Understanding maths behind DFT?

Started by dhaval_shah 8 years ago15 replieslatest reply 8 years ago269 views


I am newbee to DSP. Started reading #DFT. I came across a book of "The mathematics of DFT".

Is it really helpful to dig into mathematics of DFT before understanding DFT?


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Reply by drmikeNovember 10, 2016

There is no understanding of DFT before digging into math - it is all math.  And like the other replies here, looking at problems with the math as a root allows you to see solutions to problems far more easily. 

There are fundamentals, and then there are details.  Understanding trigonometry has to come before understanding DFT.  Why?  Because a DFT breaks things down into sines and cosines.  Then there's calculus. That's actually easy to understand after you "get it", but it can be hard for some for that to happen in the first course.  Once you have the fundamentals, the details are not so hard.  Just time consuming. And worth every second - because it allows to deal with reality in a lot of ways you didn't think were possible before.

So YES!! Dig into the math.  It will help you a lot!

Patience, persistence, truth,

Dr. mike

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Reply by Fred MarshallNovember 10, 2016

I would say YES also.  That said, it's hard to know how one's mind works.  I used to read my engineering texts and skip all the equations the first time through. I guess I was more interested in WHAT concepts I was trying to learn and accomplish and that the math could always follow in a focused way.  It's indispensable but, I believe, only as you also know what you're doing. Then, math helps.  It's a tool.

Later, I realized that I probably didn't know enough about the Fourier Transform and had some time to "fix" that.  So I spent quite a bit of time studying it.  That time spent was invaluable.  But note:

Much of what I learned during that time was helped by using what I call "cartoons".  I had the good fortune to attend a commercial lecture series that used lots of figures to show signals and spectra, etc. without much math.  So, when I wanted to think through a sampling and filtering challenge, I (still do) draw figures.  They may not be mathematically perfect or even great but they sure help my thought process.  Rick Lyon's figures are better than the ones I draw because I usually don't worry about phase.  I must say that understanding what convolution does in a constructive way is a prerequisite to drawing the cartoons.  Much later, I made some great cartoons for half-band filters.

I'm not afraid of math at all.  It just bores me much of the time.  Perhaps one might think that I lack sophistication but I will tell you that a technical paper that's full of nothing but matrices and terse notation won't hold my interest  because I know (or assume) that the objective is to impress rather than to inform.  

I love the story of the acoustics engineer who imagined that he was riding on a phonograph needle in order to understand the forces and dynamics involved.  The math could only follow.

But this leads to the question: "Can the math precede?"  Well, some math really must precede some areas of discovery.  And it would be wrong to suggest that some discovery isn't fully based on mathematics.  I just prefer, and am perhaps hindered by, a more combined physical/mathematical approach.

So get books with lots of good pictures....


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Reply by Rick LyonsNovember 10, 2016

Hi fred3, if you have an American
version of the of my "Understanding DSP"
book I can send you the appropriate errata for your
copy of my book if you can tell me two things:
(1) the Edition number, and (2) the "Printing Number"
of your copy of my book.

You can determine the "Printing Number" of the 3rd Edition
of my book by looking at the page just before the "Dedication"
page. On that page (before the Dedication) you'll see all
sorts of publisher-related information, including the
ISBN-10 number.  At the very bottom of the page you
should see lines printed something like:

       Text printed in the United States on ...
       First Printing

indicating the "First Printing" of the book.  However, for
later printings the second line above may have words
such as: "Second Printing" or maybe "Fourth Printing".

My e-mail address is: R_dot_Lyons_at_ieee_dot_org.

Best Regards,

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Reply by Caminha-MacielNovember 10, 2016

Also YES!

I think you should get some knowledge in Fourier transforms and its main applications.

This is going to open your mind for a lot of intuitive solutions to signal processing usual problems and more advanced applications.

I highly recommend the Prof. Brad Osgood's course "Fourier Transform and its applications" from Stanford Elsewhere and available on YouTube.


Best wishes.

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Reply by neiroberNovember 10, 2016


The tricky thing is that textbooks are not always the best at clearly explaining the DFT. You may need to try a few to find an approach you like.  

You might try Rick Lyon's book, Understanding Digital Signal Processing, Chapter 3.

Also, if you have Matlab, it's a really useful tool for trying out your knowledge of the DFT.



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Reply by AllenDowneyNovember 10, 2016

It depends on what you are planning to do.  If you are using DFT to do spectral analysis and you are more interested in what the results tell you about the problem domain, and less interested in how the DFT works, you don't need to know about the math at all.

As an analogy, learning about how an internal combustion engine works is very useful if you are planning to design or repair one, but if your primary goal is to be a better driver, there are better ways to spend your time.

Similarly, you can do a lot of useful things with DFT-based tools, even if you have no idea how they work.  Many people will tell you that knowing the math will (somehow) make you a better user of these tools, but that assertion is usually made without evidence.

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Reply by Rick LyonsNovember 10, 2016

Hi AllenDowney. Your comment invites meditation. In one way I agree with you. I don't know how the Parks-McClellan FIR filter design algorithm works. Yet I've used software implementations of that algorithm many times in the past to design FIR filters. So far in my work, I don't need to know how the algorithm works, I just need to know how to use it.

But when it comes to using the DFT in one of its various applications, I think knowing all the internal mathematical operations performed inside the DFT is essential for understanding the many and varied behavioral characteristics of the DFT.

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Reply by kazNovember 10, 2016

If you drive a car and think how it works you know what is going to happen.

If I use a resistor do I need to know all about electron and hole currents ...etc??

It really depends on what level of requirement you are working to get and what tools are available. If you are using a tool- and you should generally- then you need to know basic concepts of its available features but skip internal details. Don't buy or read what you don't need.

For beginners the most difficult issue is identifying what needs to be known and what should be skipped.


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Reply by Rick LyonsNovember 10, 2016

Hello dhaval_shah,

The following web page may be helpful to you. The comments on that web page were written by guys who at one time were in exactly the same situation you now find yourself.


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Reply by dhaval_shahNovember 10, 2016

Sure. I will go through the link provided and I understand to go through the set of equations for DFT + other concepts as many ppl suggests here.

I have your book "Understanding DSP" and while browsing DSPRelated.com, I came across a book "Mathematics of the DFT" by Julius O Smith III.

Later one talks about deep math for DFT like quadratic eqs, imaginary plane etc, where as first one talks about DFT as a tool.

My long term plan is to develop filters for a specific application. Do I need to digest math behind DFT before switching to DFT from your book/any other book?


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Reply by Rick LyonsNovember 10, 2016

Hello Dhaval. I do not understand the question you asked in the last paragraph of your post.  Can you clarify your question using different words?

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Reply by dhaval_shahNovember 10, 2016

Hello Rick, I am planning to work on filters in future, so started working on DSP. Started reading your book and was also browsing this site and came across a book from Julius O Smith III.

So I was bit confused about which one I pick first, maths behind DFT or DFT from your book.

Hope this helps.


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Reply by Rick LyonsNovember 10, 2016

Hi Dhaval, the DFT is the single most important topic in DSP. So learn all you are able regarding the DFT. (For your work, keep in mind that the DFT is used to determine the performance of digital filters.) I suggest you read my Chapter 3. Experiment using the DFT with your favorite software.  Experiment, experiment, and experiment again. Then dive into J. O. Smith's material.  There's a lot to learn so don't worry if it takes many hours. The more effort you expend the more benefit it will be to you.

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Reply by dhaval_shahNovember 10, 2016

Sure. Thank you Rick and all for your valuable time.


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Reply by Tim WescottNovember 10, 2016

Yes!  Much of signal processing is, at its root, about solving differential or difference equations that are both exceedingly complicated and limited to certain types.  In other words, signal processing is about math, and the more math you learn, the more you'll be able to wrap your head around this funky DSP stuff.

All the above applies to the DFT.  The Fourier Transform was invented to solve differential equations, and the DFT is simply an extension of the Fourier Transform that is exceedingly useful for solving difference equations.  The math just goes down and down and down, so the more you know, the better you'll do.