Table 1 Summary of bounds

Definition

$e\triangleq \parallel \widehat{\mathit{x}}-\mathit{x}{\parallel }_2$

(8)

$v_{\text{max}}(\widehat{\mathit{x}},\mathcal{ℬ})\triangleq \underset{\mathit{y}\in \mathcal{ℬ}}{max}\parallel \widehat{\mathit{x}}-\mathit{y}{\parallel }_2$

(8)

Upper bounds

Bound 1

(Assumption At least one measurement has positive error,

$e\le max\{\parallel \widehat{\mathit{x}}-{\mathit{a}}_1{\parallel }_2+{\widehat{d}}_1,\dots ,\parallel \widehat{\mathit{x}}-{\mathit{a}}_N{\parallel }_2+{\widehat{d}}_N\}$ (10)

i.e., ${\widehat{d}}_i\ge d_i$ for at least one i=1,2,…,N)

Bound 2

(Assumption: a reasonable lower bound on measurement noise is

$e\le \sqrt{v_{\text{sdp}}{\{{\mathit{A}}_i,{\mathit{b}}_i,c_i\}}_{i=0}^N}$ (18)

known in advance, i.e., $d_i-{\widehat{d}}_i\le {\rho }_1^i$ for known ${\rho }_i^i$ and

{A _i ,b _i ,c _i } as defined in (17)

i=1,2,…,N)

Bound 2 (relaxed)

(Assumption: a reasonable lower bound on measurement noise is

$e\le min\{\parallel \widehat{\mathit{x}}-{\mathit{a}}_1{\parallel }_2+{\tilde{d}}_1,\dots ,\parallel \widehat{\mathit{x}}-{\mathit{a}}_N{\parallel }_2+{\tilde{d}}_N\}$ (23)

known in advance, i.e., $d_i-{\widehat{d}}_i\le {\rho }_1^i$ for known ${\rho }_i^i$ and

i=1,2,…,N)

Bound 3

(Assumption: measurement error can be positive and there are

$e\le \sqrt{v_{\text{sdp}}{\{{\mathit{A}}_i,{\mathit{b}}_i,c_i\}}_{i=0}^N}$ (30)

enough distance estimates for every link, i.e., $\text{Pr}\{{\epsilon }_i^k\ge 0\}>0$,

{A _i ,b _i ,c _i } as defined in (29)

for i=1,2,…,N and k=1,2,…,K, and K is

sufficiently large.)

Bound 3 (relaxed)

(Assumption: measurement error can be positive and there are

$e\le min\{\parallel \widehat{\mathit{x}}-{\mathit{a}}_1{\parallel }_2+{\bar{d}}_1,\dots ,\parallel \widehat{\mathit{x}}-{\mathit{a}}_N{\parallel }_2+{\bar{d}}_N\}$ (31)

enough distance estimates for every link, i.e., $\text{Pr}\{{\epsilon }_i^k\ge 0\}>0$,

for i=1,2,…,N and k=1,2,…,K, and K is

sufficiently large.)