The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.
The transform preserves stability and maps every point of the frequency response of the continuous-time filter,
to a corresponding point in the frequency response of the discrete-time filter,
The bilinear transform is a first-order Padé approximant of the natural logarithm function that is an exact mapping of the z-plane to the s-plane.
is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation;[1] or, in other words, the sampling period.
That is A continuous-time causal filter is stable if the poles of its transfer function fall in the left half of the complex s-plane.
A discrete-time causal filter is stable if the poles of its transfer function fall inside the unit circle in the complex z-plane.
The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane.
Likewise, a continuous-time filter is minimum-phase if the zeros of its transfer function fall in the left half of the complex s-plane.
A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane.
Multiplying the numerator and denominator by the largest power of (z + 1)−1 present, (z + 1)-N, gives
It can be seen here that after the transformation, the degree of the numerator and denominator are both N. Consider then the pole-zero form of the continuous-time transfer function
The roots of the numerator and denominator polynomials, ξi and pi, are the zeros and poles of the system.
yielding some of the discretized transfer function's zeros and poles ξ'i and p'i
As described above, the degree of the numerator and denominator are now both N, in other words there is now an equal number of zeros and poles.
Given the full set of zeros and poles, the z-domain transfer function is then
Transforming a general, first-order continuous-time filter with the given transfer function using the bilinear transform (without prewarping any frequency specification) requires the substitution of where However, if the frequency warping compensation as described below is used in the bilinear transform, so that both analog and digital filter gain and phase agree at frequency
, then This results in a discrete-time digital filter with coefficients expressed in terms of the coefficients of the original continuous time filter: Normally the constant term in the denominator must be normalized to 1 before deriving the corresponding difference equation.
This results in The difference equation (using the Direct form I) is A similar process can be used for a general second-order filter with the given transfer function This results in a discrete-time digital biquad filter with coefficients expressed in terms of the coefficients of the original continuous time filter: Again, the constant term in the denominator is generally normalized to 1 before deriving the corresponding difference equation.
This results in The difference equation (using the Direct form I) is To determine the frequency response of a continuous-time filter, the transfer function
Likewise, to determine the frequency response of a discrete-time filter, the transfer function
is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency,
This shows that every point on the unit circle in the discrete-time filter z-plane,
Specifically, the gain and phase shift that the discrete-time filter has at frequency
is the same gain and phase shift that the continuous-time filter has at frequency
This effect of the bilinear transform is called frequency warping.
The continuous-time filter can be designed to compensate for this frequency warping by setting
These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system.
, as well as matching at DC, if the following transform is substituted into the continuous filter transfer function.
The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with Impulse invariance.