Fakesmile

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I'm learning about it in an advanced biochem protein course and have a test on it in less than 4 days. The instructor assigned readings but I don't understand them at all and I've never been so frustrated. I feel those readings assume certain high level of math/spectroscopy backgrounds that I lack (Eg. the way that NMR is talked about in those papers is not like the NMR that I know and learned from orgo and other previous related courses.) I really need to understand Fourier transform. Any great resources that are comprehensible and don't assume advanced backgrounds in math, spectroscopy, etc. would be appreciated.
 

LuciusVorenus

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Do you actually have to CALCULATE transforms? I doubt they would require that of you unless linear algebra/diff equations was a prereq for this course. Anyways, if I remember my math right (math people correct me on this :D) just think of transforms like this:

You have an equation in one variable (time) and you want to convert it to another variable (frequency). So you do a whole bunch of calc/algebra and end up with a new equation in your desired domain.

The reason this is useful in NMR is you don't want a graph of time vs. intensity, you want a graph of chemical shift vs. intensity

Here's a good picture:


As you see the NMR "machine" is actually just recording intensity vs. time and so it has to do a Fourier transform to get it into the chemical shift vs. intensity form we all know and love from o chem :p
 
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Do you actually have to CALCULATE transforms? I doubt they would require that of you unless linear algebra/diff equations was a prereq for this course. Anyways, if I remember by math right (math people correct me on this :D) just think of transforms like this:

You have an equation in one variable (time) and you want to convert it to another variable (frequency). So you do a whole bunch of calc/algebra and end up with a new equation in your desired domain.

The reason this is useful in NMR is you don't want a graph of time vs. intensity, you want a graph of chemical shift vs. intensity

Here's a good picture:


As you see the NMR "machine" is actually just recording intensity vs. time and so it has to do a Fourier transform to get it into the chemical shift vs. intensity form we all know and love from o chem :p

Awesome job explaining it Lucius.
This video: http://www.youtube.com/watch?v=ObklYbQaX24
also gives a good explanation of it at a basic match level using lots of graphics.
good luck dude!
 

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I'm learning about it in an advanced biochem protein course and have a test on it in less than 4 days. The instructor assigned readings but I don't understand them at all and I've never been so frustrated. I feel those readings assume certain high level of math/spectroscopy backgrounds that I lack (Eg. the way that NMR is talked about in those papers is not like the NMR that I know and learned from orgo and other previous related courses.) I really need to understand Fourier transform. Any great resources that are comprehensible and don't assume advanced backgrounds in math, spectroscopy, etc. would be appreciated.
For all of those folks out there who are starting to "shake" over things like this, remember, if a professor assigned some readings that you don't understand, go to the professor's office hours and get the reading assignment clarified rather than coming to a message board where you may or may not received information that will work for your situation. If you lack the background, the instructor who assigned the material should be able to explain the material. If the instructor can't explain it, it's not going to be tested.
 

thefritz

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It is very difficult to get an intuitive understanding of Fourier analysis through college classes because those classes undoubtedly get bogged down in the math and you never get the big picture. What I am going to do here is explain the concepts of Fourier analysis so that you can get a "big picture" intuitive understanding of what it is and why it's important.

Basically Fourier analysis relates functions of TIME and FREQUENCY.

The math behind Fourier analysis involves Fourier series. The general concept is that ANY function can be created with a superposition of different types of sine waves. You can take a Pink Floyd recording and recreate it by super imposing many, many different sine waves (with varying frequencies and amplitudes). This is all you need to know for the math.

The Fourier transform is a tool that uses the concept of Fourier series. WHAT THE FOURIER TRANSFORM DOES is convert a function of time to a function of frequency. There is also an inverse Fourier transform that converts a function of frequency to a function of time. Actually calculating these transforms by hand is mathematically difficult and usually on required if you are an electrical or computer engineering major. Since you are not, here's a big picture example:

Imagine a function of time, x(t) = sin(t). A simple sine wave, right? Visualize it. It is just a tone, an oscillation occurring throughout time that goes up and down in amplitude with a constant period. When you play a note on an electric keyboard, lets say A, it plays at 440 Hz. The speaker of that keyboard moves up and down 440 times each second and your ear registers those air pressure variations as a sharp tone.

Now, our objective is to convert this sine wave (oscillating at 440 times per second and continuing indefinitely throughout time) to a function of FREQUENCY instead of time. Now, what are all the spectral (frequency) components in this note? There is only 1 and it is at 440 Hz. So, to convert this function into frequency, the x-axis changes from time to frequency and instead of a sine wave, we have a single impulse at 440 on the x-axis. Additionally, there will be another impulse at -440 hz, because the Fourier transform cannot distinguish between positive and negative frequency.

You have just performed an x(t) -> X(f) conversion, a Fourier transform!

Here is a list of common transforms: http://www.astro.umd.edu/~lgm/ASTR410/ft_ref3.pdf

Scroll down to the 7th row and you can see the transform we just did. We converted a sine wave to two delta functions (impulses) located at +/- 440 Hz.

Lets look at another one of these common transforms. Look at the 2nd row, the first transform listed. We have the function x(t) = 1. This is just a horizontal line crossing the y-axis at 1. It is 1 for every value of x. What are the spectral components of this function? Well, there aren't any. The function is not changing, it is constant. The are not oscillations or variations anywhere. So the Fourier transform is simply an impulse at 0 Hz. It has NO frequency.

One more example. Look at the function below the sine example, x(t) = rect(t). This is a rectangle function (http://en.wikipedia.org/wiki/Rectangular_function). It means that there is a rectangle on the x-axis going from -.5 to +.5 with an amplitude of 1.0. Now, in the previous two examples, we have looked at functions that continue forever (sine wave and constant 1). This is only non-zero for a finite period of time. What is the spectral content of this function? You may think, it's flat - it's just like the previous example, it doesn't have any spectral content! But you are wrong. It is possible to recreate this rectangle pulse by using many, many sine waves of different frequencies and amplitudes (Fourier series). Each of these sine wave's frequnecies contribute to the spectral content of the time signal. So the Fourier transform of this signal, the frequency domain function, is actually a sinc function, which looks like this: http://en.wikipedia.org/wiki/File:Sinc_function_(both).svg

As you can see, a finite signal in the time domain (like our rectangle function), has an infinite signal in the frequency domain (that sinc function goes on forever - it has spectral content continuing out to +/- infinity Hz). But an infinite signal in the time domain (like our sine wave from the keyboard that plays forever), has a finite signal in the time domain (a simple impulse at 1 frequency). Take a look at the sinc signal in the time domain. It is infinite and goes on for all time. Consequently, it has a finite signal in the frequency domain. It is made up of exactly all the frequencies from -0.5 to 0.5 Hz, nothing else.

Now consider the case where we play our note on the keyboard at 440 Hz, but we only play it for a few seconds. after 2 seconds, the sine wave disappears and the function is 0. We now have a finite signal in the time domain. What do you think happens in the frequency domain now? Well, We still have that impulse at 440 Hz, but now because we cut the signal off), there is spectral smearing all around that 440 Hz, so there will be spectral content extending out to infinity (although it will be extremely small).

The practical applications of the Fourier Transform are most commonly used by a digital method called the Fast Fourier Transform (FFT). The FFT is what is used in your course. You can take samples in time, do an FFT on the data, and visually see how often stuff is occurring (the frequency). When you play an MP3 on winamp and are looking at the spectrum, you're seeing an FFT. You're seeing the spectral components of the song you're listening too. When you hear a bass note, you see a spike on the low end of the spectrum, and when you hear the singer screech his voice, you see a spike on the high end of the spectrum.

So, there you have an explanation of the Fourier Transform without using ANY math. I hope this has been helpful. Let me know if you need any clarification.
 

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Nice post Fritz. Now can you help me with pwelch parameters in matlab? :D please.
 

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Excellent post and responses.

Fourier transforms are typically only useful for repeating signals. That is, repeating both forward and backwards infinitely in time. "Real world" signals are often approximated as repeating in time this way to help with the math.

Think of The Fourier Domain and Time Domain as evil-twins: What happens in one is typically mathematically inverted in the other.

When you have a infinitely repeating signal in time (such as a sine wave), it is 'compactly' represented in the Fourier domain. It is tedious (understatement) to depict an infinite sine wave in the time domain. But it is a simple impulse function (a single 'blip' if you will) in the Fourier domain representation. And here's the evil twin effect: The opposite is true with an impulse function in the time domain: It would take an infinite number sine waves to represent a single impulse in the frequency domain, and thus it is tedious to do so. In fact, any non-repeating signal including the 'boxcar' example, will require an infinite number of sine waves, a.k.a. 'terms', in the Fourier domain to represent.

Another easy concept to think about is the what type of information you get in each domain: In the time domain you have absolute precision about the system or signal at any point in time, but zero frequency information. In the fourier domain, as I mentioned above, the signal is assumed to repeat forever, so you have lost all temporal information but have gained complete frequency information.

This has many similarities to the heisenberg uncertainty principle in physics: You can't have it both ways.

But can you get a little bit of both? :) For those of you who are curious about this question and wish to explore it in further detail, there is a subset of signal processing theory called subband coding and wavelets. If Time domain and Fourier Domain represent the 'extremes' of representation of the time and frequency continuum, is there a middle ground? That is, is there a way to represent data that tells me a little about time and frequency at the same time?

Just noodle on this: I take an audio recording of a slide whistle that starts at a high pitch and slowly 'slides' down to a low pitch. Then I do the reverse: low to high. I do a Fourier transform of both. Both will 'light up like Christmas trees' because there are many frequencies represented in both samples. But could you tell me which Fourier transform is which original audio sample? The answer is likely 'No', because you have lost all information about the time domain.
 

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It is very difficult to get an intuitive understanding of Fourier analysis through college classes because those classes undoubtedly get bogged down in the math and you never get the big picture. What I am going to do here is explain the concepts of Fourier analysis so that you can get a "big picture" intuitive understanding of what it is and why it's important.

Basically Fourier analysis relates functions of TIME and FREQUENCY.

The math behind Fourier analysis involves Fourier series. The general concept is that ANY function can be created with a superposition of different types of sine waves. You can take a Pink Floyd recording and recreate it by super imposing many, many different sine waves (with varying frequencies and amplitudes). This is all you need to know for the math.

The Fourier transform is a tool that uses the concept of Fourier series. WHAT THE FOURIER TRANSFORM DOES is convert a function of time to a function of frequency. There is also an inverse Fourier transform that converts a function of frequency to a function of time. Actually calculating these transforms by hand is mathematically difficult and usually on required if you are an electrical or computer engineering major. Since you are not, here's a big picture example:

Imagine a function of time, x(t) = sin(t). A simple sine wave, right? Visualize it. It is just a tone, an oscillation occurring throughout time that goes up and down in amplitude with a constant period. When you play a note on an electric keyboard, lets say A, it plays at 440 Hz. The speaker of that keyboard moves up and down 440 times each second and your ear registers those air pressure variations as a sharp tone.

Now, our objective is to convert this sine wave (oscillating at 440 times per second and continuing indefinitely throughout time) to a function of FREQUENCY instead of time. Now, what are all the spectral (frequency) components in this note? There is only 1 and it is at 440 Hz. So, to convert this function into frequency, the x-axis changes from time to frequency and instead of a sine wave, we have a single impulse at 440 on the x-axis. Additionally, there will be another impulse at -440 hz, because the Fourier transform cannot distinguish between positive and negative frequency.

You have just performed an x(t) -> X(f) conversion, a Fourier transform!

Here is a list of common transforms: http://www.astro.umd.edu/~lgm/ASTR410/ft_ref3.pdf

Scroll down to the 7th row and you can see the transform we just did. We converted a sine wave to two delta functions (impulses) located at +/- 440 Hz.

Lets look at another one of these common transforms. Look at the 2nd row, the first transform listed. We have the function x(t) = 1. This is just a horizontal line crossing the y-axis at 1. It is 1 for every value of x. What are the spectral components of this function? Well, there aren't any. The function is not changing, it is constant. The are not oscillations or variations anywhere. So the Fourier transform is simply an impulse at 0 Hz. It has NO frequency.

One more example. Look at the function below the sine example, x(t) = rect(t). This is a rectangle function (http://en.wikipedia.org/wiki/Rectangular_function). It means that there is a rectangle on the x-axis going from -.5 to +.5 with an amplitude of 1.0. Now, in the previous two examples, we have looked at functions that continue forever (sine wave and constant 1). This is only non-zero for a finite period of time. What is the spectral content of this function? You may think, it's flat - it's just like the previous example, it doesn't have any spectral content! But you are wrong. It is possible to recreate this rectangle pulse by using many, many sine waves of different frequencies and amplitudes (Fourier series). Each of these sine wave's frequnecies contribute to the spectral content of the time signal. So the Fourier transform of this signal, the frequency domain function, is actually a sinc function, which looks like this: http://en.wikipedia.org/wiki/File:Sinc_function_(both).svg

As you can see, a finite signal in the time domain (like our rectangle function), has an infinite signal in the frequency domain (that sinc function goes on forever - it has spectral content continuing out to +/- infinity Hz). But an infinite signal in the time domain (like our sine wave from the keyboard that plays forever), has a finite signal in the time domain (a simple impulse at 1 frequency). Take a look at the sinc signal in the time domain. It is infinite and goes on for all time. Consequently, it has a finite signal in the frequency domain. It is made up of exactly all the frequencies from -0.5 to 0.5 Hz, nothing else.

Now consider the case where we play our note on the keyboard at 440 Hz, but we only play it for a few seconds. after 2 seconds, the sine wave disappears and the function is 0. We now have a finite signal in the time domain. What do you think happens in the frequency domain now? Well, We still have that impulse at 440 Hz, but now because we cut the signal off), there is spectral smearing all around that 440 Hz, so there will be spectral content extending out to infinity (although it will be extremely small).

The practical applications of the Fourier Transform are most commonly used by a digital method called the Fast Fourier Transform (FFT). The FFT is what is used in your course. You can take samples in time, do an FFT on the data, and visually see how often stuff is occurring (the frequency). When you play an MP3 on winamp and are looking at the spectrum, you're seeing an FFT. You're seeing the spectral components of the song you're listening too. When you hear a bass note, you see a spike on the low end of the spectrum, and when you hear the singer screech his voice, you see a spike on the high end of the spectrum.

So, there you have an explanation of the Fourier Transform without using ANY math. I hope this has been helpful. Let me know if you need any clarification.
Did you seriously write this post or did you copy and paste? If this is your own doing then kudos, that explanation even helped me and I dry heave thinking about Fourier Transform.
 

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This thread is bringing back so many nightmares...

Former chemical engineer who viewed Fourier transforms as the monster in your closet...:scared:

Seriously, though, great explanations!!!

Definitely ask you professor what is exactly needed for the class or ask a math/engineering major to help you out.

Good luck!
 

MilkmanAl

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Did you seriously write this post or did you copy and paste? If this is your own doing then kudos, that explanation even helped me and I dry heave thinking about Fourier Transform.
With the exception of some of the practical application stuff, that's all info you'd learn pretty quickly in an intermediate level physics class. It still blows my mind that Fourier transforms are being taught in a biochem class, though.
 
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It is very difficult to get an intuitive understanding of Fourier analysis through college classes because those classes undoubtedly get bogged down in the math and you never get the big picture. What I am going to do here is explain the concepts of Fourier analysis so that you can get a "big picture" intuitive understanding of what it is and why it's important.

Basically Fourier analysis relates functions of TIME and FREQUENCY.

The math behind Fourier analysis involves Fourier series. The general concept is that ANY function can be created with a superposition of different types of sine waves. You can take a Pink Floyd recording and recreate it by super imposing many, many different sine waves (with varying frequencies and amplitudes). This is all you need to know for the math.

The Fourier transform is a tool that uses the concept of Fourier series. WHAT THE FOURIER TRANSFORM DOES is convert a function of time to a function of frequency. There is also an inverse Fourier transform that converts a function of frequency to a function of time. Actually calculating these transforms by hand is mathematically difficult and usually on required if you are an electrical or computer engineering major. Since you are not, here's a big picture example:

Imagine a function of time, x(t) = sin(t). A simple sine wave, right? Visualize it. It is just a tone, an oscillation occurring throughout time that goes up and down in amplitude with a constant period. When you play a note on an electric keyboard, lets say A, it plays at 440 Hz. The speaker of that keyboard moves up and down 440 times each second and your ear registers those air pressure variations as a sharp tone.

Now, our objective is to convert this sine wave (oscillating at 440 times per second and continuing indefinitely throughout time) to a function of FREQUENCY instead of time. Now, what are all the spectral (frequency) components in this note? There is only 1 and it is at 440 Hz. So, to convert this function into frequency, the x-axis changes from time to frequency and instead of a sine wave, we have a single impulse at 440 on the x-axis. Additionally, there will be another impulse at -440 hz, because the Fourier transform cannot distinguish between positive and negative frequency.

You have just performed an x(t) -> X(f) conversion, a Fourier transform!

Here is a list of common transforms: http://www.astro.umd.edu/~lgm/ASTR410/ft_ref3.pdf

Scroll down to the 7th row and you can see the transform we just did. We converted a sine wave to two delta functions (impulses) located at +/- 440 Hz.

Lets look at another one of these common transforms. Look at the 2nd row, the first transform listed. We have the function x(t) = 1. This is just a horizontal line crossing the y-axis at 1. It is 1 for every value of x. What are the spectral components of this function? Well, there aren't any. The function is not changing, it is constant. The are not oscillations or variations anywhere. So the Fourier transform is simply an impulse at 0 Hz. It has NO frequency.

One more example. Look at the function below the sine example, x(t) = rect(t). This is a rectangle function (http://en.wikipedia.org/wiki/Rectangular_function). It means that there is a rectangle on the x-axis going from -.5 to +.5 with an amplitude of 1.0. Now, in the previous two examples, we have looked at functions that continue forever (sine wave and constant 1). This is only non-zero for a finite period of time. What is the spectral content of this function? You may think, it's flat - it's just like the previous example, it doesn't have any spectral content! But you are wrong. It is possible to recreate this rectangle pulse by using many, many sine waves of different frequencies and amplitudes (Fourier series). Each of these sine wave's frequnecies contribute to the spectral content of the time signal. So the Fourier transform of this signal, the frequency domain function, is actually a sinc function, which looks like this: http://en.wikipedia.org/wiki/File:Sinc_function_(both).svg

As you can see, a finite signal in the time domain (like our rectangle function), has an infinite signal in the frequency domain (that sinc function goes on forever - it has spectral content continuing out to +/- infinity Hz). But an infinite signal in the time domain (like our sine wave from the keyboard that plays forever), has a finite signal in the time domain (a simple impulse at 1 frequency). Take a look at the sinc signal in the time domain. It is infinite and goes on for all time. Consequently, it has a finite signal in the frequency domain. It is made up of exactly all the frequencies from -0.5 to 0.5 Hz, nothing else.

Now consider the case where we play our note on the keyboard at 440 Hz, but we only play it for a few seconds. after 2 seconds, the sine wave disappears and the function is 0. We now have a finite signal in the time domain. What do you think happens in the frequency domain now? Well, We still have that impulse at 440 Hz, but now because we cut the signal off), there is spectral smearing all around that 440 Hz, so there will be spectral content extending out to infinity (although it will be extremely small).

The practical applications of the Fourier Transform are most commonly used by a digital method called the Fast Fourier Transform (FFT). The FFT is what is used in your course. You can take samples in time, do an FFT on the data, and visually see how often stuff is occurring (the frequency). When you play an MP3 on winamp and are looking at the spectrum, you're seeing an FFT. You're seeing the spectral components of the song you're listening too. When you hear a bass note, you see a spike on the low end of the spectrum, and when you hear the singer screech his voice, you see a spike on the high end of the spectrum.

So, there you have an explanation of the Fourier Transform without using ANY math. I hope this has been helpful. Let me know if you need any clarification.

That was fairly awesome. I've never even heard of a Fourier Transform and that made sense.
 

thefritz

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Did you seriously write this post or did you copy and paste? If this is your own doing then kudos, that explanation even helped me and I dry heave thinking about Fourier Transform.
I wrote it because I have never once read a non-mathematical, intuitive explanation of what it is, why it's important, and how it's applied. I'm sure I overlooked some important points, but I wanted to try and boil it down to something tangible.

In regards to the post that says Fourier Transforms are usually only important on periodic signals, that's not entirely true. Fourier transforms are practically applied to discrete data through signal processing. An input function is taken and sampled at regular points to produce discrete inputs. The Fourier transforms of this type of data is called the Discrete Time Fourier Transform (DTFT). It's output is continuous and periodic, and all of the information about the time-domain input signal is contained in one period. More useful is the Discrete Fourier Transform (DFT), which is also discrete in frequency. So you can take a finite, non-periodic signal (like an audio recording), sample it, perform a DFT on it, and get discrete frequency outputs. Since it has a discrete input and output, the DFT is commonly used and is most often calculated in computers with an algorithm called the FFT (fast Fourier transform). When you practically encounter the Fourier transform, it will be in the form of an FFT.

So, there you have a basic introduction to DSP (digital signal processing).

The OP asked me a follow up about negative frequency:

Fakesmile said:
Hi thefritz,

Now, our objective is to convert this sine wave (oscillating at 440 times per second and continuing indefinitely throughout time) to a function of FREQUENCY instead of time. Now, what are all the spectral (frequency) components in this note? There is only 1 and it is at 440 Hz. So, to convert this function into frequency, the x-axis changes from time to frequency and instead of a sine wave, we have a single impulse at 440 on the x-axis. Additionally, there will be another impulse at -440 hz, because the Fourier transform cannot distinguish between positive and negative frequency.
Could you explain why we get a negative frequency as well? Thanks so much for your post!
Whether a frequency is positive or negative is a relative measure. It depends on which way the object or wave is oscillating. For example, doppler radar waves that come to you have positive frequency and waves that go away from you have negative frequency. This is by convention. It is just the sign in front of the frequency. The magnitude of the frequency is the same. So if you take an infinite sinusoid (cos(wt)) in time, the Fourier transform will have spikes at + and - (w) Hz because there is no frame of reference specifying whether it is positive or negative. Basically, any function with a positive frequency component will also have a negative frequency component either the same or inverted depending if the function is even or odd. See here: http://research.opt.indiana.edu/library/fourierbook/ch12.html#T2

Mathematically, the negative frequency is clear when you look at Euler's definition of a sinusoid:

cos(wt) = (1/2)*[e^(jwt)+e^-(jwt)]

(Note that I use j to represent the imaginary number i as is typical engineering convention).

You can see here that a cos with frequency w has both a real and complex component. The real component has positive frequency w and the complex component has negative frequency -w. Both have amplitude 1/2 and when summed, they equate the complete sinusoid.

When you do a Fourier transform only on the real part of the sinusoid, e^(jwt), you get a response only at +w. But when you do a transform on the complete sinusoid, (1/2)*[e^(jwt)+e^-(jwt)], you get responses at + and - w.

Hope this helps.
 

RogueUnicorn

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What the hell are you guys talking about?
Transformers? Optimus Prime?
you know what's funny, the first Transformers movie does mention Fourier transforms, but the actress mispronounces it and also calls it a Fourier transfer. i had a small chuckle in the theater...
 

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I read through one of the links I posted earlier.

This is actually a pretty good and relatively easy to understand summary course of the topic for those who want to know more: http://research.opt.indiana.edu/library/fourierbook/toc.html
Could you please explain the difference between the Fourier Series and Fourier Transform? I thought the Fourier Series was used to represent periodic signals. The coefficients were useful in finding the power...

Thanks! your posts are very interesting.

~embarrassed senior in ee :oops:
 
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dingyibvs

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Could you please explain the difference between the Fourier Series and Fourier Transform? I thought the Fourier Series was used to represent periodic signals. The coefficients were useful in finding the power...

Thanks! your posts are very interesting.

~embarrassed senior in ee :oops:
It's been a long time since I've done any actual math involving Fourier transforms, so don't quote me. But basically, they're the same thing. The transform is just something to get you the series, whether the original time based signal is finite or infinite.

I never quite understood it completely either and I have a degree in EE :laugh: I much preferred digital logic tho! My signals class was the first B I had received in college, and that's after I breezed through all the med pre-reqs and thought I was invincible :rolleyes:
 

Fort

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Gah, I hated these things in Quantum Physics; they are a pain to compute by hand. Fourier was pretty smart in his time I'd say, his name comes up fairly often.
 

thefritz

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Could you please explain the difference between the Fourier Series and Fourier Transform? I thought the Fourier Series was used to represent periodic signals. The coefficients were useful in finding the power...

Thanks! your posts are very interesting.

~embarrassed senior in ee :oops:
Start here: http://en.wikipedia.org/wiki/Fourier_analysis#Variants_of_Fourier_analysis

The Fourier series is used to analyze infinite periodic functions whereas the Fourier transform is used to analyze finite non-periodic functions (look at the integrals). Hence the Fourier transform finds much more real world applications.

Wiki sums up the four types of Fourier analysis better than I can:

http://en.wikipedia.org/wiki/Fourier_analysis#Fourier_Transforms_Summary

Also, for a very clear example, look here at 11D: http://research.opt.indiana.edu/Library/FourierBook/ch11.html#F2

The Fourier Series has discrete terms in the frequency domain. It gives you the harmonics of the periodic signal, which are discrete frequencies. If you extend the observation period (bounds on the integral) beyond that of one period, then you get more harmonics. The Fourier transform integral goes from -infinity to +infinity, thereby creating an infinite number of harmonics in the frequency domain giving you the WHOLE spectrum.

That's the difference. Can you do a Fourier Transform of a periodic signal? Of course you can. You get essentially the same results as that of a Fourier Series. Can you do a Fourier series on an aperiodic signal? No.
 

dingyibvs

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Start here: http://en.wikipedia.org/wiki/Fourier_analysis#Variants_of_Fourier_analysis

The Fourier series is used to analyze infinite periodic functions whereas the Fourier transform is used to analyze finite non-periodic functions (look at the integrals). Hence the Fourier transform finds much more real world applications.

Wiki sums up the four types of Fourier analysis better than I can:

http://en.wikipedia.org/wiki/Fourier_analysis#Fourier_Transforms_Summary

Also, for a very clear example, look here at 11D: http://research.opt.indiana.edu/Library/FourierBook/ch11.html#F2

The Fourier Series has discrete terms in the frequency domain. It gives you the harmonics of the periodic signal, which are discrete frequencies. If you extend the observation period (bounds on the integral) beyond that of one period, then you get more harmonics. The Fourier transform integral goes from -infinity to +infinity, thereby creating an infinite number of harmonics in the frequency domain giving you the WHOLE spectrum.

That's the difference. Can you do a Fourier Transform of a periodic signal? Of course you can. You get essentially the same results as that of a Fourier Series. Can you do a Fourier series on an aperiodic signal? No.
that...actually made sense, thanks lol