Bilinear transform

The bilinear transform (also known as Tustin's method) 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 (also known as Tustin's method) 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 jomega } axis, R e [ s ] = 0   {displaystyle Re=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} ight) } 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}(jomega _{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^{jomega _{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 where T   {displaystyle 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   {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 The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, H a ( s )   {displaystyle H_{a}(s) }