Introduction
The principles behind the operation of a nuclear magnetic
resonance spectrometer and an infrared spectrophotometer were fairly easy to
explain to students fifteen years ago. The IR instrument was based on
em-radiation dispersion, much like the light prism demonstrations students had
seen all their lives. The sources and detectors were different from those used
in the visible region, but students could fit these devices into their largely
analog instrument paradigm without much difficulty. Their comprehension of
continuous wave NMR was probably more vague than that of IR devices because
chemistry students often have a meager knowledge of electronics. This prevents
them from understanding radio frequency generation and absorption circuitry and
from really understanding how the magnetic field in which the sample is
immersed can be swept linearly over a small range.
Still, if they accept that such circuitry can
exist, they can make analogies that are reasonably valid to the visible
absorption instruments;. thus, they can understand a continuous wave NMR as a
low frequency (and thus with closely spaced energy levels) absorption
instrument. The fact that these energy levels don’t exist without the magnet
creating them is a little hard to understand, but NMR can be understood
sufficiently to allow its use for solving chemical problems.
And Then There Was FFT, The Fast Fourier Transform
As NMR and IR instruments became FT-IR and
FT-NMR instruments, the understanding of these important chemical tools became
vague to nonexistent. Furthermore, the task of the teacher to explain them
became more difficult. The audio-frequency analogy presented here is offered as
a plausible technique for presenting these difficult ideas to chemistry
students.
Session 1: The Analogy
I have chosen to present my analogy as a dialog between
professor and student. The student in this scenario is a chemistry major,
junior or senior, who has an understanding of the electromagnetic spectrum,
dispersive visible UV and IR instrumentation, and some experience with analog
and digital audio (audio cassette versus CD music). The professor is, of
course, myself, sitting at a PC running audio spectral analysis software. I use
SpectraPlus FFT [1], but any similar software package will do.
Professor: Let’s talk about sound waves first. To
capture a specific example, would you please say, “moo!” into this microphone
that is connected to the computer’s sound card?
Student: moo!
Professor: Thank you. As you see on the screen,
the computer recorded that sound and is now displaying it as a squiggly-looking
wave. Think about this wave. It is a plot of sound amplitude versus time. The
amplitude, or loudness, of the sound is an arbitrary scale, but the time is
listed on the x axis as about half a second. The computer deceives us a
little bit with this graph. It looks like a continuous line (a voltage) varying
up and down as the sound varies. It looks, in other words, like an analog
signal.
It was an analog signal when it came from the microphone
into the sound card. But, the sound-card electronics immediately converted this
voltage variation with time into a bunch of numbers; that is, it sampled the
wave many times per second faster than the length of one wave. In this case, as
you can see here below the graph, the sampling rate was 11,025 Hz. So, this
sound has been converted by the microphone to a voltage variation, one that is
analogous to the pressure fluctuations in the air (analog signal). Then the
computer replaced this analog signal with a digital signal. The digital signal
is really just a list. This list consists of two columns of numbers, one the
amplitude at one instant, the other the time at that instant. Then, to make it
seem analog to our eyes, the computer has drawn a line between the points.
Let’s ask the computer to show us the data points in the vicinity of 0.4
seconds.
We can also expand the plot and see this region close up
(Figure 3).
I think you can see that the signal is now digital, just
an array of numbers. This is the important first step to using the FFT
technique to find spectral information. If you can’t digitize the squiggles,
you can’t use FFT. Remember that the frequencies that make up this squiggle of
moo! are audio frequencies. Most of them are frequencies from 100 to 5000 Hz.
It would be nice if we could do this digitizing to infrared frequencies or even
optical and ultraviolet light frequencies, but we can’t do this directly
because we don’t have
Figure 1. What moo! looks like to the computer
Figure 2. The digital nature of the power
versus time recording.
Figure 3. Expanded portion of Figure 1 to show
digital segments.
Figure 4. The spectrum of moo!
transducers (like the microphone here) to convert
the light waves into voltage variations. Another problem is that the computer
cannot sample fast enough to digitize such incredibly high frequencies; so, as
I shall show you later, we must do sneaky tricks to make FT-IR and FT-NMR
instruments.
Student: I see that this audio moo! has been
digitized, but what has that got to do with Fourier transforms?
Professor: We are almost there. Be patient. Let
me ask you to guess what the frequency of this “boo!” Signal is.
Student: I don’t understand. How am I to know?
Professor: Look at this close-up we just
generated. If there was a complete wave up and down and back to zero that you
could find, you can estimate the time for that cycle. The reciprocal of the
time for one cycle is the frequency.
Student: O.K. It looks like one cycle takes
about two milliseconds. That's 1/(2 ´ 10–3) equals 0.5 kHz; so, I guess 500 for
one of the frequencies.
Professor: Excellent! Now, look at some of the
faster squiggles and find another frequency.
Student: Well, there seems to be a kind of cycle
of about 0.0004 seconds between successive peaks. That would be 0.4
milliseconds, reciprocal about 2.5 kilosomethings. Oh, yes. 2.5 kHz. Am I
right?
Professor: It sounds reasonable. What you can now
appreciate is that this complex squiggle is made up of a mixture of lots of
frequencies generated by your larynx. What the FFT can do is what you just
tried to do, find these frequencies and their contribution to the overall
squiggle.
Let’s ask the computer to do what I asked you to do, guess
the mixture of frequencies that make up this squiggle. Incidentally, the
computer can do it faster than you can. We select the whole squiggle like
selecting a word in word processing: right click, choose average spectrum, and
behold!
Notice that there is
an approximately 500-Hz component (actually 429 Hz) and a 1280-Hz peak
corresponding to your second guess. I guess that makes you a SAAFT, a slow and
approximate Fourier transform.
Student: Very funny, but how does the computer
do it?
Professor: It does it by using some mathematical
methods that have been known for over a century, but before we get into that
(skip ahead to Session 4 if you can’t wait), I have something else to show you
about graphing our moo! data. You have seen that this data contains information
about the amplitude, the frequency, and the time. I am going to ask the program
to display this data as a three-dimensional graph for us. Of course, it can’t
really make a three-dimensional graph on a flat screen, but it will try very
hard, and we can use our imaginations for the perspective. Watch this!
On this graph, the frequency is obviously the left to
right axis. The amplitude (actually, the power) is the vertical axis, and time
is the axis perpendicular to these. Think of the time axis as an arrow starting
at the upper-left blue portion and extending to the lower-left corner of the
graph. The time, 51 ms, is at the back of the representation, and time, 462
milliseconds, is at the lower-left front.
Now, any plane parallel to the frequency–power axis will
be a spectrum plot of the frequencies present at that moment in time. A person
looking at this from the front (power-versus-frequency side) would see a series
of spectra, which, if averaged, would be the spectrum we displayed earlier
(Figure 4). A person standing to the left side of this 3-D plot would see a
series of power-versus-time plots, which, if averaged, would be the squiggle
plot we began with (Figure 1).
Student: It’s hard to imagine, but I guess so.
Professor: The viewer on the left is seeing the
signal “in the time domain,” as we say in the industry. The viewer in the
Figure 5. A three dimensional graph of the two
domains
front is seeing the signal “in the frequency
domain.” The FFT operation switches you from one view to the other.
Student: I have heard those terms before, but
they never made sense; but, why are there so many slices of the frequency
domain, not all the same?
Professor: That is because these are audio
frequencies, quite low frequencies. Our computers have become so fast that they
can do an FFT operation in a very short time. During the 411-ms duration of
this moo!, the computer can do over 100 FFT operations. These are stacked one
behind the other in the 3-D chart. When I showed you the frequency domain graph
before (the spectrum), I had asked the program to average the spectrum over the
time of the moo. The spectrum is actually different during different moments of
the moo.
Student: I suppose this speed of processing,
impressive at audio frequencies, is not going to be fast enough when we extend
this to infrared frequencies!
Professor: Very true! The sampling rate of the
computer has to be at least twice the frequency of the highest frequency in the
spectrum. Because infrared frequencies are about 1013 Hz, we can’t
do the digitizing anywhere near that fast; so, we resort to a trick to move the
IR frequencies down to a workable range. In the case of the radio frequencies
of NMR (90–500 MHz) we do, still, a different trick. But, let’s take a break
and look at FFT for these regions later.
Student: I’ll have milk instead of coffee...to
improve my moos.
Session 2: FT-IR
Professor: How was your milk?
Student: Moo!
Professor: Excellent. Now, for the infrared trick
I spoke about earlier. What do you know about radios and how they work?
Student: Radio waves come into them from
somewhere far away, and the radio circuits convert these frequencies to rock
music.
Professor: Close, but the point I want to make
here has to do with frequency conversion. Give me the frequency of your
favorite FM station.
Student: One oh three rock, of course.
Professor: I’ll try to avoid that region of the
dial, but that means 103 MHz is the carrier frequency that enters the radio
antenna coil. This frequency is not so great that it cannot be amplified and
manipulated by modern transistorized circuits, but, for reasons we won’t go
into, FM radios convert this frequency to 10.7 MHz by a circuit called a mixer
circuit. Then the rest of the radio manipulates this approximately
ten-times-lower frequency. The lower frequency (approximately 10 MHz) “carries
on its back,” so to speak, all the audio information that was carried by the
103 MHz signal from the station. The mixer circuit just has shifted it to a
lower and more easily managed signal. Are you with me?
Student: I’ve been to a few mixers myself.
Professor: Hmmm. The point is that the infrared
being too high a frequency to digitize needs to be converted to a lower, more
manageable frequency so that we can perform the Fourier transform mathematics
on it. Still, we can’t do the conversion electronically because we don’t have
the proper infrared transducer, so the trick is to use an interferometer.
This device causes the infrared waves to reflect back upon
themselves in a way that makes them either interfere with themselves
constructively (bigger wave) or destructively (smaller wave). One of the
mirrors that do the reflecting of the waves is moveable. If this mirror is
mechanically moved back and forth at a known rate, the result is that the
detector detects infrared frequencies varying in time at a much lower rate than
their true variations. The amplitude fluctuations of the detector signal are proportional to what the detector
would see if it could follow the variations of infrared itself. The factor by
which the frequencies are mixed and shifted down are much greater than the
ten-times shift of the FM radio we mentioned earlier. The interferometer shifts
the frequencies down by a factor of 1010. This shifts them into the
audio range. From there, they are analyzed by Fourier transform software
exactly as we have analyzed the moo! signal.
Student: So the “trick” you mentioned is to
shift the frequencies using an interferometer so that IR becomes audio. Then
use FFT to convert this time-domain signal to a frequency-domain signal. The
spectrum will be, I suppose, just like the spectrum obtained using an older
dispersive IR instrument?
Professor: Yes, very much like the old IR
spectrum, but better.
Student: Why better?
Professor: For two reasons: One, the energy in
the IR beam is never spread out. The full IR energy falls on the sample at all
times. Even more importantly, the time-domain signal is repeated over and over
many times per minute. Each of the generated time-domain signals (squiggles)
presumably would give identical spectra. Because of noise, they won’t be
identical, but it is simple for a fast computer to average the data from many
squiggles and print out a time-averaged spectrum. This improves the
signal-to-noise ratio of the spectrum by a factor of the square root of the
number of repetitions that are averaged; thus, FT-IR spectra are (in general)
superior to older IR spectra.
Student: I’m convinced. I’ll buy one. How does
their cost compare with the older dispersive models?
Professor: In 1985 they cost ten times more.
Today they are about the same price, and tomorrow they will, no doubt, be even
lower. The dispersive IR instruments will become part of the history of
chemistry. But if you ever buy a new FFT one be sure it has a helium–neon laser
in it.
Student: A red light laser? Why. Can it analyze
laser light? Hey, the laser would only be one frequency. A pretty dull spectrum.
Figure 6. A typical free-induction-decay
signal.
Figure 7.
The FID after being transformed to a spectrum.
Professor: The laser is used to provide a single
spectral line from the plotted graph. This spectral line is used to make sure
each spectral scan begins at the same instant in time. Otherwise, without a
reproducible standard, the spectrum of one squiggle might shift relative to the
spectrum of the next. If this were to happen, the time averaging would be of no
use.
Student: Hey! They could make a computer with a
hole in the top of it. You would just pour in a sample and see the spectrum on
the screen!
Professor: Sure, go home and try it, but be sure
to come back for FT-NMR, the next big trick.
Session 3: FT-NMR
Student: I poured some ethanol into my computer
at home and no spectrum appeared. There goes my patentable idea.
Professor: Where did you get the ethanol? Never
mind. I’m sorry I asked. Are you ready for FT-NMR?
Student: Ready, but I already know the answer.
You shift the frequencies of NMR from the radio range down to audio using a
mixer. Then do regular moo! FFT stuff.
Professor: That is certainly a reasonable guess,
but not quite right. You will recall that the energy levels involved in NMR are
very closely spaced energy levels? And further, that this means that their
relative populations (protons in the upper level/protons in the lower level)
are only slightly less than 1. In IR work, the energy levels are much further
apart, and at room temperature the upper levels are significantly less
populated than the lower levels. When you do continuous wave NMR experiments it
is very easy to apply too much rf power to the sample. If you do, you
“saturate” the sample with rf (radio-frequency) energy. That means that the
upper-level and lower-level populations can be so similar that no absorption
(or emission) takes place. In continuous wave (old-fashioned) NMR, you try to
avoid this by keeping the rf power low.
But, in FTNMR you intentionally saturate the sample by irradiating
it with a rather high-energy pulse. This pulse of radiation contains all the
frequencies across the NMR spectrum of interest. At the end of this short
pulse, the nuclei in the higher energy levels begin to relax back to their
former state; that is, more (but not much more) are in the lower state than in
the upper.
What eventually comes out of this decay of radio
frequencies is an audio-frequency squiggle that is called the “free induction
decay.” Here is an example:
When an FFT is performed on this free induction decay, the
result is a spectrum display.
Student: Wait. The free-induction decay does not
look like our audio moo sample. Why not?
Professor: You are very observant. It does look
different because of the exponential decay shape of the audio frequencies. What
are exponentially decaying are really the radio frequencies that were contained
in the pulse applied to the sample coil. What you see in the FID is not this
decaying radio frequency, but the audio component that modulates it. You remember
that after 103 Rock was converted to 10.7 MHz in your FM receiver that you
didn’t listen to this 10.7 MHz signal. You can’t hear 10.7 MHz. You listened to
the audio signal that was “riding on its back,” as we put it at the time. This
rock music audio signal was placed on the back of (modulated) the original
103-MHz signal at the radio station studio. It also modulated the shifted
frequency, 10.7 MHz. In your radio it was extracted from the radio signal by a
circuit called a detector. After that point, it was audio, amplified (probably
too much) by the audio amplifier circuits, and then transduced into sound by
your speaker system.
In a similar manner, the relaxation processes of the
nuclear magnets modulate the radio frequency decay after the pulse ends. This
radio frequency goes to a detector circuit and comes out as an audio decay, the
FID.
Student: So these FID things are just like the
*.wav files of audio sounds?
Professor: Not exactly. They are audio files, but
are in a different format than *.wav files like our moo!; therefore, this audio
analyzer we have been using will not accept them. You have to have an audio
analyzer made to handle the file type produced by your NMR instrument. In the
above FID the instrument was a Bruker, and the file format is called Bruker Aspect 2000. Fortunately, you can
download a free FFT analyzer for these and other FID files from the Web. One
such program is called MestRe-C 2.2, and is available at the URL in reference
2.
Student: So the FID can be recorded on a disk, just
like my moo! and taken to anyone’s computer to be converted into a spectrum by
FFT, integrated, and so forth?
Professor: Exactly, the FT-NMR (and its sample)
can be in one city, and the FID transmitted on the Internet to a chemist with
only a computer and the required software to analyze it.
Student: This FT stuff is cool! How did you old
folks ever get any spectra before FFT came along?
Professor: Like a porcupine makes love. Very
carefully!
Session 4: The Mathematics of FFT
Professor:You asked me earlier
howthe computer does FFT. The best way to
give you a feeling for what the computer is doing during a fast Fourier
transform is to let you make a spreadsheet to do the same thing on a small
scale. Jean Baptiste Joseph Fourier was a contemporary of Napoleon, and you may
appreciate there were no spreadsheets at that time. If you want mathematical
rigor you may wish to consult the library, but if you want a quick and dirty
FFT spreadsheet we can do it here.
Student: I want quick and dirty.
Professor: What spreadsheet are you familiar
with?
Student: I use Lotus 123, but I know most of the
other students use Excel.
Professor: Either will work, as would Quattro.
Let’s use Excel for the pop-up notes. They can be a big help in documenting the
spreadsheet. In the A column we need a time base. Fill the first column from
zero to 2.49 in 0.005 increments, first labeling it “time.” This will be the x
axis of the audio signal that we wish to analyze by FFT. In B1, enter “freq 1.”
In the B2 cell we place a cosine wave by entering =$AD$7*COS($AD$2*$A2). This
requires a little explaining.
Student: Yes please.
Professor: The $AD$7 is an absolute reference to
a cell far to the right into which cell we shall enter the amplitude value for
our first spectral peak. For example, go to cell AD7 and enter 1.
Student: Got it.
Professor: As you may have guessed, $AD$2 is a
place to enter the frequency of this
cosine wave. In AD2 enter 1000. The final $A2 is a partially absolute reference
to the time in column A. If we copy this column to other places the A will
stay, but the 2 will increment down the time axis, as we want.
Student: I save periodically. What do I call
this spreadsheet?
Professor: Call it “crudefft.” Now create
two more columns for input frequencies. You can copy down and over from column
B, then change the $AD to $AE in column C and $AD to $AF in column D. This
means that the user can pick three frequencies and three amplitudes for the
simulated spectrum. As a start, place 1100 and 11900 for the second and third
frequencies, and 2 and 0.5 for the amplitudes. These three frequencies you just
generated are shown in Figure 1 on the Excel tab. Finally, make column E be the
sum of B2+C2+D2 and copy down. Column E is now our simulated moo! See Figure 2
on the Excel tab. It is a mixture of three frequencies at three amplitudes.
Plotted versus time (column A), it is a signal in the time domain that is to be
analyzed by our crude FFT program.
Now we shall answer the original question, “how does FFT
do it?” We shall make this spreadsheet transfer this time-domain signal into a
frequency-domain signal. That is to say, the spreadsheet will figure out what
frequencies are hidden inside the “sum” signal. Ready?
Student: Ready.
Professor: Label each column from F through AB as
frequencies from 980 through 1200 in steps of 10.
Student: I’m beginning to see. The computer is
going to try each of these frequencies and see if it is there?
Professor: Absolutely. And the way it will tell
if that frequency is there is by integration. In cell F2, place the
formula =COS(980*$A2).
Student: But that will simply generate a cosine
wave of frequency 980.
Professor: Then insert E2* before the COS. This
will multiply the instantaneous value for the 0.98 signal times the time-domain
“sum” signal at each time value.
Student: OK, I copied it down the column. Now
what?
Professor: Total this column.
Student: If you say so. Hey, the total is small.
Minus 1.38.
Professor: So you have guessed a frequency that
was not included in our choice of three. Now you have multiplied one function
(the sum function) by another function (the guess frequency) for each small
increment of time, then summed the results. Does that sound like any
mathematical operation you ever heard of?
Student: Sure. It sounds like an integration.
Professor: Go to the head of the class!
Student: But I’m the only one in the class.
Professor: Then guess what’s next.
Student: I do this for all the frequencies
listed on Row 1. When they are summed, there will be something different about
the correct frequency sums to distinguish them from the other sums.
Student finishes the job at all listed frequencies.
Professor: Finally, plot the spectrum as a plot
of integral sums versus frequency. Then, you may change the frequencies and
amplitudes at will and watch the frequency domain graphs change accordingly.
Student: Enthusiastically.
I’m placing the graphs of spectra just under the table where you enter
frequencies and amplitudes. Hey, this is cool. Did you invent this?
Professor: This spreadsheet idea is not original.
I found it as an example problem in Atkin’s Physical Chemistry, 4th edition
[3]. It is not a sophisticated FFT program, but does clearly indicate what the
computer must be doing. What do you think?
Student: I think, yea, yea, yea, yea!
To the interested student:
There is an interesting and practical article in The Chemical Educator, Vol. 1, Issue4 (September 30, 1996) about FTIR [4].
Also in the Journal of Chemical Education,
there is anarticle about FFT that
explains “aliasing,” sometimes called “folded peaks” [5].
There are also many books in your library
about FT-NMR, a recent one being reference [6].
Supporting Material: Readers will be
able to download the spreadsheet as a Microsoft Excel file called crudefft.xls
(http://dx.doi.org/10.1007/s00897990427b).
The sound file moo.wav can also be downloaded should you wish to hear what is
being analyzed (http://dx.doi.org/10.1007/
s00897990427c).
References and Notes
1.
Spectra Plus, Pioneer Hill Software, 24460 Mason Rd., Poulsbo,
WA 98370, USA, phone: 360-697-3472.
2.
MestRe-C, version 2.2 is made available through the work
of the Departmento de Quimica Organica,
Facultad de Quimica, Universidad de Santiogo de Compostela, 1706 Santioago de
Compostela, SPAIN. http://qobrue.usc.es/jsgroup/MestRe-C/MestRe-C.html
(accessed October 2000).
3.
Atkins, P. W. Physical
Chemistry, 4th ed.; W. H. Freeman: New York, NY, 1990; pp 462–463.
4.
Bezoari, M. D. Chem.
Educator, [Online] 1996, 1(4); DOI: 10.1007/s00897960044a.
5.
Steel, C.; Joy, T.; Clune, T. J. Chem. Educ. 1990 ,67, 883–887.
6.
Smith B. C. Fundamentals of Fourier Transform Infrared
Spectroscopy; CRC Press, 1996.
The
Chemical Educator, Vol. 5, No. 6,
Published on Web 11/1/2000, 10.1007/s00897000427a, © 2000 Springer-Verlag
New York, Inc.