Spectral Resolution and Sensitivity Bandwidth in Spectrometers
For many of the spectrometers used at millimeter-wavelength observatories (filter bank spectrometers, correlation spectrometers, etc.), the true bandwidth of a single spectrometer channel is a convolution of the original gain response of the channel and any smoothing functions applied. This convolution of the response and smoothing functions affects two important factors:
The sensitivity bandwidth is defined as follows:
Kraus (Radio Astronomy, 2nd Edition, page 7-8) defines a similar term.
Table D.1 describes the relationship between, , and the frequency sampling for various smoothing functions F(v) assuming Nyquist sampling at . At the end of this Appendix I show the calculation for each of these integrals.
For the filter bank spectrometers at the 12m, the channel response is a second order Chebyshev bandpass filter (which approximates a pill box function), there is no smoothing, and they are not Nyquist sampled. A single channel in the Millimeter Autocorrelator (MAC), on the other hand, has a sinc frequency response which is hanning smoothed. Since the convolution of a sinc with any function that is already band-limited within the frequency response of the sinc leaves that function unchanged, we are left with a hanning function response. Therefore:
The Hanning function is non-zero only from –(N – 1) to N – 1, where N is the number of channels which are being smoothed…
Normally, we take 3 channels and give them weights 0.25,0.5,0.25. Therefore, becomes
The Hamming function is just the Hanning function with different weighting.
Copyright Arizona Radio Observatory.