Calculus of resonances in an uniform acoustic tube

We assume that the glottal end is closed, but the mouth is open. This is the configuration we are referring to:

The acoustic tube is uniform, and its length is L. The glottis, located at x=-L, is closed (infinite impedance) and the mouth, located at x=0, is open (impedance zero). Now, pressure variation p(x) along this uniform acoustic tube is expressed as:

$latex \frac{d^2p}{dx^2} + \left(\frac{2\pi f}{c}\right)^2p = 0 ~~(I)$

where f represents frequency in Hz, and c is the speed of sound: $latex 3.53 \times 10^4 cm/s$ at 37° C.

According to the boundary conditions (the impedances at both ends), the solution is:

$latex p(x) = P_m \sin{\frac{2\pi f}{c}x} ~~(II)$

where $latex P_m$ is the peak in sound pressure. On the other hand, we have a relation between pressure and volume velocity

$latex \frac{dp}{dx} = -\frac{j2\pi f \rho}{A}U ~~(III)$

$latex A$ is a constant representing the tube’s area. Now, volume velocity can be expressed as

$latex U(x) = jP_m \frac{A}{\rho c} \cos{\frac{2\pi f}{c}x} ~~(IV)$

where $latex \rho$ equals the average atmospheric density ($latex 1.14 \times 10^{-3} gm / cm ^ 3$ at 37°C).

As U(−L) = 0, resonances Fn of the acoustic tube are

$latex Fn = \frac{2n – 1}{4}\frac{c}{L} ~~(V)$

where n=1, 2, 3… And that’s it. We can see that the area function does not affect the location of resonances. Finally, remember that, in average, the male oral tract has a length of 16.9 cm, and the female tract has an average length of 14.1 cm.

Digital Signal Processing: An Introductory Note

Digital Signal Processing (DSP) comprises the techniques and algorithms for transforming, filtering and representing digital signals (DSP is a subfield of the more general Signal Processing topic).

Continuous and Digital Signal Processing

A signal is a measurement of a process, an observation of the behavior of some system. Numerically, a signal is a time-varying or spatial-varying quantity (in the following, for simplicity, we’ll assume that the independent variable is time t). Some physical signals, such as speech and image are continuous in time. For instance, the speech signal is a continuously varying acoustic pressure wave. Sometimes, continuous signals x(t) are referred to as continuous or analog waveforms (continuous and analog are typically interchangeable terms albeit analog is a kind of absolute term, and we will not be using it in the following). Continuous signals vary at an uncountable infinite number of times. On its side, digital processing units can only handle sequences of numbers, i.e., they are discrete devices. In order to harness the benefits of digital processing units, continuous signals have to be first discretized (sampled). After sampling, we get a digital signal, which we might use as a representation of the original continuous signal. This sampling process is performed by a Continuous-to-Discrete (C/D) converter.

Summarizing, sampling is the process by which a digital representation of a continuous time signal is obtained. Basically, during sampling we select a finite number of data points (in a finite time interval) to represent the infinite amount of data that the continuous signal contains (within the same interval). If sampling is periodic, we sample x(t) at uniformly spaced time instants. Sampling is by no means a trivial issue, and we have to be careful in selecting the discrete data values… how well does this discrete sequence represent the continuous signal?.

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