src_ts_wf_foutra.md

@@ -22,15 +22,159 @@ This way it computes spectral analysis with three different scaling options:
 ### Scaling options
 #### Fourier amplitude spectrum
 #### Power spectral density
-#### Spectrum of harmonics amplitudes
+#### Scaling for harmonic signals
 
-This math is inline $`a^2+b^2=c^2`$.
+For harmonic signals the FFT normalized to the duration of the available
+time window has an amplitude of the value of the maximum of the Fourier
+transform of the window function times half the amplitude of the time domain
+signal. To obtain a spectral representation with peaks of the amplitude of
+the time domain signal, the FFT must be scaled accordingly.
 
-This is on a separate line
+We understand 
 ```math
-a^2+b^2=c^2
+  \tilde{f}(\omega)=\int\limits_{-\infty}^{+\infty} 
+    f(t) \,e^{-i\omega t} \textrm{d}{t}.
+```
+as the Fourier transformation of the time domain signal
+```math
+  f(t)=\int\limits_{-\infty}^{+\infty} 
+    \tilde{f}(\omega) \,e^{i\omega t} \frac{\textrm{d}{\omega}}{2\pi}.
+```
+If we apply a time domain window function \f$w(t)\f$ to the function
+\f$f(t)\f$, we obtain the tapered function
+```math
+  g(t)=f(t) w(t)
+```
+and its Fourier transform
+```math
+  \tilde{g}(\omega)=\int\limits_{-\infty}^{+\infty}
+    \tilde{f}(\omega')\,\tilde{w}(\omega-\omega')\,\textrm{d}\omega.
 ```
 
+##### Application to harmonic signals 
+
+Let
+```math
+  f(t)=A\,\cos(\omega_0 t+\phi)
+```
+be the harmonic signal under investigation.
+Then
+```math
+  f(t)=A\left\{\cos(\omega_0 t)\cos(\phi)-
+    \sin(\omega_0 t)\sin(\phi)\right\}
+```
+and
+```math
+  \tilde{f}(\omega) =
+    \frac{A}{2}\left\{
+      \cos(\phi)
+        \left[
+          \delta(\omega-\omega_0) 
+          +
+          \delta(\omega+\omega_0) 
+        \right]
+      +i\sin(\phi)
+        \left[
+          \delta(\omega-\omega_0) 
+          -
+          \delta(\omega+\omega_0) 
+        \right]
+    \right\},
+```
+where \f$\delta(\omega)\f$ is Dirac's delta function with
+```math
+  \delta(\omega)=\left\{
+    \begin{array}{ll}
+      \infty & \textrm{if $\omega=0$ and}\\
+      0 & \textrm{otherwise}
+    \end{array}
+  \right.
+```
+and
+```math
+  \int\limits_{-\infty}^{+\infty}
+    \delta(\omega)\,\textrm{d}\omega=1
+```
+such that
+```math
+  \tilde{f}(\omega)=\int\limits_{-\infty}^{+\infty}
+    \tilde{f}(\omega')\,\delta(\omega-\omega')\,\textrm{d}\omega.
+```
+This way I obtain
+```math
+  \tilde{g}(\omega)=
+    \frac{A}{2}\left\{
+      e^{i\phi}\tilde{w}(\omega-\omega_0)
+      +
+      e^{-i\phi}\tilde{w}(\omega+\omega_0)
+    \right\}
+```
+for the Fourier transform of the tapered function.
+If we ignore interference with side-lobes from the negative frequency
+\f$-\omega_0\f$ and side-lobes of potential other harmonics at nearby
+frequencies, we can approximate
+```math
+  \tilde{g}(\omega_0)\approx\frac{A}{2}e^{i\phi}\tilde{w}(0)
+```
+and
+```math
+  \left|\tilde{g}(\omega_0)\right|\approx
+    \frac{A}{2}\left|\tilde{w}(0)\right|.
+```
+
+##### Boxcar taper function
+
+The boxcar taper is defined as
+```math
+  w(t) = 
+  \left\{
+    \begin{array}{ll}
+      1 & \textrm{if $|t|\leq T/2$ and}\\
+      0 & \textrm{otherwise}
+    \end{array}
+  \right.
+```
+with
+```math
+  \tilde{w}(\omega)
+    = T \frac{\sin(\omega T/2)}{\omega T/2}
+```
+and
+```math
+  w_{\textrm{max}}=w(0)=T.
+```
+
+##### Hanning taper function
+
+The Hanning taper is defined as
+```math
+  w(t) = 
+  \left\{
+    \begin{array}{ll}
+      \cos^2(\pi t / T)) 
+      = \frac{1}{2}\left[
+          1 + \cos(2\pi t / T)
+        \right]
+      & \textrm{if $|t|\leq T/2$ and}\\
+      0 & \textrm{otherwise}
+    \end{array}
+  \right.
+```
+with
+```math
+  \tilde{w}(\omega)
+   = T/2 \frac{\sin(\omega T/2)}{\omega T/2}
+   + T/4 \frac{\sin(\omega T/2+\pi)}{\omega T/2+\pi}
+   + T/4 \frac{\sin(\omega T/2-\pi)}{\omega T/2-\pi}
+```
+(see Blackman, R.B. and Tukey, J.W. 1958.
+     The measurement of power spectra.
+     Dover Publications, Inc., New York.
+     Section B.5)
+and
+```math
+  w_{\textrm{max}}=w(0)=\frac{T}{2}.
+```
 
 ### Usage