EE562a_Lecture_Part_2

# EE562a_Lecture_Part_2 - 2.0 Random Vectors 2.1 Deﬁnitions...

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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 cross-correlation matrix. 1 † Note: RXY = RYX . Deﬁnition: KXY = E (X − µX ) (Y − µY )† is called the cross-covariance 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 non-negative deﬁnite if for any complex vector a a† Ma ≥ 0. Claim: Correlation matrices are non-negative 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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## This note was uploaded on 05/25/2009 for the course EE 562a taught by Professor Toddbrun during the Spring '07 term at USC.

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