Input impulse response

Consider an object connected to a source tube as shown in figure 5.1. Notice that for a musical wind instrument, the input plane is the position at which the mouthpiece must be placed if the instrument is to be played. In order to carry out pulse reflectometry experiments, the mouthpiece must be removed to prevent leaks and to allow efficient transfer of acoustical energy into and out of the instrument. The length of the cylindrical sections is $l=cT/2$ where $T=1/F_s$ is the sample period such that the time for travel from the left hand side of a cylinder to the right hand side, reflection off the discontinuity and return to the left hand side will correspond to one sample in the time domain.

Figure 5.1: Travelling waves in a typical object split into cylindrical sections

We label section 1 as the plane at the end of the source tube. The forward and backward going waves here are labelled $p_+^{(1)}$ and $p_-^{(1)}$ respectively. Section 2 is then the plane immediately on the other side of the input plane discontinuity. Section 3 is a distance $l$ away at the other end of the first cylindrical section used in approximating the bore of the object. Section 4 is the plane immediately on the other side of the next discontinuity and so on.

We define the input impulse response as the sequence of reflections which return from the object under test when an ideal delta function impulse is fed into the input. We define $t=0$ as the time of arrival of the input pulse at the input plane. The forward going wave on section 1 is then an impulse

$$p_{+}^{(1)}[nT] = \delta[nT] = \left\{ \begin{array} {r@{\quad:\quad}l} 1 & n = 0, \\ 0 & n \geq 1 \end{array} \right.$$ (5.1)

where $T=1/F_s$ is the same period. $n$ is an integer running from $0$ to $N-1$ where $N$ is the total number of samples taken. The backward going wave is the input impulse response:

$$p_{-}^{(1)}[nT] = iir[nT].$$ (5.2)