Next 
Prev 
Up 
Top

REALSIMPLE Top
Dynamic Level Lowpass Filter
In real strings, the spectral centroid typically rises as
plucking/striking becomes more energetic. The EKS dynamiclevel
lowpass filter (diagrammed at the far right in Fig.4)
qualitatively models this phenomenon:^{9}
This is another unitydcgain onepole lowpass, with a pole at
set such that the gain is the same for all fundamental
frequencies [4]. Here we will derive simplified
design formulas.
Assume that the ideal continuoustime filter has the
transfer function

(3) 
where
denotes the fundamental frequency in
radians per second. This lowpass filter has unity dc gain, dB
gain at , and rolls off dB/octave for
.^{10}It also happens to be the 1storder Butterworth lowpass with cutoff
frequency set to rad/sec.
To achieve the dynamic level effect, the output of
this filter is linearly panned with its input.
If denotes the
lowpass input signal and its output, then the formula is
where the level variable may be set to achieve a desired
dynamic level at the Nyquist limit, while controls the (lesser)
attenuation at low frequencies as a function of level (e.g.,
).
At maximum level , the lowpass filter is bypassed.
Figure 7 shows a family of filter
responses at four different dynamic levels and six different
fundamental frequencies.
An example GUI specification for the calculation in Faust is as follows:
L = hslider("dynamic_level", 10, 60, 0, 1) : db2linear;
where db2linear(x) is defined in music.lib as
pow(10, x/20.0).
In [14],^{11}the impulseinvariant and bilinear
transform methods are compared for digitizing the dynamiclevel
analog filter Eq.(4), and the bilinear transform method was deemed
preferable because it gives more attenuation of high frequencies,
which helps to reduce aliasing due to later nonlinear processing. A
detailed derivation can be found there. The final digital filter so
designed has the transfer function

(4) 
with
.
Figure 6 shows a family of magnitude responses for
for 6 different fundamental frequencies
.
Figure 6:
Dynamic level lowpass filter designed
by the bilineartransform method with . The filter
amplitude response is plotted for 6 values of break frequency (50,
100, 200, 400, 800, and 1600 Hz). The sampling rate is
Hz.

Figure 7:
Dynamic level lowpass
filter responses as in Fig.6, but with ,
, , and , corresponding to desired Nyquistlimit
levels of , , , and dB, respectively. The dc
level is defined to be onethird the Nyquistlimit level.

Subsections
Next 
Prev 
Up 
Top

REALSIMPLE Top
Download faust_strings.pdf