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Very often we apply to circuits sinusoidal or constant voltages, and we study methods (like the phasors method) to analyze them. Here's a simple question to which, in many years, I have found not yet an answer: how can it be possible that a signal starts from -infinity and goes to +infinity? When studying sinusoidal circuits, every text applies a sinus or cosinus input (which never has a beginning), but there must be an exact moment in which the circuit switches on.

Another question: if I consider amplitude modulation, we know that a generic signal m(t) must be multiplied by cosinus in order to shift its spectrum, then the receiver will build again the original signal in some way, which is now not important. The questions are:

1st: the cosinus which multiplies m(t) in theory starts from minus infinity and finishes in + infinity; in reality every phisical signal must have a beginning, then: how theory would be modified if I consider a "real" signal (=a signal which has a beginning and a stop)?

2nd: In the theory of AM, the signal m(t) is already known (I mean, I can plot the graph of the function, I know m(t) in every instant), then I know for sure its spectrum. But if I think to a radio presenter, when he speaks the signal does not yet exist, I mean: m(t) is a "real time" signal, I don't know m(t) in every instant but it is "under construction". As a consequence, I don't know its spectrum and it seems to me that all the theory of AM is no more valid.

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    \$\begingroup\$ There is no paradox. The signal has a spectral limit from DC to f1 , the carrier is f2 and the result of AM is limited by the double sidebands of f1 centred on f2 and carrier/sideband ratio levels depends on modulation index \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 May 23 '18 at 13:07
  • \$\begingroup\$ Infinity is just a concept or mathematical construct for any path length that due to practical energy decay can be ignored after the medium latency or storage time constant has been transmitted. It might be seconds on earth or decades to Voyager 1 \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 May 23 '18 at 13:16
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    \$\begingroup\$ DC is a myth, the lowest frequency we can have is one over the age of the universe. Still it's a useful myth. \$\endgroup\$ – George Herold May 23 '18 at 13:31
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    \$\begingroup\$ As said to your previous question, there's no paradox here. The signal you're considering has a Fourier transform. Move on. \$\endgroup\$ – Marcus Müller May 23 '18 at 13:37
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    \$\begingroup\$ There are no signals starting at -infinity. All signals start at t=0 and, hence, we will always have a transient signal. This means: In theory, we will never have a pure (mathematical) sinewave because the transient will exist up to +infinity (or until we switch-off the system). However, for practical reasons, we neglect errors below 0.00....00 %. The reason is: Soon the error will be smaller than the noise or smaller than other unweanted contributions. \$\endgroup\$ – LvW May 23 '18 at 13:40
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For your first question

1st: the cosinus which multiplies m(t) in theory starts from minus infinity and finishes in + infinity; in reality every phisical signal must have a beginning, then: how theory would be modified if I consider a "real" signal (=a signal which has a beginning and a stop)?

Look up the Heaviside Step Function and how this can be used to model signals of finite duration. The important part though is that a time-limited signal, i.e. one that is not infinitely periodic, cannot be band-limited. So whereas a sine wave has a single spectral peak, a more realistic time-limited sine wave has a spectrum that extends infinitely. In practice, we might call where 90% of the signal energy is, the spectrum of this signal. So when you modulate this time-limited carrier wave the spectrum will be much wider and probably have some sidelobes compared to the perfect spectral peak.

With regards to this question,

2nd: In the theory of AM, the signal m(t) is already known (I mean, I can plot the graph of the function, I know m(t) in every instant), then I know for sure its spectrum. But if I think to a radio presenter, when he speaks the signal does not yet exist, I mean: m(t) is a "real time" signal, I don't know m(t) in every instant but it is "under construction". As a consequence, I don't know its spectrum and it seems to me that all the theory of AM is no more valid.

This is an excellent question that I remember being confused about before studying communications courses at university.

You are correct that it is impossible to know the exact spectrum of a signal if you don't know the signal, and in communications if everyone already knew the signal, why would you transmit it! We do know however the bandwidth of the signal, for example audio bandwidth isn't going to be more than 20kHz, and we can get away with transmitting much less. So the traditional AM theory you see presented normally just deals with the edges of the bandwidth, or a single frequency. The exact spectrum can't be predicted before the signal happens.

In digital communications, the spectrum of a modulation scheme is usually derived by assuming an infinite number of random bits. This requires a bit of knowledge of probability and stochastic processes, in particular correlation, but you can predict what the spectrum will look like as the number of bits approaches infinity. In practice, what you send will only look very similar to this theoretical result, and the exact spectrum of course depends on the exact time-domain signal you sent.

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There is no paradox.

Periodic functions have no beginning and no end. Signals always have a beginning and an end. Therefore, no signal can be exactly represented by a periodic function.

But, the mathematical techniques that we use to analyze periodic functions are both powerful and useful. So useful that we can pretend that a small segment of some actual, finite signal is a segment of some infinite periodic function, and we can analyze it as such, and learn something interesting about the signal.

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It's a mathematical assumption which is close enough to work with for steady state analysis. It's kind of like when we say noise has a Gaussian distribution, even though over time a true Gaussian distribution would result in an output voltage an actual power supply won't deliver.

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Here's a simple question to which, in many years, I have found not yet an answer: how can it be possible that a signal starts from -infinity and goes to +infinity? When studying sinusoidal circuits, every text applies a sinus or cosinus input (which never has a beginning), but there must be an exact moment in which the circuit switches on.

This is used to find the steady state solution to the problem. By considering that the signal has been present for a long time (think phasors, Fourier transform), all the transients have died out and you get the steady state response.

For this very reason, you don't get the complete solution—which you would get if you use the Laplace transform, for example. LT is for physical signals starting at t=0 (this is when the circuit 'switches on') and gives you transient terms plus the steady state solution.

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