6.2 Amplitude Manipulations

An obvious type of manipulation of spectral envelopes is amplitude manipulation . To describe that whole class of manipulations, we only need to define the two basic operations of addition and multiplication of spectral envelopes. This is straightforward if we formally define a spectral envelope as a real function v(f) with domain $0 \le f \le f_s/2$:

$$\makebox[\sul][l]{$v_{add}$ } = v_1 \makebox[\opl][c]{$\:+\:$ } v_2 \iff \makebox[\sul][l]{$v_{add}$ } (f) = v_1(f) \makebox[\opl][c]{$\:+\:$ } v_2(f) \qquad \textrm{for all\ } 0 \le f \le f_s/2$$ (6.1)

$$\makebox[\sul][l]{$v_{mul}$ } = v_1 \makebox[\opl][c]{$\:\*\:$ } v_2 \iff \makebox[\sul][l]{$v_{mul}$ } (f) = v_1(f) \makebox[\opl][c]{$\:\*\:$ } v_2(f) \qquad \textrm{for all\ } 0 \le f \le f_s/2$$ (6.2)

If the spectral envelope is in spectral representation (section 4.3), we get to the functional interpretation by linear interpolation as in equation (5.1). If it is in precise formant representation, we can evaluate the formant function (equation (4.1)) directly, if it is in filter coefficients, we convert it to spectral representation.

With these operations, we can effect manipulations such as amplification and attenuation, or spectral tilting, as explained in the following.

Amplification  and attenuation 

Amplification and attenuation of spectral envelopes is effected by a multiplication by a factor greater or less than one, respectively. Multiplication with a function over frequency (i.e. with a spectral envelope) instead of a constant factor, thus doing frequency selective amplification or attenuation , essentially amounts to applying a filter to the spectral envelope.

Tilting the spectrum

 The overall slope of the spectrum of a speech or instrument signal is called spectral tilt . For speech, it is among others responsible for the prosodic feature of accent  , in that a speaker modifies the tilt (raising the slope) of the spectrum of a vowel, to put stress on a syllable [Dog95]. For instruments, it can be dependent on dynamics, the relative force with which the instrument is played. Spectral tilt is measured in decibel per octave. To apply a given spectral tilt t to a spectral envelope v, we have to first normalize the tilt per octave to a tilt t_m over the whole range of the spectral envelope up to f_m = f_s / 2 (half the sampling rate):

$$t_m = t \frac{f_m}{\log_2 f_m}$$

Then, the additive decibel value t_m is converted to a multiplicative amplitude factor t_a:

$$t_a = 10^\frac{t_m}{20}$$

To effect the tilting of a spectral envelope v(f), we multiply by a ramp spectral envelope v_t (f) ramping from 0 to t_a:

$$v \* v_t \qquad \textrm{with}$$ tilt_t (v) = (6.3)

$$f \frac{t_a}{f_m}$$ v_t (f) = (6.4)