The bilinear transform (also known as Tustin's method or Mobius Transformation) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.
The bilinear transform is a special case of a conformal mapping (namely, the Möbius transformation), often used to convert a transfer function H a ( s ) of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function H d ( z ) of a linear, shift-invariant filter in the discrete-time domain (often called a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the j ω axis, R e [ s ] = 0 , in the s-plane to the unit circle, | z | = 1 , in the z-plane. Other bilinear transforms can be used to warp the frequency response of any discrete-time linear system (for example to approximate the non-linear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays ( z − 1 ) with first order all-pass filters.
The transform preserves stability and maps every point of the frequency response of the continuous-time filter, H a ( j ω a ) to a corresponding point in the frequency response of the discrete-time filter, H d ( e j ω d T ) although to a somewhat different frequency, as shown in the Frequency warping section below. This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a somewhat different frequency. This is barely noticeable at low frequencies but is quite evident at frequencies close to the Nyquist frequency.
The bilinear transform is a first-order approximation of the natural logarithm function that is an exact mapping of the z-plane to the s-plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed unit impulse), the result is precisely the Z transform of the discrete-time sequence with the substitution of
z = e s T = e s T / 2 e − s T / 2 ≈ 1 + s T / 2 1 − s T / 2 where T is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation; or, in other words, the sampling period. The above bilinear approximation can be solved for s or a similar approximation for s = ( 1 / T ) ln ( z ) can be performed.
The inverse of this mapping (and its first-order bilinear approximation) is
s = 1 T ln ( z ) = 2 T [ z − 1 z + 1 + 1 3 ( z − 1 z + 1 ) 3 + 1 5 ( z − 1 z + 1 ) 5 + 1 7 ( z − 1 z + 1 ) 7 + ⋯ ] ≈ 2 T z − 1 z + 1 = 2 T 1 − z − 1 1 + z − 1 The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, H a ( s )
s ← 2 T z − 1 z + 1 . That is
H d ( z ) = H a ( s ) | s = 2 T z − 1 z + 1 = H a ( 2 T z − 1 z + 1 ) . Stability and minimum-phase property preserved
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. Thus filters designed in the continuous-time domain that are stable are converted to filters in the discrete-time domain that preserve that stability.
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. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase.
As an example take a simple low-pass RC filter. This continuous-time filter has a transfer function
H a ( s ) = 1 / s C R + 1 / s C = 1 1 + R C s . If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for s the formula above; after some reworking, we get the following filter representation:
The coefficients of the denominator are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients used to implement a real-time digital filter.
It is possible to relate the coefficients of a continuous-time, analog filter with those of a similar discrete-time digital filter created through the bilinear transform process. Transforming a general, second-order continuous-time filter with the given transfer function
H a ( s ) = b 0 s 2 + b 1 s + b 2 a 0 s 2 + a 1 s + a 2 = b 0 + b 1 s − 1 + b 2 s − 2 a 0 + a 1 s − 1 + a 2 s − 2 using the bilinear transform (without prewarping any frequency specification) requires the substitution of
s ← K 1 − z − 1 1 + z − 1 where
K ≜ 2 T .
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 ω 0 , then
K ≜ ω 0 tan ( ω 0 T 2 ) .
This results in a discrete-time digital biquad filter with coefficients expressed in terms of the coefficients of the original continuous time filter:
H d ( z ) = ( b 0 K 2 + b 1 K + b 2 ) + ( 2 b 2 − 2 b 0 K 2 ) z − 1 + ( b 0 K 2 − b 1 K + b 2 ) z − 2 ( a 0 K 2 + a 1 K + a 2 ) + ( 2 a 2 − 2 a 0 K 2 ) z − 1 + ( a 0 K 2 − a 1 K + a 2 ) z − 2 Normally the constant term in the denominator must be normalized to 1 before deriving the corresponding difference equation. This results in
H d ( z ) = b 0 K 2 + b 1 K + b 2 a 0 K 2 + a 1 K + a 2 + 2 b 2 − 2 b 0 K 2 a 0 K 2 + a 1 K + a 2 z − 1 + b 0 K 2 − b 1 K + b 2 a 0 K 2 + a 1 K + a 2 z − 2 1 + 2 a 2 − 2 a 0 K 2 a 0 K 2 + a 1 K + a 2 z − 1 + a 0 K 2 − a 1 K + a 2 a 0 K 2 + a 1 K + a 2 z − 2 . The difference equation (using the Direct Form I) is
y [ n ] = b 0 K 2 + b 1 K + b 2 a 0 K 2 + a 1 K + a 2 ⋅ x [ n ] + 2 b 2 − 2 b 0 K 2 a 0 K 2 + a 1 K + a 2 ⋅ x [ n − 1 ] + b 0 K 2 − b 1 K + b 2 a 0 K 2 + a 1 K + a 2 ⋅ x [ n − 2 ] − 2 a 2 − 2 a 0 K 2 a 0 K 2 + a 1 K + a 2 ⋅ y [ n − 1 ] − a 0 K 2 − a 1 K + a 2 a 0 K 2 + a 1 K + a 2 ⋅ y [ n − 2 ] . To determine the frequency response of a continuous-time filter, the transfer function H a ( s ) is evaluated at s = j ω a which is on the j ω axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function H d ( z ) is evaluated at z = e j ω d T which is on the unit circle, | z | = 1 . The bilinear transform maps the j ω axis of the s-plane (of which is the domain of H a ( s ) ) to the unit circle of the z-plane, | z | = 1 (which is the domain of H d ( z ) ), but it is not the same mapping z = e s T which also maps the j ω axis to the unit circle. When the actual frequency of ω d is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency, ω a , for the continuous-time filter that this ω d is mapped to.
H d ( z ) = H a ( 2 T z − 1 z + 1 ) This shows that every point on the unit circle in the discrete-time filter z-plane, z = e j ω d T is mapped to a point on the j ω axis on the continuous-time filter s-plane, s = j ω a . That is, the discrete-time to continuous-time frequency mapping of the bilinear transform is
ω a = 2 T tan ( ω d T 2 ) and the inverse mapping is
ω d = 2 T arctan ( ω a T 2 ) . The discrete-time filter behaves at frequency ω d the same way that the continuous-time filter behaves at frequency ( 2 / T ) tan ( ω d T / 2 ) . Specifically, the gain and phase shift that the discrete-time filter has at frequency ω d is the same gain and phase shift that the continuous-time filter has at frequency ( 2 / T ) tan ( ω d T / 2 ) . This means that every feature, every "bump" that is visible in the frequency response of the continuous-time filter is also visible in the discrete-time filter, but at a different frequency. For low frequencies (that is, when ω d ≪ 2 / T or ω a ≪ 2 / T ), then the features are mapped to a slightly different frequency; ω d ≈ ω a .
One can see that the entire continuous frequency range
− ∞ < ω a < + ∞ is mapped onto the fundamental frequency interval
− π T < ω d < + π T . The continuous-time filter frequency ω a = 0 corresponds to the discrete-time filter frequency ω d = 0 and the continuous-time filter frequency ω a = ± ∞ correspond to the discrete-time filter frequency ω d = ± π / T .
One can also see that there is a nonlinear relationship between ω a and ω d . 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 ω a = 2 T tan ( ω d T 2 ) for every frequency specification that the designer has control over (such as corner frequency or center frequency). This is called pre-warping the filter design.
It is possible, however, to compensate for the frequency warping by pre-warping a frequency specification ω 0 (usually a resonant frequency or the frequency of the most significant feature of the frequency response) of the continuous-time system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system. When designing a digital filter as an approximation of a continuous time filter, the frequency response (both amplitude and phase) of the digital filter can be made to match the frequency response of the continuous filter at a specified frequency ω 0 , as well as matching at DC, if the following transform is substituted into the continuous filter transfer function. This is a modified version of Tustin's transform shown above.
s ← ω 0 tan ( ω 0 T 2 ) z − 1 z + 1 . However, note that this transform becomes the original transform
s ← 2 T z − 1 z + 1 as ω 0 → 0 .
The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with Impulse invariance.
