Exercise 3 of Assignment 1 (due 1/14/08)

Here we consider some basic properties of covariances. In what follows, assume for generality that all RVs and ordinary variables are complex-valued (the definition of covariance for complex-valued RVs is given in Section 2.5). All RVs are denoted by $Z$ (or subscripted versions thereof), while $c$ with or without a subscript denotes an ordinary variable. As usual, $Z^*$ denotes the complex conjugate of $Z$.

(a)

Show that $\mathop{\rm cov}\nolimits \,\{ Z, c \} = 0$.

(b)

Show that $\mathop{\rm cov}\nolimits \,\{ Z_1, Z_2 \} = \left(\mathop{\rm cov}\nolimits \,\{ Z_2, Z_1 \}\right)^*$.

(c)

Show that $\mathop{\rm cov}\nolimits \,\{ Z_1 + c_1 , Z_2 + c_2 \} = \mathop{\rm cov}\nolimits \,\{ Z_1, Z_2 \}$.

(d)

Suppose that at least one of the RVs $Z_1$ and $Z_2$ has a zero mean. Show that $\mathop{\rm cov}\nolimits \,\{ Z_1, Z_2 \} = E\{ Z^*_1 Z_2 \}$.

(e)

Show that

$$\mathop{\rm cov}\nolimits \Big\{ \sum_j c_{1,j} Z_{1,j}, \su... ...} c_{2,k} \mathop{\rm cov}\nolimits \, \{ Z_{1,j}, Z_{2,k} \},$$

where the summations are over finite sets of integers.

Note that real-valued variables can be regarded as a special case of complex-valued variables, so the results above continue to hold when some or all of the variables in question are real-valued. In particular, when $Z_1$ and $Z_2$ are both real-valued, part (b) simplifies to $\mathop{\rm cov}\nolimits \,\{ Z_1, Z_2 \} = \mathop{\rm cov}\nolimits \,\{ Z_2, Z_1 \}$. Also, when $Z_1$ and $c_{1,j}$ in parts (d) and (e) are real-valued, we can simplify $Z^*_1$ to $Z_1$, and $c^*_{1,j}$ to $c_{1,j}$.

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