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Unformatted text preview: 2.0 Random Vectors
2.1 Deﬁnitions of Correlation Matrices
Deﬁnition: Let X1 , X2 , . . . , Xn be n random variables deﬁned on (U, F, P ).
Then,
⎡
⎤
X1 (u)
⎢
⎥
⎢ X2 (u) ⎥
⎢
X(u) = ⎢ . ⎥
. ⎥
⎣
. ⎦
Xn (u)
is a random vector.
To fully characterize X(u) we need an n−dimensional joint pdf . This would
often result in very cumbersome calculations when utilized. Instead of doing
this we usually just use 1st and 2nd order statistics in our study of random
vectors. This is suﬃcient for most needs. We often write X for X(u).
Deﬁnition: µX = E [X] is called the mean vector. Note
µX = [E[X1 (u)], . . . , E[Xn (u)]]t .
Here t denotes the transpose.
Deﬁnition: RX = E XX† is called the correlation matrix. Here † denotes
the conjugate transpose, i.e.,
∗
∗
X† (u) = (X1 (u), . . . , Xn (u)) . Thus ⎡ ∗
E [X1 (u)X1 (u)]
⎢
.
.
RX = ⎣
.
∗
E [Xn (u)X1 (u)] ... ⎤ ∗
E [X1 (u)Xn (u)]
⎥
.
.
⎦.
. ...
∗
. . . E [Xn (u)Xn (u)] Deﬁnition: KX = E (X − µX ) (X − µX )† is called the covariance matrix.
Note: KX = RX − µX µ† .
X
Deﬁnition: If X(u) and Y(u) are two random vectors then RXY = E XY†
is called the crosscorrelation matrix.
1 †
Note: RXY = RYX . Deﬁnition: KXY = E (X − µX ) (Y − µY )† is called the crosscovariance
matrix.
Note: KXY = RXY − µX µ† .
Y
Let Z = [XY]t . Then the correlation matrix for Z is
RZ = E ZZ† = RX
RYX RXY
.
RY 2.2 Properties of Correlation Matrices
Deﬁnition: A matrix M is said to be Hermitian symmetric if M = M† .
Note:
†
RX = E XX† † =E XX† † =E (X† )† X† = E XX† = RX so correlation matrices are Hermitian symmetric.
Deﬁnition: A Hermitian symmetric matrix M is said to be nonnegative
deﬁnite if for any complex vector a
a† Ma ≥ 0.
Claim: Correlation matrices are nonnegative deﬁnite. 2 Proof: We just need to show a† RX a ≥ 0.
⎡⎛ ⎞ ⎤⎛ ⎞ X1
a1
⎢⎜ . ⎟
†
∗
∗
∗
∗
. ⎠ ( X 1 · · · Xn ) ⎥ ⎜ . ⎟
a RX a = (a1 . . . an ) E ⎣⎝ .
⎦⎝ . ⎠
.
Xn
an
⎡ =E⎣ n
i=1 ⎛ ⎞⎤ n a∗ Xi ⎝
i j=1 ⎡ ai Xi∗ ⎠⎦ = E ⎣
⎡ =E⎣ n
i=1 2 ⎤ n
i=1 ⎛ a∗ Xi ⎝
i n j=1 ⎞∗ ⎤ a∗ Xi ⎠ ⎦
i a∗ Xi ⎦ ≥ 0.
i Also, a† KX a ≥ 0. 2.3 Linear Transformations of Random Vectors
Y(u) is formed by a linear transformation of X(u). Here X(u) ∈ Rn and
Y(u) ∈ Rm .
n Yi(u) = j=1 hij Xj (u), i = 1, 2, . . . , m or
Y(u) = HX(u)
where ⎡ h11
⎢ .
H=⎣ .
.
hm1 ... ⎤ h1n
. ⎥
. ⎦.
...
.
. . . hmn Let us now look at the ﬁrst and second moments.
µY = E [Y(u)] = E [HX(u)] = HE [X(u)] = HµX .
RY = E Y(u)Y† (u) = E HX(u) (HX(u))† = HE X(u)X(u)† H
= HRX H† .
Also, KY = HKX H† . 3 Question: Given a vector X of n uncorrelated random variables with zero
mean and unit variance how do we transform this vector into a vector Y
with mean c and covariance KY ?
Let
˜
Y(u) = HX(u).
Then
µY = HµX = 0
˜
and KY = HKX H† .
˜ Now KX is an n × n identity matrix, In . Thus
KY = HIn H† = HH† .
˜
We now let
˜
Y(u) = Y(u) + c.
Then
Y(u) = HX(u) + c.
Hence,
µY = c
and KY = KY = HH† .
˜ Problem: We need to ﬁnd H given some KY . This is a matrix factorization
problem that we will deal with later in the course. 4 ...
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 Spring '07
 ToddBrun

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