2.4
Degree of Complex Impropriety
Complex random variables are classified as second order proper or improper. However, improper signals can take a wide range or degrees of impropriety. For example, when viewed geometrically, the circularity (propriety) of Gaussian signals can vary extensively, that is from a circular distribution to the extremely noncircular case where all the data are distributed on a line, for example when the real and imaginary parts are fully correlated the distribution is on a line. There is hence a need to quantify and measure the degrees of impropriety of complex variables. Further, the widely linear model has a larger com- putational overhead than the strictly linear model, and in some applications the degree of impropriety can determine whether the performance benefits of the widely linear model can offsets the extra computational overhead.
The circularity (or propriety) of a complex signal is preserved by linear transforma- tions, which include scaling and rotation, but not by widely linear transformations. For instance, if the complex vector z = [z1, ..., zn]T is proper then the linear transformation
G·z, where G is a nonsingular matrix, is also proper, while if z is improper, then its linear transform is also improper. However, under widely linear transformations, such as G·z+H·z∗ where both Gand Hare nonsingular, propriety is no longer preserved.
Hence, any measure of impropriety is also required to be invariant under linear transformations, but not widely linear transformations. This means that the measure must be a function of a complete set of invariants for the covariance Rz and pseudocovariance Pz under linear transformation. This has been shown to be given by the set ofcanonical correlations between z and its conjugate z∗. The canonical correlations are also known as the circularity coefficients and play a key role in independent component analysis of complex signals [11].
The first step to computing the canonical correlations, involves whitening the signal by taking the square root decomposition of the covariance matrix, that is
where the invertible matrix Rz1/2 is defined as the square root of the covariance matrix Rz. Then the vector ¯z=Rz−1/2z= [¯z1, ...,z¯n]T has covariance matrix
R¯z =E{z¯¯zH}=Rz−1/2RzRz−T /2=I (2.37) and is therefore a unit variance white random vector. The canonical correlations are de- termined from the pseudocovariance of the whitened signal ¯z, also known as the coherence matrix, that is
P¯z =E{¯z¯zT}=Rz−1/2PzRz−T /2 =M (2.38) The coherence matrix M, being a pseudocovariance matrix, is complex symmetric, M= MT, and can be decomposed usingTakagi factorisation to yield
M = FKFT
where F is a unitary matrix, that is FH = F−1, and the diagonal matrix K = diag(k1, k2, . . . , kn) contains the canonical correlations 1 ≥ k1 ≥ k2 ≥ · · · ≥ kn ≥ 0
on its diagonal.
Further, the linear transformation ´z = FH¯z = FHR
x−1/2z = [´z1, ...,z´n]T, which
simultaneously diagonalises both the covariance and pseudocovariance, is said to be given in canonical coordinates. The canonical coordinates have the special property of being white with unit variance, together with a diagonal pseudocovariance matrix of canonical correlations, that is
R´z =E{z´´zH}=I (2.39)
P´z =E{z´´zT}=K (2.40)
Vectors, such as ´z, with unit diagonal covariances, that are generally improper, are often referred to as strongly uncorrelated. The strongly uncorrelating transform is a useful framework for the analysis of complex signal processing algorithms.
2.4 Degree of Complex Impropriety 39
There are a number of plausible functions for measuring impropriety based on the canonical correlations, however, one measure stands out because it relates the entropy of a noncircular Gaussian random variable with its circular counterpart. The entropy of an improper Gaussian random vector with augmented covariance matrix Rza and the
corresponding proper Gaussian random vector with covariance matrixRzcan be expressed as [12] Himproper = 1 2ln[(πe) 2ndetR za] = ln[(πe)ndetRz] | {z } Hproper +1 2ln n Y i=1 (1−ki2) (2.41)
where det is the matrix determinant operator. This illustrates the classical result that proper Gaussian random vectors maximise entropy Himproper ≤ Hproper, while the en-
tropy difference between the proper and improper signals is a function of the canonical correlations.
The circularity measure defined as [13]
d= 1− n
Y
i=1
(1−ki2) = 1−detRza[detRz]−2 (2.42)
lends itself as the natural choice for measuring the degree of impropriety of complex random vectors. Moreover, this function is a compelling measure for several reasons:
• d is bounded as 0≤d≤1, whereby for d= 0 the signal is circular and ford= 1 the signal is maximally improper.
• It connects the entropy of the proper and improper cases.
• It is a measure of the linear dependence between zand z∗ and as such, can be used to design a generalized likelihood ratio test for impropriety.
• Tight bounds on the measure d can be obtained without the need to explicitly compute the canonical correlations, which can save on computational processing.
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 ℜ ℑ (a) Circular: η= 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 ℜ ℑ (b) Noncircular: η= 0.5 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 ℜ ℑ (c) Noncircular: η= 0.9 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 ℜ ℑ (d) Noncircular: η= 0.99
Figure 2.4: A geometric view of circularity via a real-imaginary scatter plot of white
complex Gaussian processes at different degrees of noncircularity (η), with orthogonal
real and imaginary parts.
pz=E{z2}, the measure d simplifies to:
d= |pz|
2
r2
z
(2.43)
which is essentially the square of the ratio between the pseudocovariance and covariance. This in turn motivates the ratio between the pseudocovariance and covariance, known as the circularity quotient, to be taken as an impropriety measure, that is [10]
̺= pz rz
=ηejθ (2.44)
Where η = |̺z| = √
d, 0 ≤ η ≤ 1, is the circularity coefficient4 and θ = arg(̺z) is the
circularity angle. The advantage of using the circularity quotient, over the circularity
2.4 Degree of Complex Impropriety 41
measure d, is that it preserves the phase information contained within the pseudocovari- ance, and is simpler to compute. Figure 2.4 illustrates the distributions of white complex Gaussian noise with different degrees of noncircularity. In the following chapters, the cir- cularity coefficient will be used to test the performance of algorithms at different degrees of impropriety.