudorange measurements by exploiting the high-precision relative distance information from carrier phase measurements. However, carrier-smoothing filters operate on more than just noise and multipath, and this article summarizes the response of such filters to all relevant inputs.

The GNSS carrier-smoothing filter is governed by the following finite difference equation

Equation (1) (see inset photo, above right, for article equations)

for n>0 where r_sm[n] is the filter output at the n-th epoch, whereas r_c[n] and r_p[n] are the code and phase pseudoranges, respectively. For the initial condition when n = 0, r_sm[n]= r_c[n].

In this article we will analyze the case in which parameter M, which relates to the bandwidth of the filter (larger M gives more smoothing), is constant. Figure 1 (for all figures and tables, please see the photo at the top of this article) shows a block diagram of the carrier-smoothing system. We use the following fundamental signals: the impulse signal δ[n]=1 for n = 0 and 0 for n ≠ 0, and the step signal u[n]=1 for n ≥ 0 and 0 for n < 0. The output of the block with transfer function z^-1 is x[n-1] when the input is x[n].

Taking into account the main errors, the code pseudorange can be written as

r_c[n] = r[n] + c_S[n] + c_R[n] + T[n] + I[n] + w_c[n]           (2)

where r[n] is the true range between the satellite and the GNSS receiver, c_S[n] is the error due to the bias of the satellite clock, c_R[n] is the error due to the bias of the receiver clock, T[n] is the error due to the troposphere delay, I[n] is the error due to the ionosphere delay, w_c[n] is the noise due to the receiver DLL (delay locked loop) jitter due to thermal noise and noise-like multipath. All the quantities are measured in meters.

The phase pseudorange can be written as

r_p[n] = r[n] + c_S[n] + c_R[n] + T[n] + I[n] + N[n]λ + w_p[n]           (3)

where N[n] is the integer ambiguity, λ is the wavelength, and w_p[n] is the noise due to the receiver PLL (phase locked loop) jitter plus the effect of multipath. The integer ambiguity should be constant N[n]=N₀ during the satellite visibility, but we consider here that cycle slips may occur and therefore the integer ambiguity becomes a time-varying signal.

In general, the product of the carrier wavelength and the integer ambiguity, λN[n], can be written as

Equation (4)

where n_k > 0 is the discrete-time at which the k-th cycle slip event occurs, d_k the number of cycles in the slip (integer, positive or negative), N_c.s. is the number of cycle slip events in the observation time. Parameter N₀ is the value of N[n] at n = 0, which can be set equal to zero, since it cancels out in (1) for n>0.

Both pseudoranges have a common term s[n] = r[n] + c_s[n] + c_R[n] + T[n]. They also have the term I[n] with opposite sign, and each one has its own noise component (w_p[n] and w_c[n]), which are statistically independent.

In the following sections, we analyze the outputs of the carrier-smoothing system when one and only one of the various components is present. The total system output is then given by the sum of all the individual outputs, thanks to the system’s linearity.

Tables 1–3 list all the relevant formulas and figures. Transfer functions are identified as H_x(z) (z complex number), impulse responses as h_x[n] (n integer), step responses as u_x[n] where x specifies the considered component. In the tables, H_x(f) is defined as H_x(f) = H_x(e^j2πf) where f is the normalized frequency; the true frequency is f´ = f / T (in hertz) where T is the time interval between two subsequent epochs.

Common Component
For the common signal s[n], the system has just one input and one output, as shown in Figure 2. In this case we can evaluate the transfer function in the z-domain, as the cascade of two subsystems. The first subsystem is a finite impulse response (FIR) filter with transfer function:

Equation (5)

The second subsystem is instead an infinite impulse response (IIR) filter with the following transfer function

Equation (6)

and thus, the transfer function and impulse response of the entire system with input s[n] are, respectively,

H_s(z) = H_FIR(z)H_IIR(z) =1, h_s[n] = δ[n]          (7)

as also reported in (20) and (21) of Table 1. Thus, we have r_sm,s[n] = s[n] * h_s[n] = s[n], which means that the true range, the troposphere delay, and the terms due to the clock bias are present without alterations at the output of the filter.

Initial Condition
The initial condition can be interpreted as an impulse r_c[0]δ[n] which enters the low-pass IIR filter and provides the output r_sm,i.c.[n] = r_c[0]h_i.c.[n], where h_i.c.[n] and its plots for M=30 and 100 are reported in Table 2: equation (28) and the figure at its right. We can see that a long transient exists, which has theoretically an infinite duration but can be quantified through the partial energy E_i.c.[n]. It is possible to show that

Equation (8)

with E_i.c. being the total energy of h_i.c.[n] for n → ∞. The impulse response reaches 98 percent of its total energy after approximately 2M samples. We can therefore state that, according to the 98 percent criterion, the transient ends at the discrete time 2M.

Cycle Slip
Let us consider the effects of just one cycle slip event, i.e., λN[n] = λd₀u[n – n₀] in (4). The transfer function H_c.s.(z) of the system with input λN[n] and output r_sm,c.s[n] is given in (29) of Table 2. The response to λd₀u[n – n₀] is the inverse z-transform of R_sm,c.s[z] = H_c.s(z)λd₀z^-n0 / (1- z^-1).

Equation (9)

The figure on the right of (31) in Table 2 shows the step response for M=30 and 100: as M increases, the coefficient gets closer to 1, and the duration of r_sm,c.s[n] increases. Using the 98 percent criterion, the transient is over 2M samples after the occurrence of the cycle slip.

Ionosphere Component
The analysis of the ionosphere component can be performed using the block diagram of Figure 3. In this case, the first filter has transfer function

Equation (10)

while the second filter is again the same IIR filter of Figure 2 described by equation (6).

Therefore, the transfer function for the ionosphere component I[n] is H_iono(z) = H_FIR,iono(z)H_IIR(z), given in (22) of Table 1. Equation (25) in Table 1 gives the corresponding impulse response h_iono[n], with plots for M=30 and 100 on its right. Then the carrier-smoothing filter output due to the ionosphere can be written as

r_sm,I[n] = I[n]*h_iono[n]           (11)

which unfortunately does not provide much information; so, some approximations are necessary. Filter H_iono(z) has a zero, (M-1)/(M-2), and a pole, (M-1)/M, very close to each other. The system then tends to behave as an all-pass filter, as shown in the figure on the right of (23) in Table 1, where the modulus of the transfer function H_iono(f) = H_iono(exp(j2πf)) is plotted versus the normalized frequency f. The group delay of the system is defined as

Equation (12)

where φ_iono(f) is the phase of H_iono(f), shown in the figure on the right of (24) in Table 1. We can show that, for small values of f,

Equation (13)

having defined a=(M-1)/(M-2) and b=(M-1)/M.

In nominal conditions, the ionosphere component I[n] changes slowly, which means that its bandwidth is very small, and we can approximate the transfer function in the frequency domain as

H_iono(f) ⋍ M_iono exp[−j2πτ_i f]

where M_iono = 1 is the magnitude of H_iono(f) at f = 0, and τ_i is the group delay θ_iono(f) at f = 0:

Equation (14)

Then, for a sufficiently slow ionosphere component I[n], the output of the carrier-smoothing filter is approximately equal to r_sm,I[n] I[n – 2(M-1)] with a delay τ_i = 2(M-1), which decreases as M decreases.

Noise Components
A discrete-time white Gaussian noise process w_in[n], with zero mean and variance σ_w², which enters a filter with impulse response h[n], generates an output process w_out[n] that is no longer white, but still Gaussian, with zero mean and with variance

Equation (15)

Code Pseudorange Noise. Code pseudorange noise generates an output process r_sm,c.n.[n] which is obtained by using w_c[n] as input of the filter with impulse response h_c[n]= h_IIR[n]/M. Using (15) and the total energy (obtained from (8) for n → ∞) we get the output variance σ²_c,out:

Equation (16)

where σ_c² is the variance of w_c[n]. The figure on the right of (32) in Table 3 shows how σ²_c,out decreases to zero as M increases. This is the “smoothing” aspect of the filter.

Phase Pseudorange Noise. Phase pseudorange noise generates an output process r_sm,p.n.[n] which is obtained by using w_p[n] as input of the filter with impulse response h_p[n] = (M-1){h_IIR[n]-h_IIR[n-1]/M}, which can be written as follows:

Equation (17)

where σ_p² is the variance of w_p[n]. The figure on the right of (33) in Table 3 shows how σ²_p,out increases with M with an asymptotic value equal to σ_p², which means that the system tends to behave as an all-pass filter as M increases (although since σ_p is typically much smaller than σ_c, this is not a major concern).

Summary of the Results
In the presence of all the errors, the signal at the output of the carrier-smoothing filter is then

r_sm[n] = s[n] + r_sm,i.c[n] + r_sm,c.s.[n] + r_sm,c.n.[n] + r_sm,p.s[n] + r_sm,I[n]          (18)

Table 1 lists the results valid for the common and ionosphere components s[n] and I[n]. Table 2 refers to effects due to the initial condition and the cycle slips. Table 3 summarizes the results obtained for the noise and multipath inputs w_c[n] and w_p[n].

On the one hand, we see that larger values of M decrease the output noise variance due to code pseudorange, but on the other hand these larger values increase the duration of transients in the presence of cycle slips; so, a compromise must be found. The total noise variance at the output of the carrier-smoothing filter is

Equation (19)

With this value of M, the noise variance at the output of the smoothing filter is equal to 2σ_p² and the transient durations are reasonably short. However, if cycle slips are frequent, then transients are the dominating error and M should be chosen, taking into consideration the probability and magnitude of undetectable cycle slips.