3 Second-order description of complex random vectors
3.3 Degree of impropriety
Building upon our insights from the previous section, we may now ask how to quantify the degree of impropriety of x. Propriety is preserved by strictly linear (but not widely
3.3 Degree of impropriety 71
linear) transformation. This suggests that a measure for the degree of impropriety should also be invariant under linear transformation. Such a measure of impropriety must then be a function of the circularity coefficients kibecause the circularity coefficients constitute a maximal invariant for Rx x under nonsingular linear transformation.3
There are several functions that are used to measure the multivariate association between two vectors, as discussed in more detail in Chapter4. Applied to x and x∗, some examples are
ρ1= 1 −
"r i=1
(1− ki2), (3.52)
ρ2=
"r i=1
ki2, (3.53)
ρ3= 1 n
r i=1
ki2. (3.54)
These functions are defined for r = 1, . . ., n. For full rank r = n, they can also be written in terms of Rx x and Rx x. Then, from (3.34), we obtain
ρ1= 1 − det Rx x det2Rx x
= 1 − det Q det Rx x
(3.55) and from (3.29) we obtain
ρ2=det(Rx xR−∗x x R∗x x) det Rx x
(3.56) ρ3= 1
n tr(R−1x x Rx xR−∗x xR∗x x). (3.57) These measures all satisfy 0≤ ρi ≤ 1. However, only ρ3has the two properties thatρ3= 0 indicates the proper case, i.e., ki = 0 for all i = 1, . . ., n, and ρ3 = 1 the maximally improper case, in the sense that ki = 1 for all i = 1, . . ., n. Measure ρ2 is 0 if at least one kiis 0, andρ1is 1 if at least one ki is 1.
While a case can be made for any of these measures, or many other functions of ki,ρ1
seems to be most compelling since it relates the entropy of an improper Gaussian random vector to that of the corresponding proper version through (3.35). Moreover, as we will see in Section3.4,ρ1 is also used as the test statistic in a generalized likelihood-ratio test for impropriety. For this reason, we will focus onρ1in the remainder of this section.
Example 3.1. Figure 3.3 depicts a QPSK signalling constellation with I/Q imbalance characterized by gain imbalance (factor) G > 0 and quadrature skew φ. The four equally likely signal points are{±j, ±Gejφ}. We find
Rx x = E x2 =14(j2+ (−j)2+ G2e2jφ+ (−G)2e2jφ)=12(G2e2jφ− 1), (3.58)
Rx x = E|x|2 =12(1+ G2). (3.59)
72 Second-order description of complex random vectors
I Q
f j
− j
Gejf
Figure 3.3 QPSK with I/Q imbalance.
Since x is scalar, with variance Rx x and complementary variance Rx x, the degree of improprietyρ1becomes particularly simple:
ρ1= | Rx x|2
R2x x = G4− 2G2cos(2φ) + 1 (1+ G2)2 =
(G2− 1)2
(G2+ 1)2, φ = 0
1, φ = π/2,
1
2(1− cos(2φ)), G = 1.
(3.60)
Perfect I/Q balance is obtained with G= 1 and φ = 0. QPSK with perfect I/Q balance is proper, i.e.,ρ1= 0. The worst possible I/Q imbalance φ = π/2 results in a maximally improper random variable, i.e.,ρ1= 1, irrespective of G.
3.3.1 Upper and lower bounds
So far, we have developed two internal descriptions for x: the principal components (3.14), which are found by a widely unitary transformation of x, and the canonical coordinates (3.30), which are found by a strictly linear transformation of x. Both principal components and canonical coordinates are uncorrelated, i.e.,
Eξiξ∗j = E ξiξj = 0 for i = j, (3.61) but only the canonical coordinates have unit variance. Both principal components and canonical coordinates are generally improper with
Eξi2 =
1
2(λ2i−1− λ2i), if are principal components,
ki, if are canonical coordinates. (3.62) It is natural to ask whether there is a connection between the eigenvaluesλi and the circularity coefficients ki. There is indeed. The eigenvalue spectrum{λi}2ni=1 restricts the possibilities for the circularity spectrum{ki}ni=1, albeit in a fairly intricate way. In the general setup – not restricted to the conjugate pair x and x∗– this has been explored by Drury (2002), who characterizes admissible kis for given eigenvalues{λi}. The results
3.3 Degree of impropriety 73
are very involved, which is due to the fact that the singular values of the sum of two matrices are not easily characterized. It is much easier to develop bounds on certain functions of {ki} in terms of the eigenvalues {λi}. In particular, we are interested in bounds on the degree of improprietyρ1if{λi} are known. We first state the upper bound onρ1.
Result 3.5. The degree of improprietyρ1of a vector x with prescribed eigenvalues{λi} of the augmented covariance matrix Rx x is upper-bounded by
ρ1 = 1 −
"r i=1
(1− ki2)≤ 1 −
"r i=1
4λiλ2n+1−i
(λi+ λ2n+1−i)2, r = 1, . . ., n. (3.63) This upper bound is attained when
Rx x =12Diag(λ1+ λ2n, λ2+ λ2n−1, . . ., λn+ λn+1), (3.64)
Rx x =12Diag(λ1− λ2n, λ2− λ2n−1, . . ., λn− λn+1). (3.65) This bound has been derived byBartmann and Bloomfield (1981) for the canonical correlations between arbitrary pairs of real vectors (u, v), and it holds a forteriori for the canonical correlations between x and x∗. It is easy to see that Rx x, with Rx x and Rx x as specified in (3.64) and (3.65), is a valid augmented covariance matrix with eigenvalues {λi} and attains the bound.
There is no nontrivial lower bound on the canonical correlations between arbitrary pairs of random vectors (x, y). It is always possible to choose, for instance, ExxH= Diag(λ1, . . ., λn), EyyH= Diag(λn+1, . . ., λ2n), and ExyH= 0, which has the required eigenvalues {λi} and zero canonical correlation matrix K = 0. That there is a lower bound onρ1stems from the special structure of the augmented covariance matrix Rx x, where the northwest and southeast blocks must be complex conjugates.
We will now derive this lower bound for r = n. Let x = u + jv and z = [uT, vT]T. From (2.21) and (2.22), we know that
Rx x = Ruu+ Rvv+ j(RuTv− Ruv), (3.66)
Rx x = Ruu− Rvv+ j(RuTv+ Ruv). (3.67) Since the eigenvalues are given,
det Rx x =
"2n i=1
λi (3.68)
is fixed. Hence, it follows from (3.34) that the minimumρ1 is achieved when det Rx x
is minimized. We can assume without loss of generality that Rx x is diagonal. If it is not, it can be made diagonal with a strictly unitary transform that leaves det Rx xand the eigenvalues{λi} unchanged. Thus, we have
min det Rx x = min
"n i=1
(Rx x)ii= min
"n i=1
[(Ruu)ii+ (Rvv)ii]. (3.69)
74 Second-order description of complex random vectors
Now let qi be the i th largest diagonal element of Ruu+ Rvv, and ri the i th largest diagonal element of Rzz. Then,
r with equality for r = n. We have the second inequality because the diagonal ele-ments of Rzz are majorized by the eigenvalues of Rzz (cf. Result A3.5). Since'
qi
is Schur-concave, a consequence of (3.70) is the following variant of Hadamard’s inequality:
Using this result in (3.34), we get
"n
from which we obtain the following lower bound onρ1.
Result 3.6. The degree of improprietyρ1of a vector x with prescribed eigenvalues{λi} of the augmented covariance matrix Rx x is lower-bounded by
ρ1= 1 −
Example 3.2. In the scalar case n= 1, the upper bound equals the lower bound, which leads to the following expression for the degree of impropriety:
ρ1 = k12= 1 − 4λ1λ2
(λ1+ λ2)2 =(λ1− λ2)2 (λ1+ λ2)2.
3.3 Degree of impropriety 75
Thus we also have a simple expression for the circularity coefficient k1in terms of the eigenvaluesλ1andλ2:
k1= λ1− λ2
λ1+ λ2
.
While the upper bound (3.63) holds for r = 1, . . ., n, we have been able to establish the lower bound (3.74) for r = n only. A natural conjecture is to assume that the diagonal Rx xand Rx x given by (3.75) and (3.76) also attain the lower bound for r< n. Let cibe the i th largest of the factors
4λ2 j−1λ2 j
(λ2 j−1+ λ2 j)2, j= 1, . . ., n.
Then the conjecture may be written as ρ1= 1 −
"r i=1
(1− k2i)≥ 1 −
"n i=n−r+1
ci, r = 1, . . ., n. (3.77)
We point out that the diagonal matrices Rx x and Rx x that achieve the upper bound do not necessarily give upper bounds for other functions of{ki}, such as ρ2andρ3.Drury et al. (2002) prove an upper bound forρ2and conjecture an upper bound forρ3. Lower bounds forρ2andρ3are still unresolved problems.
Example 3.3. We have seen that the principal components minimize the degree of impro-prietyρ1 under widely unitary transformation. In this example, we show that they do not necessarily minimize other measures of impropriety such asρ2andρ3.
Consider an augmented covariance matrix Rx xwith eigenvalues 100, 50, 50, and 2. The principal components have covariance matrix Rξξ = Diag(75, 26), complementary covariance matrix Rξξ = Diag(25, 24), and circularity coefficients k1= 24/26 and k2= 25/75. We compute ρ2= 0.481 and ρ3= 0.095.
On the other hand, there obviously exists a widely unitary transformation into coor-dinates xwith covariance matrix Rxx = Diag(51, 50) and complementary covariance matrix Rxx = Diag(49, 0), and circularity coefficients k1= 49/51 and k2= 0/50. The description xis less improper than the principal components when measured in terms ofρ2= 0.461 and ρ3= 0.
Least improper analog
Given a random vector x, we can produce a least improper analog = UHx, using a widely unitary transformation U. It is clear from (3.75) and (3.76) that the principal components obtained from (3.14) are such an analog, with U determined by the EVD Rx x = U Λ UH. The principal components have the same eigenvalues and thus the same power as x. They minimizeρ1 and thus maximize entropy under widely unitary transformation. We note that a least improper analog is not unique, since any strictly
76 Second-order description of complex random vectors
unitary transform will leave both the eigenvalues{λi} and the canonical correlations {ki} unchanged.
3.3.2 Eigenvalue spread of the augmented covariance matrix
Let us try to further illuminate the upper and lower bounds. Both the upper and the lower bounds are attained when Rx x and Rx x are diagonal matrices. For Rx x =
is the squared ratio of the geometric and arithmetic means of ai and bi. Hence, it is 1 if ai = bi, and 0 if ai or bi are 0, and thus measures the spread between ai and bi. Minimizing or maximizing (3.79) is a matter of choosing the subsets{ai} and {bi} from the eigenvalues{λi} using a combinatorial argument presented byBloomfield and Watson (1975). In order to minimize (3.79), we need maximum spread between the two sets{ai} and {bi}, which is achieved by choosing ai = λi and bi = λ2n−i. In order to maximize (3.79), we need minimum spread between{ai} and {bi}, which is achieved by ai = λ2i−1 and bi = λ2i. Hence, the degree of impropriety is related to the eigenvalue spread of Rx x.
3.3.3 Maximally improper vectors
Following this line of thought, one might expect a vector that is maximally improper – in the sense that K= I – to correspond to an augmented covariance matrix with maximum possible eigenvalue spread. This was in fact claimed by Schreier et al. (2005) but, unfortunately, it is only partially true. Let ev(Rx x)= [µ1, µ2, . . ., µn]Tand ev(Rx x)= [λ1, λ2, . . ., λ2n]. Let R#be the augmented covariance matrix of a maximally improper vector with K= I. Using (3.46), we may write
R#=
Rx x R1x x/2FFTRTx x/2 R∗/2x x F∗FHRHx x/2 R∗x x
(3.80) for some unitary matrix F. The matrix R#has a vanishing Schur complement and thus
R#= R1x x/2FFTRTx x/2is an extreme point in the setQ.Schreier et al. (2005) incorrectly stated that ev(R#)= [2µ1, 2µ2, . . ., 2µn, 0Tn]Tfor any extreme point R#. While this is not
3.4 Testing for impropriety 77
true, there is indeed at least one extreme point R##such that the augmented covariance matrix R##has these eigenvalues. Let Rx x = UMUHbe the EVD of Rx x. Choosing
R##= UMUT = UM1/2UHUUTU∗M1/2UT = R1x x/2FFTRTx x/2 (3.81) with F= U means that