What are critical points in bilinear transformation?
Answers
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 {\displaystyle H_{a}(s)\ } H_{a}(s)\ of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function {\displaystyle H_{d}(z)\ } 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 {\displaystyle j\omega \ } j\omega \ axis, {\displaystyle Re[s]=0\ } Re[s]=0\ , in the s-plane to the unit circle, {\displaystyle |z|=1\ } |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 {\displaystyle \left(z^{-1}\right)\ } \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, {\displaystyle H_{a}(j\omega _{a})\ } H_{a}(j\omega _{a})\ to a corresponding point in the frequency response of the discrete-time filter, {\displaystyle H_{d}(e^{j\omega _{d}T})\ } 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
Explanation:
A point at which f (z) is not conformal mapping is called critical point. 3.9 Fixed point/Invariant point: If the image of a point z under a transformation w = f(z) is itself, then the point is called a fixed point or a invariant point of the transformation. f z z − = + Then the invariant points are 1 + 2i and 1 – 2i.