digital resonators used in signal processing by making the continuous time signal into discrete form by sampling.

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

2. 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.

3. 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).

4.  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))

5. 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

6.  The magnitude of the frequency response
of the filter is shown in Figure

8. 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.

9. 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.

10.  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)

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