%[text] # A Simple Example on the Application of the Bilinear Transform %[text] %[text] Let's consider the following polynomial of the complex variable $z$ %[text] $p(z) = z^2+a\\,z+b,\\quad z \\in \\mathbb{C},\\quad a,\\,b \\in \\mathbb{R}$ %[text] It could be the characteristic polynomial of a discrete-time LTI system. %[text] We aim to infer the location of the polynomial $p(z)$ 's roots, depending on the parameters $a$ and $b$. Does any set of values for the parameters exist that guarantees obtaining polynomial $p(z)$ roots with modulus (the magnitude) less than one? Or with the modulus equal to one? Or with the magnitude higher than unity? %[text] A suitable strategy to obtain the answers to those questions relies on the application of the **Bilinear Transform** and then on the application of the Routh-Hurwitz theorem on the transformed new polynomial. %[text] %[text] ## The Bilinear Transformation: a Recap %[text] %[text]{"align":"center"} $z = \\frac{w+1}{w-1}, \\; z, \\, w \\in \\mathbb C\n\\quad \\quad \\quad \\quad\\quad\n\\begin{array}{l}\n\\vert z \\vert \< 1 \\, \\, \\Longleftrightarrow \\, \\, {\\rm Re} (w) \< 0 \\\\\n\\vert z \\vert = 1 \\, \\, \\Longleftrightarrow \\, \\, {\\rm Re} (w) = 0 \\\\\n\\vert z \\vert \> 1 \\, \\, \\Longleftrightarrow \\, \\, {\\rm Re} (w) \> 0\n\\end{array}$ %[text]{"align":"center"} %[text]{"align":"center"} ![Screenshot 2023-10-19 at 22.16.47.png](text:image:9178) %[text]{"align":"center"} %[text] Substitute %[text]{"align":"center"} $z = \\frac{w+1}{w-1}, \\; z, \\, w \\in \\mathbb C$ %[text] into $p (z) =\n\\varphi\_0 z^n + \\varphi\_1 z^{n-1} + \\cdots\n+ \\varphi\_{n-1} z + \\varphi\_n$, thus obtaining %[text]{"align":"center"} $\\begin{array}{l}\nq (w) = \\, \\left(w-1 \\right)^n \\, \\left\[\n\\varphi\_0 \\frac{\\left(w+1 \\right)^n}{\\left(w-1 \\right)^n} + \n \\varphi\_1 \\frac{\\left(w+1 \\right)^{n-1}}{\\left(w-1 \\right)^{n-1}} + \\cdots +\n \\varphi\_{n-1} \\frac{\\left(w+1 \\right)}{\\left(w-1 \\right)} \n + \\varphi\_{n}\n \\right \]\n\\end{array}$ %[text] and hence one gets %[text]{"align":"center"} $q(w) = q\_0 \\, w^n + q\_1\\,w^{n-1}+\\cdots+q\_{n-1}\\,w + q\_n$ %[text] with suitable coefficients $q\_0, \\, q\_1, \\ldots, q\_n$. %[text] %[text] Given the expression of $p(z)$, how to compute the transformed polynomial $q(w)$ using MATLAB? %% % Initialisation clear clc %% %[text] How to define the polynomial $p(z)$ in MATLAB? syms z a b w assume(a, 'real'); assumeAlso(b, 'real'); assumptions %[output:98ff1c54] %[text] Now, let's define the polynomial $p(z)$ as a symbolic expression depending on the (symbolic) variable $z$ pz = z^2+a*z+b %[output:6d8cc880] whos pz z w a b %[output:2f93fa29] %[text] %% %[text] How to perform the Bilinear Transform? %[text] According to the general formulation above, the polynomial $q(w)$ can be determined as follows %[text] $q(w) = \\big(w-1 \\big)^2 \\, \\left\[ \\left(\\frac{w+1}{w-1} \\right)^2 + a\\, \\left(\\frac{w+1}{w-1} \\right) +b \\right\]$ %[text] Using MATLAB, the polynomial is q(w) = collect(simplify(expand(((w-1)^2) *subs(pz, z, ((w+1)/(w-1))))),w) %[output:99377afc] %[appendix]{"version":"1.0"} %--- %[metadata:view] % data: {"layout":"inline","rightPanelPercent":40} %--- %[text:image:9178] % data: 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","width":444} %--- %[output:98ff1c54] % data: {"dataType":"symbolic","outputData":{"name":"ans","value":"\\left(\\begin{array}{cc}\na\\in \\mathbb{R} & b\\in \\mathbb{R}\n\\end{array}\\right)"}} %--- %[output:6d8cc880] % data: {"dataType":"symbolic","outputData":{"name":"pz","value":"z^2 +a\\,z+b"}} %--- %[output:2f93fa29] % data: {"dataType":"text","outputData":{"text":" Name Size Bytes Class Attributes\n\n a 1x1 8 sym \n b 1x1 8 sym \n pz 1x1 8 sym \n w 1x1 8 sym \n z 1x1 8 sym \n\n","truncated":false}} %--- %[output:99377afc] % data: {"dataType":"symbolic","outputData":{"name":"q(w)","value":"{\\left(a+b+1\\right)}\\,w^2 +{\\left(2-2\\,b\\right)}\\,w+b-a+1"}} %---