Shaka

Parenthesis ()

change the priorities in computing.

Square bracket []

function parameters, e.g. f[x].

Curly brace {}

list, e.g. {a, b, c}.

Double bracket

abbreviation of Part, e.g. v[[i]].

Table[expression, {i, max}]

make a list of values of expression with i from 1 to max.

Module[{a, b, c}, expression1; expression2;…]

a procedure with local variables a, b, c return value of last expression.

Do[expression, {i, min, max, di}]

evaluate expression with i run from min to max in steps of di.

While[test, body]

evaluate body repeatedly, so long as test is True.

For[star,test,inc,body]

evaluate start, then repeatedly evaluate body and inc, until test fails.

If[test, then, else]

evaluate then if test is True, and else if it is False.

Which[test1,value1,test2,value2,…]

give the value associated with the first test that is True.

Switch[expression, form1, value1, form2, value2, …]

give the value associated with first form matching expression.

Function[x, body]

specify a pure function.

Nest[f, x, n]

apply the function f nested n times to x.

Apply[f, {a, b, c}]

f(a, b, c).

Map[f, {a, b, c}]

apply f to each elements, {f(a), f(b), f(c)}.

Given a sparse array, the locations of its component antennas/sensors can be normalized to a minimum inter-antenna distance, denoted by \(d\), e.g., typically \(d = \lambda / 2\). Accordingly, a sparse array can be simply denoted by an integer set \(\mathbb{S}\), and the locations of the antennas are \(nd\), \(n \in \mathbb{S}\).

Given a sparse array \(\mathbb{S}\), its difference coarray (DCA) is defined as \[\begin{aligned} \mathbb{D} \triangleq \{n_1 - n_2 \mid n_1, n_2 \in \mathbb{S}\}. \end{aligned}\]

• Each entry in \(\mathbb{D}\) is termed a spatial lag.
• The cardinality of DCA, i.e., \(|\mathbb{D}|\), is termed degree of freedom (DoF) of the sparse array \(\mathbb{S}\).
• Clearly, DCA is symmetric, i.e., \(\forall n \in \mathbb{D}\), \(-n \in \mathbb{D}\).

Let \(\mathbb{U}\) denote the central ULA segment of \(\mathbb{D}\). Its cardinality \(|\mathbb{U}|\) is termed uniform degree of freedom (UDoF). Then, the number of uncorrelated sources which can be identified is \(\dfrac{|\mathbb{U}| - 1}{2}\).