A Conversation with Mr. Fourier

A Conversation with Mr. Fourier

By Kurt Steffensen

Mr. Fourier, are you there...? - Mr. Fourier..?

Fourier :
"Yes, I am here"

Me :
" Where ? "

Fourier :
"Right here young man", - tapping my back.....

Me:
"Oh, you frightened me, Sir....Hmm....Cool clothes... " :-)

"Mr. Fourier, all this talk about sine waves, and squares built of sine waves, is that not just something you said many years ago, in order to gain a name ? "

Fourier:
" Listen, young man.. A little respect, would not be misplaced. A square is composed of an eternal amount of sine waves.That is the truth"

Me:
"Can you prove that ? "

Fourier:
"Certainly.- Square = 4/Pi ( eternal ) Epsilon times sine times 2n - 1 x 2 Pi frequency t divided with 2n-1."

Me:
- not really understanding the formula: " ehhh.. well , yes, I see. But wouldn't an f₂ round the flat top ? "

Fourier:
" Hmm. Not if they are not there. I mean it is composed of entire infinite ODD frequencies"

Me:
" Oh, yes, that makes more sence. But as the phase changes , wouldn't some of these compare to a second related frequency ? "

Fourier:
- a little irritated by now - "No, I most certainly assume that they are all in phase, from the same starting point"

Me:
"I see, Sir. Yes, I understand it now, but can you PROVE that...." - blink - blink :-)

Fourier:
"Of course, I can young man. 1f + 1/3 ( 3f) + 1/5 ( 5f) + ..... and so on in eternity

Me:
That seems right, Sir, I believe you. But please, come over here and see my tone generator. I have removed the top.- It is an old Trio, but it is real good"

Fourier:
"Trio ? ", takes a good and long look in to the PCB.. "What are all these small black things ?"

Me:
"Oh, that would be transistors, Sir, and ICs"

Fourier:
"Hmm...Where are all the transformers ? "

Me:
" That will be a long tale, Sir ... Lets just get down to the basic. We do not have all time".

Fourier:
" I'm in no hurry. I kind of like being here. You have so many interesting things , and you talk funny"

Me:
" - Funny..?... Hmm.. yes...Well, this is a generator , as you know, and it produces sine waves and square waves"

Fourier:
"Yes, yes, I know that. I just could not spot the spark gap relay"

Me:
" Spark..?...No, - but what I mean, is I do not have an eternal amount of sine wave generators in here, I only have one. And yet it is capable of producing perfect square waves"?

Fourier:
"Ha, ha.. You are so funny young man. You only need two batteries, and a shift relay to produce a square wave. It is a DC value, don't you get it ? " - Smiling from his head to his knees...

Me:
"Ahem, yes, I hope I do.... At least to some degree , but what about your statement before, with all the formulas ? "

Fourier:
"Oh, I see what you mean, silly young man. Do you actually imply that I claim we need a infinite amount of odd signals to produce a single square ? Of course, not, we can make it SO many other ways. Ha . ha. ha.... I really like being here" :-)

Me:
"But, Sir, I am only to...."

Fourier:
"Now, hush, young man, and I will explain it to you.. Ha, ha....This is SO funny.. - What I am proving with the square is that my analysis about the mathematical relation of sine waves signals, holds water. ANY signal can be composed and resolved in to sinewaves..Well, at least it goes for PERIODIC signals. The discrete sine wave is the fundamental signal, that can not be parted further. - According to the math that is..."

Me:
"Periodic signals ? "

Fourier:
"Yes, you know something that is repeated again and again. - A rotation, like a sine wave"

Me:
"Ah, I see. So it does not covers for music signals, like an orchestra ?"

Fourier:
"No, of course not. I never said that.The definition of a sine wave is 2 pi at constant seconds, that is electronically constant amplitude Vpeak sinus times 2 Pi. f seconds + constant phase angle. Certainly no music instrument signal at all is composed of such, young man. Musical instruments, as well as any other natural sound, may not even complete the entire cycle ( = sine wave ), and the fundamental is always gradually falling in amplitude. - Think of all the harmonics coming from say a violin string, the wood mechanics and reflections ? They bend and move in and out of phase, frequency and amplitude... Not to speak of all the reflections and intermodulation and tone difference that takes place... No, you would be very lucky indeed, if you as much as found one single entire pure sine wave, or even two periods repeating to become a cycle, son. - And do remember, that it takes at least two 100% identical periods to define/identify a cycle."

Me:
"Thank you very much Mr. Fourier. I am glad you would attend here to night and help us all with the questions we have discussed. I am sure we are all better informed now. At least I understand the matters better now"

- Goodnight, Mr. Fourier"

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