Summations and integrals

Note that

(\sum_{n = 1}^{N} x [n]) (∑_{n = 1}^{N} y [n]) = ∑_{n = 1}^{N} ∑_{m = 1}^{N} (x [n] y [m])

because, e.g., $(a + b + c) (d + e + f) = ad + ae + af + bd + be + bf + cd + ce + cf$ . Similarly, in the continuous-case

(\int_{⟨ T_{0} ⟩} x (t) dt) (\int_{⟨ T_{0} ⟩} y (t) dt) = \int_{⟨ T_{0} ⟩} \int_{⟨ T_{0} ⟩} (x (t) y (s)) dtds,

where one should note the adoption of distinct integration variables $t$ and $s$ . This result allows to express

{(\int_{⟨ T_{0} ⟩} x (t) dt)}^{2} = \int_{⟨ T_{0} ⟩} \int_{⟨ T_{0} ⟩} (x (t) x (s)) dtds,

(A.22)

which is an useful expression. Note that, in general,

{(\int_{⟨ T_{0} ⟩} x (t) dt)}^{2} \neq \int_{⟨ T_{0} ⟩} x^{2} (t) dt .