Lecture 13: Applications o Fourier transorms (Recipes, Chapter 13 There are many applications o FT, some o which involve the convolution theorem (Recipes 13.1: The convolution o h(t and r(t is deined by s(t = h* r = dt h(t r(t t = dt h(t r(t t It is trivial to show that S( = [h* r](t exp (πit dt = H( R( 93
Instrumental response unction A classic example o a convolution results rom the inite resolution o a measuring instrument. Example: a spectrograph has inite wavelength resolution, which smears the observed spectrum: even a delta unction is smeared out to a eature o inite width Flux λ 94
Instrumental response unction The instrumental response is characterized by a resolution unction r, which describes how an intrinsically very narrow unction will be broadened The unction r is usually peaked at zero r 0 95
Convolution o a signal with a resolution unction Suppose we have discretely sampled our resolution unction r(t and our signal h(t to obtain r k and h k Suppose h is periodic and has been sampled at N points: k = 0, N 1 r is usually sampled (almost symmetrically about 0: r k = r(kδt with k = ½N+1,, 0,, ½N 96
Convolution o a signal with a resolution unction The convolution can then be written N / (h* r k = r j h k j j= 1 N / To avoid aliasing, r j must also be periodic with period N r j = r j+n so we can shit the indexing rom k = ½N+1,, 0,, ½N to k = 0 to N 1 97
Convolution o a signal with a resolution unction The reordering changes the r k rom to 0 0 N and the convolution becomes N 1 (h* r k = r j h k j j= 0 98
Convolution o a signal with a resolution unction The convolution theorem tells us that the discrete Fourier transorm o s k = (h * r k is S n = H n R n suggesting that we can compute the convolution by obtaining the discrete FTs o h and r, multiplying them, and taking the inverse Fourier transorm o the product 99
Computational expense o convolution Thanks to the Fast Fourier Transorm (FFT, this can save a lot o computational expense N 1 Direct calculation o (h* r k = r j h k j takes N j= 0 operations 300
Computational expense o convolution Calculation with FFT method takes N log N operations or H n N log N operations or R n N operations to multiply H n and R n N log N operations to take the inverse FT o S n or a total o N(1+3 log N operations.. 301
Computational expense o convolution. which can yield a huge savings e.g. or N = 10 5 : N = x 10 10 N(1+3 log N = 5 x 10 6 (although note that i the resolution unction is narrowly peaked, r k may have only M non-zero values, where M << N, and the direct calculation only costs NM 30
Deconvolution A key application, o course, is the deconvolution: i.e. given the measurements s k, and knowing the resolution unction r k, we estimate the signal h k A direct solution o s k = r j h k j h, requires O(N 3 j 0 operations, given s and (or O(N M in the limit where r k has only M nonzero elements N 1 = 303
Deconvolution Using FFT, the procedure entails Compute S n : Compute R n : Compute H n = S n /R n : Inverse FT o H n : N log N operations N log N operations N operations N log N operations again or a total o N(1+3 log N operations.. 304
Problems with deconvolution Two key issues arise: 1 R n (and S n may be zero (or very small at some requencies so that H n is undeined (smearing leads to unavoidable loss o inormation at high requencies Noise can enter and spoil the precision 305
Eect o noise Instead o measuring a perect convolved signal, s(t = h * r, we actually measure c(t = s(t + n(t, where n(t is the noise at time t How do we extract a good estimate o h in the presence o noise? 306
Filtering (Recipes 13. We can attempt to correct or the presence o noise by multiplying C( by a ilter unction, Φ(, which we will take as real: i.e. we write our best estimate o H( ~ H ( = C( Φ( / R( [instead o H( = S( / R(] Q: what ilter unction yields the best estimate o H(? 307
308 Filtering The optimal ilter unction is chosen to minimize the mean square deviation d R N S R S d H H dt t h t h P MSD ( ] ( ( [ ( ( ( ( ~ ( ( ~ ( + Φ = = =
Filtering This becomes P MSD 1 Φ( S( = R( + Φ( N( ] d plus terms involving N*(S(, which average to zero i the noise is uncorrelated with the signal 309
Wiener iltering The mean square deviation is minimized or 0 = dp MSD / dφ = [1 Φ( ] S( R( + Φ( N( ] d Φ( = S( S( + N( 310
Wiener iltering O course, S( is not directly measurable All we measure is c(t = s(t + n(t, or which the Fourier transorm is C( I the signal and the noise are uncorrelated, the power spectrum o c(t is C( = S( + N( = S( + N( 311
Power spectrum o c(t The power spectrum o the real data then looks like this: 31
Power spectrum o c(t rom which Φ( = S( / ( S( + N( can oten be inerred Alternatively, N( can be characterized independently in a separate measurement where the signal is absent 313