Summations and Integrals

Note that

(∑_{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

(∫_{⟨ T_{0} ⟩} x(t)dt) (∫_{⟨ T_{0} ⟩} y(t)dt) = ∫_{⟨ T_{0} ⟩} ∫_{⟨ 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

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

(A.18)

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

{(∫_{⟨ T_{0} ⟩} x(t)dt)}^{2} ≠ ∫_{⟨ T_{0} ⟩} x^{2} (t)dt.