One of the fundamental limitations of the pulse reflectometry is the bandwidth of the source of acoustic energy. The compression driver loudspeakers used at present give little energy above 10 kHz, due to their limited frequency response and due to losses in the source tube. This means that sharp steps in the bore are reconstructed relatively poorly. More of a problem, however, are the very low frequencies (50 Hz and below) which have long wavelengths, so will have an influence on the general upward and downward trend of the reconstructed bore. In particular, no energy may be produced by the loudspeaker at the zero of frequency, hence the zero frequency bin in the frequency domain response is not accurately measured. This means a lack of accuracy in the dc offset in the time domain input impulse response, which is why the dc offset must be calculated by averaging the response from a cylindrical tube section (an actual or virtual dc tube) and subtracted in post-processing.
An alternative is to deduce the value of the zero frequency bin from first principles. The zero frequency bin of the input impedance must be zero for all open ended instruments because pressure will not build up due to a steady velocity. Studying the equations relating the frequency domain input impedance and input impulse response reveals that the zero frequency bin of the input impulse response will have a value of -1. Setting the zero frequency bin of the frequency domain input impulse response to -1 before inverse Fourier transforming to the time domain should then remove the dc offset. While much effort has been expended in removing the dc offset in more complicated ways, preliminary results suggest that this simple technique is effective for dc offset removal. It should be noted that this method has not been implemented previously due to bandwidth problems which we will now discuss.
There is an inevitable discontinuity between the end of the source tube and the object under test. Ideally this would show as a small impulse at the start of the input impulse response. Because of the limited bandwidth of the measurement this impulse is spread over several samples in the time domain, both before and after the sample. There are no samples before , so the measurement is in some way corrupted. A signal introduced ``before'' the first sample is actually observed in the final few samples of the signal because Fourier transforms are inherently periodic. This apparently non-causal signal represents part of the energy of the reflection from the first step however, and should be present after the first sample in the input impulse response if a bore reconstruction is to be accurate. We may achieve this by using the virtual dc tube method to delay the object reflections. Because the dc offset was calculated by averaging the response over the first couple of milliseconds, the dc tube was previously around 40 cm long. The method suggested here would only require a very short virtual dc tube whose length corresponds to the distance sound propagates in a few time samples.
As mentioned previously, the bandwidth is partly controlled by losses. High frequencies are attenuated by travel in the source tube much more strongly than low frequencies. The use of longer source tubes to enable measurement of longer objects accentuates this problem. A solution currently being investigated is to use multiple microphones. This enables the forward and backward going waves to be separated, so removing the problem of interference between the input signal, the reflected signal and the source reflections. The source tube can then be made far shorter, decreasing the losses at high frequency and increasing the bandwidth. A source tube with five microphones is currently being developed to enable large bandwidth measurement. Preliminary results for two microphones have been presented recently by van Walstijn et al. .
The formula used in deconvolution of the object reflections and calibration pulse to produce the input impulse response is another area of useful study. A constraining factor is included in the frequency domain division to prevent high frequency noise occurring. The effect of this is to filter out any high frequencies in the input impulse response. This is not mathematically rigorous; the answer depends slightly on the arbitrary choice of constraining factor. Work on applying truncated singular value decomposition, a more advanced method of deconvolution, to pulse reflectometry data is discussed by Forbes et al. .
A more minor improvement to the single microphone method of acoustic pulse reflectometry which may be applied to increase the speed at which experiments can be performed is to measure the calibration pulse simultaneously with the object reflections. At present, the calibration pulse (the reflection from closing the end of the source tube) is deconvolved from the object reflections to calculate the input impulse response of the object. We may improve upon this by recording the full system impulse response, including the pulse passing the microphone on the way to the object under test. The calibration pulse may then be calculated by applying a loss filter to the recording of the pulse passing the microphone the first time, to account for the difference in propagation distance between the pulse passing the microphone initially and the reflections which return from the end of the source tube. The object reflections may be isolated as normal. This method will be especially easy to apply with MLS excitation because full system impulse response measurements have already been performed.
The derivation of an expression for the multimodal radiation impedance of an open ended pipe without a baffle would be a productive area of research as it would enable increased realism, and therefore accuracy, in input impedance calculations. Further work is also necessary to include higher modes in a pulse reflectometry bore reconstruction algorithm using the approaches discussed in chapter 6. Optimisation methods are currently being studied with a view to correcting the plane wave bore reconstruction algorithm results by minimising the difference between the theoretical input impulse response of the bore reconstruction and the measured input impulse response.