Traitement du signal, Shannon, et distributions.

∼
L2(]a, b[) −∞ < a < b < ∞
L2(]a, b[)
δ0
L2(R)
L2(R)
L2(R)
L2(R)
t∈R
f:t∈R→f(t)∈Rf:t∈R→f(t)∈C
f(t) = α0cos(2πν0t+θ0) = α0cos(2π
Tt+θ0) = α0cos(ω0t+θ0),
α0, ν0, ω0, T ∈R

L2(]a, b[) −∞ < a < b < ∞
ν0T=1
ν0
ω0= 2πν0θ0
|α0|
cos(a+b) = cos acos b−sin asin b
f(t) = β0cos(2πν0t) + γ0sin(2πν0t),
β0=α0cos θ0γ0=α0sin θ0
f(t) = α0cos(2πν0(t+t0)) t0=θ0
2πν0g(t) = f(t−t0)
g(t) = α0cos(2πν0t) = α0cos(2π
Tt) = α0cos(ω0t).
f(t) = α0ei(2πν0t+θ0)=α0ei(2π
Tt+θ0)=α0ei(ω0t+θ0),
<(f)(t) = α0cos(ω0t+θ0)
f
L2(]a, b[) −∞ <a<b<∞
L2(]a, b[)
z∈Cz|z|=√zz
z=a+ib a, b ∈Rz=a−ib |z|=√a2+b2
f:R→C|f|:R→Rf:R→C
f2:R→C
|f|(x) = |f(x)|, f(x) = f(x), f2(x) = (f(x))2.
sin2(x) = (sin(x))2
−∞ ≤ a < b ≤ ∞
L2(]a, b[; C) = {f: ]a, b[→CZb
a|f(t)|2dt < ∞} =L2(]a, b[)
T=b−a f :]a, b[→R
||f||L2(]a,b[) =³Zb
a|f(t)|2dt´1
2=||f||L2,
]a, b[
t→eiαt α∈RL2(]a, b[) ||eiα·||2
L2=Rb
a1dt =
b−a < ∞L2(R)
L2(]a, b[) (·,·)L2:L2(]a, b[) ×
L2(]a, b[) →R
(f, g)L2=Zb
a
f(t)g(t)dt
||f||L2=p(f, f )L2=³Zb
a|f(t)|2dt´1
2.
(L2(]a, b[),(·,·)L2)
L2(]a, b[) R→C
(·,·)L2

L2(]a, b[) −∞ < a < b < ∞
eik2πν0·t→eik2πν0t
a, b ∈Ra < b T =b−a ν0=1
Tω0=ν0 2π
(eik2πν0·)k∈Z(= (eik2π·
T)k∈Z= (eikω0·)k∈Z)
(L2(]a, b[),(·,·)L2)
1
√T(eik2πν0·)k∈Z(= ( 1 √Teik2π·
T)k∈Z=1
√T(eikω0·)k∈Z)
L2(]a, b[)
(eikω·, ei`ω·)L2=Zb
a
ei(k−`)ωt dt =(b−a=T k =`,
0k6=`.
T
R
eik2πν0teik2πν0·:t∈R→eik2πν0t
eik2πν0t
t
(eik2πν0t)k∈Z
(eik2πν0·)k∈Z
f∈L2(]a, b[)
ck∈Ck∈Zf
t∈]a, b[
f(t) = X
k∈Z
ckeik2πν0t(= X
k∈Z
ckeik2πt
T=X
k∈Z
ckeikω0t).
f
ck=1
T(f, eik2πν0·)L2(]a,b[) =1 TZb
a
f(t)e−ik2πν0tdt,
||f||2
L2=TX
k∈Z
c2
k.
(1
√Teik2πt T)k∈ZL2(]a, b[)
(f(t), ei`2πν0t)L2=X
k
ck(eik2πν0t, ei`2πν0t)L2=X
k
ckT δk` =c`T,
||f||2
L2= (f, f)L2=Pk` ckc`(eik2πν0t, ei`2πν0t)L2=Pk` ckc`δk`T

L2(]a, b[) −∞ < a < b < ∞
t∈RS(f) : [a, b]→R
S(f)(t) = X
k∈Z
ckeik2πν0t
f t
f∈L2(]a, b[) S(f)
S(f) = f .
f:]0,2π[→Rf(t) = 0 t∈]0, π[f(t) = 1 t∈]π, 2π[
f(π) = 36 1 Sf (π) = 1
2
ν1=ν0
νk=kν k ≥2
(ck, νk)k∈Zf
L2
f t0g
(g(t0)6=f(t0),
g(t) = f(t)t6=t0,)ck(f) = ck(g)f g
S(f) = S(g)f6=g
||f||2
L2=||g||2 L2f g
f L2(]a, b[) S(f)(t)
f(t)t f g f
S(f) = f f
f: [a, b]→C
f(t) : [a, b]→Rf(t)∈Rt
Sf (t) = Pkcke2ikπν0tSf =f
Sf (t)∈RSf(t) = Sf(t)
Sf (t) = Pkcke−2ikπν0t=Pkc−ke+2ikπν0t=Sf(t)f
ck=c−k
k∈Zc0∈R
k > 0 : ak=ck+c−k∈R
k > 0 : bk=i(ck−c−k)∈iR
a0=c0,
k > 0 : ck=ak
2−ibk 2
k > 0 : c−k=ak
2+ibk 2
c0=a0,
eiθ = cos θ+isin θ
Sf (t) = a0+∞
X
k=1
akcos(2kπν0t) + ∞
X
k=1
bksin(2kπν0t),
f

L2(R)
δ0
δ0L2
L2
δ0
δ0
]−π, π[
δ0=1
2πX
k∈Z
eikx,
δ0
ck=1
2πk
PN(x) = 1
2π
N
X
k=−N
eikx
δ0N
PNL2(] −π, π[) PNR]−π, π[
PNPN
PNck=1
2πk|k|< N ck= 0
k|k|> N
PN(x) = 1
2π(1 + 2 cos(x) + ... + 2 cos(Nx)) PN
cos(kx) 2π
k6= 0
Zπ
−π
Pn(x)dx =1
2πZπ
−π
1dx = 1,
PN
2πPN(x) = e−iNx P2N
k=0(eix)kx6= 0
2πPN(x) = e−iNx 1−(eix)2N+1
1−eix =e−i(N+1
2)x−ei(N+1 2)x
e−i1
2x−ei1 2x=sin(N+1
2)x
sin 1
2x,
n x
0 2πPN(0) = 2N+1 PNPN(0) → ∞ N→ ∞
(PN)n∈NRπ
−πPN(x)f(x)dx →f(0)
[−π, π]
δ0
L2(R)
t=−∞ +∞
R
L2(R)
L2(R) = {f:R→CZ∞
−∞ |f(t)|2dt < ∞}
f:R→R
L2(R) (·,·)L2:L2(R)×L2(R)→C
(f, g)L2=ZR f(t)g(t)dt