Now we will use an inverse Fourier transform of the plane reflectance to
give the plane component of the input impulse response of our simple
discontinuity between cylindrical pipes.
A similar calculation was performed by Boone et al. [58] for the input
impulse response of a lightly damped rectangular cavity.
In order to work out the
inverse Fourier transform we need to calculate the reflectance for
a number of equally spaced frequencies. The zero frequency component
cannot be worked out with the present multimodal method because means
that
has elements which go to infinity. It may, however, be
noted that the plane reflectance tends to the plane wave approximation
reflection coefficient at low frequency, so the
plane reflectance is
taken from equation (2.20) in what follows.
When the Fourier transform of a real signal is performed the result is a conjugate symmetric spectrum. That is, the real part of the spectrum is symmetric and the imaginary part anti-symmetric. The point of symmetry is called the Nyquist frequency, and has a value half that of the sample frequency. We require that there are no significant components in the signal above the Nyquist frequency to prevent numerical errors due to aliasing. Similarly, when we calculate an inverse Fourier transform, we require that the spectrum decays to zero by the Nyquist frequency. From figure 6.2 we note that if we go to very high frequencies the plane reflectance does indeed decay to zero, meaning that numerical problems may be avoided if the Nyquist frequency chosen is high enough. The vector of the plane reflectance must then be made conjugate symmetric for the inverse Fourier transform to be calculated [59].
The result of an inverse Fourier transform of the plane reflectance is
shown in figure 6.5 for a radius ratio of . Progressively
higher values for the Nyquist frequency
in the calculation of the plane reflectance were taken
until the inverse transform showed convergence on a final answer.
The appearance of the time response is that of a
negative pulse. This is to be expected as we are calculating the reflection
of an impulse from a expansion which has a negative reflection coefficient.
The finite width of the pulse shows how the high frequency components
have been removed and the oscillations in the time domain response correspond
to the peak in the plane reflectance due to the lowest frequency cut-off.
The results presented so far are for scattering from a junction between
two infinite pipes. In order to treat physically realisable systems we
want to have a method of calculating the input impulse response of any
musical instrument approximated by a series of cylinders. This is
achievable by first calculating the input impedance matrix
by the multimodal method described in section 2.6.
The reflectance matrix can then be found using equation (6.8).
The plane wave component of the input impulse response can be calculated
by taking the inverse Fourier transform of the plane reflectance
in the same manner as in section 6.4. It should be noted that this
method involves calculating the multimodal radiation impedance; this can
be readily done if the instrument is assumed to have an infinite baffle
termination.
It is logical that the input impulse response should not depend on the
radiation impedance for the first seconds taken for sound to
reflect from the radiating end and arrive back at the input where
is
the length of the instrument.