S. Boyd EE102Lecture 5Rational functions and partial fractionexpansion• (review of) polynomials• rational functions• pole-zero plots• partial fraction expansion• repeated poles• nonproper rational functions5–1Polynomials and rootspolynomialsa(s) = a0+ a1s + ··· + ansn• a is a polynomial in the variable s• aiare the coefficients of a (usually real, but occasionally complex)• n is the degree of a (assuming an6= 0)roots (or zeros) of a polynomial a: λ ∈ C that satisfya(λ) = 0examples• a(s) = 3 has no roots• a(s) = s3− 1 has three roots: 1, (−1 + j√3)/2, (−1 − j√3)/2Rational functions and partial fraction expansion 5–2factoring out roots of a polynomialif a has a root at s = λ we can factor out s − λ:• dividing a by s − λ yields a polynomial:b(s) =a(s)s − λis a polynomial (of degree one less than the degree of a)• we can express a asa(s) = (s − λ)b(s)for some polynomial bexample: s3− 1 has a root at s = 1s3− 1 = (s − 1)(s2+ s + 1)Rational functions and partial fraction expansion 5–3the multiplicity of a root λ is the number of factors s − λ we can factorout, i.e., the largest k such thata(s)(s − λ)kis a polynomialexample:a(s) = s3+ s2− s − 1• a has a zero at s = −1•a(s)s + 1=s3+ s2− s − 1s + 1= s2− 1 also has a zero at s = −1•a(s)(s + 1)2=s3+ s2− s − 1(s + 1)2= s − 1 does not have a zero at s = −1so the multiplicity of the zero at s = −1 is 2Rational functions and partial fraction expansion 5–4Fundamental theorem of algebraa polynomial of degree n has exactly n roots, counting multiplicitiesthis means we can write a in factored forma(s) = ansn+ ··· + a0= an(s − λ1) ···(s − λn)where λ1, . . . , λnare the n roots of aexample: s3+ s2− s − 1 = (s + 1)2(s − 1)the relation between the coefficients aiand the λiis complicated ingeneral, buta0= annYi=1(−λi), an−1= −annXi=1λiare two identities that are worth rememberingRational functions and partial fraction expansion 5–5Conjugate symmetryif the coefficients a0, . . . , anare real, and λ ∈ C is a root, i.e.,a(λ) = a0+ a1λ + ··· + anλn= 0then we havea(λ) = a0+ a1λ + ··· + anλn= (a0+ a1λ + ··· + anλn) = a(λ) = 0in other words:λ is also a root• if λ is real this isn’t interesting• if λ is complex, it gives us another root for free• complex roots come in complex conjugate pairsRational functions and partial fraction expansion 5–6example:PSfrag replacements=<λ1λ2λ3λ4λ5λ6λ3and λ6are real; λ1, λ2are a complex conjugate pair; λ4, λ5are acomplex conjugate pairif a has real coefficients, we can factor it asa(s) = anÃrYi=1(s − λi)!ÃmYi=r+1(s − λi)(s − λi)!where λ1, . . . , λrare the real roots; λr+1, λr+1, . . . , λm, λmare thecomplex rootsRational functions and partial fraction expansion 5–7Real factored form(s − λ)(s − λ) = s2− 2(<λ) s + |λ|2is a quadratic with real coefficientsreal factored form of a polynomial a:a(s) = anÃrYi=1(s − λi)!ÃmYi=r+1(s2+ αis + βi)!• λ1, . . . , λrare the real roots• αi, βiare real and satisfy α2i< 4βiany polynomial with real coefficients can be factored into a product of• degree one polynomials with real coefficients• quadratic polynomials with real coefficientsRational functions and partial fraction expansion 5–8example: s3− 1 has rootss = 1, s =−1 + j√32, s =−1 − j√32• complex factored forms3− 1 = (s − 1)³s + (1 + j√3)/2´³s + (1 − j√3)/2´• real factored forms3− 1 = (s − 1)(s2+ s + 1)Rational functions and partial fraction expansion 5–9Rational functionsa rational function has the formF (s) =b(s)a(s)=b0+ b1s + ··· + bmsma0+ a1s + ··· + ansn,i.e., a ratio of two polynomials (where a is not the zero polynomial)• b is called the numerator polynomial• a is called the denominator polynomialexamples of rational functions:1s + 1, s2+ 3,1s2+ 1+s2s + 3=s3+ 3s + 32s3+ 3s2+ 2s + 3Rational functions and partial fraction expansion 5–10rational function F (s) =b(s)a(s)polynomials b and a are not uniquely determined, e.g.,1s + 1=33s + 3=s2+ 3(s + 1)(s2+ 3)(except at s = ±j√3. . . )rational functions are closed under addition, subtraction, multiplication,division (except by the rational function 0)Rational functions and partial fraction expansion 5–11Poles & zerosF (s) =b(s)a(s)=b0+ b1s + ··· + bmsma0+ a1s + ··· + ansn,assume b and a have no common factors (cancel them out if they do . . . )• the m roots of b are called the zeros of F ; λ is a zero of F if F (λ) = 0• the n roots of a are called the poles of F ; λ is a pole of F iflims→λ|F (s)| = ∞the multiplicity of a zero (or pole) λ of F is the multiplicity of the root λof b (or a)example:6s + 12s2+ 2s + 1has one zero at s = −2, two poles at s = −1Rational functions and partial fraction expansion 5–12factored or pole-zero form of F :F (s) =b0+ b1s + ··· + bmsma0+ a1s + ··· + ansn= k(s − z1) ···(s − zm)(s − p1) ···(s − pn)where• k = bm/an• z1, . . . , zmare the zeros of F (i.e., roots of b)• p1, . . . , pnare the poles of F (i.e., roots of a)(assuming the coefficients of a and b are real) complex poles or zeros comein complex conjugate pairscan also have real factored form . . .Rational functions and partial fraction expansion 5–13Pole-zero plotspoles & zeros of a rational functions are often shown in a pole-zero plotPSfrag replacements=<1j(× denotes a pole; ◦ denotes a zero)this example is forF (s) = k(s + 1.5)(s + 1 + 2j)(s + 1 − 2j)(s + 2.5)(s − 2)(s − 1 − j)(s − 1 + j)= k(s + 1.5)(s2+ 2s + 5)(s + 2.5)(s − 2)(s2− 2s + 2)(the plot doesn’t tell us k)Rational functions and partial fraction expansion 5–14Partial fraction expansionF (s) =b(s)a(s)=b0+ b1s + ··· + bmsma0+ a1s + ··· + ansnlet’s assume (for now)• no poles are repeated, i.e., all roots of a have multiplicity one• m < nthen we can write F in the formF (s) =r1s − λ1+ ··· +rns − λncalled partial fraction expansion of F• λ1, . . . , λnare the poles of F• the numbers r1, . . . , rnare called the residues• when λk= λl, rk= rlRational functions and partial fraction expansion 5–15example:s2− 2s3+ 3s2+ 2s=−1s+1s + 1+1s + 2let’s check:−1s+1s + 1+1s + 2=−1(s + 1)(s + 2) + s(s + 2) + s(s + 1)s(s + 1)(s + 2)in partial fraction form, inverse Laplace transform is easy:L−1(F ) = L−1µr1s − λ1+ ··· +rns − λn¶= r1eλ1t+ ···