The Poisson summation formula, the sampling theorem, and Dirac combs

Transcription

1 The Poisson summation ormula, the sampling theorem, and Dirac combs Jordan Bell Department o Mathematics, University o Toronto April 3, 24 Poisson summation ormula et S be the set o all ininitely dierentiable unctions on such that or all nonnegative integers m and n, ν m,n = sup x m n x x is inite For each m and n, ν m,n is a seminorm This is a countable collection o seminorms, and S is a Fréchet space One has to prove or v k S that i or each ixed m, n the sequence ν m,n k is a Cauchy sequence o numbers, then there exists some v S such that or each m, n we have ν m,n k I mention that S is a Fréchet space merely because it is satisying to give a structure to a set with which we are working: i I call this set Schwartz space I would like to know what type o space it is, in the same sense that an p space is a Banach space For S, deine ˆξ = e iξx xdx, and then x = e ixξ ˆξdξ et T = /Z For φ C T, deine and then ˆφn = φt = e int φtdt, ˆφne int

2 For S and, deine φ : T C by φ t = j Z t + j, where x = x We have φ C T For n Z, φ n = One checks that ξ = ˆ e int j Z t + jdt = e int t + jdt j Z = e inx xdx = n ξ, so φ n = ˆ n Thus or t T, φ t = ˆ n e int Thereore, i S then or each t T, j Z t + j = ˆ n e int, where x = x This is the Poisson summation ormula 2 Nyquist-Shannon sampling theorem et S Suppose that there is some such that x = i x > namely, is spacelimited We can in act choose so that all terms with j on the let-hand side o the Poisson summation ormula are zero, which then expresses t in terms o discrete samples o the Fourier transorm o I there is an such that ˆξ = or ξ >, we say that is bandlimited For : T C this correspond to being a trigonometric polynomial We can prove a dual Poisson ormula and apply it to bandlimited unctions, but instead I am going to give a direct argument ollowing Charles Epstein, Introduction to the Mathematics o Medical Imaging, second ed, which is a good reerence or sampling Suppose that S and ξ = or ξ > For ξ, deine In act, i, ˆ and is bandlimited then it ollows that is ininitely dierentiable, although it does not ollow that is Schwartz 2

3 F : T C by 2 For n Z, F ξ = ˆ ξ F n = = = 2 e inξ F ξdξ e inξ ˆ ξ dξ t in e ˆtdt On the other hand, or x, using the act that is bandlimited, Thereore or n Z, x = e ixξ ˆξdξ = e ixξ ˆξdξ As F C T, or t T we have Hence or ξ, F t = ˆξ = F F n = n F ne int = ξ = n n For η <, let φ : satisy { ξ η ˆφξ = ξ > e int e inξ For instance, or η = let ˆφ be the characteristic unction on [, ] But i η > we can choose ˆη to be smooth rather than a characteristic unction I ˆξ = or ξ > η rather than just or ξ >, then ˆ = ˆ ˆφ, because φ 2 We are deining a periodic unction F rom a unction ˆ that is not periodic, but rather which becomes ater 3

4 is equal to on the support o ˆ Then or x, x = e ixξ ˆxi ˆφξdξ = e ixξ n e inξ ˆφξdξ = n inξ ixξ+ e ˆφξdξ 2 = n φ x + n 2 = n φ x + n ˆφ can be chosen so that φ x + n quickly goes to as n This means that ewer terms o the above series need to be calculated to get a good approximation o x For ˆφ the characteristic unction on [, ], we have where sincx = sin x x or φx = e ixξ dξ = sinx x Then we obtain, or x, x = n x = n = sincx, sinc x + n, sinc x n Thus i is bandlimited, we can sample at a sequence o points and using these values reconstruct x or any x This act is the Nyquist-Shannon sampling theorem, and the actual ormula above is the Whittaker-Shannon interpolation ormula A pleasant exposition o sampling is given in Sampling: What Nyquist DidnÕt Say, and What to Do About It, by Tim Wescott, 2 3 Dirac combs, pulse trains Deine a tempered distribution X on by Xt = δt n This is called a Dirac comb or pulse train I S, X = n 4

5 The convolution o and X is the unction X t = t n Deine X T t = T X t T Because δt/a = a δt the scaling property o the Dirac delta distribution; we have an absolute value sign because i a is negative then doing a change o variables in the deinition o δt/a we get two negative signs, i T > then X T t = T δ t T n = δt nt, and we have and X T = nt X T t = t nt X T t + T = X T t, so X T is a periodic distribution on It ollows that X T is a distribution on T Folland talks about periodic distributions around p 298 o his eal Analysis, second ed et γ Cc such that γxdx =, and let ω = γ χ [, I we have a -periodic distribution F on and ψ C T, we deine F ψ to be F ωψ ψ is viewed in this instance as a -periodic unction on For ψ C T we have, viewing ψ as a -periodic unction on thus ψ2n = ψ, X ωψ = δt 2nωψ = ω2nψ2n = ψ γxχ [, 2n xdx = ψ γx χ [, 2n xdx = ψ γx = ψ Thus as a distribution on T, or ψ C T we have X ψ = ψ For n Z, X n = X e int = Thus the Fourier series o X is e int 5