As I studied it converts a time domain into a frequency domain. What does this really mean?
Here is an oversimplification from NMR. Consider a slowly spinning top that has 1 red dot on the side. As the top spins the red dot slowly comes into view until it is right in front of you then slowly goes out of view. We can consider the amount of the red dot in view as the signal. So as the top spins you get a wave (signal over time). This can be transformed into a frequency (how fast is the top spinning). Now if you have several tops all spinning at different speeds, The signal will be the sum of all the tops. That signal can still be decomposed into individual frequencies and this is what the FT does.
@ Julio da Silva , Sir I don't have advance math in my background but still I want to know about this please help if possible. Thank you very much.
In it's simplicity, when applied to crystallography, it allows the transformation of structural factors (as recipricoal space) into the electron density equation p(x,y,z) (as real space). Why?
To explain it as a pipeline: a crystal (real space) is x-ray diffracted to produce diffraction Intensities on a detector plate along with a recipricoal space "pattern " of these intensities. Thus nature does a "natural FT" of the crystal into so-called recipricoal space diffraction spots". So how do we extract information from these spots to produce a chemical model of the crystal?
Well, form these spots we can produce a periodic function (just like time is a periodic function as you mentioned) of the crystal as a complex 3D wave equation known as the Structural Factor equation that incorporates the AMPLITUDE of the wave as intensities, the FREQUENCY of the wave as hkl Miller indices of of the xyz positions of the atoms as part of Euler's exponent coefficients, but the position (PHASE) of the wave is missing from the spots (thus the so called "wave problem").
Thus now we bring Joseph Fourier back to life from the 18th century, because the FT allows us to decompose any periodic function into the sum of an infinite set of frequency functions as sines and cosines (thanks to Euler's Formula). The "..calculations are less complicated .." now as Julio mentions above. Thus now we can do a "back FT" of the Structural Factor Equation (recipricoal space) into the electron density equation, p(x,y,z), as back into REAL space coordinates of the crystals' atoms. We can now add the missing PHASES back into (via MR, experimental phases, etc) the p(x,y,z) equation, and finally graph this equation as a 3D "mesh" as in COOT model building program. Finally we can now build an atom by atom model into this 3D mesh via simple pattern recognition.
Note, any reffereal by me of the word "simple" is an understatement that can only be measured by "light years". Of course none of this probably makes sense, until you finally can "follow the math". To start read "Crystallography Made Crystal Clear" by Gale Rhodes, then brush up on linear algebra, some calculus, advanced statistics ( Bayesian, another theist back back to life from the 18th century), and finally jump, "dive into if you can swim", into Bernhard Rupp's "Biomolecular Crystallography" . Note: It took me a number of years to finally "grasp" a solid understanding of this fascinating and challenging field of crystallography (note reffered by some as "XTREME SCIENCE').
My way to understand, considering a signal : in the "time domain" you will pick a value every t time, if t is small enough you can plot your data and see (for example) the signal as a sinusoid ; in the "frequency domain" you can see the signal as a composition of simple entities that *represent* sinusoid.
It almost like in one hand you have (0,0.48,0.84,0.99,0.90) and in the other and "sin".
In a very, very simple way, Fourier Transform (FT) means: a generic wave can be seen as a superposition of many sinoidal waves and each sinoidal has a frequency. Using FT, we obtain a spectral analysis, i.e., all the sinoidal frequencies. If we have only one sinoidal wave, we obtain a unique result, i.e., the value of wave frequency.
Here is an oversimplification from NMR. Consider a slowly spinning top that has 1 red dot on the side. As the top spins the red dot slowly comes into view until it is right in front of you then slowly goes out of view. We can consider the amount of the red dot in view as the signal. So as the top spins you get a wave (signal over time). This can be transformed into a frequency (how fast is the top spinning). Now if you have several tops all spinning at different speeds, The signal will be the sum of all the tops. That signal can still be decomposed into individual frequencies and this is what the FT does.
It is really Euler's formula (complex exponential function) that is the beauty of why FT "works so to say", because now you can treat any sinusoidal wave as "these sines and cosines". Referring to the nice spinning top analogy by Andrew, now you can capture the red dot as the "cosine part as frequency and amplitude. In X-ray diffraction, you can't capture the phase (sine part) .
I'd like to add that a crystal is not really sinusoidal (it doesn't vary over time), although it is periodic in the "repetitive meaning sense". Thus the sinusoidal pattern (Time domain) in the diffraction pattern spots comes from the x-rays ,which as an electromagnetic wave is sinusoidal, in combination with the periodic "repetitive" unit cells of the crystal.
Of cours you can capture the phase from "special" anomalous (heavy atom) diffraction differences.
Thanks a lot Dr. Wes. Now it is somehow understandable to me. It is quite valuable information you gave ! Thank you very much.
When I introduce Fourier Transform to Under graduates, not much interested in Maths how do I start
Fourier transform measures the frequency content in a signal. For example, if we apply FT to a signal f about the point "10" the result will give us information about the quantity of components vibrating at the frequency of 10 in our signal.
- Badges
- Science topic
- Similar topics
- Bioinformatics
- Bioinformatics and Computational Biology