hotpaw2

You posted a mathematical identity, one side used negative numbers, one side did not. How can only one side of an identity be the correct description?

Might just be a mix or real cosine waves and real sine waves, no imaginary signal generator boxes required.

The OP likely wanted a description that did not involve implementation details, such as what type of multiplication might be involved.

That's one possible description (likely preferred by anybody who's studied complex algebra), but not the only description (as typically used by "normal" humans). The assumption is that the OP is part of the majority of humans.

I sell strobe tuner software. The new apps use complex FFTs. The really old versions just used strictly real sine and cosine math. The new ones use far less code, and the code is far more readable. BUT, they worked equally well.

You seem to be confusing frequency with frequency differences. Two people with different heights do not have negative heights on their medical records. Even when comparing them back to back.

I think math professors sometime are confused that because a certain formulation is far more elegant, and very importantly uses far less chalk on the chalkboard, when using complex exponentials instead of a mess of sines and cosines, that only their form of description is correct.

Actually, professional piano tuners do tune by listening for beats, the phase dependance against the reference oscillator. And using fairly expensive, not cheap, tuning forks and strobe devices.

Only for signals that are asymmetric with reference to whatever you consider to be T0.

I note your own answer above does not say the the radio also transmits an equal amount of RF at -14.3 MHz.

No one tunes their guitar to e^-jx, even if it shows up that way on some chalkboard. Two different domains.

A 1 kHz audio signal is a 1 kHz audio signal, not a -1 kHz audio signal. The frequency of an audio spectrum is described differently than the frequency of audio (what you set on the dial of your tone generator box), and can include stuff that doesn't exist in the real world.

Any finite duration signal (less than the lifetime of the universe and beyond) has sidebands (it's part of the mathematics of the support of distributions, and of transforms). The shorter the signal (even a "pure" sine wave), the wider the sideband support.

If you are an electron, how, at a modulation peak, can you tell whether you were hit with 2 Volts of EMF, or the sum of a 1 Volt field plus two 0.5 Volt fields at the same time? Or at the modulation trough, hit with nothing, or a 1 Volt field going one way, plus two half Volt fields going the other way? So how can you, the electron, say you weren't hit by one combo, but were by the other possible combo ???

Due to both QM and a non-zero temperature, the electrons are not moving together. Their motions average together to the same as 3 types of motions, or one changing type motion, which statistically look the same to any observer. e.g. at any given time, some percentage of electrons are probably going the "wrong" way.

You can't say there aren't 3 waves (in superposition) being emitted if no physical experiment (or mathematical calculation) can tell the difference. Similar to special relativity or General Relativity.

It's not "not" if the transmitter electrons can't tell the diff between 1 varying force, and 3 forces that sum to the same (varying) electron acceleration. They are clueless. Same with the reception of the RF EM photons by the electrons in the receiver antenna.

@Dan : your assumption still appears to be that you (or the electrons) can tell the difference between 1 modulated wave, and 3 unmodulated waves mixed/summed. With AM modulation you can't. Neither can the electrons.

Now with quantum mechanics, you might ask what frequency RF photons are being emitted by the antenna. But the QM uncertainty principle probably says that you can't statistically tell the difference between 3 slightly different energies of photons, sometimes cancelling each other out, due to phase, and 1 energy of photon coming at a varying rate. Likely, the electrons in your antenna are similarly confused.

@Dan: don't think of what's being generated. Think of what's being received.

Headphones on a softrock are well after a lot of filtering in the signal chain, which obscures the details of the quadrature offset. Instead look at the voltage measured at each analog switch closure, I and Q channel, of a Tayloe mixer (an analog sampler) while varying the frequency of the receiver's RF input signal.

You are confusing trigonometry with actual voltages and analog switch closures (or diode forward bias periods).

And rotation occurs between the peaks of the two sampling waveforms. So no longer in 90 degree quadrature at those 2 sampling peaks (or switch close zero crossings). But still rotating in circles.

One of your blog posts mentioned also acquiring a voltage field probe. You might want to use the voltage field probe around the transmitter to see if there is any difference in "RF in the shack" between center and end fed configurations. With and without counterposes of varying lengths (including 0.05 and quarter wave).

Excellent measurement setup to remove potential feedline radiation and support parasitics from your test setup (without a multi-million dollar chamber). A better question to ask in your question heading is what might cause any other sources of experimental error in your test setup, such as what might be causing the dip at the far end of the dipole. I would also measure the field strength all around the transmitting Pi as well to see how much it became part of the antenna. Maybe by probing with the sensor on the end of a really long fiberglass pole.

Maybe not 13 years old. More like 16 or 17, back in the days before electronic calculators, where almost every high school student taking college prep courses for physics or math had to learn to use a slide rule and/or log tables.

Read up on the works of William Kahan, who was a professor in both the Mathematics and EECS departments when I was at UC Berkeley.

Any Turing-complete computer architecture can emulate any other, given enough memory and time. So a stack can be emulated, if needed, perhaps (very) slowly.

@grgarside : the reported clock speed may or may not be correct. Hard to tell since it's a private API in iOS. A better question would be about actual iPhone behavior, not some possibly random number.

@grgarside : you trust geekbench measurements that much? I don't. AFAIK, the real clock speed number on iOS devices is reported by a private API.

As a comment, because this answer isn't really CS related: Quaternary, in practice, with the 4th symbol less well defined. People learning to transmit or receive code usually first learn individual letters as dots and dashes with 1 unit spacing (Ternary code per letter). They then often send messages with somewhat arbitrary spacing between letters, sometimes with run together words or very long thought pauses between words, or with purposeful extra long word/letter gap Farnsworth timing to make it easier for less skilled individuals to mentally decode higher WPM dot timing.