Author Topic: Reconstructing a waveform from its harmonics.  (Read 7981 times)

0 Members and 2 Guests are viewing this topic.

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #75 on: March 11, 2018, 08:46:03 am »
Simon, with your course, ideally you do want fantastic teachers, but if you are expecting to learn this stuff by just listening, it will not stick anyway. Listening to a lecture will not make you feel comfortable or at home using the things you were taught now in 10 years time. If you can dive in and have some fun playing with the new ideas now, then in 10 years, you will think - "Great - Fourier Transforms - love it!".

You have to get in and start crunching numbers for yourself - just like people have done in this thread. We weren't even doing the course, but we had some fun. Build and test circuits. Every time you go into areas like Fourier Transforms, Laplace Transforms, Maxwell's Equations, Classical Filter theory, Bode plots and stability, Semiconductor Theory and so on it gives a new perspective to your understanding so you can start to see how electronics is working from new directions.

No matter how good or bad the lectures seem to you, if you can take the ideas from the current subject home and start to have some fun, you do start having some really big "Wow!" moments along the way and you do feel far more confident. You never want to let any course limit how much you learn. There is no reason why you can't be better then the lecturers - any lecturers. It is the same as thinking that no football player can ever be better then the coach.

There are plenty of Wow! moments with Fourier Transforms. If you put a pure sinewave into a circuit with terrible distortion, you get a mess out. But it looks very different when you look at a Fourier transform of the distorted output.

The lecturers are leading you on a path up a mountain, but it is up to you to look out at the ever-expanding view for yourself. It is your journey, not the lecturers. If you are staring at your feet the whole time you climb the mountain, you never see anything for yourself and you will never get much out of the course other then a piece of paper.

Richard

What teachers? what lectures? I have what appears to be lecturers note for module material and I study at home alone with no help. I have never been to the university itself. Apparently in the UK you can take a yea at uni at home around a day job.
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #76 on: March 11, 2018, 08:51:11 am »
So attached is the monkey see monkey do version. Next I will try the same with the corrected frequencies on a 50Hz fundamental.

Blue is using sine on the real parts, green is using cosine on the real parts, the blue seems to be right but out of phase

SPOILER (you should have realized this by now):
For the correct final product, use sine for the fundamental, negative sine for the 5x tone, and sine for the 7x tone.

These phase differences are understood because:

3   0.848      -7463.29135403605-14445.5266079862i            15.87850   3.98478   52.734375
                                at 52hz, a - multiplied by a - equals a positive, so for the 50hz, add a sine.

14   1.126      -5940.59149956592+6735.79475621755i            8.77068   2.96153   246.093750
                                at 246hz, a - multiplied by a + equals a negative, so for the 246hz, subtract a sine.

20   -0.568      -1429.15358218379-6241.73751283557i            6.25319   2.50064   351.562500
                                at 351hz, a - multiplied by a - equals a positive, so for the 351hz, add a sine.

so, to construct the exact final waveform, this is all you need:
(50hz Sine at 15.9 magnitude) - (250hz Sine at 8.8 magnitude) + (350hz Sine at 6.3 magnitude)


I'm confused. the instuctions I was given tell me to divide the complex answers by the total amount of samples and use sin on the complex part and cosine on the real part and add the waves together....
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Offline BrianHG

  • Super Contributor
  • ***
  • Posts: 3141
  • Country: ca
Re: Reconstructing a waveform from its harmonics.
« Reply #77 on: March 11, 2018, 09:27:10 am »
I'm confused. the instuctions I was given tell me to divide the complex answers by the total amount of samples and use sin on the complex part and cosine on the real part and add the waves together....

Yes, putting a sine and cosine together, with identical periods in each like this, will manipulate the phase of your output waveform.

Ok, looking at only one Fourier component line in your Excel spread sheet, the 52.7hz line #3, tell me exactly how you would plot this one and only sine tone with the Fourier real & imaginary component which is currently in your spread sheet at sample #3:

Samp#  Data        Fourier component                                                Magnitude i^2                    Frequency
3           0.848      -7463.29135403605-14445.5266079862i            15.87850          3.98478   52.734375

I will add another column with your sine generator in my spreadsheet and additional color line on the reconstruction plot to compare.  (I'm off to bed now, I'll do it first thing when I get up.)  (LOL, we'll force Excel to do more...)
__________
BrianHG.
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #78 on: March 11, 2018, 09:32:07 am »
If I was using just a sine I'd have to calculate the angle from the complex result and use the modulus so I get the combined magnitude and the correct angle from one sine wave instead of combining the sine and cosine waves as shown in the course.
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Offline BrianHG

  • Super Contributor
  • ***
  • Posts: 3141
  • Country: ca
Re: Reconstructing a waveform from its harmonics.
« Reply #79 on: March 11, 2018, 09:31:51 pm »
instead of combining the sine and cosine waves as shown in the course.
This is exactly what I asking you to give me, the combination of the 2.  I will plug in the formula you used into Excel to generate a new addition to the waveform display.  The idea here is to plot your courses method of reconstruction, for 1 frequency band, to see if that 1 sine result sits on the corresponding 1 frequency band which I have decoded with perfect phase.  Basically, since I have already perfectly decoded the 3 primary bands, I can turn off the other 2 bands and plot my 50hz in Excel on-top of your course's method to see if they are aligned.  Once done, we can switch to the 2 other sines individually and see how they line up, or skew, or be completely wrong.  The complex waveform you are regenerating can't give you this insight.

Summing the complete sine/cosine Fourier components should give the same results as other bands have no signal strength, except for those 3 spikes right up at the 18000hz.  You can also try your reconstruction ignoring all the bands above 9000hz.
__________
BrianHG.
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #80 on: March 11, 2018, 09:43:02 pm »
The course suggests using individual cosine for real parts and sini for imaginary parts, there is a mention of calculating the modulus and phase angle as well. The problem here is that due to the lack of resolution in the frequency analysis there is that inaccuracy in pinpointing the peak frequencies so replotting with those will lead to a non perfect waveform with each cycle being different. I have described it in my answer as heterodyning.
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Offline BrianHG

  • Super Contributor
  • ***
  • Posts: 3141
  • Country: ca
Re: Reconstructing a waveform from its harmonics.
« Reply #81 on: March 11, 2018, 10:47:28 pm »
So attached is the monkey see monkey do version. Next I will try the same with the corrected frequencies on a 50Hz fundamental.

Blue is using sine on the real parts, green is using cosine on the real parts, the blue seems to be right but out of phase

The construction of the your Blue line beginning at 0.25 is correct.  What you see to the left is like looking back in time.  This is still considered correct as the Fourier requires some number of samples in time to correlate the proper phase.

However, the amplitude is wrong by exactly half.  Either you reference sample data is 2x what it should be, or the Fourier was done wrong, or, when decoding magnitude, you need to double you output result.

Hint: The vertical axis title in your 'Waveform Spectrum' table in your Excel file is " i2  /amps2 ".  What do you think that '2' is after the amps.  I didn't write that, you did.  It is still possible that the mistake lies somewhere else.
__________
BrianHG.
 

Offline amspire

  • Super Contributor
  • ***
  • Posts: 3783
  • Country: au
Re: Reconstructing a waveform from its harmonics.
« Reply #82 on: March 11, 2018, 11:18:39 pm »
What teachers? what lectures? I have what appears to be lecturers note for module material and I study at home alone with no help. I have never been to the university itself. Apparently in the UK you can take a yea at uni at home around a day job.
Are you saying you are not in contact with tutors? That would be tough. Still, for me, I always found courses really hard when I was just trying to catch up. Whenever I got ahead of the subject matter in my own time just through self interest, the course was easy and fun.

Are you going to attend the university after this year?
 

Offline rhb

  • Super Contributor
  • ***
  • Posts: 2663
  • Country: us
Re: Reconstructing a waveform from its harmonics.
« Reply #83 on: March 12, 2018, 01:47:46 am »
Export column 2 as a CSV to a.csv
Export column 4 asa CSV to b.csv
 
Start Octave.

a=csvread('time.csv');
A=fft(a);
plot( abs(A));
hold
# look at the plot
B=csvread('frequency.csv');
plot(abs(B));
# the amplitude spectra overlay to within roundoff.

You will see that the amplitude spectrum is symmetric about the 512th sample in the complex series.  The frequency values above the 512th value are wrong in the spreadsheet. 

The truncation of the series, the fact that the sine waves are not multiples of the Fourier series frequencies and wraparound effects make exactly reproducing the time series from 3 sine waves impossible.  I suggest you sort the Fourier series by magnitude and then build up a plot adding the next lower magnitude frequency.  You can probably stop after 7 or 8 series.  Use 2*Pi*k/1024 for the frequencies where k is the sample index.  Those are the correct frequencies.  You can't actually convert to Hz unless you know what dT is.

I did all this in Windows 7 Pro using the official Octave binary and the Windows snip tool. Plus a bit of awk under Cygwin to extract the columns from the full CSV file.

The last plot is the 3 largest components of the Fourier series.  Each curve adds another component.  The individual components are not shown except the first.  The rest of the significant components are left as an exercise.

Hint:  Set all but the 4 largest components to zero and inverse transform.
« Last Edit: March 12, 2018, 04:42:41 am by rhb »
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #84 on: March 12, 2018, 08:47:30 am »
What teachers? what lectures? I have what appears to be lecturers note for module material and I study at home alone with no help. I have never been to the university itself. Apparently in the UK you can take a yea at uni at home around a day job.
Are you saying you are not in contact with tutors? That would be tough. Still, for me, I always found courses really hard when I was just trying to catch up. Whenever I got ahead of the subject matter in my own time just through self interest, the course was easy and fun.

Are you going to attend the university after this year?

I can email but that is not the same as being there and they can change with no warning. No I do not plan to actually go to uni. I am doing a HNC which is a year 1 uni course, then I can do a HND which is year 2, the third year gets you a degree, I dont know if I need to actually go there to do that but I plan to stop at a HND at most.
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #85 on: March 12, 2018, 08:50:07 am »
So attached is the monkey see monkey do version. Next I will try the same with the corrected frequencies on a 50Hz fundamental.

Blue is using sine on the real parts, green is using cosine on the real parts, the blue seems to be right but out of phase

The construction of the your Blue line beginning at 0.25 is correct.  What you see to the left is like looking back in time.  This is still considered correct as the Fourier requires some number of samples in time to correlate the proper phase.

However, the amplitude is wrong by exactly half.  Either you reference sample data is 2x what it should be, or the Fourier was done wrong, or, when decoding magnitude, you need to double you output result.

Hint: The vertical axis title in your 'Waveform Spectrum' table in your Excel file is " i2  /amps2 ".  What do you think that '2' is after the amps.  I didn't write that, you did.  It is still possible that the mistake lies somewhere else.

The amps they give is the square. I made a column of amps. I used the Fourier transformed data for the graph, no idea that is without looking at it again.
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Offline rhb

  • Super Contributor
  • ***
  • Posts: 2663
  • Country: us
Re: Reconstructing a waveform from its harmonics.
« Reply #86 on: March 12, 2018, 12:44:41 pm »
The spectral plot in the spreadsheet has been interpolated  with a sin(x)/x interpolator.  Many graphing programs do that when there are only a small number of samples in the display.  The displays I posted have not been.  The spreadsheet plot is actually closer to the true picture.  The actual peaks in the spectrum fall between the frequency domain samples.  You can also produce that result by zero padding the series and doing the transform.  Try padding the time series with 10240 - 1024 zeros and do the transform.

In any case the sums of 1, 2 & 3 terms are *exact*.  If you continue the result will converge to the time series to within round off error.

I've handed you 3/4ths of the answer.  I did so only because you were being misled by well meaning people who did not know what they were doing. I'm a retired PhD level  oil industry reflection seismologist.  I made a large part of my living by  knowing the Fourier transform *really* well.

Once you know the transform array layout and understand that sampling rate controls the maximum frequency of the result and the series length controls the spacing of the transform values all the rest is practice which I've not had since Octave went full MATLAB clone and the syntax I'm used to changed.
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #87 on: March 12, 2018, 01:06:00 pm »
I used microsoft mathematics to do my graph so data points should not be a problem as I simply plotted sine and cosine curves with thos amplitudes and frequencies, obviously the lack of data points in the first place will make perfect reconstruction a challenge and require human input.

I was hoping to learn Fourier properly on this course bu in reality they have tried to come up with material that is a cut down version of the full story and then assignments that are easy to do so that I pass. My problem is that I overthink everything and expect to learn something. On future modules I probably wont waste my time on the material and buy my own books. As soon as I am done with this module all 1600 pages of it are going in the waste paper, I have or can buy better books that if anything have indexes.....
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Online IanB

  • Super Contributor
  • ***
  • Posts: 9590
  • Country: us
Re: Reconstructing a waveform from its harmonics.
« Reply #88 on: March 12, 2018, 02:37:31 pm »
I was hoping to learn Fourier properly on this course bu in reality they have tried to come up with material that is a cut down version of the full story and then assignments that are easy to do so that I pass. My problem is that I overthink everything and expect to learn something. On future modules I probably wont waste my time on the material and buy my own books. As soon as I am done with this module all 1600 pages of it are going in the waste paper, I have or can buy better books that if anything have indexes.....

I think you might be expecting too much from an HNC level course. Learning Fourier transforms properly would be quite painful for students on a degree course. At HNC level I imagine they are just trying to get across an appreciation of the concept with one or two practical examples to show how it works.

It's admirable that you are trying to learn it more deeply, but I doubt that is what is being expected of you.
I'm not an EE--what am I doing here?
 

Offline rhb

  • Super Contributor
  • ***
  • Posts: 2663
  • Country: us
Re: Reconstructing a waveform from its harmonics.
« Reply #89 on: March 12, 2018, 04:51:23 pm »
I used microsoft mathematics to do my graph so data points should not be a problem as I simply plotted sine and cosine curves with thos amplitudes and frequencies, obviously the lack of data points in the first place will make perfect reconstruction a challenge and require human input.

I was hoping to learn Fourier properly on this course bu in reality they have tried to come up with material that is a cut down version of the full story and then assignments that are easy to do so that I pass. My problem is that I overthink everything and expect to learn something. On future modules I probably wont waste my time on the material and buy my own books. As soon as I am done with this module all 1600 pages of it are going in the waste paper, I have or can buy better books that if anything have indexes.....

The time series is reconstructable to numerical precision from the coefficients in the Fourier coefficients in the spreadsheet.  I verified that in the FFT plots I made.  The spreadsheet values exactly overlay the FFT done with Octave.  Therefore, if you do the inverse transform with all the spreadsheet coefficients you will exactly reconstruct the time series. Of the 1024 Fourier coefficients, only about 20 are needed to get very close to the time series which is the entire point of the exercise aside from the THD stuff which seems quite out of place to me, but I don't know what the course purports to teach.

I'm confused as to the origin of the spectrum plot in the spreadsheet.  It is *not* sampled the same as the Fourier coefficients.  Did you make the plot or was that provided by the course?

I think you will be way ahead if you buy 2-3 books on a subject, read them and *then* enroll in the course. 
 

Offline rhb

  • Super Contributor
  • ***
  • Posts: 2663
  • Country: us
Re: Reconstructing a waveform from its harmonics.
« Reply #90 on: March 12, 2018, 05:09:30 pm »
Here are some plots from Ronald Bracewell's 2nd edition which is arguably the classic textbook on the Fourier transform.

If you know the major transform pairs well you can usually do the transform in your head and draw it on a cocktail napkin by breaking the series into a sum of things for which you know the transforms.
 

Offline rhb

  • Super Contributor
  • ***
  • Posts: 2663
  • Country: us
Re: Reconstructing a waveform from its harmonics.
« Reply #91 on: March 12, 2018, 05:13:11 pm »
And some more.  There's another page of them in the book.  Bracewell presumes a good understanding of calculus, but it's well written.
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #92 on: March 12, 2018, 06:04:57 pm »

I'm confused as to the origin of the spectrum plot in the spreadsheet.  It is *not* sampled the same as the Fourier coefficients.  Did you make the plot or was that provided by the course?

I think you will be way ahead if you buy 2-3 books on a subject, read them and *then* enroll in the course. 

They provided the graph, it populates itself when you run the excel tool (didn't I say something about "monkey see monkey do" ? yes it's that bad).

I have a couple of books now but no time to read them in order to keep up with the timescales I am already behind on and they seem to be closing an eye on it as kicking me off means losing a few thousand pounds, about 1000 a year.
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #93 on: March 12, 2018, 06:13:56 pm »
I think you might be expecting too much from an HNC level course. Learning Fourier transforms properly would be quite painful for students on a degree course. At HNC level I imagine they are just trying to get across an appreciation of the concept with one or two practical examples to show how it works.

It's admirable that you are trying to learn it more deeply, but I doubt that is what is being expected of you.

The problem is I dont know how much I am supposed to learn, i was given a ton of math I struggled with and was then shown how to use excel. The problem is it feels a bit like a smoke screen and if I give an adequate smoke screen back i get a pass.
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Offline rstofer

  • Super Contributor
  • ***
  • Posts: 6540
  • Country: us
Re: Reconstructing a waveform from its harmonics.
« Reply #94 on: March 12, 2018, 07:02:08 pm »

I think you might be expecting too much from an HNC level course. Learning Fourier transforms properly would be quite painful for students on a degree course. At HNC level I imagine they are just trying to get across an appreciation of the concept with one or two practical examples to show how it works.

It's admirable that you are trying to learn it more deeply, but I doubt that is what is being expected of you.

Yup!  Slide rules didn't help that much...  The problem wasn't in solving the integral, it was in visualizing the results.  3Blue1Brown does a magnificent job of this.

I had my grandson watch the video linked above.  He was stoked at the magic of being able to extract frequency components, which leads to using my signal generator and DS1054Z to display the FFT of a square wave and I'll probably duplicate the setup with my Analog Discovery 2 (better presentation).

My signal generator will add harmonics - I need to set up an experiment with that.  I can turn a harmonic frequency spike on/off at will!  I can even adjust the magnitude of the harmonics as well as the phase.  Wait till he sees this!

Simon, your homework problems are great fun, keep posting them!
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #95 on: March 12, 2018, 07:19:28 pm »
Yes I remember explaining the basis of fuorier to a friend and showing him a sine and its 3rd harmonic added together with a sig gen and scope to show the start of a square wave. then I went on to show it in a graphical calculator to the 40+ harmonic.
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 

Offline BrianHG

  • Super Contributor
  • ***
  • Posts: 3141
  • Country: ca
Re: Reconstructing a waveform from its harmonics.
« Reply #96 on: March 12, 2018, 07:38:41 pm »
They provided the graph, it populates itself when you run the excel tool (didn't I say something about "monkey see monkey do" ? yes it's that bad).

I thought you created the spreadsheet, but this explains why things have turned out the way they did.  Either those who made the question deliberately thrown everything slightly out of whack on purpose, or, they thought that this example was the best choice of sample set & fourier size can achieve.  I doubt they were expecting you to sum all the first 100 or so fourier values unless they provided you with the instructional tools and software to do such a large set of points.

Very 'sneakily' within the hidden comment in cell 'J3' in the spreadsheet, there is a red arrow which only shows this 'comment' when your mouse is held on that point for a second, which then will exclusively tell you that your waveform's base is EXACTLY 50Hz.  I completely missed that and might have arrived at my solution much sooner if I knew that.  If I was doing a fourier here with this info, I'd do the fourier with only the first 720 samples instead of the full 1024 and would have had a perfect textbook 'monkey see - monkey-do' reconstruction first shot as the fourier frequency bands would hit perfectly on 50hz, 250hz, 350hz exactly.

« Last Edit: March 12, 2018, 08:08:47 pm by BrianHG »
__________
BrianHG.
 

Offline rstofer

  • Super Contributor
  • ***
  • Posts: 6540
  • Country: us
Re: Reconstructing a waveform from its harmonics.
« Reply #97 on: March 12, 2018, 07:39:44 pm »
Here are some plots from Ronald Bracewell's 2nd edition which is arguably the classic textbook on the Fourier transform.

Is this "The Fourier Transform and Its Applications"?

The year might be important because, according to Alibris, there are several editions starting in 1965 and working up to 2005 with prices from $20 to $400:

https://www.alibris.com/The-Fourier-Transform-and-Its-Applications-Ronald-N-Bracewell/book/2429291?matches=90
 

Offline rhb

  • Super Contributor
  • ***
  • Posts: 2663
  • Country: us
Re: Reconstructing a waveform from its harmonics.
« Reply #98 on: March 12, 2018, 10:40:26 pm »
Transform graphs to memorize:

boxcar <=> sinc

Gaussian <=> Gaussian

sine <=> positive and negative spikes

cosine <=> pair of spikes with same sign

squarewave <=> decaying series of odd order harmonic spikes

closely spaced spikes <=> widely spaced spikes

triangle <=> sinc**2

Hilbert operator

These are the things you have to deal with the most..  Other transforms popup from time to time, but generally you can get a pretty close approximation from some combination of these.  So if you're discussing it in the cafeteria or a bar after work you can convey what is going on.

I have The Fourier Transform and Its Application" 2nd ed.  I'm pretty sure they all have the dictionary of transforms.  It's one of the reasons it was so popular.  It's mathematically rigorous, though not at the level of "Operational Mathematics" by Churchill.  The latter consumed much of my life for two semesters. 

Book prices have gotten crazy.  Publishers like to push out new editions even though they are not justified in order to obsolete older editions.  But it makes the old edition cheap.  If you're building a personal library that's the way to go.

As 720 is not prime, there are FFTs which exist.  However, the most common transforms are the radix 2 transforms popularized by Cooley and Tukey and those are restricted to powers of 2 lengths.  Prime factor routines will handle 720.  My favorite is an algorithm attributed to Glassmann. The FORTAN version is only two pages with lots of white space.  Probably the best is Dave Hale's prime factor algorithm which uses a series of tests to select the fastest transform larger than the input series.

I don't mean to beat up on anyone, but the proper way to address the coarse sampling in the frequency domain is to pad the end of the series with zeros.   Adding 19,456 zeros will give sub Hertz resolution in the frequency domain.  That will also make clear the implicit sinc function imposed by truncating the series to 1024 values.

I did 3/4ths of the problem because Simon was being misled. both by the person who prepared the spreadsheet and forum members who don't quite know as much as they think they do.

I ran an "orphan home for lost problems" at large oil companies.  My tools of the trade for a lot of it were Octave, gnuplot, awk and CWP/SU.  The latter is a seismic processing package.  I supported it for many years but got weary of John's renaming my contributions. I thought authors were entitled to choose the names of their own programs.   It's somewhat buggy, but it's the best of the lot. Madagascar has potential if Sergey ever gets after documenting and debugging it.  The function fit routine in gnuplot is the best Marquardt-Levenberg L2 solver  I've ever used.  If you know what you're doing it will fit things nothing else will short of a sparse L1 pursuit.  The latter being the current sate of the art method for a vast array of problems.  However, it makes the traditional Fourier-Wiener-Shannon math look simple.
 

Offline Simon

  • Global Moderator
  • *****
  • Posts: 13848
  • Country: gb
  • Did that just blow up? No? might work after all !!
    • Simon's Electronics
Re: Reconstructing a waveform from its harmonics.
« Reply #99 on: March 16, 2018, 04:50:57 pm »
Well I am sure you will all be surprised to know that I got a distinction on this assignment. Standards are really much lower than I thought. One question went unanswered and I blagged a few by explaining how I arrived at the answer without using the expected techniques.

My new module "digital and analogue electronics" looks like a bit of a joke from the assignments that I can mostly do without the study material.
https://www.simonselectronics.co.uk/shop
Varied stock of test instruments and components including EEVblog gear and Wurth Elektronik Books.
Also, if you want to get ripped off: https://www.ebay.co.uk/usr/simons_electronics?_trksid=p2047675.l2559
 


Share me

Digg  Facebook  SlashDot  Delicious  Technorati  Twitter  Google  Yahoo
Smf