4.1 Fourier Transforms and the Parseval Identity

Transcription

1 Chapter 4 ransfer Function Models his chapter introduces models of linear time invariant (LI) systems defined by their transfer functions(or, in general, transfer matrices). he subject is expected to be familiar to the reader, especially in the case of rational transfer functions, which are suitable for describing systems defined in terms of ordinary differential equations and difference equations. he main objective of the chapter is to build a mathematical framework suitable for handling the non-rational transfer functions resulting from partial differential equation models which are stabilizable by finite order LI controllers. 4.1 Fourier ransforms and the Parseval Identity Fourier transforms play a major role in defining and analyzing systems in terms of nonrational transfer functions. his section requires some background in the theory of integration of functions of a real argument (measureability, Lebesque integrabilty, completeness of L 2 spaces, etc.), and presents some minimal technical information about Fourier transforms for finite energy functions on Z and R Background: Spaces of Square Integrable Functions his chapter deals extensively with complex vector-valued square integrable functions defined on the sets R, jr (the imaginary axis {s C : Re{s} = 0), (the unit circle {z C : z = 1}), and Z (all integers). he sets of all such functions (traditionally called spaces ) will be denoted by L m 2 (X), where X {R,jR,,Z}. While there are some technical distinctions caused by the differences between the four domains X of interest (R, jr,, and Z), the sets L m 2 (X) share important abstract 35

2 (Fall 2011) ransfer Functions features, which allow one to treat them in a way similar to how ordinary vector spaces C n are treated. Specifically, all these sets are complex vector spaces, i.e. operations of addition and scaling by a complex scalar are defined on L m 2 (X), and satisfy the usual commutative and distributive laws. Moreover the spaces have an inner product operation, mapping every ordered pair (f,g) of functions f,g L m 2 (X) to a complex number c denoted by f g. he inner prodiuct satisfies the conditions of Hermitean symmetry (g f is the complex conjugate of f g), linearity with respect to the second argument (f (cg) = c(f g)forc C,andf (g 1 +g 2 ) = f g 1 +f g 2 ),andpositivesemidefiniteness(f f 0). he function f = f f is used as a measure of length of a function, and satisfies the triangle inequality f+g f + g (or, equivalently, thecauchyinequality f g f g ), sothat one can talk about convergence in L m 2 (X): a sequence {f k } k=0 of functions f k L m 2 (X) is said to converge to f L m 2 (X) when f k f 0 as k. With respect to this definition of convergence, the sets L m 2 (X) have the completeness property: for every sequence {f k } k=0 of functions f k L m 2 (X) satisfying the Cauchy property ( f k f i 0 as k,i ) there exists a function f L m 2 (X) such that f k f 0 as k. For example, L m 2 (Z) will denote the set of all functions u : Z C m (essentially, double-sided sequences of vectors from C m ) such that u(t) 2 <, t Z with the inner product u v def = u(t) v(t). t Z Similarly, L m 2 (R) will be the set of all measureable functions u : R C m such that with the inner product u v def = u(t) 2 dt <, u(t) v(t)dt. Note that a function f L m 2 (R) such that f = 0 can take non-zero values, but in this case the Lebesque measure of the set {t : f(t) = 0} is zero. A function F : C k is called integrable when the function F : ( π,π) C m defined by F(Ω) = F(exp(jΩ)) is Lebesque integrable, in which case we use the shortcut notation F(z)dm(z) def = 1 2π π π F(exp(jΩ))dΩ.

3 6.245 (Fall 2011) ransfer Functions 37 he set L m 2 () contains integrable functions U : C m such that U(z) 2 dm(z) <. he inner product on L m 2 () is defined by U V = U(z) V(z)dm(z). Finally, L m 2 (jr) contains functions U : jr C m such that the function g : R C m defined by g(ω) = U(jω) is square integrable. he inner product on L m 2 (jr) is defined by U V = 1 2π Fourier ransform on Z U(jω) V(jω)dω. For a function (sequence) u : Z C m, one expects the Fourier transform to be a function U : C m given by U(z) = u(t)z t. (4.1) t Z his definition is straightforward for absolutely summable sequences, i.e. those satisfying the condition u(t) <, Z t Z which guarantees that the sum in (4.1) converges uniformly over z. Moreover, the simple observation that { z i z k def 1, i = k, dm(z) = δ ik = (4.2) 0, i k, for all i,k Z yields the ever helpful Parseval identity u(t) 2 = U(z) 2 dm(z). (4.3) t Z he problem is, the class of summable seqences u is too narrow to be useful in many applications. he objective of this subsection is to introduce Fourier transform for square summable sequences, i.e. elements of L m 2 (Z).

4 (Fall 2011) ransfer Functions According to a proper definition, the Fourier transform U of u L m 2 (Z) is any U L m 2 () such that U U 0 as, where U (z) = t= u(t)z t. (4.4) According to this definition, a given sequence u L m 2 (Z) can have multiple Fourier transforms, but this non-uniqueness is not really essential, as any two Fourier transforms U,V of the same square summable sequence u must satisfy U V = (U U ) (V U ) U U + V U 0 as, which means that U V = 0, i.e. U(z) V(z) = 0 for almost all z, so that, for all practical purposes, U and V can be considered equal. he main statement about Fourier transforms of square summable sequences is their existence, coupled with the Parseval identity. heorem 4.1. Every u L m 2 (Z) has a Fourier transform U L m 2 (). Moreover, every Fourier transform U L m 2 () of u L m 2 (Z) satisfies (4.3). Proof. For arbitrary positive integers τ < the Parseval identity yields 1 U U τ 2 2 = u(t) 2 t=τ u(t) 2 0 as τ, where U are defined in (4.4). Hence {U } is a Cauchy sequence, and, by completeness of L m 2 (), has a limit U. By definition, U is a Fourier transform of u. o prove the Parseval identity, note that U U = (U U )+U U U U + U U 0 as, U U = U U +U U U U + U U 0 as. t=τ Hence U 2 = lim U 2 = lim u(t) 2 = u 2. t=

5 6.245 (Fall 2011) ransfer Functions Approximations by rigonometric Polynomials in L m 2 () A trigonometric polynomial (of a single argument) is a continuous function P : C m which can be expressed in the form P(z) = t= p(t)z t, where p(t) C m are some vectors of coefficients. (he name is likely due to the fact that z n is a polynomial with respect to cos(ω) and sin(ω) when z = exp(jω).) For the development of Fourier transforms, we need the classical theorem establishing that trigonometric polynomials can approximate continuous functions V : C m arbitrarily well. heorem 4.2. For every continuous function V : C m and every ǫ > 0 there exists a trigonometric polynomial P : C m such that max P(z) V(z) < ǫ. z Proof. For every positive integer n define function P n : C m by P n (z) = h n (z,w)v(w)dm(w), (4.5) where h n (z,w) = 1 n 1+ w z + w2 z 2 wn z n 1 Since for every fixed n and w the function z h n (z,w) is a trigonometric polynomial, the result P n (z) of integration in (4.5) is a trigonometric polynomial, too. he kernel function h n (z,w) has several useful properties: (a) h n (z,w) 0; (b) lim n sup z: z w >δ h n (z,w) = 0 for all δ (0,π); (c) h n(z,w)dm(w) = 1 for all n. hese properties guarantee that lim max P n(z) V(z) = 0. (4.6) n z Indeed, since V is a continuous function, for every ǫ > 0 there exists δ (0,π) such that V(w) V(z) < 0.5ǫ whenever w z < δ..

6 (Fall 2011) ransfer Functions For this δ > 0, property (b) of the kernel guarantees existence of n such that h n (z,w) < 0.5ǫ 1+2γ for z w > δ, where γ = max z V(z). Hence for every z P n (z) V(z) = = ǫ. h n (z,w)(v(z) V(w))dm(w) h n (z,w) V(z) V(w) dm(w)+ z w <δ 0.5ǫdm(w) + 0.5ǫ 1+2γ 2γdm(w) z w >δ h n (z,w) V(z) V(w) dm(w) Fourier Coefficients For every U L m 2 () and t Z the Fourier coefficients integral u(t) = U(z)z t dm(z) (4.7) defines a function (sequence) u : Z C m. he following statement, which relies on the result of the previous subsection, shows that Fourier coefficients deefine a function U L m 2 () almost completely. heorem 4.3. If all Fourier coefficients (4.7) of a function U L m 2 () are zero, then U = 0 (i.e. U(z) = 0 for almost all z ). Proof. he theorem can be proven component-wise, i.e. it is sufficient to consider the case m = 1. For 0 < θ < ν < 2π consider the function F θ,ν : C defined by 1, 0 Ω < θ, F θ,ν (exp(jω)) = 1 Ω θ ν θ, θ Ω < ν, 0, ν Ω < 2π. Since F θ,ν is continuous, there exists a sequence of trigonometric polynomials P n such that lim n max z F θ,ν(z) P n (z) = 0.

7 6.245 (Fall 2011) ransfer Functions 41 Hence Since by assumption lim (F θ,ν (z) P n (z))v(z)dm(z) = 0. n P(z)V(z)dm(z) = 0 for every trigonometric polynomial P, we conclude that F θ,ν V(z)dm(z) = lim P n (z)v(z)dm(z) = 0. n For 0 < θ < 2π let Since we conclude that i.e. F θ (exp(jω)) = { 1, 0 Ω < θ, 0, θ Ω < 2π.. F θ (z) F θ,ν (z) 1 and lim ν θ F θ,ν (z) = F θ (z), F θ (z)v(z)dm(z) = lim F θ (z)v(z)dm(z) = 0, ν θ θ 0 V(exp(jΩ))dΩ = 0 for all θ (0,2π). According to the basic properties of Lebesque integration, this means that V(exp(jΩ)) = 0 or almost all Ω (0,2π) Inverse Fourier ransform on As stated by the following theorem, mapping (4.7) of a function U L m 2 () to the sequence of its Fourier coefficients serves as the inverse of the Fourier transform (4.1). heorem 4.4. For every U L m 2 () the sequence u : Z C m defined by (4.7) has finite energy (i.e. u l m 2 ). Moreover, U is a Fourier transform of u. Proof. First prove that u is square summable. For positive integers > 0 define functions U : C m as in (4.4). Multiplying equality (4.7) by u(t) on the left and summing from t = to t = yields u(t) 2 = U (z) U(z)dm(z). t=

8 (Fall 2011) ransfer Functions Applying the Cauchy inequality U U U U to the inner product on the left side yields U 2 U U, hence U 2 U 2 for all > 0, and therefore u L m 2 (Z). Let V denote the Fourier transform of u. Since U V 0 as, and z t U (z)dm(z) = u(t) for t, Z U and V have same Fourier coefficients. Hence U V = 0 (heorem 4.3), which means that U is a Fourier transform of u as well Fourier ransform on R he derivation of Fourier transform on L m 2 (R) follows a path similar to that of L m 2 (Z). A key result is the following limited version of the Parseval identity. heorem 4.5. he identity where u(t) 2 dt = 1 U (jω) 2 dω, (4.8) 2π U (jω) = e jωt u(t)dt, holds for every > 0 and every function u L m 2 (R). Proof (a sketch). It is actually easier to prove the slightly more general identity where U (jω) = u(t) v(t)dt = 1 2π e jωt u(t)dt, V (jω) = U (jω) V (jω)dω, (4.9) e jωt v(t)dt, and u,v L m 2 (R) (proving the identity involves showing that U,V L m 2 (jr)). A good starting point is to consider functions u,v of the form u(t) = exp(jπa/), v(t) = exp(jπb/), (4.10) where a, b are arbitrary integer numbers. After some manipulations with integrals, this boils down to computing the integrals { sin 2 (ω) ω(ω +dπ) dω = π, d = 0, 0, d Z, d 0.

9 6.245 (Fall 2011) ransfer Functions 43 (Would be easy to do this using the Parseval identity, but this would make for a circular argument: Parseval identity is exactly what is being proven right here!) Once (4.9) is established for individual exponents (4.10), it extends easily to finite linear combinations of such exponents. According to the results on inverse discrete Fourier transform established in the previous subsection, every square integrable function on the interval [, ] can be approximated arbitrarily well (in terms of L 2 convergence) by linear combinations of functions (4.10), which allows one to prove the general case of (4.9). heorem 4.5 shows that, for a given function u L m 2 (R) the sequence of incomplete Fourier transforms U is a Cauchy sequence in L m 2 (jr). Hence, there exists U L m 2 (jr) such that U U 0 as Any such function U L m 2 (jr) is called a Fourier transformofu. Naturally,twodifferentFouriertransformsofthesamefunctionu L m 2 (R) must be equal almost everywhere on jr. he derivation of inverse Fourier transform on L m 2 (jr) relies on minor modification of identity (4.8): 1 Ω U(jω) 2 dω = u Ω (t) 2 dt, (4.11) 2π Ω where u Ω (t) = 1 Ω e jωt U(jω)dω. 2π Ω Once again, the inverse Fourier transform u L m 2 (R) of U L m 2 (jr) is a limit of the Cauchy sequence u Ω as Ω. heorem 4.6. Function U L m 2 (jr) is a Fourier transform of u L m 2 (R) if and only if u is an inverse Fourier transform of U, in which case u(t) 2 dt = 1 2π U(jω) 2 dω. (4.12) 4.2 Hardy Classes: H 2 and H Hardy classes H 2 are one-sided Fourier transforms of square integrable signals. hey are introduced as subsets of L m 2 () (in a discrete time framework) or L m 2 (jr) (in a continuous time framework). However, the functions from H 2 are most interesting because of the possibility of extending them into the outside of the unit circle (for L m 2 ()) or into the right half plane (for L m 2 (jr)), where they become analytical functions of complex argument. he main objective of this section is to introduce functions from class H, as a minor modification of class H 2, as they match exactly transfer functions of L2 gain stable causal

10 (Fall 2011) ransfer Functions LI systems, and also serve as building blocks for transfer functions of stabilizable causal LI systems Analytical Functions A function F : Ω C defined on an open subset Ω of C is called analytical if the limit F(s) = lim δ 0 F(s+δ) F(s) δ exists at all points s Ω. As a rule, elementary functions, such as polynomials or exponents, or logarithms (within limited domains), or their compositions, turn out to be analytical. For example, function F(s) = 1/s is analytical on Ω = {s C : s 0}, and function F(s) = log(s) def = log s +arg(s), arg(s) ( π,π) is analytical in the right half plane C + = {s : Re{s} > 0}. In contrast, functions which require separating real and imaginary parts of s, such as F(s) = Re{s} or F(s) = s are not analytical. For example, the expression Re{s+δ} Re{s} δ converges to 1 when δ approaches zero along the real axis, but converges to zero when δ approaches zero along the imaginary axis. he following classical theorem gives an alternative description of analytical functions. heorem 4.7. For some positive numbers R 0,R 1 such that 0 < R 0 < R 1 let Ω = {s C : R 0 < s < R 1 }. A function F : Ω C m is analytical if and only if there exists a function f : Z C m such that r t f(t) < for R 0 < r < R 1, and F(s) = s t f(t) s Ω. k=

11 6.245 (Fall 2011) ransfer Functions Class H m 2 () Formally speaking, H m 2 () (or simply H 2 () when m = 1) is the subset of L m 2 () consisting of those functions U for which the Fourier coefficients u(t) equal zero for t < 0. In other words, H m 2 () is the set of Fourier transforms of signals of finite energy. Most significantly, for an arbitrary U H m 2 () the infinite sum U(w) = t=z u(t)w t = u(t)w t t=0 converges absolutely on the open set D + = {w C : w > 1}. Allowing some abuse of notation, we will denote the resulting analytical function on D + by the same letter, i.e. as U : D + C. he following theorem shows that, combined with analiticity, boundedness of the energies U r 2 for U r (z) = U(rz) over r > 1 yields a complete description of extensions of functions from class H m 2 () to D +. heorem 4.8. Let U : D + C m be a function. he following conditions are equivalent: (a) U is an extension of a function U H m 2 (); (b) U is analytical on D +, and sup U(rz) 2 dm(z) <. (4.13) r>1 Moreover, when conditions (a),(b) are satisfied, the functions U r : C m defined for r > 1 by U r (z) = U(rz), converge to U, in the sense that U U r 0 as r 1. One useful corollary of heorem 4.8 is a criterion for a function U from class H2 m to have zero Fourier coefficients u(t) = 0 for all t <, where > 0. Since this is equivalent to V(s) = s U(s) being an extension of a function from class H 2, the necessary and sufficient condition is sup r 2 U(rz) 2 dm(z) <. (4.14) r>1

12 (Fall 2011) ransfer Functions Example 4.1. he function defined for w > 1 by U(w) = log(1 1/w) (where the values of the logarithm with imaginary part from ( π,π) are used) is an extension of an H 2 () class function. o establish this, note that that U is analytical on D +, and use the inequality 1 1 z rz z, r 1. Since the imaginary part of log(1 1/w) for w D + is bounded (lies within the ( π,π) interval), and the real part cannot be larger than log(2), we have log(1 1 rz ) π +log(2)+ log 1 1 z, which allows one to bound the integrals log(1 1/(rz)) 2 dm(z) π +log(2)+ log(1 1/z) 2 dm(z) < (the last inequality holds because log( Ω ) is square integrable over Ω ( π, π)). Naturally, the function U(z) = log(1 1/z) (defined arbitrarily for z = 1) provides the values of U to the unit circle. Since we have already established an integrable majorant for U(z) U(rz), the convergence lim r 1 U(z) U r (z) 2 dm(z) = 0 follows from the point-wise convergence U r (z) U(z) as r 1 for z 1. Proof of heorem 4.8, (a) (b). Let u L m 2 (Z) be the inverse Fourier transform of U. By assumption u(t) = 0 for t < 0. Hence the expansion U(s) = t Z u(t)s t satisfies the absolute convergence condition u(t) r t <, t Z which makes U analytical on D +. Since U r (z) = U(rz) is the Fourier transform of u r (t) = r t u(t), the Parseval identity yields r 2 U(rz) 2 dm(z) = r 2t u(t) 2 U 2. t=0

13 6.245 (Fall 2011) ransfer Functions 47 Moreover, a similar use of the Parseval identity yields r 2 U(rz) U(z) 2 dm(z) = (1 r t ) 2 u(t) 2 0 as r 1. Proof of heorem 4.8, (b) (a). By heorem 4.7 there exists an expansion U(s) = u(t)s t t Z t=0 which converges absolutely for s > 1. Hence the sequence u r (t) = r t u(t) is square summable for all r > 1, and, by the Parseval identity, r 2 U(rz) 2 dm(z) = r 2t u(t) 2 for all r > 1. Since the sum on the right converges to + as r, unless u(t) = 0 for all t < 0, we conclude that u(t) = 0 for t < 0. Subject to this, the sum on the right converges to u 2 as r 1, which proves that U H m 2 () Class H m,k () Functions from class H m,k () will serve as transfer matrices of causal L2 gain stable LI models (with m inputs and k outputs) in discrete time. Each element G H m,k () can be viewed as a k-by-m matrix with entries from H () = H 1,1 (). For the purpose of this section, it is sufficient to study the properties of the scalar functions U H (). Formally, the class H () is the subset of those functions G H 2 () = H2() 1 which are uniformly bounded on, i.e. satisfy the condition t=0 sup U(z) <. z he following statement describes the extensions of functions from H () to D +. heorem 4.9. Let U : D + C be a function. he following conditions are equivalent: (a) U is an extension of a function U H (); (b) U is analytical, and γ = sup z D + U(z) <. (4.15)

14 (Fall 2011) ransfer Functions Moreover, if conditions (a),(b) are satisfied then γ equals the essential supremum of U(z) over, i.e. the minimal upper bound of those h 0 for which the set of z satisfying U(z) h has positive Lebesque measure. Example 4.2. Out of the two functions G 1 (z) = exp ( 1 z 1+z ), G 2 (z) = exp ( ) z 1, z +1 the first belongs to class H () because it has an extension which is analytical and bounded in the domain z > 1. In contrast, function G 2 L 2 () does not belong to any of the classes H 2 () or H (), because it grows very large as z approaches 1 from the outside of the unit circle. Proof of heorem 4.9, (a) (b). For every s D + consider the continuous function R s : R + defined by ( z z s) R s (z) = 1+ s + = s 2 1 s z 2. By construction, Since R s 0 and identity (4.17) implies U(s) = for all s D +. U(s) = k=1 R s (z)u(z)dm(z) s D +. (4.16) R s (z)dm(z) = 1, R s (z)u(z)dm(z) R s (z) U(z) dm(z) γ Proof of heorem 4.9, (a) (b). For every s D + consider the continuous function R s : R + defined by ( z z s) R s (z) = 1+ s + = s 2 1 s z 2. By construction, Since R s 0 and U(s) = k=1 R s (z)u(z)dm(z) s D +. (4.17) R s (z)dm(z) = 1,

15 6.245 (Fall 2011) ransfer Functions 49 identity (4.17) implies U(s) = for all s D +. R s (z)u(z)dm(z) R s (z) U(z) dm(z) γ Proof of heorem 4.9, (a) (b). By heorem 4.8, U is an extension of an H 2 () function V, and U r V 0 as r 1, where U r (z) = U(rz). It is sufficient to show that the essential supremum of V is not larger than γ. If this is not true, i.e. for some γ 1 > γ the inequality V(z) γ 1 holds on a subset Z of measure δ > 0, then U r V 2 = U r (z) V(z) 2 dm(z) U r (z) V(z) 2 dm(z) δ(γ 1 γ) 2 > 0, which contradicts the convergence U r V 0 as r Class H m 2 (jr) Z Formally speaking, H m 2 (jr) (or simply H 2 (jr) when m = 1) is the subset of L m 2 (jr) consisting of those functions U for which an inverse Fourier transform u(t) equals zero for t < 0 (which is equivalent to every inverse Fourier transform u of U satisfying the condition u(t) = 0 for almost all t < 0). In other words, H m 2 (jr) is the set of Fourier transforms of C signals of finite energy. Most significantly, for an arbitrary U H m 2 (jr) the Laplace transform integral U(s) = u(t)e st dt = u(t)e st dt t=0 converges absolutely on the open set C + = {s C : Re{s} > 0}. Allowing some abuse of notation, we will denote the resulting analytical function on C + by the same letter, i.e. as U : C + C m. he following theorem shows that, combined with analiticity, boundedness of the energies U σ 2 for U σ (jω) = U(jω + σ) over σ > 0 yields a complete description of extensions of functions from class H m 2 (jr) to C +. heorem Let U : C + C m be a function. he following conditions are equivalent: (a) U is an extension of a function U H m 2 (jr);

16 (Fall 2011) ransfer Functions (b) U is analytical, and sup U(jω +σ) 2 dω <. (4.18) σ>0 Moreover, when conditions (a),(b) are satisfied, the functions U σ : jr C m defined for σ > 0 by U σ (jω) = U(σ +jω), converge to U, in the sense that U U σ 0 as σ 0. One useful corollary of heorem 4.10 is a criterion for a function U from class H m 2 (jr) to be a Fourier transform of u L m 2 (R) such that u(t) = 0 for all t <, where > 0. Since this is equivalent to V(s) = exp(s)u(s) being an extension of a function from class H 2, the necessary and sufficient condition is supe 2σ U(σ +jω) 2 dω <. (4.19) σ>0 Example 4.3. he function defined for Re{s} > 0 by U(s) = log(s)/(s+1) (where the values of the logarithm with imaginary part from ( π,π) are used) is an extension of an H 2 (jr) class function Class H m,k (jr) Functions from class H m,k (jr) will serve as transfer matrices of causal L2 gain stable LI models (with m inputs and k outputs) in continuous time. Each element G H m,k (jr) can be viewed as a k-by-m matrix with entries from H (jr) = H 1,1 (jr). For the purpose of this section, it is sufficient to study the properties of the scalar functions U H (). Formally, theclassh (jr)canbedefinedasthesubsetofthosefunctionsg : jr C which are uniformly bounded on jr, and can be represented in the form G(jω) = (1+jω) G(jω) (4.20) for some G H 2 (jr). Extending H 2 class function G from (4.20) into the right half plane C + defines a natural extension for G H (jr), via G(s) = (1 + s) G(s). he following statement describes the extensions of functions from H (jr) to C +. heorem Let U : C + C be a function. he following conditions are equivalent: (a) U is an extension of a function U H (jr); (b) U is analytical, and γ = sup s C + U(s) <. (4.21)

17 6.245 (Fall 2011) ransfer Functions 51 Moreover, if conditions (a),(b) are satisfied then γ equals the essential supremum of U(s) over s jr, i.e. the minimal upper bound of those h 0 for which the set of ω R satisfying U(jω) h has positive Lebesque measure. 4.3 L2 Gain Stable ransfer Matrix Models Defining LI system models in terms of their transfer functions is supposed to be straightforward: apply Fourier transform to the input, multiply the result by the transfer function, and then apply inverse Fourier transform to the product. here are several important questions to resolve, though: (a) What are the boundary conditions for the resulting system? (b) What to do when the signals involved are not square integrable (c) Which conditions guarantee that the resulting system is well defined(e.g. the inverse Fourier transform involved does exist)? In this section we use functions from Hardy classes H as transfer matrices L2 gain stable transfer matrix models. he following shortcut notation (applicable to signals of arbitrary dimension d), where X = Z, Y =, and Z = l for discrete time systems, or X = R, Y = jr, and Z = L for C systems) will be used in this section: F: Fourier transform F : L d 2(X) L d 2(Y); F 1 : inverse Fourier transform F 1 : L d 2(Y) L d 2(X); P : past projection P : Z d L d 2(X) L d 2(X), P f = g means g(t) = { f(t), 0 t, 0, otherwise; P + : future projection P + : L d 2(X) L d 2(X), P + f = g means g(t) = { f(t), t 0, 0, t < 0;

18 (Fall 2011) ransfer Functions Real Symmetry So far, we have applied Fourier transforms to general complex-valued functions and sequences. Since we use real-valued functions to describe actual real world signals, a description of Fourier transforms of real-valued functions will be useful. If U L m 2 () is a Fourier transform of u L m 2 (Z) then the function U c L m 2 () defined by U c (z) = U( z) is a Fourier transform of u c L m 2 (Z) defined by u c (t) = u(t). Since U c (z) = U(z) for almost all z if and only if u c (t) = u(t) for all t Z, inverse Fourier transform of a function U L m 2 () is real-valued if and only if U( z) = U(z) for almost all z. Similarly, inverse Fourier transform of a function U L m 2 (jr) is real-valued if and only if U( jω) = U(jω) for almost all ω R. his observation suggests that transfer matrices of LI systems should also have the real symmetry property, which leads us to the definition of class H k,m Re (X) (where X = or X = jr) as the subset of those functions G from H k,m (X) (i.e. k-by-m matrices with entries from the class H (X)) which satisfy the real symmetry condition G( z) = G(z). Recall that a rational function satisfies the real symmetry condition if and only if it has real coefficients. Naturally, one expects transfer matrices of physically implementable systems to have real symmetry Discrete ime Stable ransfer Matrix Models We will use the following preliminary observation. heorem For every G H k,m Re () and w lm there exists a unique y def = L G w l k such that P y = P F 1 GFP w 0. (4.22) Moreover, if w(t) = 0 for t τ then y(t) = 0 for t τ. he function L G defined in heorem 4.12 generates the zero-state response of the transfer matrix model defined by G. Its definition follows the familiar Fourier tranform, multiplication, inverse Fourier transform pattern, with projections P inserted to ensure legality of the associated operations. Also, for G H k,m Re (), let X G denote the subset of sequences x 0 L k 2(Z) which can be approximated arbitrarily well (in the sense of x 0 y 0 0 by sequences y 0 of the form y 0 = P + F 1 GFw 0 (i.e. those obtained by taking w 0 L m 2 (Z), applying Fourier transform, multiplying the result by G, applying inverse Fourier transform, and then zeroing out the samples with t < 0), where w 0 L m 2 (Z) is such that w 0 (t) = 0 for t 0, and w 0 (t) R m for t < 0. Due to the real symmetry assumption, all sequences in X G

19 6.245 (Fall 2011) ransfer Functions 53 are real. he set X G will be used to represent the boundary conditions or zero input response of the LI system associated with G. Finally, the transfer matrix model SD M (G) defined by G Hk,m Re () is the discrete time system SD M (G) Sm,k D (X G), such that S M D (G)(w,x 0 ) = {x 0 +L G w} w l m, x 0 X G. (4.23) Example 4.4. Let G : C be defined by G(z) = 1/z. Since G is the Fourier transform of real-valued signal { 1, t = 1, g(t) = 0, t 1, we conclude that G H Re (). What kind of LI system is SD M(G)? Let us begin by studying time-domain implications of the Fourier transform relation Y(z) = G(z)W(z), where Y,W L 2 (). Since Y(z) = G(z)W(z) means zy(z) = W(z), the corresponding inverse Fourier transforms y,w satisfy the difference equation y(t+1) = w(t) for all t Z. he original observation proves that y = L G w means { w(t 1), t > 0, y(t) = 0, t = 0. Similarly, sequences of the form y 0 = P + F 1 GFw 0 are such that { w 0 (t 1), t 0, y 0 (t) = 0, t < 0, where w 0 (t) = 0 for t 0, which means that y 0 (t) = 0 for all t except at t = 0, while y 0 (0) can be arbitrary. herefore the set X G consists of all sequences x 0 : Z R such that x 0 (t) = 0 for all t 0. While, formally speaking, X G is a set of sequences, its elements can be parametrized linearly by a single real parameter! Finally, combining these descriptions of L G and X G, we conclude that the set of input-output pairs (w,y) in system SD M (G) is defined by equation y(t+1) = w(t), where the initial value y(0) can be an arbitrary real number, and is determined by the boundary condition vector x 0 X G. No surprises here! Proof of heorem 4.12 o simplify notation and terminology, the proof is given for the case m = k = 1. For every u l and > 0 the Fourier transform U of u = P u is real symmetric, and has an analytical extension Ũ : D + C such that lim sup r 1 z Ũ(rz) U (z) = 0.

20 (Fall 2011) ransfer Functions As an element of H (), function G also has an analytical extension G : D + C which is uniformly bounded, and such that G(rz) G(z) 2 dm(z) = 0. lim r 1 Hence the function Ỹ : D + C defined by Ỹ(z) = G(z)Ũ(z) is an analytical extension of Y = GU, and Y (z) Ỹ(rz) 2 dm(z) = G(z)U (z) G(rz)Ũ(rz) 2 dm(z) = 2 { G(z)(U (z) Ũ(rz)) 2 + (G(z) G(rz))Ũ(rz) 2 }dm(z) 2max U (z) Ũ(rz) 2 G(z) 2 dm(z) z +2max z G(z) 2 G(z) G(rz) 2 dm(z) 0 as r 1. Hence Y H 2 () has inverse Fourier transform y. o complete the proof, it is sufficient to show that for every τ > the equality y (t) = y τ (t) holds for all t. Indeed, since u (t) = u τ (t) for t we have supr 2 Ũ(rz) Ũτ(rz) 2 dm(z) <. r>1 Since G(z) is uniformly bounded in D +, this implies supr 2 Ỹ(rz) Ỹτ(rz) 2 dm(z) = supr 2 G(rz)(Ũ(rz) Ũτ(rz)) 2 dm(z) <, r>1 r>1 and hence y (t) = y τ (t) for t Continuous ime Stable ransfer Matrix Models he development follows the D case closely. We start by defining zero state response. heorem For every G H k,m Re (jr) and w Lm there exists a unique y def = L G w L k such that P y = P F 1 GFP w 0. (4.24) Moreover, if w(t) = 0 for t τ then y(t) = 0 for t τ.

21 6.245 (Fall 2011) ransfer Functions 55 For G H k,m Re (jr), let X G denote the subset of sequences x 0 L k 2(R) which can be approximated arbitrarily well (in the sense of x 0 y 0 0 by sequences y 0 of the form y 0 = P + F 1 GFw 0 (i.e. those obtained by taking w 0 L m 2 (R), applying Fourier transform, multiplying the result by G, applying inverse Fourier transform, and then zeroing out the samples with t < 0), where w 0 L m 2 (Z) is such that w 0 (t) = 0 for t 0, and w 0 (t) R m for t < 0. Due to the real symmetry assumption, all sequences in X G are real. Finally, thetransfermatrixmodels M C time system SC M (G) Sm,k C (X G), such that (G)definedbyG Hk,m Re (jr)isthecontinuous S M C (G)(w,x 0 ) = {x 0 +L G w} w l m, x 0 X G. (4.25) Example 4.5. Let G : jr C be defined by G(s) = e τs, where τ > 0 is a parameter. Since G is uniformly bounded on jr, and G(s)/(s+1), as the Fourier transform of { 0, t < τ, g(t) = e τ t, t τ, is from class H 2 (jr), we conclude that G H (jr). Moreover, since G has real symmetry, G H Re (jr). What kind of LI system is SC M(G)? We begin by studying time-domain implications of the Fourier transform relation Y(jω) = G(jω)W(jω), where Y,W L 2 (jr): y(t) = w(t τ) for all t R. he original observation proves that y = L G w means { w(t τ), t τ, y(t) = 0, t < τ. Similarly, functions of the form y 0 = P + F 1 GFw 0 are such that { w 0 (t τ), t 0, y 0 (t) = 0, t < 0, where w 0 (t) = 0 for t 0, which means that y 0 (t) = 0 for all t 0, and the restriction of y 0 to the interval (0,τ) can be arbitrary square integrable function. herefore the set X G consists of all square integrable functions x 0 : R R such that x 0 (t) = 0 for all t τ. Note that the set of boundary conditios does not have a finite basis: this is what we expect when the transfer function is not rational. Finally, combining these descriptions of L G and X G, we conclude that the set of input-output pairs (w,y) in system SC M (G) is defined by equation y(t+τ) = w(t), where the value y(t) for t < τ can be an arbitrary, as long as they are real, and the resulting function is square integrable over (0,τ).