Notes on the Fourier Transform

Transcription

1 Noes on he Fourier Transform The Fourier ransform is a mahemaical mehod for describing a coninuous funcion as a series of sine and cosine funcions. The Fourier Transform is produced by applying a series of "Tes Frequencies" o deermine he conribuion of each es funcion o he observed signal. This documen will ouline he fundamenals of how he Fourier Transform (FT) works. For addiional deails, a series of mahcad documens designed as homework exercises are available. As an example, sar wih a 2 Hz cosine signal wih an ampliude of. This signal is "aquired" by a compuer using an A-D (analog o digial) board in he compuer. For his experimen he compuer samples he signal once every.953 ms for a oal of ond. This is a sampling rae of 52 Hz and i gives a daa file wih 52 poins. Sample and signal parameers: Signal frequency in Hz. ν signal ( 2Hz ) Signal frequency in radians per ond. ω signal 2 π ν signal Signal ampliude Time beween poins (Dwell Time) A DT.953 ms ms 3 Toal acquisiion Calculaed Values from Above: Sampling Rae Number of poins (floor is a funcion ha reurns an iniger, used since you can' have a fracion of a measuremen) rae DT rae = Hz N N = 52 floor( rae ) Equaion for he signal a and o define he of each poin. signal( ) A cos ω signal, N.. Signal Waveform signal This is jus a cosine wave wih an ampliude of and a frequency of 2 Hz. noice ha here are wo complee cycles in he ond ha daa is aquired. Remember his curve is made from 52 poins.

2 Now ha we have a signal, he fourier ransform can deermine he frequency componens in he signal. This is done by applying a series of es frequencies. For each es frequency, he signal is muliplied by he es frequency o produce a new waveform. The inegraion of he new (produc) waveform is he signal a ha es frequency Firs le's use a Hz es frequency o deermine he inensiy of he Hz componen in he signal. ν es Hz ω es 2 π ν es Generae he es wave: es( ) cos ω es Tes Waveforms.5 es Tes and Signal Waveforms signal es

3 Nex he produc wave is produced by muliplying he wo waveforms (signal and es) Muliply he wo waveforms, poin by poin: produc( ) es( ) signal() Produc Waveform produc Inegrae he produc funcion o deermine he area of he wave. This is he inensiy of he signal a Hz (he es frequency): Analyically (wih calculus) Numerically (add he poins) produc() d = N i = produc i N N = Also inigrae he funcion by looking closely a i. Noice ha all he posiive areas (above ) and he negaive areas (below zero) cancel. This resul should make sense. The es frequency was Hz, bu he signal was 2 Hz. There is no signal a Hz.

4 Now le's ry i again wih a 2 Hz es frequency. (Wha do you expec he answer o be?) ν es 2Hz ω es 2 π ν es Generae he es wave: es( ) cos ω es Tes and Signal Waveforms signal es Nex he produc wave is produced by muliplying he wo waveforms (signal and es) Muliply he wo waveforms, poin by poin: produc( ) es( ) signal() Produc Waveform produc Inegrae he produc funcion o deermine he area of he wave. This is he inensiy of he signal a 2 Hz (he es frequency): Analyically (wih calculus) Numerically (add he poins) produc() d =.5 N i = produc i N N =.5 Also inigrae he funcion by looking closely a i. Noice ha he enire wave is above so nohing cancles his. The produc waveform inigraes o a posiive value. This should make sense since he signal and he es frequency are boh 2 Hz.

5 Finally le's ry i again wih a 3 Hz es frequency. (Wha do you expec he answer o be?) ν es 3Hz ω es 2 π ν es Generae he es wave: es( ) cos ω es Tes and Signal Waveforms signal es Nex he produc wave is produced by muliplying he wo waveforms (signal and es) Muliply he wo waveforms, poin by poin: produc( ) es( ) signal() Produc Waveform produc Inegrae he produc funcion o deermine he area of he wave. This is he inensiy of he signal a 3 Hz (he es frequency): Analyically (wih calculus) Numerically (add he poins) produc() d = N i = produc i N N = Also inigrae he funcion by looking closely a i. Noice ha once again he negaive and posiive porions of he waveform all cancel. This should make sense since he signal is a 2 Hz and he es frequency is 3 Hz. In a "real" experimen he signal will conain MANY differen frequencies. All a differen inensiies. The FT can handel his jus fine.

6 Nex we will examine he idea of "phase". The orignal "heory" of he FT is ha any coninuous funcion may be described as a series of cosine and sine waves. The cosine par is called he "real" and he sine par is called he "imaginary". This is analogous o real and imaginary numbers used in mahemaics. This concep will be very imporan for FT-NMR so a bi of deail is relevan here. (Noe i is also imporan in FT-IR, bu you do no usually realize i). A Hz cosine and sine waves looks like his: ν Hz ω 2 π ν Hz waveform cos( ω) sin( ω) The cosine wave may also be shown vs he angle (A Hz wave jus goes around he circle once each ond). This angle is in radians or degrees. angle,... 2 π cycle cos( angle) π 2 π sin( angle) angle rad The wave may also be displayed wih he angle in degrees. Cycle cos( angle) 9 8 sin( angle) angle deg

7 Noice ha he sine and cosine waves are separaed by a 9 degree (or π/2 radian) "phase angle. Wach wha happens o a sine wave if a π/2 degree phase angle is added o i. cycle sin angle+ π 2 π 2 π angle rad Now i is he same as a cosine wave. So a sine and cosine wave are separaed by 9 degrees. If you subrac 9 degrees from a cosine wave i will look jus like a sine wave. Nex we can relae his o he phase of a signal waveform. If he signal waveform is a pure cosine wave (like he signal we used above). The signal is "real". Alernaively, if he signal is a pure sine wave i is an "imaginary" signal. NOTE: These are jus names, hey do no imply anyhing abou he "realiy" of he signal. We can show his by now using wo es waves. One cosine and one sine. Wach wha happens when we look a a Hz "real" signal, wih Hz cosine and Hz sine es waves. Generae he waveforms: signal( ) cos 2 π Hz es real ( ) cos 2 π Hz es im ( ) sin 2 π Hz Tes and Signal Waveforms signal es real es im

8 Nex he produc wave is produced by muliplying he wo waveforms (signal and es) produc real ( ) es real ( ) signal() produc im ( ) es im ( ) signal() Produc Waveforms produc real.5 produc im Inegrae he produc funcion o deermine he "real" and "imaginary" signal. Inegrae he real signal: Inegrae he imaginary signal: produc real () d =.5 produc im () d = This resul is consisen wih wha you expec. Since he signal is a pure cosine wave i should be all "real". I also is consisen wih visual inspecion of he produc waveforms.

9 Nex le's see wha happens if he signal is a pure sine wave. Generae he waveforms: signal( ) sin 2 π Hz es real ( ) cos 2 π Hz es im ( ) sin 2 π Hz Tes and Signal Waveforms signal es real es im Nex he produc wave is produced by muliplying he wo waveforms (signal and es) produc real ( ) es real ( ) signal() produc im ( ) es im ( ) signal() Produc Waveforms produc real.5 produc im Inegrae he produc funcion o deermine he "real" and "imaginary" signal. Inegrae he real signal: Inegrae he imaginary signal: produc real () d = produc im () d =.5 This resul is consisen wih wha you expec. Since he signal is a pure sine wave i should be all "imaginary". I also is consisen wih visual inspecion of he produc waveforms.

10 Nex wha happens if he signal has a phase beween (cosine or real) and 9 (sine or imginary). Generae he waveforms: signal( ) cos 2 π Hz es real ( ) cos 2 π Hz es im ( ) sin 2 π Hz + π 4 A 45 degree phase angle Tes and Signal Waveforms signal es real es im Nex he produc wave is produced by muliplying he wo waveforms (signal and es) produc real ( ) es real ( ) signal() produc im ( ) es im ( ) signal() Produc Waveforms produc real produc im Inegrae he produc funcion o deermine he "real" and "imaginary" signal. Inegrae he real signal: produc real () d =.354 Inegrae he imaginary signal: produc im () d =.354 This resul is consisen wih wha you expec. Since he signal is midway beween a sine and a cosine wave. The inensiy of he real signal is.5 cos( 45 deg) =.354 and he inensiy of he imaginary signal is.5 sin( 45 deg) =.354. The negaive sign comes from he angle relaive o he cosine and sine es waves (he signal is ahead of one and behind he oher.

11 One more angle jus o be cerain everyhing fis ogeher. Generae he waveforms: signal( ) cos 2 π Hz es real ( ) cos 2 π Hz es im ( ) sin 2 π Hz + π 6 A 3 degree phase angle Tes and Signal Waveforms signal es real es im Nex he produc wave is produced by muliplying he wo waveforms (signal and es) produc real ( ) es real ( ) signal() produc im ( ) es im ( ) signal() Produc Waveforms produc real produc im Inegrae he produc funcion o deermine he "real" and "imaginary" signal. Inegrae he real signal: produc real () d =.433 Inegrae he imaginary signal: produc im () d =.25 This resul is consisen wih wha you expec. Since he signal is midway beween a sine and a cosine wave. The inensiy of he real signal is.5 cos( 3 deg) =.433 and he inensiy of he imaginary signal is.5 sin( 3 deg) =.25. The negaive sign comes from he angle relaive o he cosine and sine es waves (he signal is ahead of one and behind he oher.

12 This documen was developed by: Sco E. Van Bramer Deparmen of Chemisry Widener Universiy Cheser, PA 93 hp://science.widener.edu/~svanbram