Source reflection cancellation method

The increase of the length of the source tube to measure longer objects is clearly limited by the resulting larger losses in the source tube. Since the aim is to isolate the backward going object reflections from the waves going forward from the source, the use of multiple microphones is a possible technique. This has been attempted by Louis et al. [63]. Active real-time cancellation of the source reflections has also been attempted by Sharp [64] with limited success. Here we discuss a method of cancelling the source reflections by post-processing [62].

We define the backward travelling calibration pulse as $I_i$. This is shown in the first 40ms of figure 7.3. The last 40ms of figure 7.3 shows the forward going reflections of the calibration pulse from the source which we will refer to as $I_j$.

Figure 7.3: Calibration pulse including source reflections

The filter $H$ representing the source reflection function can be derived from these signals by deconvolution:

$$H(\omega) = \frac{I_{j}(\omega)I^{*}_{i}(\omega)} {I_{i}(\omega)I^{*}_{i}(\omega) + q}.$$ (7.1)

Now consider the reflections from a extended object. We define $R_{i}$ as the first part of the object reflections, consisting entirely of backward travelling waves. $R_{j}$, the second part of the object reflections, however, is in general $R_{j} = R^{+}_{j}+R^{-}_{j}$ where $R^{+}_j$ is the forward travelling reflection of $R_i$ off the loudspeaker and $R^{-}_j$ is the remains of the backward travelling wave. In order to reconstruct the bore we want to isolate $R^{-}_j$ by calculating $R^{+}_{j}$ and subtracting:

$$R^{+}_{j}(\omega) = H(\omega) \times R_{i}(\omega)$$ (7.2)

and $R^{-}_j = R_j - R^{+}_{j}$.