Fourier Deconvolution

Problem description

The convolution that links the impulse response \(h(z)\) to the logarithmic time-constant spectrum \(R(\zeta)\)

(18)\[h(z)\;=\;\bigl(R \;\ast\; w_z\bigr)(z) \;=\;\int_{-\infty}^{\infty} R(\zeta)\, e^{\,z-\zeta-\exp(z-\zeta)} \,\mathrm{d}\zeta\]

turns into a simple product in the Fourier domain:

(19)\[\mathcal{F}\{h\}(\Phi) \;=\; V(\Phi)\,W(\Phi),\]

where

  • \(\mathcal{F}\{f\}\) denotes the Fourier transform of function \(f\),

  • \(V(\Phi) = \mathcal{F}\{R\}(\Phi)\), is the Fourier transform of the logarithmic time-constant spectrum \(R(\zeta)\),

  • \(W(\Phi) = \mathcal{F}\{w_z\}(\Phi)\), is the Fourier transform of the kernel \(w_z(z)\).

With measured data the recorded signal contains noise,

(20)\[m(z)=h(z)+n(z) \;\Longrightarrow\; M(\Phi)=H(\Phi)+N(\Phi).\]

Division by \(W(\Phi)\) therefore yields

(21)\[V'(\Phi) \;=\; \frac{M(\Phi)}{W(\Phi)} \;=\; V(\Phi) \;+\; \frac{N(\Phi)}{W(\Phi)}.\]

Whenever \(|W(\Phi)|\) approaches zero the noise term explodes; high- frequency whitening is unavoidable. Practical NID hence applies a frequency-domain window that suppresses harmful bands before the inverse FFT reconstructs \(R(\zeta)\).

Padding

Because the FFT assumes circular convolution, the impulse response is normally zero-padded to at least twice its original length. Padding helps to avoid wrap-around artefacts and to match the length demanded by the chosen window function.

Frequency-domain windows

Let the filtered spectrum be

(22)\[V_\text{filtered}(\Phi) \;=\; V'(\Phi)\,F(\Phi),\]

with the window

(23)\[\begin{split}F(\Phi)= \begin{cases} 0, & |\Phi|>\Phi_c,\\ F_\text{window}(\Phi), & |\Phi|\le\Phi_c, \end{cases}\end{split}\]

where \(\Phi_c\) is the cut-off. Below are the most common window profiles (index \(n=0\dots N_W\) spans the non-zero part).

  • Rectangular

    (24)\[F_\text{Rect}[n]=1.\]

  • Hann

    (25)\[F_\text{Hann}[n] \;=\; \sin^2\!\Bigl(\pi n/N_W\Bigr).\]

  • Gaussian – width controlled by \(\sigma\le0.5\)

    (26)\[F_\text{Gauss}[n] \;=\; \exp\!\Bigl[ -\tfrac12 \bigl(\tfrac{n-N_W/2}{\sigma\,N_W/2}\bigr)^2 \Bigr].\]

  • Nuttall – four-term cosine series

    (27)\[F_\text{Nuttall}[n] = 0.355768 \;-\;0.487396\cos\Bigl(\tfrac{2\pi n}{N_W}\Bigr) \;+\;0.144232\cos\Bigl(\tfrac{4\pi n}{N_W}\Bigr) \;-\;0.012604\cos\Bigl(\tfrac{6\pi n}{N_W}\Bigr).\]

  • Fermi–Dirac (no finite \(\Phi_c\))

    (28)\[F_\text{Fermi}(\Phi) \;=\; \frac{1}{ \exp\!\bigl(\tfrac{|\Phi|-\mu}{\beta}\bigr)+1 } \;=\; \frac{ \exp\!\bigl[-(|\Phi|-\mu)/\beta\bigr] }{ 1+\exp\!\bigl[-(|\Phi|-\mu)/\beta\bigr] }.\]

    Parameters

    • \(\mu\) – half-width of the transition band,

    • \(\beta\) – slope (small \(\beta\) ⇒ sharper edge).