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MIT EECS 6.241 (FALL 2006) LECTURE NOTES BY A. MEGRETSKI

9 Laplace Transform and z-Transform

This lecture presents basic properties of Laplace transform needed to work with non- rational transfer matrices. The discrete time analog, z-transform, is also discussed.

9.1 Laplace Transform When studying Laplace transform, it would be very inconvenient to limit one’s attention to piecewise continuous functions only. In this lecture, it will be appled to measureable m functions f : IR+ 7→ IR which are locally square integrable, in the sense that the

Z T |f(t)|2dt 0

2,loc are finite for all T > 0. The set of all such functions will be denoted by Lm (simply L2,loc when m = 1). For example, the

 0, t = 0, f(t) = ta, t > 0, belongs to L2,loc if and only if a > −0.5.

2,loc Definition 9.1 A function f ∈ Lm is said to admit a Laplace transform if there exist h such that Z ∞ e−ht|f(t)|dt < ∞. 0 Let r ∈ {−∞} ∪ IR be the largest lower bound of such h. The corresponding right halp-plane Cr = {s ∈ C : Re(s) > r} is called the region of convergence of Laplace transform of f. The Z ∞ F (s) = f(t)e−st t=−∞

m converges absolutely for all s ∈ Cr, and the resulting function F (s):Cr 7→ C is called the Laplace transform of f.

1 m 2,loc Laplace transform F :Cr 7→ C of a function f ∈ Lm is analytical in the sense that the limit ˙ F (s) − F (s0) F (s0) = lim s∈C ,s6=s0,s→s0 s − s0 exists for all s0 ∈ Cr. Note that an analytical function F0 :Ω 7→ Cm defined on an open subset Ω of Cr and such that F0(s) = F (s) at an uncountable number of points satisfies F0(s) = F (s) for all s ∈ Ω, and hence will also be referred to as a Laplace transform of f, even when Ω is not the region of convergence for the Laplace transform of f. Since f takes values in IR (not C), the Laplace transform F is also real symmetric, in the sense that F (s0) = F (s0) m for all s0 ∈ Cr. However, it is not true that every real analytical function F :Cr 7→ C is a Laplace transform of some f : IR 7→ IRm.

Example 9.1 F (s) ≡ 1 is not Laplace transform of a f ∈ L2,loc.

The following theorem provides necessary and sufficient conditions for a real analytical m 2,loc function F :C r 7→ C to be Laplace transform of some f ∈ L .

m Theorem 9.1 Let F :Cr 7→ C be a real symmetric analytical function. The following conditions are equivalent:

(a) F is Laplace transform of some function f ∈ L2,loc;

(b) for every R > r the integrals

Z ∞ 1 2 Jh(F ) = |F (jω + h)| dω 2π −∞ are uniformly bounded for h ∈ (R, ∞), i.e.

sup Jh(F ) < ∞. R

When conditions (a),(b) are satisfied, and h > r, the identity

1 Z ∞ f(t) = F (h + jω)eh+jωtdω (9.1) 2π −∞ takes place, where (9.1) means that

Z 0 Z W 2 jωt lim F (h + jω)e dω dt = 0 W →+∞ −∞ −W

2 and Z ∞ Z W 2 −ht jωt lim f(t)e − F (h + jω)e dω dt = 0. W →+∞ 0 −W Moreover, Z ∞ −Rt 2 e f(t) = sup Jh(F ) (9.2) 0 h>R for all R ≥ r. Equality (9.1) can be viewed as a formula for the inverse Laplace transform. Similarly, (9.2) is an equivalent form of the well known Parceval identity.

Example 9.2 Function F :C 0 7→ C , defined by F (s) = log(s)/(s + 1), (using the main branch of the , i.e. log(1) = 0), is Laplace transform of a signal 2,loc f ∈ L , while the function √ s F (s) = s + 1 is not.

2,loc Theorem 9.1 can also be used to figure out whether a function f ∈ Lm defined by m its Laplace transform F :Cr 7→ C satisfies the condition f(t) = 0 for all t < T, (9.3) T > 0 is a given number. Indeed, (9.3) is equivalent to F (s) = e−sT G(s), where Z G(s) = e−stf(t + T )dt 0 2,loc is the Laplace transform of function g ∈ Lm defined by g(t) = f(t + T ). Hence (9.3) holds if and only if Z ∞ sup e2hT |F (jω + h)|2dω < ∞. (9.4) h>R −∞ In other words, the integrals of |F |2 over the lines Re(s) = h in the should converge to zero exponentially as h → +∞. √ −2s s Example 9.3 Function F :C 0 7→ C defined by F (s) = e (s+1)2 is the Laplace transform 2,loc of a signal f ∈ L such that f(t) = 0 for t < 2. In contrast, function F :C 0 7→ C defined by √ √ ea s − e−a s F (s) = Fa(s) = √ √ e s − e− s 2,loc is the Lapalace transform of some f = fa ∈ L when 0 ≤ a < 1, but for all a ∈ [0, 1) there is no T > 0 such that fa(t) = 0 for all t < T .

3 9.2 in Laplace Transform Domain An important operation used in modeling of distributed LTI systems is multiplication in 2,loc the Laplace transform domain. This means taking a “time domain” function f ∈ Lm , a k×m k×m “Laplace domain” function G :Cr 7→ C (where C denotes the set of all complex 2,loc k-by-m matrices), and defining y ∈ Lk as the function for which the Laplace transform equals Y (s) = G(s)F (s), where F is the Laplace transform of f. 2,loc 2,loc According to Theorem 9.1, y ∈ Lm will be well defined for all fL which admit Laplace transform when there exists R > 0 such that G is analytical, real symmetric, and bounded in the right halp plane CR. Moreover, the resulting transformation f 7→ y is causal in the sense of the following statement.

k×m Theorem 9.2 Let G :CR 7→ C be analytical, real symmetric, and uniformly 2,loc bounded. Let f ∈ Lm be a function such that f(t) = 0 for all t < T , where T > 0. m If f has Laplace transform F :Cr 7→ C then Y = GF is Laplace transform of some 2,loc y ∈ Lk such that y(t) = 0 for all t < T .

The proof follows easily using condition from (9.4) as a test of existence of inverse Laplace transform satisfying (9.3). An important implication of Theorem 9.2 is that multiplication in the Laplace trans- 2,loc form domain is available even for those functions f ∈ Lm which do not have Laplace 2,loc T 2,loc transform. Indeed, for every f ∈ Lm and T > 0 the function f ∈ Lm defined by  f(t), t < T, f (t) = T 0, t ≥ T,

2,loc admits Laplace transform FT = FT (s), and hence yT ∈ Lk is well defined as the inverse def k×m Laplace transform of YT (s) = G(s)FT (s) when G :CR 7→ C satisfies the conditions of Theorem 9.2. Since the equality fT (t) = fτ (t) is satisfied for τ > T > t ≥ 0, it follows that yT (t) = yτ (t) for all τ > T > t ≥ 0. Therefore the condition y(t) = yT (t) for t < T 2,loc defines a unique function y ∈ Lk , which can be naturally described as the result of multiplication by G in the Laplace transform domain. 2,loc The notation y = G f will be used to indicate that y ∈ Lk is the result of multi- 2,loc plying f ∈ Lm by a k-by-m complex matrix-valued function in the Laplace transform domain. The following statement describes representation of continuous time LTI state space models in the Laplace transform domain.

Theorem 9.3 Let A, B, C, D be real matrices of dimensions n-by-n, n-by-m, k-by-n, and 2,loc 2,loc k-by-m respectively. Then a pair (f, y) ∈ Lm × Lk satisfies equations x˙(t) = Ax(t) + Bf(t), y(t) = Cx(t) + Df(t) (9.5)

4 n for some x : IR+ 7→ IR if and only if y can be represented in the form y = G f + y0, where

G(s) = D + C(sI − A)−1B, and the Laplace transfororm Y0 of y0 is given by

−1 Y0(s) = C(sI − A) a for some a ∈ IRn.

n Proof. For a given solution (f, y, x) of (9.5) let a = x(0). Define x0 : IR+ 7→ IR as the solution of the ODE x˙ 0(t) = Ax0(t), x0(0) = a, obtained by substituting f ≡ 0 into the first equation in (9.5). It is known that such solution k at x0 = x0(t) is a liear combination of functions of the form φa,b,c(t) = t e cos(bt + c), and hence admits Laplace transform X0 = X0(s). Integration by part shows that the Laplace transform Z(s) of z = Ax0 equals Z ∞ Z ∞ −st −st e x˙(t)dt = s e x(t)dt − x(0) = sX0(s) − a. 0 0

On the other hand, due to of Laplace transform, Z(s) = AX0(s). Hence

−1 AX0(s) = sX0(s) − a, i.e. X0(s) = (sI − A) a.

Following the way multiplication in the Laplace transform domain was defined, for T > 0 let

 f(t), t < T, f (t) = T 0, t ≥ T.

Let yT (t) be defined by equations

x˙ T (t) = AxT (t) + BfT (t), xT (0) = 0, yT (t) = CxT (t) + DfT (t), obtained by replacing f with fT and x by xT in (9.5). Then y(t) = yT (t) + Cx0(t) for t < T . On the other hand, xT and fT (and hence yT ) admit Laplace transforms XT , FT , and YT . Due to an argument similar to the one used in computing X0(s), it follows that

sXT (s) = AXT (s) + BFT (s),YT (s) = CXT (s) + DFT (s).

Hence YT (s) = G(s)FT (s), which means that y − Cx0 = G f.

5 9.3 z-Transform An analog of the Laplace transform used in discrete time is the z- transform. Its use is less technically involved, as it does not require integration.

Definition 9.2 A discrete time signal f ∈ `m is said to admit z-transform if there exists h > 0 such that ∞ X h−t|f(t)| < ∞. (9.6) t=0 If r is the largest lower bound of such h, the area

def Dr = {z ∈ C : |z| > r} is called the region of convergence of z-transform of f, and the function F : Dr 7→ Cm defined by ∞ X z−tf(t) (9.7) t=0 is called the z-transform of f.

An analog of Theorem 9.1 for z-transforms is given below.

Theorem 9.4 Let F : Dr 7→ Cm be an analytical real symmetric function. The following conditions are equivalent:

(a) F is z-transform of a signal f ∈ `m;

(b) F is uniformly bounded in Dh for all h > r. Moreover, if conditions (a),(b) are satisfied then for every h > r:

Z π −t 1 jωt jω h f(t) = e F (he )dω for t ∈ ZZ+, (9.8) 2π −π 1 Z π 0 = ejωtF (hejω)dω for t = −1, −2,..., (9.9) 2π −π ∞ X 1 Z π |h−tf(t)|2dt = |F (hejω)|2dω. (9.10) 2π t=0 −π

Multiplication in the z-transform domain means in the time domain.

6 k×m Theorem 9.5 Let G : Dr 7→ C be a complex matrix-valued function which is the z-transform of a sequence g = g(t) of real k-by-m matrices, in the sense that

∞ X G(z) = z−tg(t) t=0 for |z| > r. Let t X y(t) = g(τ)f(t − τ). τ=0

For T ∈ ZZ+ let yT `k be defined as the signal with z-transform YT (z) = G(z)FT (z), where FT (z) is z-transform of  def f(t), t < T, f (t) = T 0, t ≥ T.

Then yT (t) = y(t) for T > t ≥ 0.

The notation y = G f and y = g ∗ f will be used to indicate that y ∈ `k is the result of multiplying f ∈ `m by a complex k-by-m matrix-valued function G in the z-transform domain, or, equivalently, the result of convolution of f with a real k-by-m matrix-valued k×m function g : ZZ+ 7→ IR . The following statement describes representation of discretre time LTI state space models in z-transform domain.

Theorem 9.6 Let A, B, C, D be real matrices of dimensions n-by-n, n-by-m, k-by-n, and k-by-m respectively. Then a pair (f, y) ∈ `m × `k satisfies equations

x(t + 1) = Ax(t) + Bf(t), y(t) = Cx(t) + Df(t) (9.11) for some x ∈ `n if and only if y can be represented in the form

y = G f + y0 = g ∗ f, where  D, t = 0, G(z) = D + C(zI − A)−1B, g(t) = CAt−1B, t > 0, and z-transform of y0 is −1 Y0(z) = C(zI − A) a n t−1 for some a ∈ IR , i.e y0(0) = 0 and y0(t) = CA a for t > 0.

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