Derivation of the Fourier Inversion Formula, Bochner s Theorem, and Herglotz s Theorem

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1 1 Derivatio of the Fourier Iversio Formula, Bocher s Theorem, ad Herglotz s Theorem Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 Respose of Liear Time-Ivariat Systems to Siusoidal Iputs 1 2 The Delta Fuctio 2 3 Summability Kerels 3 4 Fourier Iversio Theorems The L 2 Theory 8 5 Bocher s Theorem 9 6 Herglotz s Theorem The Discrete Bocher Theorem 12 Refereces 14 Idex 15 Abstract I Sectio 1 the Fourier trasform is show to arise aturally i the study of the respose of liear, time-ivariat systems to siusoidal iputs. I Sectio 2, the Dirac delta fuctio is itroduced. Sice the delta fuctio caot exist as a ordiary fuctio, it is approximated i Sectio 3 usig summability kerels. I Sectio 4, summability kerels are used to prove Fourier iversio theorems. I Sectio 5, positive-semidefiite fuctios are itroduced, ad a weak versio of Bocher s theorem is proved. This result is the used to motivate Bocher s theorem itself, which is proved usig results from probability theory about characteristic fuctios. Sectio 6 derives Herglotz s theorem, which is a versio of Bocher s theorem for positivesemidefiite sequeces. If you fid this writeup useful, or if you fid typos or mistakes, please let me kow at Joh.Guber@wisc.edu 1. Respose of Liear Time-Ivariat Systems to Siusoidal Iputs The Fourier trasform arises aturally whe cosiderig the respose of a liear, time-ivariat system to a siusoidal iput. If the system has impulse respose h,

2 2 the the respose to the complex siusoid x(t) = e jωt is the covolutio where h(τ)x(t τ)dτ = h(τ)e jω(t τ) dτ ( ) = h(τ)e jωτ dτ e jωt = H(ω)e jωt, H(ω) := h(τ)e jωτ dτ is the Fourier trasform of h. 12 We see the that system respose to e jωt is proportioal to e jωt. I other words, H(ω) is a eigevalue ad e jωt is the correspodig eigefuctio. ad 2. The Delta Fuctio Recall that the Dirac delta fuctio is defied by the two properties δ(t) = 0, t 0, (1) Sice δ(t) = 0 for t 0, we see that for ay fuctio g, Thus, δ(t)dt = 1. (2) g(t)δ(t) = g(0)δ(t), for all t. g(t)δ(t)dt = g(0)δ(t)dt = g(0) δ(t)dt = g(0), by (2). 1 We assume that the system uder cosideratio is stable; i.e., its impulse respose is itegrable i the sese that h(t) dt <. Sice h is itegrable ad sice x is bouded, it is clear that the product h(τ)x(t τ) is a itegrable fuctio of τ. Thus, the covolutio itegral is well defied. Similarly, the product h(τ)e jωτ is itegrable, ad the itegral for H(ω) is well defied. We also ote that H(ω) is bouded sice H(ω) h(τ)e jωτ dτ = h(τ) dτ <. 2 Give 1 p <, if h(t) p dt <, we say h L p ; hece, a impulse respose h is stable if h L 1. We say h has fiite eergy if h L 2.

3 3 It ow follows that the Fourier trasform of δ is (ω) := δ(t)e jωt dt = e jω 0 = Summability Kerels Sice the delta fuctio caot exist as a ordiary fuctio, we approximate it with a summability kerel. We say that a family of fuctios {δ (t)} is a summability kerel if each δ is such that δ (t)dt = 1; (3) there is a fiite costat M with δ (t) dt M, for all, (4) ad for all ν > 0, lim δ (t) dt = 0. (5) { t ν} Clearly (3) is equivalet to (2). A little thought shows that (5) is a attempt to approximate (1). If each δ (t) is oegative, the (4) is implied by (3) with M = 1. I geeral, (4) is just a techicality. Sice the Fourier trasform of the delta fuctio is oe, a great way to defie δ (t) is by iverse Fourier trasformig some (ω) such that (ω) 1. For example [2, Sec. 9.7], if (ω) := e ω /, the clearly (ω) 1 for all ω. It is also easy to calculate δ (t). Sice (ω) is real ad eve, δ (t) = 1 (ω)e jωt dω 2π = 1 (ω)cos(ωt)dω 2π = 1 π = 1 π 0 = Re 1 π 0 (ω)cos(ωt)dω e ω/ cos(ωt)dω 0 = /π 1 + (t) 2. e ω/ e jωt dω

4 4 We ow verify (3) (5). To establish (3), write δ (t)dt = 2 1 π (t) 2 dt = 2 1 π θ 2 dθ = 2 π ta 1 θ = 1. 0 Next, sice δ is oegative, (4) holds with M = 1. Fially, for (5), write ν /π 1 + (t) 2 dt = 1/π dθ 0 as. 1 + θ 2 ν We also ote that our particular kerel has oe additioal property. Observe that for t ν > 0, δ (t) = 1 2 /π 1 + (t) /π 2 t πν 2. Hece, where B ν := (πν 2 ) 1. δ (t) B ν, for t ν, (6) Theorem 1 (Siftig). Let δ be a summability kerel that also satisfies (6). If h is itegrable ad cotiuous at t, the Proof. Write By (3) we ca write ad so lim h(t τ)δ (τ)dτ = h(t). e := h(t τ)δ (τ)dτ h(t). h(t) = h(t)δ (τ)dτ, e = [h(t τ) h(t)]δ (τ)dτ. Sice h is cotiuous at t, for every ε > 0, there is a ν > 0 such that Now write e { τ <ν} τ < ν h(t τ) h(t) < ε. h(t τ) h(t) δ (τ) dτ + h(t τ) h(t) δ (τ) dτ. { τ ν}

5 5 Let I ad J deote these two itegrals. The I ε δ (τ) dτ ε δ (τ) dτ εm, { τ <ν} by (4). Next, J is upper bouded by h(t τ) δ (τ) dτ + h(t) δ (τ) dτ. (7) { τ ν} { τ ν} The itegral o the right is equal to h(t) δ (τ) dτ, { τ ν} which goes to zero by (5). Usig (6), the itegral o the left i (7) is upper bouded by B ν h(t τ) dτ B ν h(t τ) dτ = B ν h(θ) dθ. { τ ν} 4. Fourier Iversio Theorems We begi with the well-kow fact that covolutio i the time domai correspods to multiplicatio i the frequecy domai. Theorem 2 (Covolutio). If h is itegrable with Fourier trasform H, ad if G is itegrable with iverse Fourier trasform g, the h(t τ)g(τ)dτ = 1 H(ω)G(ω)e jωt dω. 2π Proof. 3 By a chage of variable, h(t τ)g(τ)dτ = h(θ)g(t θ)dθ. Sice g is a iverse trasform, g(t θ) = 1 G(ω)e jω(t θ) dω. 2π 3 Both itegrals i the Covolutio Theorem are well defied. Sice g is the iverse Fourier trasform of the itegrable fuctio G, g is bouded (recall footote 1). Sice h is itegrable so is the product h(t τ)g(τ). Similarly, sice H is bouded (by footote 1) ad G is itegrable, so is H(ω)G(ω)e jωt.

6 6 Substitutig this formula shows that the covolutio is equal to [ 1 ] h(θ) G(ω)e jω(t θ) dω dθ, 2π or 4 1 [ ] G(ω) h(θ)e jωθ dθ e jωt dω, 2π which is the desired result. Theorem 3. Uder the coditios of the Siftig Theorem, if δ is the iverse trasform of some itegrable, the 1 h(t) = lim H(ω) (ω)e jωt dω. (8) 2π Proof. Use the Siftig ad Covolutio Theorems. Theorem 4 (Iversio). If h is itegrable ad cotiuous at t, ad if its trasform H is also itegrable, the h(t) = 1 H(ω)e jωt dω. 2π Proof. We first apply the previous theorem to a bouded sequece (ω) 1. Sice H(ω) is itegrable, the domiated covergece theorem allows us to brig the limit iside the itegral i (8). The result follows because (ω) 1 for all ω. Example. Take h(t) = (1 t )I [ 1,1] (t). Clearly, h is cotiuous ad itegrable. Sice h is real valued ad eve, a easy calculatio shows that 1 ( siω H(ω) = 2 (1 t)cos(ωt)dt = 2 0 ω Sice H is also itegrable, ad sice h is cotiuous, h(t) = 1 ( siω ) 2e 2 jωt dω. 2π ω I particular, if we take t = 0 ad use the fact that h(0) = 1, we fid that ) 2. ( siω ) 2 dω = π. (9) ω 4 Chagig the order of itegratio is justified by the Toelli ad Fubii Theorems, ad uses the fact that both h ad G are itegrable.

7 7 Aother popular summability kerel is obtaied by takig It is the easy to see that ( (ω) := δ (t) = π 1 ω ) I [,] (ω). ( si(t) ) 2. O accout of (9), we see that (3) holds. Sice δ is oegative, (4) holds automatically with M = 1. It is also easy to see that (5) holds. Property (6) obviously holds with B ν = (πν 2 ) 1. By Theorem 3, if h is itegrable ad cotiuous at t, the 1 h(t) = lim 2π ( t 1 ω ) H(ω)e jωt dω. (10) Both of the summability kerels we have itroduced are oegative ad their trasforms satisfy 0 (ω) +1 (ω) 1. This fact ca be used to prove the followig useful result. Theorem 5 (Positive Iversio). If h is itegrable ad cotiuous at t = 0, ad if its trasform H is oegative, the H is itegrable. For all t at which h is cotiuous, h(t) = 1 H(ω)e jωt dω. 2π Proof. We ca apply Theorem 3 to either of our bouded, oegative summability kerels. Takig t = 0 i (8) yields 1 h(0) = lim H(ω) (ω)dω. 2π Sice the itegrads are oegative ad odecreasig, the mootoe covergece theorem allows us to brig the limit iside the itegral. Hece, h(0) = 1 H(ω) dω, 2π ad we see that H(ω) is itegrable. The, as argued i the proof of the Iversio Theorem, we ca brig the limit iside the itegral i (8) for arbitrary cotiuity poits t.

8 The L 2 Theory Lemma 6. If g L 2 L 1, the its autocorrelatio fuctio h(t) := g(t + τ)g(τ)dτ is a bouded, itegrable, cotiuous fuctio, ad the trasform of the autocorrelatio fuctio is H(ω) = G(ω) 2. Proof. Sice g L 2, we ca write h(t) = g(t + ),g. The by the Cauchy Schwarz iequality, h(t) g 2 2. To show that h L1, write 5 h(t) dt = A similar calculatio shows that [ H(ω) = g(τ) g(t + τ)g(τ)dτ dt g(t + τ)g(τ)dτ g(t + τ) dt dτ = g 2 1. ] e jωt dt = G(ω) 2. To establish the cotiuity of h at ay poit t, suppose that t t. The g(t + ) g(t + ) 2 0 by cotiuity of traslatio [2, p. 196, Th. 9.5]. By cotiuity of the ier product, g(t + ),g g(t + ),g ; i.e., h(t ) h(t). Sice the sequece t was arbitrary, h is cotiuous at t. Lemma 7. If g L 2 L 1, the G L 2 ad g 2 2 = G 2 2 2π. Proof. Apply the Positive Iversio Theorem with t = 0 to the autocorrelatio fuctio h of the precedig lemma, ad ote that h(0) = g 2 2 ad H(ω)dω = G(ω) 2 dω = G 2 2. For arbitrary g L 2, put g (t) := g(t)i [,] (t). The g L 2 L 1 by Hölder s iequality. Further, sice g g 2 0, g is Cauchy i L 2. By the precedig lemma, G G m 2 0 as,m ; i.e., G is Cauchy i L 2. Hece, there exists The material i this subsectio is ot eeded for Bocher s Theorem. 5 Chagig the order of itegratio is justified by Toelli s Theorem.

9 9 a G L 2 with G G 2 0. This limit G is the Fourier trasform of g L 2. Furthermore, 6 which is Parseval s equatio. g 2 = lim g 2 = 1 2π lim G 2 = 1 2π G 2, 5. Bocher s Theorem A fuctio h is positive semidefiite if for ay times t 1,...,t ad ay complex costats c 1,...,c, i=1 k=1 Takig = 1 ad c 1 = 1 shows that h(0) 0. c i h(t i t k )c k 0. (11) Lemma 8. If h is a cotiuous, positive-semidefiite fuctio, the for T > 0, T ( H T (ω) := 1 t T ) h(t)e jωt dt 0. (12) Proof. The proof cosists of two parts. We first show that for ay T > 0, T T h(t θ)e jω(t θ) dt dθ 0. (13) We the show that this double itegral is equal to 2T H 2T (ω). It follows that H T (ω) 0 for all T > 0. Sice h is cotiuous, the itegral i (13) is equal to a limit of fiite Riema sums of the form i h(t i t k )e jωt i t i e jωt k t k. k These sums are oegative because h is positive semidefiite. This establishes (13). It remais to simplify the double itegral. Write it as T I [,T ] (t)h(t θ)e jω(t θ) dt dθ. Make the chage of variable τ = t θ ad obtai T I [,T ] (τ + θ)h(τ)e jωτ dτ dθ. 6 By the triagle iequality, x + y x + y, it follows that x y x y. Hece, if if x x 0, the x x.

10 10 Now chage the order of itegratio to get T h(τ)e jωτ I [,T ] (τ + θ)dθ dτ. To evaluate the ier itegral with respect to θ, make the chage of variable s = θ. The the ier itegral becomes T I [,T ] (τ s)ds = I [,T ] (s)i [,T ] (τ s)ds. I other words, we have the covolutio of two rectagular pulses. Hece, { T 2T τ, τ 2T, I [,T ] (τ + θ)dθ = 0, otherwise. We have thus show that (13) is equal to 2T H 2T (ω). Theorem 9 (Bocher, weak versio). Let h be a itegrable, cotiuous, positivesemidefiite fuctio. The H is itegrable, ad H(ω) := h(t)e jωt dt 0, (14) h(t) = 1 H(ω)e jωt dω. (15) 2π Proof. Sice h is itegrable, i (12) we ca let T = ad apply the domiated covergece theorem to obtai (14). The itegrability of H ad (15) follow immediately from the Positive Iversio Theorem. To motivate Bocher s Theorem, observe that if h(0) > 0, the (15) ca be rewritte as h(t) h(0) = e jωt H(ω) 2πh(0) dω. The quotiet o the right is oegative. Takig t = 0 shows that it itegrates to oe, ad is therefore a probability desity. If we deote the correspodig cumulative distributio fuctio by F, the or, if K(ω) := 2πh(0)F(ω), h(t) h(0) = e jωt df(ω), h(t) = 1 e jωt dk(ω). 2π

11 11 Usually K is called the spectral distributio fuctio. Bocher s theorem asserts the existece of such a distributio fuctio eve if h is ot itegrable. Theorem 10 (Bocher). Let h be a cotiuous, positive-semidefiite fuctio with h(0) > 0. The there is a cumulative distributio fuctio F such that or, writig K(ω) := 2πh(0)F(ω), Proof. Let h(t) h(0) = e jωt df(ω), (16) h(t) = 1 e jωt dk(ω). 2π ( q T (t) := 1 t T so that the itegral i (12) ca be writte as ) I [,T ] (t) H T (ω) = q T (t)h(t)e jωt dt. Sice the product q T h is itegrable ad cotiuous ad has oegative trasform H T, the Positive Iversio Theorem tells us that H T is itegrable, ad which we ca rewrite as q T (t)h(t) = 1 H T (ω)e jωt dω, 2π q T (t)h(t) h(0) = e jωt H T (ω) 2πh(0) dω. We thus see that the left-had side is the characteristic fuctio of the probability desity H T (ω)/2πh(0). The left-had side coverges to the cotiuous limit h(t)/h(0). It is well-kow [1, pp , Corollary 1] that uder these coditios, there exists a limitig cumulative distributio fuctio F whose characteristic fuctio is the limit h(t)/h(0); i.e., (16) holds. Remark. From (16) we see that h( t) = h(t) ad h(t) h(0). I fact, these are implied directly by the positive semidefiiteess of h. Let Q deote the matrix with elemets h(t i t k ), ad let c = [c 1,...,c ]. The (11) says that c Qc 0 for all c. Takig c = x + λy ad cosiderig the cases λ = j ad λ = 1 shows that y Qx = y Q x for all x ad y. This implies that Q = Q. For the case = 2 with

12 12 t 1 = t ad t 2 = 0, Q = Q implies h( t) = h(t). Let c = [e jθ,e jϕ ]. The c Qc 0 is equivalet to h(0) [ e j(ϕ θ) h(t) + e j(ϕ θ) h( t) ] /2. Usig the fact that h( t) = h(t) ad writig h(t) = h(t) e jζ, where ζ = argh(t), we fid that h(0) cos(ϕ θ + ζ ). h(t) Sice ϕ ad θ are arbitrary, we must have h(0)/ h(t) 1, or h(0) h(t). 6. Herglotz s Theorem The Discrete Bocher Theorem If h() is a discrete-time positive semidefiite sequece, the we would like to write i aalogy with (14) ad (15), H(ω) = = h()e jω 0, (17) ad h() = 1 π H(ω)e jω dω. (18) 2π π The aalog of Bocher s Theorem would be h() = 1 π e jω dk(ω), (19) 2π π where K(ω) = 2πh(0)F(ω) for some cumulative distributio fuctio F. We begi by otig that a argumet aalogous to that i the proof of Lemma 8 shows that H N (ω) := Hece, if h is summable; i.e., if N = N ( 1 N + 1 h() <, = ) h()e jω 0. (20) the H N H i (17) ad so H is well defied ad oegative. Furthermore, the series i (17) is absolutely ad uiformly coverget. It follows that H is cotiuous ad therefore itegrable o ay fiite iterval. Thus, the itegral i (18) is well defied. Substitutig the series for H ito the itegral i (18) ad iterchagig the order of summatio ad itegratio (justified by the uiform covergece) shows that (18) holds.

13 13 To prove the discrete aalog of Bocher s Theorem is more puzzlig because we caot talk about the cotiuity of a discrete-time sequece. However, i the proof of Bocher s Theorem, all that the cotiuity of h(t) really bought us was the tightess of the sequece of cumulative distributios correspodig to the desities H T (ω)/2πh(0) [1, p. 304, Proof of Corollary 1]. Fortuately, the aalogous cumulative distributios i the discrete-time problem all live o a fiite iterval ad are therefore always tight. Theorem 11 (Herglotz). If h() is a positive semidefiite sequece with h(0) > 0, the there is a cumulative distributio fuctio F such that if K(ω) := 2πh(0)F(ω), the (19) holds. Proof. With H N defied i (20), it is easy to see that 7 for k N, ad i particular, 1 π ( H N (ω)e jωk dω = 1 k ) h(k) 2π π N π H N (ω)dω = h(0). 2π π Cosider the cumulative distributio fuctio ω H N (θ) F N (ω) := dθ, π ω π. π 2πh(0) For ω < π, put F N (ω) := 0 ad for ω > π, put F N (ω) := 1. Sice the correspodig desities are supported o a commo fiite iterval, the cumulative distributios are tight [1, p. 290], ad so there exists a subsequece N i ad a limitig cumulative distributio fuctio F such that for all bouded ad cotiuous fuctios g [1, pp , Theorem 25.8], lim g(ω)df Ni (ω) = g(ω)df(ω). i Sice F Ni (ω) = 0 for ω < π ad F Ni (ω) = 1 for ω > π, the limit F(ω) iherits these properties. Hece, the above formula reduces to π π lim g(ω)df Ni (ω) = g(ω)df(ω), i π π 7 Sice H N is defied by a fiite sum, we do ot eed a discrete-time versio of the Positive Iversio Theorem. Hece, o prelimiary results o Fourier series are eeded here.

14 14 ad we ca write π π Formula (19) ow follows. e jω df(ω) = lim π i π π = lim i π e jω df Ni (ω) ( = lim 1 i N i + 1 = h() h(0). Refereces e jω H N i (ω) 2πh(0) dω ) h() h(0) [1] P. Billigsley, Probability ad Measure. New York: Wiley, [2] W. Rudi, Real ad Complex Aalysis, 2d ed. New York: McGraw-Hill, 1974.

15 15 Bocher s theorem, 9, 11 Dirac delta fuctio, 2 eigefuctio, 2 eigevalue, 2 Fourier trasform of L 1 sigals, 2 of L 2 sigals, 9 Hergloz s theorem, 12, 13 impulse respose, 1 stable, 2 itegrable, 2 Parseval s equatio, 9 positive semidefiite, 9 spectral distributio fuctio, 11 stable (impulse respose), 2 summability kerel, 3 tightess, 13 Idex