acoustic-phase-localization/Acoustic Localization Using Carrier-Phase Measurements

### A Pluto.jl notebook ###
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using Markdown
using InteractiveUtils

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md"""
# Acoustic Localization Using Carrier-Phase Measurements

## Introduction

TODO!

## Problem Statment

!!! info "This text is copied from the provided document."
	We assume an acoustic localization system with $M$ beacons transmitting signals of a certain carrier frequency. The signals are received by a mobile device (MD) located at ``x = [x, y, z]^T``, while the beacons are located at ``b_i = v = [x_i, y_i, z_i]^T``, ``i = 1, 2, \ldots, M``. The beacons and the MD are all synchronized.

	By processing the received signal from each beacon, the MD estimates its range from each beacon. In addition, the MD can also obtain the phase of the carrier signal received from each beacon. The range measurements for beacon ``i`` can be modelled as 
	```math
	r_i =\|x − b_i\|_2 + w_i,
	```

	where wi is additive noise and ``\|\cdot\|_2`` is the Euclidean norm. On the other hand, the phase data can be described as
	```math
	\varphi_i = \|x − b_i\|_2 + n_i + v_i,
	```

	where ``v_i`` is additive noise and ``n_i \in \mathbb{Z}`` is an integer phase ambiguity. In (1) and (2), we assume that the range and phase are
	measured in units of cycles or wavelengths. Hence, ``\varphi_i \in [−0.5, 0.5)``.
	The objective of this work is to use both sets of measurements in (1) and (2) to estimate the MD position ``x``. To motivate the
	use of the carrier phase data, we assume generally that ``\|v_i\|_2 \ll \|w_i\|_2,\, \forall i \in {1, 2, · · · , M}``. However, this is not a necessary condition for the carrier-phase use to be beneficial. To proceed further, we assume that ``w_i`` and ``v_i`` are independent for all ``i``
	values, and that these noise processes are of zero mean and variances ``\alpha_i^2``  and ``\sigma_i^2``, respectively.

"""

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md"""
# Outline of the solution

!!! info "This text is copied from the provided document."

	To formulate the estimation problem, we can define the objective function:
	```math
	f(\mathbf{x}, \mathbf{n}) = \sum_{i=1}^M
			      \alpha_i^{-2} (r_i - \|x-b_i\|_2)^2
			    + \sigma_i^{-2} (\varphi_i - \|x-b_i\|_2 - n_i)^2
	```
where n ≜ [n1, n2, · · · , nM]T. Hence, we can obtain a MD position estimate by solving the following mixed integer optimization
problem:
ˆx, ˆn =
arg min
x∈R3,n∈ZMf(x, n)
"""

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# Appendex
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