Activity 01 - Audio Synthesis

What is Audio Synthesis

Audio Synthesis is technologically modelling or replicating elements of sound.

Following model was used

a0 y[n] = b0 x[n] + b1 x[n-1] + b2 x[n-2] ............- a1 y[n-1] - a2 y[n-2] - a3 y[n-3] .............

The model can be either linear or non-linear.

The system stability depends on the coefficients.

In the activity following coefficients were tried out to get the observations.

a0 =1

a1 = -2rcos(w)

a2 = (r^2)

where r = decaying rate

w= angular velocity

w=2*pi*f0/fs

f0=frequency

fs=sampling frequency

Then observations were taken by changing the values of r and f0 to observe their effect on the system.

As the input signal a unit impulse was used.

Initial parameters

fs=10000 Hz

1. Changing r from 0 to 1, while keeping f0 constant.

Observations

When r<1, the signal decays and when r>1 the signal grows exponentially.

When r<1,

When r is small the signal rapidly decays.

When r is increased the decaying rate reduces and system decays slowly.

When r = 1, the signal sustains as a sinusoidal wave.

Conclusion

We know that poles of a system affect its stability.

We can observe that when r increases the decaying rate decreases. This is because when r increases the poles move near the unit circle and when r is close to zero the poles are more inside the unit circle, close to zero. When r>1, the poles are outside the unit circle, hence the system becomes unstable.

2. Changed the value of f0 while keeping r constant at 0.995.

Observations

The decaying rate remains the same even though the frequency of the signal is varied.

Conclusion

The values of r and f0, significantly affects the response and the stability of the system since the directly affects the coefficients.

Appendix

%audio engineering activity 01a

%Audio Synthesis

fs = 10000;

Ts = 1 / fs;

duration = 0.1;

t = [0 : Ts : duration];

%creating an impulse

x = zeros(1, (duration * fs+1));

x(50) = 1;

r= 0.995;

f0 = 0;

w = 2*pi*f0/fs

a = [1, -2*r*cos(w), r^2]

b = [1]

y = filter(b, a, x);

plot(t,y)

xlabel("time/S");

ylabel("amplitude");