12M Manual Appendix D


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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 full-width to half-power, or -3dB, bandwidth, which is the true spectral resolution.
The sensitivity bandwidth, which is the channel bandwidth used in calculations of the rms sensitivity in a spectrum.

 

The sensitivity bandwidth is defined as follows:

 

  (D.1)

Text Box: (D.1)

 

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.

Table D.1: Spectral Resolution and Sensitivity Bandwidth

F(v)
sinc 1.207 1.000
gaussian 2.000 1.505
hanning 2.000 2.667
hamming 1.820 2.935

 is the frequency sampling interval.

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:

 

= (D.2)
       
       
= (D.3)
       
       
= (D.4)
       
       
= (D.5)

 

 

D.1 Function Integrals

D.1.1 Sinc

  (D.6)

 

D.1.2 Gaussian

  (D.7)

 

D.1.3 Hanning

The Hanning function is non-zero only from –(N – 1) to N – 1, where N is the number of channels which are being smoothed…

  (D.8)

Normally, we take 3 channels and give them weights 0.25,0.5,0.25.  Therefore,  becomes

 

 

  (D.9)
 

D.1.4 Hamming

 

The Hamming function is just the Hanning function with different weighting.

 

  (D.10)
   

 


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Last updated: 11/08/11.