Filter bandwidth

Figure 7. Modulation index (m), as a function of filter centre wavelength, assuming the reference and measurement cells contain 100% C02 gas at 1 Bar, 20 °C and are of lm length. An optical filter with 2 nm FWHM bandwidth is assumed. This shows that, with this filter bandwidth, the maximum modulation index occurs at a filter centre wavelength of 2.004 pm.

The maxima in the modulation index correspond to the use of an unrealistically narrowband optical filter. Figure 9 is a 3D plot, showing modulation index as a function of optical selection filter bandwidth and centre wavelength. Optical filters with centre wavelengths between 1.9 pm and 2.1 pm, and bandwidths between 0.01 nm and 80 nm were considered. 

Figure 9. C02 Detection variation of modulation Index (m), with optical filter centre wavelength and bandwidth. The broad range of absorption lines causes a very complex variation of modulation indices when using narrow filters (not all peaks at narrow filter bandwidths are shown, as this would obscure the behaviour with wider filter bandwidths). Reference and measurement cells are assumed to he of 1 m length and contain 100% C02 gas at 1 Bar/20 °C.

As can be seen, the modulation index response is highly complex, with several other narrow-bandwidth maxima in the response. As already discussed, and as will be shown by the SNR analysis later, peaks corresponding to the use of a very narrow bandwidth filter (2 nm) are unsuitable, and unfortunately no further clear peaks are seen as the filter bandwidth is increased. To optimize the system, it was therefore necessary to take account of other system parameters, such as SNR in measurements, which will now be considered. 

Figure 10 shows the dependency of the modulation index on the measurement gas cell concentration (%v/v), assuming dilution by nitrogen gas, at a pressure of 1 Bar and a temperature of 20 °C. This shows that there is a significant non-linearity in the modulation index response, particularly at higher CO2 gas concentrations in the measurement cell. As before, an optical filter bandwidth of 100 nm was assumed. 

Figure 11 plots the SNR versus filter bandwidth, at 3 levels of received optical intensity. It may be observed that the SNR is not very dependent on filter centre wavelength, but is more strongly related to the bandwidth of the optical filter. Optimum SNR is attained with an optical filter bandwidth of approximately 80-100 nm, i.e. significantly wider than the very narrow bandwidth that was found to maximise the modulation index. 

It may also be observed that small variations in the received light intensity at the detector have very little effect on the choice of best filter bandwidth to maximise SNR. Note, in these results, the very poor SNR obtained with narrow filter bandwidths (Figure 12). 

Figure 16. The variation (crosstalk) of modulation index with filter bandwidth, when the measurement cell contains a high concentration (0.05 Bar partial pressure) of H20 vapour impurity (both cells are 1 m in length cell at 1 Bar and 20 °C, and the reference cell contains 100% C02 gas).

The time-bandwidth product, constraining the minimum analysis filter bandwidth to be inversely proportional to the observation time interval, must also be confronted. 

Increase the filter bandwidth by 500-1,000 Hz if there are signals near the edges of the spectral window. 

The Morlet wavelet can be understood to be a linear bandpass filter, centred at frequency m = coo/a with a width of /(aoa). Some Morlet wavelets and their Fourier spectra are illustrated in Fig. 4.4.4. The translation parameter b has been chosen for the wavelet to be centred at time f = 0 (top). With increasing dilatation parameter a the wavelet covers larger durations in time (top), and the centre frequency of the filter and the filter bandwidths become smaller (bottom). Thus depending on the dilatation parameter different widths of the spectrum are preserved in the wavelet transform while other signals in other spectral regions are suppressed. 

Here, y k) represents the estimate of the true signal. Further, / is the filter constant, or, in other words, the filtering bandwidth. By a judicious choice of / , one can remove high-frequency noise components from the signal and retain the relevant signal characteristics. Figure 6.10 shows the frequency response of a first-order filter and how the bandwidth changes as a function of / . 

Moore, B. C. J., and Glasberg, B. R. Suggested formulae for calculating auditory-filter bandwidths and excitation patterns. Journal of the Acoustical Society of America 74 (1983), 750-753. 

FIGURE 20.66 Typical spectrum analyzer display of a single-cavity filter. Bandwidth (BW) = 458 to 465 MHz = 7 MHz. Filter Q = fcr/BW = 463 MHz/7 MHz = 66. The trace shows 1-dB insertion loss. 

The noise index NI in dB is measured by a band-pass filter (bandwidth of 1 kHz between 618 Hz and 1618 kHz) [51]. A noise index NI = 0 dB corresponds to 1 pV noise voltage per 1 V DC voltage. For ruthenate-based TFRs, the noise index exhibits an increase proportional to the sheet resistivity and a decrease proportional to the resistor area [47,52]. 

B. Same as above, but filter bandwidth adjusted to 100 - 1000 Hz to isolate the oscillatory potentials. 

---