Maxwell's Rainbow — Question 8

Question 8

Octave-spaced channels have equal logarithmic but unequal absolute bandwidths

A spectral channel bank has edges $f_j=f_0 2^j$ for j=0 through 6, with $f_0=1.0$ GHz. Use $c=3.00\times10^8$ m/s. Each adjacent pair defines one channel.

1. Find the full frequency range, wavelength range, and number of channels.

2. Find the bandwidth of channel j from fj to f(j+1), and compare the sixth and first bandwidths.

Question 8 - Solution

Octave-spaced channels have equal logarithmic but unequal absolute bandwidths

(a) Method and calculation

$$f_6=64f_0=64\,\mathrm{GHz},\quad\boxed{6\text{ channels over }1\text{--}64\,\mathrm{GHz}}.$$$$\boxed{\lambda_{\max}=0.300\,\mathrm m,\quad\lambda_{\min}=4.6875\,\mathrm{mm}}.$$

(b) Interpretation and further calculation

$$\Delta f_j=f_{j+1}-f_j=f_0 2^j,\quad\Delta f_j/f_j=1.$$$$\boxed{\Delta f_5/\Delta f_0=2^5=32}.$$

Physical check

Equal spacing on a logarithmic axis represents equal ratios; absolute channel widths grow with frequency.