Index modulation

The modulation index is related to the transmission spectra of the gas-filled reference cell, TRef( ), the measurement cell, with its unknown gas concentration, Ty Ca,(/.), and the optical filter, F(/.), all of which are shown in Equation 1. If required, the source spectra can also be taken into account, by using additional spectral functions (although this will usually have little spectral variation over the linewidth of the gas band) ... 

Using the modelling approach above, the predicted response of a CO2 correlation spectroscopy system will now be shown. This requires knowledge of the transmission spectrum of CO2 gas, so that predictions from the model can be made, for the expected modulation index, SNR (signal to noise ratio) and cross-sensitivity to a contaminant gas. 

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.

Prediction of Expected Modulation Index ( Optical Contrast )... 

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.

Figure 10. Predicted modulation index (m), as a function of C02 gas concentration (%v/v) in the measurement cell. Reference and measurement cells were of lm length and the reference cell contained 100% C02 gas at lBar/20 °C. A choice of optical filter, having a bandwidth of 100 nm and a centre wavelength of 2.004 pm, was assumed.

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. 

Dependence of Modulation Index on Measurement Cell Gas Temperature... 

Figure 13. Expected dependence of modulation index on total gas measurement cell pressure, over the likely possible range of variation (0.8 to 1.2 Bar) of terrestrial atmospheric pressure, as calculated at 293 °K. It was assumed that the reference cell contained 100% C02 gas at 1 Bar 20°C, and that the optical fdter had a centre wavelength of 2.004 pm, with a bandwidth of 100 nm.

Figure 14. Expected variation of modulation index with temperature of the measurement cell, assuming both cells contained 100% C02 at 1 Bar and were 1 m long. The reference cell was held at 20 °C and the optical filter had a centre wavelength of 2.004 pm and a bandwidth of 100 nm.

The cross-sensitivity of the C02 sensor to water vapor is shown in Figure 16, where the expected modulation index, with the measurement cell filled... 

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).

Figure 17. Modulation index as a function of applied C02 concentration. 0%, 15.6%, 33.0%, 52.5%, 74.7% and 100% C02 concentration was applied to the measurement cell. The 90 cm long reference cell, contained 100% C02, and the 30 cm long measurement gas cell, contained 100% C02, were at 1 Bar and 20 °C. The optical LED emission spectrum was centred at 2.04 pm and had a 150 nm FWHM bandwidth.

Figure 17 shows recent experimental results for modulation index changes, in response to changing C02 gas concentration in the measurement cell18. 

In equation 1, bmin is the minimum feature size transferable, A is the wavelength of light, s is the separation between the mask and the substrate, and d is the thickness of the resist layer. In projection printing, a series of undulating maxima and minima are produced. Because of mutual interference, the dark region is never completely dark, and the maximum brightness does not correspond to 100% transmission. The quality of transfer can be conveniently indicated by the modulation index, M, which is defined as follows ... 

The maximum instantaneous frequency deviation A (Umax is therefore given by A (i)max I(i>m. When the modulation index/is nonzero, side frequencies occur above and below the carrier 00c, and the number of side frequencies increases with increasing I. 

Figure 9.29 Spectral dynamics of FM synthesis with linearly changing modulation index/( ). (Reprinted with permission from [Moorer, 1977], 1977, IEEE)...

This signal representation with only odd harmonics is an approximate model for a clarinet as with a uniform tube closed at one end and open at the other. In order to capture the time-varying envelope and bandwidth, one applies a A(n ) with a fast attack and slow release, and also makes the modulation index I(n ) inversely proportional to this envelope, thus emulating the decreasing bandwidth as a function of time. 

In percussive sounds, such as a bell, gong, drum and other nonperiodic sounds, spectral components are typically aharmonic and can be simulated by forming an irrational relation between ooc and b)m (e.g., (Om = /2C0c). In addition, the envelope is characterized by a sharp (almost instantaneous) attack and rapid decay, and the bandwidth moves from wide to narrow. Bell-like sounds, for example, can be made by making the modulation index proportional to an amplitude envelope which has exponential decay. For a drum-like sound, the envelope decay is even more rapid than the bell, and also has a quick overshoot giving a reduced initial bandwidth, followed by a widening and then narrowing of the bandwidth. 

Generalizations. An important generalization of the basic FM model is the introduction of dynamics. From Equation (9.83) an interesting property of FM synthesis is that the bandwidth of the spectrum increases with increasing modulation index I. Therefore, making the modulation index a function of time will allow spectra with... 

In this case uq is the carrier frequency, ft is the modulation frequency and r is the modulation index which describes the region over which the modulation takes place. 

We have developed FM lasers based on a commercial ring laser (Coherent 699-21). In this case all the intracavity etalons are removed and replaced by a lithium niobate phase modulator. This modulator can be resonantly driven at a frequency close to the cavity mode spacing. A simple theory of FM operation of a laser suggests that the modulation index is given by [12]... 

This represents another FM oscillation centred at a carrier frequency 2uq with modulation index 2rcos(0/2). When 0 is chosen to equal t we obtain a single frequency at twice the carrier frequency. 

We can make the spectral component at tog + n 2 reasonably large by making p close to unity or the modulation index Aoi/D close to n. This is the primary practical problem for n large. 

A simple differential saturation method has been proposed (63) in which the differential saturation effects are obtained at the same overall irradiation level due to an audio-frequency modulation of the resonance frequency in the continuous-wave mode of operation. This results in the appearance of sidebands in addition to the centre band of the spectrum. By a judicious selection of the modulation index, the saturation of signals in the sidebands may be made to amount to 10 — 0-1% of those in the centre band. The lineshape of such a combination of the centre band and the two closest sidebands may be adjusted to that corresponding to the function ... 

The index of modulation, 7, is defined as A /. Carson s rule (a rule of thumb) states that the number of significant bands on each side of the carrier frequency (sidebands) is roughly equal to 7+2. For example, a carrier sinusoid of frequency 600 Hz, a modulator sinusoid of frequency 100 Hz, and a modulation index of three would produce sinusoidal components of frequencies 600, 700, 500, 800,400, 900, 300, 1000,200, 1100, and 100 Hz. Inspecting these components reveals that a harmonic spectrum with 11 significant harmonics, based on a fundamental frequency of 100 Hz, can be produced by using only two sinusoidal generating functions. Figure 10.12 shows the spectrum of this synthesis. 

By setting the modulation index high enough, huge numbers of sidebands are generated, and the aliasing and addition of these (in most cases) results in noise. Figure 10.14 shows the synthesized waveform and spectrum of an FM... 

Modulation index (m) or depth of modulation affects the overall energy transfer into the implant. At a given rf signal amplitude, less energy is transferred into the implanted device when 100% modulation is used m = 1) as compared to 10% modulation (m = 0.053). However, retrieval of 100% modulated signal is much easier than retrieval of a 10% modulated signal. 

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