082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

II The Greek alphabet 981
f(t)
T
0 t 0 T 2T kT t
FigureAI.3
Thearea under f (t)can be approximated byaseriesofrectangular areas.
Now compare Equations (AI.1) and (AI.2). If we multiply the series of delta functions
byT we see that
∞∑
T f[k]δ(t −kT)
k=0
will bound the same area as our approximation to the area under the graph of f (t). In
thissense wecan regard
∞∑
T f[k]δ(t −kT)
k=0
as anapproximation tothe function f (t).We will denote thisapproximation by ˜f(t).
In summary, a continuous function f (t) can be represented as a sum of weighted
impulses each ofstrength f[k]occurring att =kT:
∑ ∞
˜f(t)=T f[k]δ(t −kT)
k=0
Thisrepresentationcanalsobethoughtofasawayofexpressingadiscretesequenceof
values, f[k],asacontinuousfunction ˜f (t).Thisisusefulwhenstudyingtheztransform
andthediscreteFouriertransform.Sometimesitwillbeconvenienttoworkwithoutthe
factorT, inwhich casewedefine
∞∑
f ∗ (t) = f[k]δ(t −kT)
k=0
It should be remembered that when using this form, f ∗ needs to be multiplied by the
factorT inordertoapproximatethe function f (t).
AppendixII THEGREEKALPHABET
A α alpha I ι iota P ρ rho
B β beta K κ kappa σ sigma
Ŵ γ gamma λ lambda T τ tau
δ delta M µ mu Y υ upsilon
E ε epsilon N ν nu φ phi
Z ζ zeta ξ xi X χ chi
H η eta O o omicron ψ psi
θ theta π pi ω omega