A Quadrature Signals Tutorial: Complex, But Not Complicated

Understanding the 'Phasing Method' of Single Sideband Demodulation

Complex Digital Signal Processing in Telecommunications

Introduction to Sound Processing

Introduction of C Programming for DSP Applications

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The ideal *plucked string* is defined as an initial string
displacement and a zero initial velocity distribution [317]. More
generally, the initial displacement along the string and the
initial velocity distribution
, for all , fully determine the
resulting motion in the absence of further excitation.

An example of the appearance of the traveling-wave components and the resulting string shape shortly after plucking a doubly terminated string at a point one fourth along its length is shown in Fig.6.7. The negative traveling-wave portions can be thought of as inverted reflections of the incident waves, or as doubly flipped ``images'' which are coming from the other side of the terminations.

An example of an initial ``pluck'' excitation in a digital waveguide string model is shown in Fig.6.8. The circulating triangular components in Fig.6.8 are equivalent to the infinite train of initial images coming in from the left and right in Fig.6.7.

There is one fine-point to note for the discrete-time case:
We cannot admit a sharp corner
in the string since that would have infinite bandwidth which would alias
when sampled. Therefore, for the discrete-time case, we define the ideal
pluck to consist of an arbitrary shape as in
Fig.6.8 *lowpass filtered* to less than half
the sampling rate. Alternatively, we can simply require the initial
displacement shape to be bandlimited to spatial frequencies less than
. Since all real strings have some degree of stiffness which
prevents the formation of perfectly sharp corners, and since real plectra
are never in contact with the string at only one point, and since the
frequencies we do allow span the full range of human hearing, the
bandlimited restriction is not limiting in any practical sense.

Note that acceleration (or curvature) waves are a simple choice for
plucked string simulation, since the ideal pluck corresponds to an initial
*impulse* in the delay lines at the pluck point. Of course, since we
require a bandlimited excitation, the initial acceleration distribution
will be replaced by the impulse response of the anti-aliasing filter
chosen.
If the anti-aliasing filter chosen is the ideal lowpass filter cutting off
at , the initial acceleration
for the
ideal pluck becomes

sinc | (7.13) |

where is amplitude, is the pick position, and, as we know from §4.4.1, sinc is the ideal, bandlimited impulse, centered at and having a rectangular spatial frequency response extending from to . (Recall that sinc). Division by normalizes the area under the initial shape curve. If is chosen to lie exactly on a spatial sample , the initial conditions for the ideal plucked string are as shown in Fig.6.9 for the case of acceleration or curvature waves. All initial samples are zero except one in each delay line.

Aside from its obvious simplicity, there are two important benefits of
obtaining an impulse-excited model: (1) an extremely efficient ``commuted
synthesis'' algorithm can be readily defined
(§8.7),
and (2) linear prediction (and its relatives) can be
readily used to calibrate the model to recordings of normally played tones
on the modeled instrument.
Linear Predictive
Coding (LPC) has been used extensively in speech modeling [296,297,20].
LPC estimates the model filter coefficients under the
assumption that the driving signal is *spectrally flat.* This
assumption is valid when the input signal is (1) an impulse, or (2) white
noise. In the basic LPC model for voiced speech, a periodic impulse train
excites the model filter (which functions as the vocal tract), and for
unvoiced speech, white noise is used as input.

In addition to plucked and struck strings, simplified *bowed
strings* can be calibrated to recorded data as well using LPC
[428,439]. In this simplified model, the
bowed string is approximated as a periodically plucked string.

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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