Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every seconds so will a plo of he echocardiogram hus, we can inerpree an echocardiogram as a plo of a cerain periodic funcion of wih period Our goal is develop a mahemaically meaningful way of characerizing such a funcion, in a way ha would aid he diagnosis of hear paiens Our mahemaical saring poin will be Fourier heory We do no need o know oo much abou Fourier heory in order o exploi i he fundamenal principle of his heory is ha every coninuous period funcion wih period can be represened in he following form (1 f ( a + where he coeffi ciens a, a 1,, and b 1, b, are deermined by he following inegral formulas f ( d, n, 1,, b n f ( d, n 1,, 3, Noe ha, osensibly, in order o compue he coeffi ciens a i and b i you need o know wha he funcion f (x is (oherwise, how would you calculae he inegrals? On he oher hand, as I will show below, he mere exisence of formula (1 acually provides an effecive means of compuing he coeffi ciens Now an expansion of he form (1 is called a Fourier expansion (or, a lile more specifically, a Fourier e-ine expansion, and he various rignomeric funcions ha appear are referred o as he Fourier modes and heir coeffi cien a i or b i aached o a specific mode is referred o as he ampliude of ha mode his language comes from a paricular applicaion of Fourier heory in which he Fourier modes are used o describe he modes of a vibraing rod 1 In wha follows, we shall assume ha once we reach he erms wih n N, he remaining erms in (an oherwise infinie Fourier series are insignifican In our paricular medical applicaion, his assumpion seems naural, as i seems likely ha he oscillaions of a hear can no aain arbirarily high frequencies Bu more pracically, we shall assume ( f ( a N + because we need o reain only a finie number of erms in order o keep he problem of deermining he coeffi ciens racable So henceforh we shall assume ha he periodic funcion represening a paricular echocardiograph is of he form ( (say wih N around or so Our problem is o deermine a means o deermine he N + 1 coeffi ciens a, a 1,, a N, b 1, b,, b N from an echocardiograph graph he basic idea we ll pursue is simple, we have N + 1 unknowns, so we need N + 1 equaions o solve for hem Such equaions are easily had Divide he inerval [, ] up ino N + 1 subinervals; ug n n N + 1, n,, N + 1 1 However, in he heory of vibraions one ofen uses a slighly differen choice of Fourier modes For example, he naural modes of a vibraing rod of lengh where boh of is endpoins are held fixed are πnx L 1
as he endpoins of he subinervals Nex, evaluae boh sides of formula ( a each n f ( a N + f ( 1 a N 1 + 1 f ( N a N N + f ( N+1 a N N+1 N + N+1 he erms ha appear of he lef hand side, f (, f ( 1,, can be deermined by looking a he echocardiograph graph and idenifying he verical y" coordinae of he graph when he horizonal "x" coordinae is a paricular i he erms of righ hand side are jus linear sums of he unknowns" a,, a N, b 1, b N wih cerain numerical coeffi ciens obained by evaluaing he various Fourier mode funcions a, 1,, N+1 Now if you re really saring meiculously a sysems, you migh noice ha we acually have a sysem of N + equaions in N + 1 unknowns However, ha las equaion, for f ( N+1 is acually redundan wih he firs his is because we are assuming f ( is periodic wih period hus, he lef hand sides of he firs and las equaion are he same, f ( f ( f f ( N+1 ( f ( N+1 f (N + 1 f f ( (N + 1 and, he righ hand sides also evaluae o idenical expressions: a N + N a N+1 + N+1 a N a N ( (N + 1 + ( (N + 1 a N So if we drop he equaion corresponding o N+1 we ll have as many equaions as unknowns and so presumably a unique soluion he cavea I m hedging here is ha i is possible o wrie down a se of N + 1 equaions in N + 1 knowns ha has no soluion However, his does no happen curren siuaion because he Fourier modes are known o provide a se of linearly independen funcions; his in urn assures us ha we have a se of N + 1 independen equaions in N + 1 unknowns and so a unique soluion exiss he problem hen becomes, how can we solve he following linear sysem
3 f ( a N + f ( 1 a N 1 + 1 f ( N N + N S I Le me firs ge rid of some of he cluer in our Fourier e and ine funcions We have ( (3 j j N + 1 N + 1 j and, similarly, ( (4 I claim (5 (6 j N + 1 j N + 1 j m 1,, N { N + 1 if m N + 1 j if m and ha (5 and (6 lead o hree more ideniies (7 N + 1 j N + 1 j (8 (9 { N+1 if m n if m n πj N + 1 n N + 1 n i, j {,, N} N + 1 j N + 1 j Problem 1 Prove hese ideniies (5 - (9 N + 1 if m n N+1 if m n if m n Remark 1 he sums over j, 1,, N corresponds o a sum over he daa poins", 1,, N in he inerval [, he inegers m, n, on he oher hand, are used o label paricular Fourier mode funcions: ( n h Fourier e funcion ( n h Fourier ine funcion he n h Fourier e and ine funcions repea hemselves exacly n imes in he inerval [, and correspond o n h order harmonics of he fundamenal e and ine funcions
4 3 D C, b n Suppose we have chosen a se of N + 1 equally disribued poins, 1,, N+1 along he horizonal axis of our echocardiograph, and hen for each of hese j we found he corresponding value f j of echocardiagraph reading: We seek o find he coeffi ciens a,, a N, b 1,, b N so ha he following equaions are saisfied f j f ( j a N j + j, j, 1,, N or, via (3 and (4, (1 f j a N N + 1 j + N + 1 j, j, 1,, N
5 Le s now fix a paricular ( m,, N creae a whole series of equaions by sysemaically muliplying he above equaion by N+1, j leing j run from o N : f N + 1 ( f 1 N + 1 (1 f j N + 1 (j a N + 1 ( + a N + 1 (1 + a N + 1 (j + N + 1 ( N + 1 ( + N + 1 (1 N + 1 (1 + N + 1 j N + 1 (j + N + 1 ( N + 1 (1 N + 1 (j f N N + 1 (N a N + 1 (N + Now le s add up hese equaions, summing over j We ll have (31 f j N + 1 j a + + N + 1 (j N b n N + 1 j N + 1 (N + N + 1 j N + 1 j ( N + 1 j N + 1 j Noe ha for each m, he lef hand side of (11 explicily ug he daa poins ( j, f j on he echocardiograph Le s look now examine equaion (11 more carefully When m, we ll have, on he lef hand side, f j N + 1 j f j while, he righ hand side will be a sum of he following hree erms a N + 1 (j a N + 1 j N + 1 j 1 a (N + 1 ( by (6 N + 1 j ( N + 1 (N
6 and N ( N + 1 j N + 1 j And so, when m, equaion (11 reduces o N ( N + 1 j b n ( by (5 or f j a (1 a N + 1 (N + 1 f j hus, he coeffi cien a can be deermined direcly from he graph daa {( j, f j j,, N} Now suppose m While he lef hand side of (11 doesn simplify, he righ hand side does For when m a N + 1 (j a ( N + 1 j N + 1 j hus, for m, equaion (11 reduces o or (13 a m N ( N + 1 j N + 1 j N + 1 f j N + 1 j f j N + 1 j N + 1 δ m,n N + 1 a m b n (, m 1,, N N + 1 a m Noe ha equaion (13 now allows us o compue he coeffi ciens a 1,, a N direcly from he echocardiagraph daa I remains o deermine he coeffi ciens b 1,, b N Our mehod will be very similar o wha we did above We( fix an m 1,, N, generae N + 1 equaions from (1 by muliplying boh sides of (1 by facors N+1, j j,, N, and hen sum hese equaions over j from o N his produces f j N + 1 j a N + 1 j + N + 1 j N + 1 j + N b n ( ( N + 1 j ( N + 1 j
7 By virue of he ideniies (5, (7 and (8, he equaion above reduces o N + 1 f j N + 1 j + + b m hus, (14 b m N + 1 f j N + 1 j, m 1,, N Equaions (1, (13, and (14 hus allow us o deermine all he Fourier coeffi ciens of he funcion f ( ha corresponding o our echocardiograph