How to extand the spectral analysis to non periodic continuous time funtions ?
Fourier Transform
Fourier transform
Let f∈L1(R)∩L2(RR). The Fourier transform of f, denoted by f^ is given by
f^(ν)=∫−∞+∞f(t)e−i2πνtdt.
Other definitions of the Fourier transforms exists. By replacing the frequency ν in Herz by the pulsation ω=2πν in radian per second (or “pulsation”)
f^(ω)=∫−∞+∞f(t)e−iωtdt
and by weighting by 2π1f^(ω)=2π1∫−∞+∞f(t)e−iωtdt
If f∈L1(R), ie. f is summable, the quantity f^ is well defined for all ν∈R. We then have the following decreasing property
If f∈L1(R), then f^ is continuous, bounded and
ν→+∞limf^(ν)=0.
The more regular the function f is, the faster f^ goes to 0.
The Fourier transform ensures the Energy preservation, thanks to the Plancherel-Parseval theorem
Plancherel-Parseval
Let f∈L1(R)∩L2(R) and f^ its Fourier transform. We have then the energy preservation
∥f∥22=∥f^∥22
ie.
∫R∣f(t)∣2dt=∫R∣f^(ν)∣2dν
Let g∈L1(R)∩L2(R) and g^ its Fourier transform. We have then the inner product preservation
⟨f,g⟩=⟨f^,g^⟩
ie.
∫Rf(t)g(t)dt=∫Rf^(ν)g^(ν)dν
Proof
We proof the inner product conservation. Energy conservation is then a direct consequence.
Let f,g∈L1(R)∩L2(R). Let gˇ such that gˇ(t)=gˉ(−t). We denote by u(t)=(f∗gˇ)(t). Then
u^(ν)=f^(ν)gˇ(ν)=f^(ν)g^ˉ(ν)
We obtain
u(t)=∫Rf^(ν)g^ˉ(ν)ei2πνtdν
and more particularly
u(0)=∫Rf^(ν)g^ˉ(ν)dν
By definition of u thanks to the convolution product, we have
u(τ)=∫Rf(t)gˇ(τ−t)dt=∫Rf(t)gˉ(t−τ)dt
leading to
u(0)=∫Rf(t)gˉ(t)dt
hence the result.
Spectrum
As for the periodic functions, we can define the spectrum of a continuous function
Spectrum
Let f∈L1(R)∩L2(RR) and f^ its Fourier transform. The spectrum of f is given by
spectrum(f)={∣f^(ν)∣2}
In practice, we are interested by the bandwidth of a signal, and more particularly by band limited signals
Bandwidth
Let f∈L2(R) an analogical signal and f^ its Fourier transform. supp{f^} is the {\em bandwidth} of the signal. It corresponds to the frequency spreading of the signal.
Band limited signal
Let f∈L2(R) an analogical signal and s^ its Fourier transform. f is said to be band limited iff it exists B>0 such that
supp{f^}⊂[−B,B].
An important result in signal processing is given by the Paley-Wiener’s theorem
Paley-Wiener
Let f∈L2(R) a non null function with compact support. Then its fourier transform cannot vanish on an interval. Likewise, if f^ is with compact support, then f cannot vanish on an interval.
This theorem says that an analogical signal cannot be both bandlimited and with a limited time support!
Inverse Fourier Transform
Inversion of the Fourier transform
Let f∈L1(R)∩L2(R) such that f^∈L1(R). If f is
continuous in t, then
f(t)=∫Rf^(ν)ei2πνtdν.
Proof
We give here an abusive proof of this result, supposing that we can write the Fourier transform of the Dirac as
δ^(ν)=∫Rδ(t)e−i2πνtdt=1
and more generally, denoting by \delta_\tau(t) = \delta(t-\tau)$
δ^τ(ν)=∫Rδ(t−τ)e−i2πνtdt=e−i2πτ
Conversely, the constant function e−i2πτ admit for invert Fourier transform the Dirac at the time τ. Such a transform and its inverse is an abuse of notation, but can be defined using Distribution theory.
This leads to
∫Rf^(ν)ei2πνtdν=∫R∫Rf(τ)e−i2πντdτei2πνtdν=∫R∫Rf(τ)ei2πν(t−τ)dτdν=∫Rf(τ)∫Re−i2πντei2πνtdνdτ=∫Rf(τ)δ(t−τ)dτ=f(t)
The rigorous proof use a sequence of function eεnν2, and then makes the sequence εn tends to 0 (eεnν2 then tends to Dirac). Using the dominated convergence theorem one can conclude the proof.
With the two other definitions, reconstruction formula becomes
f(t)=2π1∫Rf^(ν)eiωtdωf(t)=2π1∫Rf^(ω)eiωtdω
Calculus Properties
Let f,g∈L1(R)∩L2(R) and f^,g^ their Fourier transforms.
Convolution:
f∗g(ν)=f^(ν)g^(ν)
Multiplication:
f.g(ν)=(f^∗g^)(ν)
Translation: let ga(t)=f(t−a)g^a(ν)=e−i2πaνf^(ν)
Modulation: let gθ(t)=ei2πθtf(t)g^θ(ν)=f^(ν−θ)
Scaling: let gs(t)=f(t/s)g^s(ν)=∣s∣f^(sν)
Time derivation:
f(p)(ν)=(i2πν)pf^(ν)
Frequency derivation: let gp(t)=(−i2πt)pf(t)g^p(t)=f^(p)(ν)
Complexe conjugation
fˉ(ν)=f^(−ν)
Hermitian symetry
f(t)∈R⇒f^(−ν)=f^(ν)
Proof
We show the convolution property (1.), (2.) is a direct consequence. Let u(t)=(f∗g)(t). We have
u^(ν)=∫Ru(t)e−i2πνtdt=∫R(f∗g)(t)e−i2πνtdt=∫R∫Rf(τ)g(t−τ)dτe−i2πνtdt=∫R∫Rf(τ)g(x)e−i2πν(τ+x)dτdx=∫Rf(τ)e−i2πντdτ∫Rg(x)e−i2πνxdx=f^(ν)g^(ν)
Remaining proof are similar to the properties of Fourier series coefficients. Points 3, 4 and 5 can be shown by a simple change of variable. T montrent par simple changement de variable. The temporal derivative property can de proven by integration by part and an immediate induction.