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Making synthetic formants

We now analyze the spectrum of the periodic signal generated in Figure 2 in the special case where the table $ p(t)$ (still regarded as holding a continuous function of the parameter $ t$ ) contains a sinusoid, either the fundamental or a harmonic:

$\displaystyle p(t) = {e^{2 \pi h t i}}
$

where the integer $ h$ is the number of the harmonic. Since the output is a linear function of the waveform, we can then infer the spectrum for an arbitrary choice of $ p(t)$ .

Figure 3: Spectrum of the output of Figure 2: a. $ hS = 3$ , $ T=1$ ; b. $ hS = 3.5$ , $ T=1$ ; c. $ hS = 3.75$ , $ T=2$ .
\begin{figure}\psfig{file=fig3.ps}\end{figure}

For simplicity of analysis we will suppose the fundamental is of the form $ \omega = 2\pi/N$ so that the resulting period $ N$ is an integer; this only amounts to a possible sample rate adjustment which does not affect the end result. For $ -N \le n < N$ the output of the diagram of Figure 1 (with period $ 2N$ ) is:

$\displaystyle x[n] = w(nT/2N) p(nS/N)
$

This may be expressed as a Fourier series:

$\displaystyle x[n] = \sum_{k=0}^{N-1} {b_k} {e^{2 \pi k t i/ (2N)}}
$

where the partial amplitudes $ b_k$ are real-valued (phase zero at $ n=0$ ) and equal to:

$\displaystyle {b_k} = {1 \over T}W({{k - 2hS}\over{T}})
$

$\displaystyle W(k) = {1\over 4} \mathrm{sinx}(k-1) +
{1\over 2} \mathrm{sinx}(k) +
{1\over 4} \mathrm{sinx}(k+1)
$

$\displaystyle \mathrm{sinx}(k) = \sin(2\pi k)/(2 \pi k) \; \;$   $\displaystyle \mbox{(or $1$\ when $k=0$)}$

Here $ W(k)$ is the (suitably normalized) Fourier transform of the Hann window function. Overlap-adding as in Figure 2 extracts the even-numbered partials:

$\displaystyle y[n] = x[n] + x[n+N]
$

which gives:

$\displaystyle y[n] = \sum_{k=0}^{N-1} {c_k} {e^{2 \pi k t i/ N}}
$

$\displaystyle {c_k} = {2 \over T}W({{2(k - hS)}\over{T}})
$

Figure 3 shows the result for three $ (hS,T)$ pairs. The spectrum has one formant. Changing the parameter $ S$ has the effect of rescaling the spectral envelope along the frequency axis. The parameter $ T$ controls bandwidth.

When the minimum bandwidth is selected by setting $ T=1$ , as in parts (a) and (b) of the figure, the result is a mixture of consecutive partials; if the center frequency lies directly on a partial, only the one corresponding partial is present in the result. This is the same result as that of setting the bandwidth to zero in the PAF generator.


next up previous
Next: Using recorded sounds Up: Synthesis algorithm Previous: Synthesis algorithm
Miller Puckette 2006-03-30