Fourier Transform question

Hey, we just began to learn about the Fourier transform. I have a few questions which nobody ever addresses it seems.

First you have to understand I'm not a person who thinks in terms of math. What I do is understand the concept and then I use the math as a tool to find a solution to the problem.

So having said that, I understand the Fourier series conceptually. It means that any periodic signal is composed of a sum of an infinite number of sinusoids of frequencies which are multiple integers of the original periodic signal.

Taking speech as an example. A speech signal itself is not periodic, but its composed of a sum of periodic signals, each of which in turn has a Fourier series.

Now the Fourier transform says that a NON-periodic signal can be transformed to the frequency domain. First of all, how the heck does that make sense? If a signal is not periodic where does the frequency come from?

If say we took f(t) = t^2. This is just a parabola. If we took the F transform of this, what frequency are you talking about? Where?

It makes sense to me mathematically, and yea the calculus works and all that. But if the fourier transform is used so much to deal with real signals in circuits, it has to mean something in real life, not just mathematically.

Can someone enlighten me on this? Thank you.

I'm not sure what you're asking, but back when I was in college I had more than a passing interest. Speech and RF is a good example where one application lies, the bandwidth of the RF signal you allow goes a long way to determining how accurately you can reproduce a high quality audio waveform. This includes non-traditional modulation methods such as compressed FM or PWM.

Another mechanism where it is important (also audio, sort of) was modems. Ever wonder how they got 56K out of a phone connection that was 300Hz - 3.4Khz? I still haven't got that one quite figured out.

My feeling is a lot of this analysis lets you know what you can get by with.

I suspect you have to look for examples of complex repeating or chaotic waveforms, such as video, to find even better examples.

You're college courses are coming along nowadays, aren't they? What is the ideal job you are aiming for?

Yea, I am getting there. Junior now. Personally I am interested in electronics. I am interested in communications and signal processing also but the math kills me. If I can figure it out, then maybe.

Well I hope someone can understand what I'm asking. How can there be a frequency in a signal which is not periodic? The word frequency itself means something is periodic. Sure as the textbook says, you can make it periodic and let the period go to infinity, but this is all math. I'm trying to understand how this makes sense in real life.

One can think of a related "companion question" which might raise the level of appreciation of what the Fourier transform can deliver in terms of analysis. Suppose an amplifier has to faithfully reproduce an input waveform which is approaching an ideal pulse - let's say a single 10V square pulse of duration 10usec with an arbitrary but comparatively slow repetition rate - say >>1 second. What bandwidth must the amplifier have to do this?

To 'perfectly reproduce' the input, the amplifier presumably requires an infinite bandwidth - which implies an infinite continuous harmonic content in the input signal spectrum.

count_volta said:

Now the Fourier transform says that a NON-periodic signal can be transformed to the frequency domain. First of all, how the heck does that make sense? If a signal is not periodic where does the frequency come from?

If say we took f(t) = t^2. This is just a parabola. If we took the F transform of this, what frequency are you talking about? Where?

It makes sense to me mathematically, and yea the calculus works and all that. But if the fourier transform is used so much to deal with real signals in circuits, it has to mean something in real life, not just mathematically.

Can someone enlighten me on this? Thank you.

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This is a good question, but probably difficult to answer to your satisfaction. There is one way to gain more insight, but you'll have to do a bit of work to see it through.

If you are willing, try the following. Consider aperiodic signals that start and end with zero value. For eample f(t)=0 for t<-1 and t>+1. In between those limits you can have any function, for example f(t)=1 for -1<t<+1. This is an aperiodic signal and you can take the Fourier transform.

Now, use that basic function to build a periodic signal with large period. For example from t=-100 to +100 this function is valid, but put an infinite number of of these stacked along the time axis. This is then a periodic signal with period of 200, and you can find the Fourier series of this function.

Now keep increasing the period of this "stacking" arrangement of your aperiodic function and consider the limit as the period goes to infinity. This is a way to make a seemless transitions from periodic signals to aperiodic signals. You will see that the Fourier series of the periodic signals begins to approach the Fourier Transform of the aperiodic signal.

The question you are asking is similar to the ones we all ask when learning basic calculus. We have to trust that the derivative is a limiting form of approximating slope, and integrals are a limiting form of approximating area.

Similarly, the Fourier Transform is akin to a limiting form of the Fourier Series; however, it's harder to see it. The odd thing here is that the discrete function that is the Fourier series, turns itself into a continuous function that is the Fourier Transform. This is much more radical an idea than numbers (such as slope and area) approaching a limiting value.

steveb said:

This is a good question, but probably difficult to answer to your satisfaction. There is one way to gain more insight, but you'll have to do a bit of work to see it through.

If you are willing, try the following. Consider aperiodic signals that start and end with zero value. For eample f(t)=0 for t<-1 and t>+1. In between those limits you can have any function, for example f(t)=1 for -1<t<+1. This is an aperiodic signal and you can take the Fourier transform.

Now, use that basic function to build a periodic signal with large period. For example from t=-100 to +100 this function is valid, but put an infinite number of of these stacked along the time axis. This is then a periodic signal with period of 200, and you can find the Fourier series of this function.

Now keep increasing the period of this "stacking" arrangement of your aperiodic function and consider the limit as the period goes to infinity. This is a way to make a seemless transitions from periodic signals to aperiodic signals. You will see that the Fourier series of the periodic signals begins to approach the Fourier Transform of the aperiodic signal.

The question you are asking is similar to the ones we all ask when learning basic calculus. We have to trust that the derivative is a limiting form of approximating slope, and integrals are a limiting form of approximating area.

Similarly, the Fourier Transform is akin to a limiting form of the Fourier Series; however, it's harder to see it. The odd thing here is that the discrete function that is the Fourier series, turns itself into a continuous function that is the Fourier Transform. This is much more radical an idea than numbers (such as slope and area) approaching a limiting value.

Click to expand...

Thank you, you understand what I'm getting at which is great. But the question is even more deep than that.

Say we have a speech signal. Not mathematical abstract representation. A real speech signal. It consists of individual sounds like for example in the first 3 microseconds of the speech signal, someone uttered the letter A (when starting to say the word Apple)

That sound (uttering letter A) consists of a Fourier series of basic sinusoids, but a huge number of them. Any sound a human being makes is a Fourier series. Any sound is a Fourier series. A pure tone has just one term in the Fourier series (the sinusoid at the frequency of the tone). A more complicated sound has many more Fourier series terms. And human speech is an incredibly complicated combination of sounds.

The overall signal doesn't look periodic. But if you look closely, its composed of a summation of periodic signals. The entire human speech signal that is.

So even though this signal is not periodic it has a frequency hidden within it. Many frequencies actually. The frequencies change at every instant in time.

So is the Fourier transform meant to be used with signals like that? Because a signal like f(t) = t^2, has no frequency no matter how you look at it, and if you take its Fourier transform and transform it to the frequency domain it doesn't really make sense, since its just a parabola which doesn't repeat.

Man I hope you understand this nonsense I'm asking about here.

The laplace transform for example, I just think of as (Do this and you solve differential equations easier) Or do this, and you find the poles and zeros.

The Fourier transform is nothing more than the Laplace transform with s = jw, and yet because its connected with something which is conceptually easy (Fourier series) I can't help but try to understand what it actually means (not just mathematical tool)

I should probably stop this, because its making life harder than it needs to be. LOL.

I think you're taking the Fourier transform much too seriously.

A theoretical pure sine wave of a single frequency has no beginning and no end, because either would represent a deviation from the pure sine.

So, to be mathematically correct, in order to match pure sines with the actual transient nature of real signals, the Fourier transform needs to multiply the real-world signal by a "windowing" function. There are dozens of different windowing functions in use; Google "windowing function" to learn more.

But most descriptions of Fourier analysis just sort of skip over that, accepting that real-world signal analysis include lots of approximations.

Take a "square wave" for example. Theory would say it can be decomposed into a sine wave of the fundamental frequency and an infinite number of its odd harmonics. But in real life, circuits don't have infinite bandwidth, so the square waves we actually work with limit the upper frequencies.

It's still very useful to understand things like the Fourier analysis of the square wave, so you have a sense of allowing for the higher frequencies if you need to preserve the squareness of the wave, but nobody considers the windowing function, or expects an infinite number of harmonics, or expects infinite bandwidth when they talk about the Fourier transform of the square wave.

So while those signals you're describing do have a formal Fourier analysis, the formal analysis can get so theoretical and ugly that they're not very useful if you insist on applying the full mathematical expression of the transform.

davebee said:

I think you're taking the Fourier transform much too seriously.

A theoretical pure sine wave of a single frequency has no beginning and no end, because either would represent a deviation from the pure sine.

So, to be mathematically correct, in order to match pure sines with the actual transient nature of real signals, the Fourier transform needs to multiply the real-world signal by a "windowing" function. There are dozens of different windowing functions in use; Google "windowing function" to learn more.

But most descriptions of Fourier analysis just sort of skip over that, accepting that real-world signal analysis include lots of approximations.

Take a "square wave" for example. Theory would say it can be decomposed into a sine wave of the fundamental frequency and an infinite number of its odd harmonics. But in real life, circuits don't have infinite bandwidth, so the square waves we actually work with limit the upper frequencies.

It's still very useful to understand things like the Fourier analysis of the square wave, so you have a sense of allowing for the higher frequencies if you need to preserve the squareness of the wave, but nobody considers the windowing function, or expects an infinite number of harmonics, or expects infinite bandwidth when they talk about the Fourier transform of the square wave.

So while those signals you're describing do have a formal Fourier analysis, the formal analysis can get so theoretical and ugly that they're not very useful if you insist on applying the full mathematical expression of the transform.

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Ok that does help. Thanks.

I understand what you're asking and it's a reasonable question.

First, you can numerically experiment with a computer program (I recommend python, numpy, and matplotlib because they are easy to use and freely available) to see the effect of taking a periodic signal and making the period larger and larger. This process takes you to a non-periodic signal. For a concrete example, suppose we have a 1 volt pulse that is 1 μs wide. Suppose it's a pulse train with period 1 ms. Now, start increasing the pulse train's period. When the period gets to 1 day, 1 month, 1 year, etc., you're effectively dealing with an isolated pulse for most practical purposes. Yet this pulse will have a frequency spectrum that's well-approximated through the Fourier integral.

Another way to see this experimentally is to grab a bandpass filter and send isolated pulses through it at various center frequencies of the filter (i.e., a manual wave analyzer). You'll theoretically see measureable signals at all frequencies (practically, some will be down in the noise).

Hopefully, these two experiments (or Gedankenexperiments) can help your intuition see where the "frequencies" are coming from. There's no clear dividing line between a periodic and aperiodic signal in a practical sense. But the notion of letting the period get arbitrarily large lets the Fourier series be replaced by an integral; that, in turn, led to lots of elegant mathematics of wide applicability. So, in one sense, you can view the Fourier integral as a highly convenient fiction from the practical standpoint.

If you need more understanding, you may have to head to the department discussing self-adjoint linear operators in Hilbert spaces. Your choice of a parabola is a good "bad" example, as it's not square-integrable over the real line, so it technically doesn't have a Fourier transform. Practical folk get around this by defining things to have finite extent. Physics majors get "square integrable" pounded into their heads in the typical quantum mechanics class -- which is reasonable, for a particle to be modelled by a wave packet, it's position or momentum has to be (fairly) localized in space and time and things like total energy have to have finite values.

Thanks someonesdad. From reading more about the Fourier transform and doing example problems with filters and modulation I am starting to understand much better. Its much easier to treat the F transform as a mathematical tool.

For example a speech signal will have a certain bandwidth, and the F transform will help you see what this bandwidth is. Then you can use it in filters to pass the bandwidth or allow a portion of the bandwidth to pass, or to block other parts.