Digital Resonators-PRAKASH KUMAR [13209]-EED ( III YEAR )

Digital Resonators

A digital resonator is a special two-pole bandpass filter with a pair of complex-conjugate poles located very near the unit circle, as shown in Figure.Digital resonators are second-order recursive filters.

Digital resonators

In many applications it is of interest to generate a sinusoidal signal,y_sin(n) = sin(ω0n).A sinusoidal signal is needed in digital tone generation.

It is not practical to use the definition directly, as n may become very large and because the computation of the trigonometric function is time-consuming. Therefore, it is preferable to generate a sinusoidal signal by constructing a dynamic model for it by expressing y_sin(n) in terms of previous values, y_sin(n − 1) and y_sin(n − 2).

From previous slide----Y_sin(n − 1) and y_sin(n − 2) can be achieved by using the trigonometric identities

sin(α + β) = sin α cos β + cos α sin β;

sin α cos β = (½) sin(α + β) +(1/2) sin(α − β)

Then we can write

sin(ω0n) = sin(ω0(n − 1) + ω0) = cos ω0 sin(ω0(n − 1)) + sin ω0 cos(ω0(n − 1))

Hence the signal ysin(n) = sin(ω0n) satisfies the difference equation

ysin(n) − 2cos ω0 ysin(n − 1) + ysin(n − 2) = 0

In the same way it can be shown that the signal cos(ω0n) satisfies the difference equation It follows that the general sinusoidal signal

ys(n) = asin(ω0n) + bcos(ω0n) can be generated by the difference equation

y(n) − 2cos ω0 y(n − 1) + y(n − 2) = x(n)

where the input x(n) is used to set the initial signal values

The magnitude of the frequency response of the filter is shown in Figure

The name resonator refers to the fact that the filter has a large magnitude response in the vicinity of the pole position. The angle of the pole location determines the resonant frequency of the filter.

The amplitude response of such filters are characterized by single peaks (the resonance frequency) of variable width (bandwidth). The impulse response is a damped sinusoid. Within the source-filter model of speech, each formant of the vocal tract filter can be treated as a digital resonator with a particular frequency and bandwidth. Thus, each formant can be specified with two filter coefficients.

In general, the transfer function of a digital resonator can be obtained from the expression in . Note that this is just a special case of equation (9) in the Recursive Filter handout, in the special case where M=1 and L=2.

H(z) = b(1)1 + a(1)z-1 + a(2)z-2

The frequency and bandwidth characteristics of a digital resonator are determined by the values of a(1) and a(2)

Applications :- Simple bandpass filtering and speech generation. Modal synthesis involves the use of second-order digital resonators to

synthesize sounds with a relatively small number of decaying resonant modes.

A ``bank'' of resonance filters can be combined in parallel to simulate all the resonant modes of a system. Each resonator should have its own amplitude, center frequency, and rate of decay.