Convert From Discrete to Continuous Domain

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 bilinear transform is a special case of a conformal mapping (namely, a Möbius transformation), often used to convert a transfer function H a ( s ) {\displaystyle 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 ) {\displaystyle 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 {\displaystyle j\omega } axis, R e [ s ] = 0 {\displaystyle \mathrm {Re} [s]=0} , in the s-plane to the unit circle, | z | = 1 {\displaystyle |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 ) {\displaystyle \left(z^{-1}\right)} 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 ) {\displaystyle H_{a}(j\omega _{a})} to a corresponding point in the frequency response of the discrete-time filter, H d ( e j ω d T ) {\displaystyle H_{d}(e^{j\omega _{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.

Discrete-time approximation [edit]

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 {\displaystyle {\begin{aligned}z&=e^{sT}\\&={\frac {e^{sT/2}}{e^{-sT/2}}}\\&\approx {\frac {1+sT/2}{1-sT/2}}\end{aligned}}}

where T {\displaystyle T} is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation;[1] or, in other words, the sampling period. The above bilinear approximation can be solved for s {\displaystyle s} or a similar approximation for s = ( 1 / T ) ln ( z ) {\displaystyle 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 {\displaystyle {\begin{aligned}s&={\frac {1}{T}}\ln(z)\\&={\frac {2}{T}}\left[{\frac {z-1}{z+1}}+{\frac {1}{3}}\left({\frac {z-1}{z+1}}\right)^{3}+{\frac {1}{5}}\left({\frac {z-1}{z+1}}\right)^{5}+{\frac {1}{7}}\left({\frac {z-1}{z+1}}\right)^{7}+\cdots \right]\\&\approx {\frac {2}{T}}{\frac {z-1}{z+1}}\\&={\frac {2}{T}}{\frac {1-z^{-1}}{1+z^{-1}}}\end{aligned}}}

The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, H a ( s ) {\displaystyle H_{a}(s)}

s 2 T z 1 z + 1 . {\displaystyle s\leftarrow {\frac {2}{T}}{\frac {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 ) . {\displaystyle H_{d}(z)=H_{a}(s){\bigg |}_{s={\frac {2}{T}}{\frac {z-1}{z+1}}}=H_{a}\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right).\ }

Stability and minimum-phase property preserved [edit]

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.

Transformation of a General LTI System [edit]

A general LTI system has the transfer function

H a ( s ) = b 0 + b 1 s + b 2 s 2 + + b Q s Q a 0 + a 1 s + a 2 s 2 + + a P s P {\displaystyle H_{a}(s)={\frac {b_{0}+b_{1}s+b_{2}s^{2}+\cdots +b_{Q}s^{Q}}{a_{0}+a_{1}s+a_{2}s^{2}+\cdots +a_{P}s^{P}}}}

The order of the transfer function N is the greater of P and Q (in practice this is most likely P as the transfer function must be proper for the system to be stable). Applying the bilinear transform

s = K z 1 z + 1 {\displaystyle s=K{\frac {z-1}{z+1}}}

where K is defined as either 2/T or otherwise if using frequency warping, gives

H d ( z ) = b 0 + b 1 ( K z 1 z + 1 ) + b 2 ( K z 1 z + 1 ) 2 + + b Q ( K z 1 z + 1 ) Q a 0 + a 1 ( K z 1 z + 1 ) + a 2 ( K z 1 z + 1 ) 2 + + b P ( K z 1 z + 1 ) P {\displaystyle H_{d}(z)={\frac {b_{0}+b_{1}\left(K{\frac {z-1}{z+1}}\right)+b_{2}\left(K{\frac {z-1}{z+1}}\right)^{2}+\cdots +b_{Q}\left(K{\frac {z-1}{z+1}}\right)^{Q}}{a_{0}+a_{1}\left(K{\frac {z-1}{z+1}}\right)+a_{2}\left(K{\frac {z-1}{z+1}}\right)^{2}+\cdots +b_{P}\left(K{\frac {z-1}{z+1}}\right)^{P}}}}

Multiplying the numerator and denominator by the largest power of (z + 1)-1 present, (z + 1)-N , gives

H d ( z ) = b 0 ( z + 1 ) N + b 1 K ( z 1 ) ( z + 1 ) N 1 + b 2 K 2 ( z 1 ) 2 ( z + 1 ) N 2 + + b Q K Q ( z 1 ) Q ( z + 1 ) N Q a 0 ( z + 1 ) N + a 1 K ( z 1 ) ( z + 1 ) N 1 + a 2 K 2 ( z 1 ) 2 ( z + 1 ) N 2 + + a P K P ( z 1 ) P ( z + 1 ) N P {\displaystyle H_{d}(z)={\frac {b_{0}(z+1)^{N}+b_{1}K(z-1)(z+1)^{N-1}+b_{2}K^{2}(z-1)^{2}(z+1)^{N-2}+\cdots +b_{Q}K^{Q}(z-1)^{Q}(z+1)^{N-Q}}{a_{0}(z+1)^{N}+a_{1}K(z-1)(z+1)^{N-1}+a_{2}K^{2}(z-1)^{2}(z+1)^{N-2}+\cdots +a_{P}K^{P}(z-1)^{P}(z+1)^{N-P}}}}

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

H a ( s ) = ( s ξ 1 ) ( s ξ 2 ) ( s ξ Q ) ( s p 1 ) ( s p 2 ) ( s p P ) {\displaystyle H_{a}(s)={\frac {(s-\xi _{1})(s-\xi _{2})\cdots (s-\xi _{Q})}{(s-p_{1})(s-p_{2})\cdots (s-p_{P})}}}

The roots of the numerator and denominator polynomials, ξi and pi , are the zeros and poles of the system. The bilinear transform is a one-to-one mapping, hence these can be transformed to the z-domain using

z = K + s K s {\displaystyle z={\frac {K+s}{K-s}}}

yielding some of the discretized transfer function's zeros and poles ξ'i and p'i

ξ i = K + ξ i K ξ i 1 i Q p i = K + p i K p i 1 i P {\displaystyle {\begin{aligned}\xi '_{i}&={\frac {K+\xi _{i}}{K-\xi _{i}}}\quad 1\leq i\leq Q\\p'_{i}&={\frac {K+p_{i}}{K-p_{i}}}\quad 1\leq i\leq P\end{aligned}}}

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. The multiplication by (z + 1) -N means the additional zeros or poles are [2]

ξ i = 1 Q < i N p i = 1 P < i N {\displaystyle {\begin{aligned}\xi '_{i}&=-1\quad Q<i\leq N\\p'_{i}&=-1\quad P<i\leq N\end{aligned}}}

Given the full set of zeros and poles, the z-domain transfer function is then

H d ( z ) = ( z ξ 1 ) ( z ξ 2 ) ( z ξ N ) ( z p 1 ) ( z p 2 ) ( z p N ) {\displaystyle H_{d}(z)={\frac {(z-\xi '_{1})(z-\xi '_{2})\cdots (z-\xi '_{N})}{(z-p'_{1})(z-p'_{2})\cdots (z-p'_{N})}}}

Example [edit]

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 . {\displaystyle {\begin{aligned}H_{a}(s)&={\frac {1/sC}{R+1/sC}}\\&={\frac {1}{1+RCs}}.\end{aligned}}}

If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for s {\displaystyle 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.

Transformation for a general first-order continuous-time filter [edit]

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, first-order continuous-time filter with the given transfer function

H a ( s ) = b 0 s + b 1 a 0 s + a 1 = b 0 + b 1 s 1 a 0 + a 1 s 1 {\displaystyle H_{a}(s)={\frac {b_{0}s+b_{1}}{a_{0}s+a_{1}}}={\frac {b_{0}+b_{1}s^{-1}}{a_{0}+a_{1}s^{-1}}}}

using the bilinear transform (without prewarping any frequency specification) requires the substitution of

s K 1 z 1 1 + z 1 {\displaystyle s\leftarrow K{\frac {1-z^{-1}}{1+z^{-1}}}}

where

K 2 T {\displaystyle K\triangleq {\frac {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 {\displaystyle \omega _{0}} , then

K ω 0 tan ( ω 0 T 2 ) {\displaystyle K\triangleq {\frac {\omega _{0}}{\tan \left({\frac {\omega _{0}T}{2}}\right)}}} .

This results in a discrete-time digital filter with coefficients expressed in terms of the coefficients of the original continuous time filter:

H d ( z ) = ( b 0 K + b 1 ) + ( b 0 K + b 1 ) z 1 ( a 0 K + a 1 ) + ( a 0 K + a 1 ) z 1 {\displaystyle H_{d}(z)={\frac {(b_{0}K+b_{1})+(-b_{0}K+b_{1})z^{-1}}{(a_{0}K+a_{1})+(-a_{0}K+a_{1})z^{-1}}}}

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 + b 1 a 0 K + a 1 + b 0 K + b 1 a 0 K + a 1 z 1 1 + a 0 K + a 1 a 0 K + a 1 z 1 . {\displaystyle H_{d}(z)={\frac {{\frac {b_{0}K+b_{1}}{a_{0}K+a_{1}}}+{\frac {-b_{0}K+b_{1}}{a_{0}K+a_{1}}}z^{-1}}{1+{\frac {-a_{0}K+a_{1}}{a_{0}K+a_{1}}}z^{-1}}}.}

The difference equation (using the Direct Form I) is

y [ n ] = b 0 K + b 1 a 0 K + a 1 x [ n ] + b 0 K + b 1 a 0 K + a 1 x [ n 1 ] a 0 K + a 1 a 0 K + a 1 y [ n 1 ] . {\displaystyle y[n]={\frac {b_{0}K+b_{1}}{a_{0}K+a_{1}}}\cdot x[n]+{\frac {-b_{0}K+b_{1}}{a_{0}K+a_{1}}}\cdot x[n-1]-{\frac {-a_{0}K+a_{1}}{a_{0}K+a_{1}}}\cdot y[n-1]\ .}

General second-order biquad transformation [edit]

A similar process can be used for a general second-order 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 . {\displaystyle H_{a}(s)={\frac {b_{0}s^{2}+b_{1}s+b_{2}}{a_{0}s^{2}+a_{1}s+a_{2}}}={\frac {b_{0}+b_{1}s^{-1}+b_{2}s^{-2}}{a_{0}+a_{1}s^{-1}+a_{2}s^{-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 {\displaystyle H_{d}(z)={\frac {(b_{0}K^{2}+b_{1}K+b_{2})+(2b_{2}-2b_{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})+(2a_{2}-2a_{0}K^{2})z^{-1}+(a_{0}K^{2}-a_{1}K+a_{2})z^{-2}}}}

Again, the constant term in the denominator is generally 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 . {\displaystyle H_{d}(z)={\frac {{\frac {b_{0}K^{2}+b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}+{\frac {2b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-1}+{\frac {b_{0}K^{2}-b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-2}}{1+{\frac {2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-1}+{\frac {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 ] . {\displaystyle y[n]={\frac {b_{0}K^{2}+b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n]+{\frac {2b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n-1]+{\frac {b_{0}K^{2}-b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n-2]-{\frac {2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot y[n-1]-{\frac {a_{0}K^{2}-a_{1}K+a_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot y[n-2]\ .}

Frequency warping [edit]

To determine the frequency response of a continuous-time filter, the transfer function H a ( s ) {\displaystyle H_{a}(s)} is evaluated at s = j ω a {\displaystyle s=j\omega _{a}} which is on the j ω {\displaystyle j\omega } axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function H d ( z ) {\displaystyle H_{d}(z)} is evaluated at z = e j ω d T {\displaystyle z=e^{j\omega _{d}T}} which is on the unit circle, | z | = 1 {\displaystyle |z|=1} . The bilinear transform maps the j ω {\displaystyle j\omega } axis of the s-plane (of which is the domain of H a ( s ) {\displaystyle H_{a}(s)} ) to the unit circle of the z-plane, | z | = 1 {\displaystyle |z|=1} (which is the domain of H d ( z ) {\displaystyle H_{d}(z)} ), but it is not the same mapping z = e s T {\displaystyle z=e^{sT}} which also maps the j ω {\displaystyle j\omega } axis to the unit circle. When the actual frequency of ω d {\displaystyle \omega _{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 {\displaystyle \omega _{a}} , for the continuous-time filter that this ω d {\displaystyle \omega _{d}} is mapped to.

H d ( z ) = H a ( 2 T z 1 z + 1 ) {\displaystyle H_{d}(z)=H_{a}\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)}
H d ( e j ω d T ) {\displaystyle H_{d}(e^{j\omega _{d}T})} = H a ( 2 T e j ω d T 1 e j ω d T + 1 ) {\displaystyle =H_{a}\left({\frac {2}{T}}{\frac {e^{j\omega _{d}T}-1}{e^{j\omega _{d}T}+1}}\right)}
= H a ( 2 T e j ω d T / 2 ( e j ω d T / 2 e j ω d T / 2 ) e j ω d T / 2 ( e j ω d T / 2 + e j ω d T / 2 ) ) {\displaystyle =H_{a}\left({\frac {2}{T}}\cdot {\frac {e^{j\omega _{d}T/2}\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)}{e^{j\omega _{d}T/2}\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)}}\right)}
= H a ( 2 T ( e j ω d T / 2 e j ω d T / 2 ) ( e j ω d T / 2 + e j ω d T / 2 ) ) {\displaystyle =H_{a}\left({\frac {2}{T}}\cdot {\frac {\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)}{\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)}}\right)}
= H a ( j 2 T ( e j ω d T / 2 e j ω d T / 2 ) / ( 2 j ) ( e j ω d T / 2 + e j ω d T / 2 ) / 2 ) {\displaystyle =H_{a}\left(j{\frac {2}{T}}\cdot {\frac {\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)/(2j)}{\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)/2}}\right)}
= H a ( j 2 T sin ( ω d T / 2 ) cos ( ω d T / 2 ) ) {\displaystyle =H_{a}\left(j{\frac {2}{T}}\cdot {\frac {\sin(\omega _{d}T/2)}{\cos(\omega _{d}T/2)}}\right)}
= H a ( j 2 T tan ( ω d T / 2 ) ) {\displaystyle =H_{a}\left(j{\frac {2}{T}}\cdot \tan \left(\omega _{d}T/2\right)\right)}

This shows that every point on the unit circle in the discrete-time filter z-plane, z = e j ω d T {\displaystyle z=e^{j\omega _{d}T}} is mapped to a point on the j ω {\displaystyle j\omega } axis on the continuous-time filter s-plane, s = j ω a {\displaystyle s=j\omega _{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 ) {\displaystyle \omega _{a}={\frac {2}{T}}\tan \left(\omega _{d}{\frac {T}{2}}\right)}

and the inverse mapping is

ω d = 2 T arctan ( ω a T 2 ) . {\displaystyle \omega _{d}={\frac {2}{T}}\arctan \left(\omega _{a}{\frac {T}{2}}\right).}

The discrete-time filter behaves at frequency ω d {\displaystyle \omega _{d}} the same way that the continuous-time filter behaves at frequency ( 2 / T ) tan ( ω d T / 2 ) {\displaystyle (2/T)\tan(\omega _{d}T/2)} . Specifically, the gain and phase shift that the discrete-time filter has at frequency ω d {\displaystyle \omega _{d}} is the same gain and phase shift that the continuous-time filter has at frequency ( 2 / T ) tan ( ω d T / 2 ) {\displaystyle (2/T)\tan(\omega _{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 {\displaystyle \omega _{d}\ll 2/T} or ω a 2 / T {\displaystyle \omega _{a}\ll 2/T} ), then the features are mapped to a slightly different frequency; ω d ω a {\displaystyle \omega _{d}\approx \omega _{a}} .

One can see that the entire continuous frequency range

< ω a < + {\displaystyle -\infty <\omega _{a}<+\infty }

is mapped onto the fundamental frequency interval

π T < ω d < + π T . {\displaystyle -{\frac {\pi }{T}}<\omega _{d}<+{\frac {\pi }{T}}.}

The continuous-time filter frequency ω a = 0 {\displaystyle \omega _{a}=0} corresponds to the discrete-time filter frequency ω d = 0 {\displaystyle \omega _{d}=0} and the continuous-time filter frequency ω a = ± {\displaystyle \omega _{a}=\pm \infty } correspond to the discrete-time filter frequency ω d = ± π / T . {\displaystyle \omega _{d}=\pm \pi /T.}

One can also see that there is a nonlinear relationship between ω a {\displaystyle \omega _{a}} and ω d . {\displaystyle \omega _{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 ) {\displaystyle \omega _{a}={\frac {2}{T}}\tan \left(\omega _{d}{\frac {T}{2}}\right)} 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 {\displaystyle \omega _{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 {\displaystyle \omega _{0}} , as well as matching at DC, if the following transform is substituted into the continuous filter transfer function.[3] This is a modified version of Tustin's transform shown above.

s ω 0 tan ( ω 0 T 2 ) z 1 z + 1 . {\displaystyle s\leftarrow {\frac {\omega _{0}}{\tan \left({\frac {\omega _{0}T}{2}}\right)}}{\frac {z-1}{z+1}}.}

However, note that this transform becomes the original transform

s 2 T z 1 z + 1 {\displaystyle s\leftarrow {\frac {2}{T}}{\frac {z-1}{z+1}}}

as ω 0 0 {\displaystyle \omega _{0}\to 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.

See also [edit]

  • Impulse invariance
  • Matched Z-transform method

References [edit]

  1. ^ Oppenheim, Alan (2010). Discrete Time Signal Processing Third Edition. Upper Saddle River, NJ: Pearson Higher Education, Inc. p. 504. ISBN978-0-13-198842-2.
  2. ^ Bhandari, Ayush. "DSP and Digital Filters Lecture Notes" (PDF). Archived from the original (PDF) on 3 March 2022. Retrieved 16 August 2022.
  3. ^ Astrom, Karl J. (1990). Computer Controlled Systems, Theory and Design (Second ed.). Prentice-Hall. p. 212. ISBN0-13-168600-3.

External links [edit]

  • MIT OpenCourseWare Signal Processing: Continuous to Discrete Filter Design
  • Lecture Notes on Discrete Equivalents
  • The Art of VA Filter Design

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Source: https://en.wikipedia.org/wiki/Bilinear_transform

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