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An FFT creates what is known as a Bin Histogram, from Bin 0, at the far left which stores the amplitude of any DC the waveform may have, to Bin N/2 (N being the number of samples you present to the FFT) on the right. Bin N/2 is known as the Nyquist Frequency, which is Fs/2 (Sampling Frequency) Between these two bins are (N/2)-2 bins that store the amplitude of each frequency between DC and the Nyquist Frequency. The frequency resolution of each bin is equal to Fs/N or Ft/M where Ft is the test frequency and M is the number of cycles of the test frequency acquired during sampling. I explain this often and have explained it to tens of thousands of people, both in real life and on CZcams, Matthew Mahoney wrote a book on this (DSP Based Testing) but here is my most recent video on the topic. czcams.com/video/KScRCkFWVE0/video.html

@@DanBullard thanks..doing some relearning...

This is not for a 1980s distortion analyzer, or anything as primitive as a spectrum analyzer. This is for VLSI testers that can source and capture analog waveforms using Coherent Sampling. I speak about Coherent Sampling often, perhaps you should look around my site on this topic. As to your assertion that "energy from overlapping bins leak into each other" is totally false when using Coherent Sampling. If I source and capture a wave with coherent sampling, every femtovolt of the wave I sourced goes into the bin I selected (with my M/N ratios), not one single attovolt goes into an adjacent bin. You are living in the past if you think like that. Spectrum analyzers, scopes, distortion analyzers are all garbage. Only a VLSI tester such as a Nextest Lightning, Credence VIstaVision/Duo/Quartest, Sclumberger S9000 EXA, LTX Fusion, Teradyne A5XX can do what I am talking about. Interestedly, Raytheon used a Teradyne A585 on this F16 part, and even set up the timing between the AWG (source) and ACP (capture) to be coherent, but failed to use an FFT to extract the waveform, allowing noise to become a huge part of the resulting measurement, which is why they had to take 4 million samples and average them. I got infinitely better results by taking 8192 samples 500 times fewer samples, and shaving 10 seconds off the test time. Which really doesn't matter to Raytheon, test time can be in days as far as they're concerned, they get paid by the second, so they are more than happy to waste time.

I made a new video to address your concerns. czcams.com/video/pLS6_wXnKGQ/video.htmlsi=t3dBblfo69ciMrS-

Exactly!

You have 3K Ohms spread between 3 volts. There is no other current path through, or around R1, R2 and R3, because, as I state very clearly there will be no current flowing into the op amp, so R4 is a trillion ohm resistor as far as we are concerned. 3V/3K = 1mA, 1 volt will be dropped across each resistor, R1, R2 and R3. There is no 1.5V anywhere, unless you somehow got current to flow into the op amp, which is an absolute impossibility.

Think of it as the reference point (i.e. 0 Volts) for the voltages in the circuit. If you're using "ground" as "the reference point", then, yes. I've learned over the years to use "ground" carefully, because it has different meanings, depending on who you talk to (i.e. big difference if you're an electrician wiring up a house vs. an electronic tech fixing an amp).

The thing to remember is that all of these old school companies are dinosaurs, they keep plodding along using the same mentality for decades. When I was at Tektronix, which was a world leader decades ago, I tried to get them to adopt the FFT and other DSP based solutions. They thought I was crazy. And here we are, they are selling off their campus in Beaverton to pay the bills because they ignored the path to the future. Some gambles, pay off, Xerox didn't exist until Chester Carlson went to company after company demonstrating his dry copying process. Even Kodak turned him down. But the Haloid Photographic Company had nothing to lose, they adopted his process and the company Xerox was born. But decades later, Xerox handed their greatest inventions, the GUI WYSIWYG desktop and the mouse to Apple, and the rest is history.

@@DanBullard Hey thanks for replying, considering your expertise in the industry, you could make videos on these kinds of topics, on how small innovation can lead to huge changes. I think such videos will do well.

@@HarshitBujarBaruah Good idea!

@@HarshitBujarBaruah i think it will all be digital, if you want analog you’ll need to generate it digitally. A neighbor showed me an AI representation of his voice, completely indistinguishable from his own voice.

It's my opinion that the energy is still there for the Even harmonics, but just like a BMW Boxer engine with opposed pistons, the energy is cancelled by the other piston, so there is no Even harmonics when you have symmetry between the two polarities, or, between the two halves of the wave in TIME. That's why a 50% duty cycle square wave has no Even harmonics and a 50% Rise/50% Fall triangle wave has no Even harmonics. This reference to Odd and Even harmonics gets tedious without any differentiation, that's why I had my books printed in color with the Odd harmonics in blue and the Even harmonics in red. Can't do that in black and white, which is why it took me so long to find someone to print them, Luckily KDP does it pretty cheap.

@@DanBullard Thanks, that the energy is continuously there makes perfect sense to me (guitar player, and college windsurfer back in the day). When multiple waves' amplitudes add together or subtract people call it "interference" but the word is a bit misleading. I know that superposition is not harmonics, but back to harmonics, when I lift my finger off the fret board of the guitar, and place it softly on a harmonic node - basically where the vibrating string is not actually vibrating at that exact node but just for that particular harmonic - it silences the tuned note of the string, and other harmonics which don't also have zero displacement at that node, and sounds out only the harmonic with a stationary node where the finger is lightly pressing. I started reading through some of your online harmonics stuff...then I saw you have experience in dsp after being a Navy radio technical expert, and guess you know a thing or two about signals! I tinker around with ATmega328p with RFM95 LoRa transceivers, so it was serendipitous to bump into you in the internets and I plan to read a bit more of your stuff. Incredible and kind of you to publish so much useful stuff that can help people like me who are interested but don't have your experience or knowledge. -a

Dan, may I email you about a tricky harmonics question related to energy absorbed by a guitar string? I saw your email address on your website. On the other hand, you look like you are enjoying your retirement, you’ve already helped me and clearly thousands of people over your life who will forever learn from your publications, and I didn’t want to write uninvited. Also seems like we have this comment thread to ourselves, but don’t want to overuse it!

Can you hear the difference or not? I have no theory as to why you can hear it, but unless you are deaf (or dishonest) everyone can hear the difference.

And for the record, I've never used a spectrum analyzer. I do real time spectral analysis using the FFT.

​@@DanBullard A human (and animal) ear doesn't do DFT. It's a bunch of analog filters and amplitude detectors. Whatever artifacts you see in comparing DFT of different signals doesn't apply to an analog filterbank. In an analog filterbank (always with infinite impulse response), each amplitude detector produces continuous reading of amplitude. In DFT spectrum analysis, you get discrete samples of uniformly spaced FIR filters. A Dirac Delta function pulse will produce uniform DFT readings across all bands, same as white noise (except that white noise will produce random readings). The difference is that white noise produces random readings continuously with time, while a delta pulse will do that just once. You can apply an appropriate window to white noise, to generate just a single DFT worth of readings, this will sound kinda close to a delta function, just as a wide band click. To summarize: continuous white noise will be easily distinguishable from a Dirac pulse, while a time-windowed piece of white noise can sound close, if the windowing function length matches the bandwidth of filters in the ear's filterbank. Hope this helps.

​@@Alexagrigorieff Your summary says it all, "a time-windowed piece of white noise..." Only idiots do windowing, and how can you avoid windowing? Coherent sampling baby, originally popularized by Matthew Mahoney in his world famous book DSP Based Testing. I NEVER "window" anything, I never filter ANYTHING. But it's clear you have never heard of Coherent Sampling which removes all error from DSP based waveform capture and analysis. White noise is identical to an impulse spectrally, but if you look at the phase (no windowing needed unless you are an idiot and don't know how to coherently capture a wave) the phases of all the harmonics of an impulse have a 0 degree offset relative to the fundamental, they are all cosine waves, whereas the phases of white noise are random relative to the fundamental tone, and you can hear the difference even though the two spectra are identical. If you don't know how to measure phase, you need to relearn complex numbers and the arctan2() function. You can throw out all the theory you want, I have used this stuff in the real world, the WHOLE world, from Taiwan and South Korea to Austin Texas to Munich Germany to Glasgow Scotland and my favorite, Analog Devices in Limerick Ireland. In fact I guarantee that you have one of my inventions on your person at this time. It's in your wallet, it's the smart card that allows you to buy things. That card was tested with my PATENTED invention, look it up.

@@DanBullard I hope you are aware that Coherent Sampling is only applicable for a known input signal frequency. You cannot apply coherent sampling to a random signal, especially white noise.

How accurate a squarewave source were you using? Was it accurate to 1 femtosecond? You can't do this with 20 year old leftover Radioshack instruments, even modern instruments are not accurate enough in most cases. Watch my Applicos video to see how this works. czcams.com/video/CHfeMGQC6WI/video.html

@DanBullard the rise time of the signal is around 5ns but it's hard to know the transient properties well from my 150 MHz probes. It's coming from a Teensy 4.0 so as good as that thing can output I guess plus like 12 inches of unshielded wire. The two loads might be coupling in the device and that's what I think might be causing the asymmetry

@@jacobrandall4891 I suspect you're trying to disprove the speed of light by pumping a signal down a coax cable. The speed of a signal in RG58 is like 0.6C, but that doesn't disprove Einstein, it simply means that a practical experiment in non-ideal conditions doesn't disprove a general principle. If you watched my Applicos video, you can see that it doesn't take much asymmetry to get Even harmonics to pop up, 4mV in a 5V signal was all it took to reveal the Even harmonics. I seriously doubt you can do this experiment with an O'scope no matter what your signal source. When I look at square waves I digitize them with a single bit (watch "Using Aliasing to change the world") so there is no asymmetry from the amplitude, since the only voltage information is wrapped up in 1 bit, 0 or 1.

@DanBullard yea ok that makes sense. Thanks for the input. I wanted an insightful explanation to my boss why even harmonics are showing up in measurements and not simulation instead of instead of just saying an "asymmetrical distortion in the system". Thanks for the video referenc. I learned a little more about the Fourier Transform than I did before. BTW I was capable of measuring these even harmonics of an S-band wireless system on a cabled digital signal oscilloscope sampling at RF, but a second lower bandwidth scope with 150MHz maximum. That's why I could measure the even harmonics that had a power of like -90dBm, not on a $ 200 scope 😂

Thanks, very nice to hear your opinion.

This circuit is used to test an interviewee's knowledge, so the purpose of R4 and R5 is to see if the interviewee knows that no current ever flow into or out or the inputs.

@@DanBullard Aha! Thanks for the response. I hope you will keep doing these videos, good teachers with good knowledge are rare.

Perhaps the first person to get that.

Thank you Matt! Great to hear your kind words. If you want more info on this I would suggest you follow the lead of a guy named Tino Pfützenreuter who recently asked me for a copy of my transfer function in Excel. I gave him one of my Excel spreadsheets that includes the transfer function as well as all the other tabs to do the FFT and display the harmonics. If you would like that please email me at dan@danbullard.com and I will send you a zip file of it (I promise there are no viruses).

No equations here, there are plenty of videos that explain these with equations. Watch those.

3V across 3K is 1mA with a clear path to ground, NOT ZERO. That's a simple voltage divider. Does that do it?

@@DanBullard yes perfect sense now that I look at it like that, I was doing the math on each separately on each resistor to find the current instead of the whole series to ground. Been a while, just getting back into this stuff so I gotta shake off the cobwebs. Thank you so much.

@@DanBullard my next question is what is the point of R4?

@@matthewlocke1942 To let you prove you know how Op Amps work. This is a classic interview circuit, it doesn't actually do anything, but employers use it to see if you really understand Op Amps. I developed this video for a friend who was interviewing for a job at General Electric in Reno NV. Long story short, she got the job.

@@DanBullard ok so it’s just there to show there is no actual current flow, and if you wanted to prove you know how it works you would show 0 voltage drop, got it. Sorry I’m trying to learn more about op amp circuits in my own time and I didn’t catch that part in the intro. Thank you for your time, I really appreciate it.