Description

These functions are designed to facilitate approximation of integrals such as

$$\int g(f) h(f) df$$

when $g(f)$ and $h(f)$ are sampled with different frequency resolutions. If the frequency resolution were the same for both functions, e.g., a frequency spacing of $\delta f$ and a start frequency of $f_0$, so that the $k$th element corresponded to a frequency $f_k = f_0 + k\delta f$, the approximation would be defined as

$$\int g(f) h(f) df \approx \delta f \sum_k g_k h_k$$

whose contribution from the $k$th element is^27.1

$$\int_{f_k-\delta f/2}^{f_k+\delta f/2} g(f) h(f) df \approx \delta f g_k h_k .$$

The central idea in our definitions of coarse graining will thus be the correspondence

$$h_k \approx \frac{1}{\delta f} \int_{f_k-\delta f/2}^{f_k+\delta f/2} h(f) df$$ (27.1)

The purpose of this function is to obtain a frequency series $\{h_k\}$ with start frequency $f_0$ and frequency spacing $\delta f$ from a finer-grained frequency series $\{h'_\ell\}$ with start frequency $f'_0$ and frequency spacing $\delta f'$. Focussing on the $k$th element of the coarse-grained series, which represents a frequency range from $f_k-\delta f/2$ to $f_k+\delta f/2$, we consider the elements of the fine-grained series whose frequency ranges overlap with this. (Fig. )

Figure: Coarse graining a frequency series

We define $\ell^{\scriptstyle{\rm min}}_k$ and $\ell^{\scriptstyle{\rm min}}_k$ to be the indices of the first and last elements of $h'_\ell$ which overlap completely with the frequency range corresponding to $h_k$. These are most easily defined in terms of non-integer indices $\lambda^{\scriptstyle{\rm min}}_k$ and $\lambda^{\scriptstyle{\rm max}}_k$ which correspond to the locations of fine-grained elements which would exactly reach the edges of the coarse-grained element with index $k$. These are defined by

$$\begin{eqnarray*} f_0 + \left(k-\frac{1}{2}\right) \delta f &=& f'_0 + \left(... ...(\lambda^{\scriptstyle{\rm max}}_k+\frac{1}{2}\right) \delta f' \end{eqnarray*}$$

or, defining the offset $\Omega=(f_0-f'_0)/\delta f'$ and the coarse graining ratio $\rho = \delta f / \delta f'$,

$$\begin{eqnarray*} \lambda^{\scriptstyle{\rm min}}_k &=& \Omega + \left(k-\fra... ...&=& \Omega + \left(k+\frac{1}{2}\right) \rho - \frac{1}{2} . \end{eqnarray*}$$

Examination of Fig.  shows that $\ell^{\scriptstyle{\rm min}}_k$ is the smallest integer not less than $\lambda^{\scriptstyle{\rm min}}_k$ and $\ell^{\scriptstyle{\rm min}}_k$ is the largest integer not greater than $\lambda^{\scriptstyle{\rm min}}_k$.

With these definitions, approximating the integral in () gives

$$h_k = \frac{1}{\rho} \left( (\ell^{\scriptstyle{\rm min}}_k... ... max}}_k) h'_{\ell^{\scriptscriptstyle{\rm max}}_k+1} \right)$$ (27.2)

In the special case $f_0=f'_0$, we assume both frequency series represent the independent parts of larger frequency series $\{h_k\vert k=-(N-1)\ldots(N-1)\}$ and $\{h'_\ell\vert\ell=-(N-1)\ldots(N-1)\}$ which obey $h_{-k}=h_k^*$ and $h'_{-\ell}{}=h'_\ell{}^*$ (e.g., fourier transforms of real data). In that case, the DC element of the coarse-grained series can be built out of both positive- and implied negative-frequency elements in the fine-grained series.

$$h_0 = \frac{1}{\rho} \left[ h'_0 + 2 \mathrm{Re} \left... ... h'_{\ell^{\scriptscriptstyle{\rm max}}_0+1} \right) \right]$$ (27.3)