Signal Processing - September 2017 - 77

signal. Importantly, two-steps solution can be trapped in such
local optimum, yielding to potentially large biases in range
and velocity estimation. Considering that position is jointly
estimated with the information of all visible satellites, a sort
of diversity is introduced in this estimation as the propagation
path for each satellite link is independent.

Practical aspects of DPE-based GNSS receivers

1

1

0.8

0.8

0.6

0.6

Λ(γ )

Λ(υ)

The previous section delved into the foundations of DPE
receivers, their formulae and bounds. This section provides
a discussion of advanced topics, either necessary for DPE's
implementation or to further enhance DPE's capabilities.
There are several ways to implement a DPE-based receiver, a fact that defines the final receiver scheme. First, pre- or
postcorrelation approaches are allowed. Whereas the former
might involve a large amount of resources to compute (5) due

to the potentially high number of in-phase and quadrature
(I&Q) samples, the latter implies adjusting a larger number
of design parameters such as the number of correlation outputs, their spacing, and spanning. Second, another design
choice is whether to go for open- or closed-loop schemes,
in which case the data stream is processed either batch or
sequentially, respectively. Open-loop schemes, such as those
followed in HS-GNSS involve optimization of (5), using portions of the data stream that are not necessarily concatenated, thus avoiding error propagation. A coarse optimization is
typically performed, and a refined search or interpolation is
applied afterward to increase accuracy. In contrast, closed-loop
schemes process data sequentially through PLLs and DLLs
(in conventional receivers), which alleviate the computational
burden and avoid buffering large amounts of samples since
the stream is only processed once. This article, as per (5),

0.4
0.2
0
2

0.4
0.2

1

0

-1

0

y - y [m]

0

500

1,000

-500
-500
-1,000 -1,000
x - x [m]
Q

-2,000
fdi -fdi [Hz]

500

Q

-2 -4,000

Q

Q

ti - ti [Tc ]

0

2,000

0
1,000

4,000

(a)

(b)

FIGURE 2. A comparison of cost functions for (a) two-steps and (b) DPE optimization problems. In two-steps, the cost function is a function of synchroni-

1

1

0.8

0.8
Λ(γ )

Λ (υ)

zation parameters of the locally generated code of the ith satellite, y. In DPE, the cost function is parameterized by the PVT parameters gathered in c.

0.6

0.6

0.4

0.4

0.2

0.2

0
2

1

0

-1

4,000

500

0

0

500

1,000

-500
-500
-1,000 -1,000
y - y [m]
x - x [m]

-2,000
fdi -fdi [Hz]

Q

Q

-2 -4,000

2,000

Q

Q

ti - ti [Tc ]

0

0
1,000

(a)

(b)

FIGURE 3. A comparison of cost functions for (a) two-steps and (b) DPE optimization problems in the presence of a replica for one of the satellites. The echo
has 3-dB lower power than the corresponding loss. In two-steps, a strong secondary optimum appears, whereas DPE cost function is virtually unaltered.
IEEE SIGNAL PROCESSING MAGAZINE

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September 2017

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77
Table of Contents for the Digital Edition of Signal Processing - September 2017
