Continuous time

How to extand the spectral analysis to non periodic continuous time funtions ?

Fourier Transform

Fourier transform Let fL1(R)L2(RR)f\in L^1(\mathbb{R})\cap L^2(\mathbb{RR}). The Fourier transform of ff, denoted by f^\hat f is given by f^(ν)=+f(t)ei2πνtdt . \hat f(\nu) = \int_{-\infty}^{+\infty} f(t) e^{-i2\pi\nu t} \mathrm{d} t\ .

Other definitions of the Fourier transforms exists. By replacing the frequency ν\nu in Herz by the pulsation ω=2πν\omega = 2\pi\nu in radian per second (or “pulsation”) f^(ω)=+f(t)eiωtdt \hat f(\omega) = \int_{-\infty}^{+\infty} f(t) e^{-i\omega t} \mathrm{d} t and by weighting by 12π\frac{1}{\sqrt{2\pi}} f^(ω)=12π+f(t)eiωtdt \hat f(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} f(t) e^{-i\omega t} \mathrm{d} t

If fL1(R)f\in L^1(\mathbb{R}), ie. ff is summable, the quantity f^\hat f is well defined for all νR\nu\in\mathbb{R}. We then have the following decreasing property

If fL1(R)f\in L^1(\mathbb{R}), then f^\hat f is continuous, bounded and limν+f^(ν)=0 . \lim_{\nu\rightarrow +\infty} \hat f(\nu) = 0\ . The more regular the function ff is, the faster f^\hat f goes to 00.

The Fourier transform ensures the Energy preservation, thanks to the Plancherel-Parseval theorem

Plancherel-Parseval Let fL1(R)L2(R)f\in L^1(\mathbb{R})\cap L^2(\mathbb{R}) and f^\hat f its Fourier transform. We have then the energy preservation f22=f^22 \Vert f\Vert_2^2 = \Vert\hat f\Vert_2^2 ie. Rf(t)2dt=Rf^(ν)2dν \int_{\mathbb{R}} |f(t)|^2 \mathrm{d} t = \int_{\mathbb{R}} |\hat f(\nu)|^2 \mathrm{d} \nu Let gL1(R)L2(R)g\in L^1(\mathbb{R})\cap L^2(\mathbb{R}) and g^\hat g its Fourier transform. We have then the inner product preservation f,g=f^,g^ \langle f,g\rangle = \langle \hat f,\hat g\rangle ie. Rf(t)g(t)dt=Rf^(ν)g^(ν)dν \int_{\mathbb{R}} f(t) \overline{g(t)} \mathrm{d} t = \int_{\mathbb{R}} \hat{f}(\nu) \overline{\hat{g}(\nu)} \mathrm{d} \nu

Proof

Spectrum

As for the periodic functions, we can define the spectrum of a continuous function

Spectrum Let fL1(R)L2(RR)f\in L^1(\mathbb{R})\cap L^2(\mathbb{RR}) and f^\hat f its Fourier transform. The spectrum of ff is given by spectrum(f)={f^(ν)2} \text{spectrum(f)}=\lbrace |\hat f(\nu)|^2 \rbrace

In practice, we are interested by the bandwidth of a signal, and more particularly by band limited signals

Bandwidth Let fL2(R)f\in L^2(\mathbb{R}) an analogical signal and f^\hat f its Fourier transform. supp{f^}\text{supp}\lbrace\hat f\rbrace is the {\em bandwidth} of the signal. It corresponds to the frequency spreading of the signal.

Band limited signal Let fL2(R)f\in L^2(\mathbb{R}) an analogical signal and s^\hat s its Fourier transform. ff is said to be band limited iff it exists B>0B>0 such that supp{f^}[B,B] . \text{supp}\lbrace\hat f\rbrace\subset [-B,B]\ .

An important result in signal processing is given by the Paley-Wiener’s theorem

Paley-Wiener Let fL2(R)f\in L^2(\mathbb{R}) a non null function with compact support. Then its fourier transform cannot vanish on an interval. Likewise, if f^\hat f is with compact support, then ff 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 fL1(R)L2(R)f\in L^1(\mathbb{R})\cap L^2(\mathbb{R}) such that f^L1(R)\hat f\in L^1(\mathbb{R}). If ff is continuous in tt, then f(t)=Rf^(ν)ei2πνtdν . f(t) = \int_{\mathbb{R}} \hat f(\nu) e^{i2\pi\nu t} \mathrm{d} \nu\ .

Proof

With the two other definitions, reconstruction formula becomes

f(t)=12πRf^(ν)eiωtdω f(t) = \frac{1}{2\pi}\int_{\mathbb{R}} \hat f(\nu) e^{i\omega t}\mathrm{d}\omega f(t)=12πRf^(ω)eiωtdω f(t) = \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} \hat f(\omega) e^{i\omega t} \mathrm{d}\omega

Calculus Properties

Let f,gL1(R)L2(R)f,g\in L^1(\mathbb{R})\cap L^2(\mathbb{R}) and f^,g^\hat f, \hat g their Fourier transforms.

  1. Convolution: fg^(ν)=f^(ν)g^(ν)\widehat{f* g}(\nu) = \hat f(\nu) \hat g(\nu)
  2. Multiplication: f.g^(ν)=(f^g^)(ν)\widehat{f . g}(\nu) = (\hat f * \hat g ) (\nu)
  3. Translation: let ga(t)=f(ta)g_a(t) = f(t-a) g^a(ν)=ei2πaνf^(ν)\hat g_a(\nu) = e^{-i2\pi a\nu} \hat f(\nu)
  4. Modulation: let gθ(t)=ei2πθtf(t)g_{\theta}(t) = e^{i2\pi \theta t}f(t) g^θ(ν)=f^(νθ)\hat{g}_\theta(\nu) = \hat f(\nu - \theta)
  5. Scaling: let gs(t)=f(t/s)g_{s}(t) = f(t/s) g^s(ν)=sf^(sν)\hat g_s(\nu) = |s|\hat f(s \nu )
  6. Time derivation: f(p)^(ν)=(i2πν)pf^(ν)\widehat{f^{(p)}}(\nu) = (i2\pi\nu)^p\hat f(\nu)
  7. Frequency derivation: let gp(t)=(i2πt)pf(t)g_p(t) = (-i2\pi t)^pf(t) g^p(t)=f^(p)(ν)\hat g_p(t) = \hat f^{(p)}(\nu)
  8. Complexe conjugation fˉ^(ν)=f^(ν)\widehat{\bar f}(\nu) = \overline{ \hat{ f}(-\nu)}
  9. Hermitian symetry f(t)Rf^(ν)=f^(ν)f(t)\in\mathbb{R} \Rightarrow \hat f(-\nu) = \overline{\hat{ f}(\nu)}
Proof