Fourier Analysis. f () t = + cos[5 t] + cos[10 t] + sin[5 t] + sin[10 t] x10 Pa

Transcription

1 Fourier Aalysis I our Mathematics classes, we have bee taught that complicated uctios ca ote be represeted as a log series o terms whose sum closely approximates the actual uctio. aylor series is oe very powerul applicatio o this idea. I the case o aylor series, the uctio is approximated by a costat value o the uctio at a particular poit added to successive derivatives evaluated at that same poit ad multiplied by speciic costats or coeiciets. Aother type o series is the Fourier series. Here speciic costats are multiplied by sie ad cosie terms to geerate the series that approximates the uctio. As a example, cosider the ollowig series o ive terms that represet the oscillatig pressure sesed by a hypothetical detector as a soud passes by: 5 () t = + cos[5 t] + cos[ t] + si[5 t] + si[ t] x Pa Notice some thigs about this series. he irst term is a costat, sometimes called the DC term usig a aalogy to electrical voltages ad currets. he secod ad third are cosie terms. 5 he agular requecy o the secod term is 5 rad/sec ad the amplitude is /3 x Pa. he third term has twice the agular requecy so it oscillates twice as ast, but has ad amplitude o 5 oly /4 x Pa. he ourth ad ith terms have the same requecy ad amplitude as the secod ad third but are shited i phase by 9 degrees relative to the cosies. Whe plotted or 5 sec, this series looks like this: 5 (t) x Pa t (sec) Most ote i experimetal acoustics, we have a detector to receive a sigal like this oe ad it is our purpose to work backwards ad determie the requecies ad the amplitudes o the toes (terms i the series) that make up the periodic sigal. he method o idig these toes is called Fourier Aalysis. Fidig the requecies is simply a matter o determiig the overall period o the repeatig sigal. he udametal requecy, or requecy o the irst sie or cosie term i the series (i Hertz), is simply the reciprocal o that requecy. Higher requecy terms are just multiples or harmoics o the udametal requecy. Geerally the requecy is give i rad/sec istead o Hz. Fidig the coeiciets or amplitude o each term occurs usig a very clever bit o mathematics discovered by Fourier. his method is sometimes called Fourier s Hammer because it is used 7-

2 to hammer out each o the coeiciets (amplitudes) i the series. We ll study this method i some detail below. I act, may souds are combiatios o discrete requecy compoets that we hear as oe soud. I class, we will use spectrum aalyzers ad digital oscilloscopes which use digital sigal processig algorithms to id the magitude (proportioal to the Fourier Series Coeiciet) ad requecy o each compoet o a sigal. Calculatig Coeiciets Startig with a periodic uctio (such as a soud wave), we ca breakdow this uctio ito separate requecy compoets by usig Fourier Aalysis. Note that we must KNOW the period o the wave ad BE ABLE O DEFINE the uctio, (t), over that period to be able to use Fourier Aalysis. Ote the uctio will be zero, a costat, or a straight lie with costat slope. Whatever it is, we must be able to write a math expressio (or a good approximatio) or the uctio over the etire period. First let us be very speciic about the requecy i rad/sec. Oce we have idetiied the period over witch the uctio repeats, the agular requecy is: ω= I the example plot o the periodic uctio above, the period is approximately.5 sec by ispectio o the time scale. his is cosistet with the equatio we plotted sice rad ω= = 5rad / sec..5sec Other terms i the Fourier series will have requecies that are multiples o 5 rad/sec, e.g. rad/sec, 5 rad/sec, rad/sec,.. Calculatig the amplitudes is somewhat more complicated. First cosider the equatio we plotted above (where I have dropped the uits ad costat 5) : () t = + cos[5 t] + cos[ t] + si[5 t] + si[ t] Eve though we kow the amplitude o the irst cosie term is /3, let s try to develop a method to umask it. First, multiply each term by cos(5t). t cos 5t cos 5t + cos[5 t]cos 5t + cos[ t]cos 5t si[5 t]cos( 5t ) + si[ t]cos( 5t) 3 4 () ( ) = ( ) ( ) ( ) 7-

3 Next, we id the time average o each term i the series usig the ormal deiitio or the time average o a uctio. his is a reasoable approach because we are lookig or a represetative value or the amplitude averaged over at least oe cycle, ot a istataeous value. () t = () t dt he result looks complicated ad log but will quickly simpliy. () t cos( 5t) dt cos( 5t) dt + cos[5 t]cos( 5t) dt + cos[ t]cos( 5t) dt = si[5 t]cos ( 5t) dt + si[ t]cos( 5t) dt 3 4 A quick ispectio o the let side o the equal sig reveals that most o the terms itegrate to zero. I act all but oe term are zero sice, si ωt cos mω tdt = si ωt si mω tdt = cos ωt cos mω tdt = uless m=. I that case, (sie would be idetical) cos t cos m tdt cos tdt ω ω = ω = his leaves us with the ollowig: () t cos( 5t) dt + cos [5 t]dt+++ = = 3 3 Rearragig slightly shows that the coeiciet we were tryig to id, i.e. the /3, must be calculated as ollows: = () t cos ( 5t ) dt = a 3 he ame we will give to this coeiciet is a. We arbitrarily decide to call all the coeiciets or cosie terms a ad or sie terms b. he subscript tells us which harmoic o the udametal requecy the coeiciet is associated with. I this case, = is the udametal term. Hopeully you see that this approach ca be used to id ay coeiciet (ay value o a or b ). All we have to do is multiply the series by either cosωt or siωt ad time average the result. Sice most o the terms average to zero, the result ca be summarized i the ollowig set o rules. I truth, idig Fourier coeiciets ca be a very mechaical procedure that you ca perorm simply by learig these rules. Let us start with ay time varyig sigal, (t). I (t) is periodic over the iterval t, it ca be broke dow ito a series o requecy compoets (coeiciets) where: 7-3

4 ω= the coeiciets are calculated by: a = () t cos( ω t) dt or =,,, 3,... = () t si ( ω t) dt or =,, 3,... Note that goes rom to or a but goes rom to or b. hat is because there is o b term. he si o (ωt) where = is always, thus b is always. he coeicets a, a, a,, ad b, b, b 3, are the Fourier coeiciets o the uctio, (t). Now the origial uctio (t), ca be described as the summatio o may dieret sie ad cosie uctios. () t = a + acos( ω t) + a cos( ω t) + a3cos( 3ω t) + + b si ω t + b si ω t + b si 3ω t + or, ( ) ( ) ( ) 3 a t a cos t b si t = () = + ( ω ) + ( ω ) Example whe < t < Give the periodic uctio : () t = whe < t < which repeats every secods. A sketch o the uctio would look like: (t) t (sec) - he uctio ca be expaded ito a series o sie ad cosie terms that whe added together, replicate the origial uctio. It is our job to id the coeiciets o those terms. 7-4

5 First we must idetiy the period o the repeatig uctio. Hopeully it is obvious that = secods. From this we id the agular requecy, ω. rad ω= = = rad / sec. sec his is a coveiet result sice the agular requecy o harmoic terms is just ω = rad/sec. he coeiciets are the oud as ollows. Notice that we break the itegral up ito pieces where the uctio has two dieret costat values, zero ad oe. a = () t cos( t) dt = cos( t) dt + cos( t) dt a = cos t dt = si x ( ) ( ) a = si( ) si( ) = ad = () t si ( t) dt = si ( t) dt + si ( t) dt = si t dt = cos x ( ) ( ) = cos( ) cos( ) = ( cos( )) or = odd umbers otherwise = b = or = odd umbers ad a = () t cos( *t) dt = *dt *dt + a = dt = thus, the origial uctio ca be expaded to: ( ) ( ) ( ) a si t si 3t si 5t () t = + a cos( t) si( t )... ω + ω = =

6 I we added up all the terms o the Fourier Expasio, a graphical represetatio would look like this: (t) Fourier Aalysis.5.5 a a+b a+b+b3 a+b+b3+b5 a+b+b3+b5+b7 a+b+b3+b5+b7+b9 -.5 time he importat thig to ote is that the origial square wave uctio ca be composed rom addig compoets o multiple sie ad cosie uctios with requecies that are multiples o the base requecy. he base requecy o the compoets is the same as the base requecy o the square wave. Odd or Eve Fuctios By lookig at the orm o the iput sigal, (t), we ca come up with some shortcut rules or derivig the coeiciets. I we ca determie i the (t) is a odd or eve uctio, we ca determie whether the a or b coeiciets are equal to zero as i the last example. A uctio is odd or eve based o the ollowig: Eve Fuctio : Odd Fuctio : (- t) = ( t) (- t) (t = ) Eve uctios are thus uctios that are symmetric about the y-axis. Odd uctios are uctios that are symmetric about the x-axis AND are mirror images o each other (symmetric about the origi). May uctios are either odd or eve, but uderstadig this characteristic uctio type lets us aticipate which Fourier coeiciets might be zero. 7-6

7 Some samples o eve ad odd uctios. A odd uctio (t) = si (ωt) A eve uctio (t) = cos(ωt) A odd uctio (t) = t 7-7

8 A eve uctio (t) = t Sice cosies are eve, other eve uctios are made up oly o cosies. O the other had, odd uctios are made up oly o sies. hus the coeiciets or the dieret type uctios are: a = I ( x ) is odd the = () t si ( ω t) dt or =,, 3,... () I t is eve the a = () t cos( t) dt or,,, 3,... ω = = Remember, some uctios are either eve or odd i which case you must simply calculate all the Fourier coeiciets ad see what results are obtaied. 7-8

9 Problems. Give the ollowig pressure uctio, p(t), which ca be described as a square wave o Pa or /3 sec, ad Pa or /3 sec show below where = sec : P(t) Pa t(s) /3 /3 4/3 5/3 Pa, < t < 3 pt () = Pa, < t < 3 a) Is this uctio odd, eve, both, or either? How do you kow? b) What is the base or udametal requecy o the square wave? c) Perorm the itegratios to calculate the coeiciet, a o. d) Perorm the itegratios to calculate the coeiciet, a coeiciets. e) Perorm the itegratios to determie the b coeiciets. 7-9

10 ) Fill out the ollowig table or 9: a (Pa) b (Pa) = / (sec) = / (Hz) N/A N/A N/A g) What is the patter here? List the requecies o the irst ie o-zero harmoics o the udametal that go make up the irst ie terms o the Fourier Expasio. 7-

11 Lesso 7 Fourier Series Periodic Fuctios a () t = a cos( t) si( t ) or, + ω + ω = () t = a + acos( ω t) + acos( ω t) + + b si ω t + b si ω t + or a uctio = () t where : ω = the coeiciets are calculated by : a = () t cos( ωt) dt or =,,, 3,... () t si( ωt) dt or =,, 3,... ( ) ( ) () t Example whe < t < = whe < t < (t) t - rad Note: = sec ω= = sec Coeiciets Example a = () t cos( t) dt = cos( t) dt + cos( t) dt a = cos( t) dt = si( x) a = [ si( ) si( )] = ime Domai Frequecy Domai Fourier rasorm o a Square Wave = () t si ( t) dt = si ( t) dt + si ( t ) dt = si ( t) dt = cos( x) = cos( ) cos( ) = ( cos( )) or = odd umbers otherwise = b = or = odd umbers (t) - () t whe < t < = whe < t < t Amplitude Frequecy (rad/s) si () ( t) si( 3t) si( 5t) t = Demos Mathematica Logger Pro Odd ad Eve Fuctios Eve Fuctio : Odd Fuctio : (- t) = ( t) (- t) = ( t) Eve Odd

12 Lesso 7 Odd ad Eve Fuctios Eve Eve Fuctio : Odd Fuctio : (- t) = ( t) (- t) = ( t) Odd a = () t cos( ωt) dt or =,,, 3,... I () t is eve the b = a = I () t is odd the = () t si ( ω t) dt or =,, 3,...