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Author Topic: Fourier transform of a reversed signal?  (Read 3946 times)

jimmyjazz

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Fourier transform of a reversed signal?
« on: February 01, 2006, 01:06:42 PM »

Curiousity killed the cat . . .  can any of you tell me what happens to the amplitude and phase response of a finite signal when you reverse it in time?
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danlavry

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Re: Fourier transform of a reversed signal?
« Reply #1 on: February 01, 2006, 03:21:57 PM »

jimmyjazz wrote on Wed, 01 February 2006 18:06

Curiousity killed the cat . . .  can any of you tell me what happens to the amplitude and phase response of a finite signal when you reverse it in time?


The FFT is a filter. It does "averaging". So does the ear behave like an FFT?
Generally the ear does not behave like an FFT. Say we take a 1 second long FFT, for 2 cases:

Case 1: A 1KHz tone that grows from 0V to 1V linearly in 1 second.  

Case 2: A 1KHz tone that starts at 1V and linearly decreases to 0 in 1 second.

The 2 cases will yield the same plot - the energy content of the signal over the FFT duration. Of course we expect to see energy at 1KHz, but we will also see some energy due to the modulation (the increase or decrease of amplitude).

Note: In practice, we will likely window the FFT to help deal with sudden start and stop conditions of such signal which requires infinite bandwidth.

The point is: Running a signal backwards is not a practical issue, but will yield the same amplitude vs frequency plot.
Running the "amplitude envelope" backwards is practical and will also yield the same FFT plot.

Another case: Say you take a 10 second FFT (very long), in which the signal is all very quite (9 seconds of no signal) but there is a very loud 1 second gun shot say at the "middle of the signal". The FFT will not show that the energy was concentrated over a short time (1 second). It will seem as if the gunshot sound was 10 second long at 1/10 the amplitude...

So when does the ear behave "sort of like an FFT"? That is a psychoacoustic question. The ear has it's own time window (frequency bandwidth) of response. The ear "knows" that a 1 second gun shot is not "smeared" over 10 seconds, but the ear does some averaging over short time periods (in the milliseconds or more).

But that is not my area of expertize. People that deal with audio data compression know a lot more about it...

Regards
Dan Lavry
www.lavryengineering.com
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jimmyjazz

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Re: Fourier transform of a reversed signal?
« Reply #2 on: February 01, 2006, 04:17:20 PM »

Thanks, Dan.  You touched on part of what I was looking for, but honestly, I was expecting a more theoretical answer.

Maybe if I eliminated the "finite" time restraint, it would clarify things.  Imagine a reasonably complicated, but infinitely repetitive signal.  I dunno, a sawtooth wave.  Reverse it.  Perform Fourier integrals on both signals.  Are the amplitude and phase responses identical?  What accounts for the fact that the finite-sloping portion of one sawtooth has a positive slope and one has a negative slope?  The phase response?  It must be.  The amplitude versus frequency content has to be the same in both signals, doesn't it?

The thing that prompted this is a sound effect I want to create for a film I'm working on.  I thought about taking a mechanical noise (say, from an HVAC compressor) and combining it with its reversed signal.  That got me to thinking "comb filtering", and that got me to thinking spectral analysis, and pretty soon I was back in grad school, and pissed off about it.  So I decided to ask the experts (which I am certainly not)!
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danlavry

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Re: Fourier transform of a reversed signal?
« Reply #3 on: February 01, 2006, 06:33:44 PM »

jimmyjazz wrote on Wed, 01 February 2006 21:17

Thanks, Dan.  You touched on part of what I was looking for, but honestly, I was expecting a more theoretical answer.

Maybe if I eliminated the "finite" time restraint, it would clarify things.  Imagine a reasonably complicated, but infinitely repetitive signal...!


An infinity repetitive signal is a much simpler case then I was talking about. It is the case of Fourier series, and in such a case, you will get the same amplitude vs frequency plot. You do have to be carfull about phase. A cosine wave is symmetrical and going back in time will yield the same result as going forward.
But a sine wave going forward starts by rising. A sine wave going back starts at a negative cycle. You have to be careful and pay attention to the phase...

I can be as theoretical as you wish, but try to be understood by as many here as possible. What is that you need?

Regards
Dan Lavry
www.lavryengineering.com
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jimmyjazz

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Re: Fourier transform of a reversed signal?
« Reply #4 on: February 01, 2006, 07:08:47 PM »

What's the difference between a cosine and a sine for an infinite signal?  It's just a 90 degree shift, right?  There's no "t = 0", so there's no reference point.  They're one and the same as I see it.

I'm just too lazy to get out my engineering math text.  Well, first I'd have to find it.

I started out wondering if I could get crazy cancellation effects and create "non-physical" noise by adding a signal to its reverse (NOT inverse) with an arbitrary phase shift.  I was wondering if the two would be correlated.  Of course they would have to be, wouldn't they?  Seems to me that correlation should result in more extreme cancellation effects than the combination of, say, an HVAC compressor and a car motor.

I should also try it by adding it to itelf with a phase shift.  That would get me the classic comb filtering effect, I suppose.

Sorry for the mental freestyling, Dan.  I figured you would know the signal math stuff, though, so I just went straight to the top of the mountain.  Feel free to let this slide to the bottom (and past)!
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Ronny

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Re: Fourier transform of a reversed signal?
« Reply #5 on: February 01, 2006, 10:36:59 PM »




With out getting to technical about it, you could just try it with the intended material and see if it works.

This reminds me of the reversed lead guitar on Hendrix' "Are You Experienced" song on the same titled album. That was pretty cool.

"Hey, Mr. Kramer, I still ain't satisfied with this lead and it's the 45th take",

"no problem James, we'll run it backwards, it worked for the Beatles."  Very Happy
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------Ronny Morris - Digitak Mastering------
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dcollins

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Re: Fourier transform of a reversed signal?
« Reply #6 on: February 01, 2006, 11:23:20 PM »

danlavry wrote on Wed, 01 February 2006 12:21


So when does the ear behave "sort of like an FFT"? That is a psychoacoustic question. The ear has it's own time window (frequency bandwidth) of response.


This paper has more on the Cochlea and its "filter bank."

http://www.aes.org/sections/pnw/ppt/jj_aes04_ts1.pdf

DC

danlavry

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Re: Fourier transform of a reversed signal?
« Reply #7 on: February 02, 2006, 02:10:35 PM »

dcollins wrote on Thu, 02 February 2006 04:23

danlavry wrote on Wed, 01 February 2006 12:21


So when does the ear behave "sort of like an FFT"? That is a psychoacoustic question. The ear has it's own time window (frequency bandwidth) of response.


This paper has more on the Cochlea and its "filter bank."

http://www.aes.org/sections/pnw/ppt/jj_aes04_ts1.pdf

DC



Hi Dave,

Thanks for the link. I certainly recommend reading it!!!
The author James D. Johnston (J.J.) has a lot of expertize in the area, much of it based on many years of research and experiments at Bell Labs. J.J. is very strong in math, but he made this presentation comprehensible to a wide audience (people with little math background).

Regards
Dan Lavry
www.lavryengineering.com
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mpd

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Re: Fourier transform of a reversed signal?
« Reply #8 on: February 11, 2006, 03:14:36 PM »

You asked for theory... Smile

If x(n) is a N-point sequence, and X(k) is the DFT of that sequence, then we usually write:

x(n) <-> X(k)

then

x(n) <-> X(k)
x(N-n) <-> X(N-k)

In other words, reversing the sequence results in a reversal of the DFT.

However, if x(n) is real (suach as an audio signal), then

X(k) = X*(N-k)

In words, if x(n) is real then the DFT is conjugate symetrical.

Therefore if x(n) is real

x(N-n) <-> X*(k)

So, if you reverse x(n), then you end up with the complex conjugate of the original DFT.  The magnitude of the DFT is the same, but you end up with the negative of the phase response.

I think I did this right...


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jimmyjazz

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Re: Fourier transform of a reversed signal?
« Reply #9 on: February 11, 2006, 11:07:32 PM »

mpd wrote on Sat, 11 February 2006 15:14

You asked for theory... Smile

The magnitude of the DFT is the same, but you end up with the negative of the phase response.




Outstanding!  It seems intuitively correct . . . I still haven't had time to try it myself.  I suppose I could cram an audio snippet and its reverse through MatLab, but that's a rather crude solution.  (But it would be convincing enough for my concerns.)

Thank you very much!
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