Mixed Square Waves Filtered vs. Sinewaves

Thank you for your detailed explanation. I have printed it out for future reference.

Klaus Jensen

square wave into the sum of series of sinusoids, starting with a sine wave of the same frequency as the square wave and the same amplitude, and p rogressing through the odd harmonics of that sine wave, with the amplitude of each harmonic inversely proportional to the harmonic number - that is the first odd harmonic, which is third, and thus at three times the freque ncy of the square wave, has one third of the amplitude.

, in the same direction, so there's no necessity for complex components.

l.

The time waveforms can always be real. One way of representing a phase diff erences between sinsoidal components of different frequencies is to split t hem into into real and imaginary components.

Declaring that the frequency domain is COMPLEX without spelling out what's going on is unhelpful. Pedagogues get attached to their favourite teaching scheme, and fail to remember that their job is to get the student to unders tand what's going on, as opposed to getting the students into a state where they can regurgitate their lessons under exam conditions.

It's more 'unhelpfull' to make completely erroneous statements, like the Fourier Transform of a 50% duty cycle waveform is REAL.

What is really important, and I sadly glossed over, is a good explanation of the 'replicating' function.

split them into into real and imaginary components.

what's going on is unhelpful. Pedagogues get attached to their favourite teaching scheme, and fail to remember that their job is to get the student to understand what's going on, as opposed to getting the students into a state where they can regurgitate their lessons under exam conditions.

Fourier Transform of a 50% duty cycle waveform is REAL.

A "Fourier transform" isn't necessarily real or complex - it's just a question of how you chose to represent the sinusoidal components that are summed to reproduce the arbitrary waveform being transformed into a "sum of sinusoids" representation.

of the 'replicating' function.

If you say so.

Huh? I don't 'choose' to arbitrarily represent the sinusoidal components, they are just there. Perhaps, you are saying how I choose to define t=0? which then sets the sinusoidal components. [I'd hate to be a student of yours]

Given time waveforms, Cosine transforms are REAL. Sq Wave transforms, of ANY duty cycle *IF* centered at t=0, are REAL.

You have still left the uncorrected impression from your original statement that the Fourier Transform for a 50% duty cycle waveform is always REAL, that type of declaration could mislead a 'student'.

they are just there. Perhaps, you are saying how I choose to define t=0? which then sets the sinusoidal components. [I'd hate to be a student of yours].

You'd certainly take more teaching than most, so the feeling is mutual.

ANY duty cycle *IF* centered at t=0, are REAL.

It has clearly mislead you, but you aren't acting much like a student - more like the pre-programmed exam passer who has mastered one way of thinking about Fourier transforms without having got his head around the basic idea.

Trying thinking about Walsh transforms as well as Fourier transforms and try to pick out the common basic idea (and it isn't complex numbers).

You are here to learn, not to pontificate, particularly when you aren't remotely pontif-candidate material.

I agree, that most engineers really haven't really got a handle on the bigger, general picture of Fourier, or Walsh expansions for that matter.

I view FT and such like, really from the general expansion of functions on an orthogonal vector basis set. You miss a lot from just attacking FT from a one off process rather than generalised dot products and expansion of vectors. You can derive all sorts of stuff from the general without worrying about the specifics. There are really nice links between Sturm?Liouville theory and Quantum Mechanics' Hermitian operators and Hilbert spaces. A FT being just one specific trivial detail of the theory.

Kevin Aylward

I do/did learn. YOU on the other hand made an erroneous statement. After the error 'gently' being pointed out to you, instead of correcting your error for others, you actually changed the subject and went into some kind of show off mode. That's Ok. At that time, no comment from me regarding your avoiding the subject and your not correcting your statement [maybe I did, not germaine]. I did not, as I rightly should have, in any way chide you for your methods of discussion - that is your methodology of demeaning others by name calling and changing the subject in order to move to something you can succeed at. Now, when only brought BACK to the subject STILL would not correct your statement, instead start on personal attack. and more 'show-off' subject changing.

I would point out much more to you by asking rhetorical questions, but I'm afraid you would answer them in a manner that requires more of my time. I'll pass.

Again, the frequency spectrum of a 50% duty cycle time waveform is NOT necessarily REAL. Depending on where one selects t=0 determines that. If t=0 is in the exact middle, all is REAL. If t=0 is at the rising edge, all is COMPLEX. [actually all imaginary].

BACK TO OUR SUBJECT: All time waveforms by their very nature are REAL [they can be represented in all kinds of ways using complex numbers, but the result is ALWAYS REAL] and the frequency spectrum, a mathematical manipulation, is a complex plane. Fortunately, since time waveforms are always REAL, the frequency spectrum from 0 to +F is the complex conjugate of the frequency spectrum from 0 to -F; sometimes a very useful feature.

I've pointed all this out without once calling you an idiot. And, that's the way to discuss.

You are welcome. Wish could be better detail and have better step by step. If you're into books, Ron Bracewell of Stanford wrote "Fourier Transform" or such. it's a great reference with an appendix of pictures showing time/frequency domain for many functions.

Do you have octave? If so, it's very useful to explore your original subject comparing pure sinusoids and various shaped functions, etc. then add filtering, etc. ...back to everybody's suggestion of just simulating to gain understanding.

But nowhere near enough.

I didn't. Try to learn enough to be able realise why it wasn't erroneous.

I quote from your original post: On Sun, 24 Aug 2014 09:53:44 -0700, Bill Sloman wrote: "It can be, but it isn't in this case. Taking the Fourier transform of a

If t=0 occurs at the rising edge, YES, it actually does generate complex components.

As said earlier, you should correct your erroneous statement, many who read these posts may be left with the impression that your statement had value as being true. It was wrong and misleading.

If all of the components have a magnitude that can be represented by a real number, is it complex? That is, if all components have a phase of 0 or 180, is the series complex?

s.

50% duty cycle square wave doesn't actually generate any complex components."

x components.

And you think this is true? Try to explain why.

read these posts may be left with the impression that your statement had value as being true. It was wrong and misleading.

I'm afraid the boot is on the other foot. There's no need add the labels "r eal" and "imaginary" to any of the sinusoidal components that can be seen a s adding up to form a 50% duty cycle square wave, and your trying to do so was just adding an unnecessary and unhelpful complication to the exposition .

It did allow you to advertise that you'd done a course on Fourier transform s. What you don't seem to have realised is that you never got your head aro und the underlying concepts, and you've advertised that fact at length.

All the components are indeed 0 or 180 in reference to the fundamental, BUT, depending on where one selects t=0 completely determines that 'initial' angle.

For example, t=0 is centered on the positive pulse, the terms are all REAL consisting of what you describe, 0, 180, 0, 180, or in terms of real, +,

-, +, - and, yes, are all real.

However, *IF* t=0 is selected at the rising edge of the pulse the terms are again relative to the fundamental 0, 180, 0, 180 with the first term at -1i [I think] So that would be -90, +90, -90, +90 And, are all COMPLEX.

What I was not agreeing with was the sweeping concept that a 50% duty cycle square wave somehow automatically makes the frequency spectrum only REAL. Just not the case.

Again, you should correct your erroneous statement. Not change subject for the sole purpose of showing off. And, in keeping in the vein of not making more erroneous statements, avoid unfounded personal attacks.

It was a very specific comment about a very specific waveform. Where your "sweeping concept" comes from eludes me.

And the use of "real" and "imaginary" components to document the phase relationships between a set of functions being used to represent an arbitrary waveform is not fundamental.

You can equally well use polar coordinates to convey the same information - and a phase-shifted harmonic is a real waveform, and an equally valid way of representing the information you can extract with a Fourier transform.

OK, if all of the components have an angle of 45 and 225, is the series really complex or are you just doing more work than you need to do? It seems a constant (time) will make everything right. It's all relative.

QED.

Who cares? You're doing the sampling. Why make life difficult?

That is the case.

s "real" and "imaginary" to any of the sinusoidal components that can be se en as adding up to form a 50% duty cycle square wave, and your trying to do so was just adding an unnecessary and unhelpful complication to the expo sition.

sforms. What you don't seem to have realised is that you never got your h ead around the underlying concepts, and you've advertised that fact at leng th.

AS you should have been able to work out by now, it wasn't erroneous.

How was the subject changed?

d unfounded personal attacks.

Persistently telling me that I've posted an erroneous statement - when I ha ven't - is definitely a form of personal attack. Your persistence in this l unacy makes it necessary for me to explore the defect in your understanding that leads you to keep on making a false claim. You may experience this as a personal attack, but you'd be in better shape if you treated it as conti nuing education (which does seem to be necessary).

The proper reaction to being caught posting an error is to apologise. I've done it from time to time. Denial just makes you look even more stupid.

The

Not quite true. There will remain the 150 Hz component of the 50 Hz square wave in the filtered result. Then you have unstated phase issues to consider.

?-)