Test: Bilinear Transformations

Explanation: The bilinear transformation can be regarded as a correction of the backward difference method. The bilinear transformation is used for transforming an analog filter to a digital filter.

Explanation: Bilinear transformation uses trapezoidal rule for integrating a continuous time function.

Explanation: In bilinear transformation of an analog filter to digital filter, using the trapezoidal rule, the substitution for ‘s’ is given as

Explanation: The given integral is approximated by the trapezoidal rule. This rule states that if T is small, the area (integral) can be approximated by the mean height of x(t) between the two limits and then multiplying by the width. That is

Explanation: We know that the derivate equation is
dy(t)/dt=x(t)

Explanation: We know that
y(n)-y(n-1)= [(x(n)+x(n-1))/2]T
Taking z-transform of the above equation gives
=>Y(z)[1-z^-1]=([1+z^-1]/2).TX(z)
=>H(z)=Y(z)/X(z)= T/2[(1+z^-1)/(1-z¹ )].

Explanation: In bilinear transformation, the z to s transformation is given by the expression
z=[1+(T/2)s]/[1-(T/2)s].
Thus unlike the backward difference method, the left-half s-plane is now mapped entirely inside the unit circle, |z|=1, rather than to a part of it.

Explanation: Unlike the backward difference method, the left-half s-plane is now mapped entirely inside the unit circle, |z|=1, rather than to a part of it. Also, the imaginary axis is mapped to the unit circle. Therefore, equation is a true frequency-to-frequency transformation.

Explanation: We know that if =σ+jΩ and z=re^jω, then by substituting the values in the below expression

When r<1 => σ < 0.

Explanation: We know that if =σ+jΩ and z=re^jω, then by substituting the values in the below expression

=>σ = 2/T[(r²-1)/(r²+1+2rcosω)] When r=1 => σ = 0.
This shows that the imaginary axis in the s-domain is mapped to the circle of unit radius centered at z=0 in the z-domain.