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Author Topic: what is the second dimension of a sinusoidal signal ?  (Read 791 times)
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Hi, This must be one of those basic questions, but..

in my text books sinuses are claimed to be defined by a real and imaginary part, i.e. two dimensions.
Now the real part is the amplitude, but the imaginary part seems to be just a delayed version of the
amplitude. the imaginary part is said to be the phase, but phase is only a difference (delay) between two signals.

please enlighten me : why must a sinus be described with two dimensions/entities/parts and not
simply with it's amplitude (over time) ? and then : what does the second dimension describe ?

Thanks!
 


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There are rectangular and polar representation:
http://www.dspguide.com/ch8/8.htm
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thanks, nice link.
But I know the maths, in general.
my question is more about the fundamentals underneath. 

I just do not get what information I loose when only recording (amplitude) samples from my ADC.
what else should I record ?  why bother with that Q stuff ?


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Data array from your ADC is two dimensional , amplitude and time. May be time axis is not so obvious, each data sample has "time-stamp" - cell address in the array, this is why phase information would be preserved.
 What is Q stuff?
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Quote
why bother with that Q stuff ?
Are you referring to I/Q used in polar representation?
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Sinusoid signals are entirely real.  There's a bunch of math that works out to be easier to do if you express the trigonometric functions as complex exponentials instead ("Phasors"), but IIRC the imaginary parts are supposed to disappear or be ignored in the final result...
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Thanks for your answers.
I was indeed referring to the I/Q representation. 
it seems indeed that phase sort of relates to time delay, only that phase
is expressed in (a fraction of) number of waves (and hence varies with frequency for the same amount of delay..)

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The reason complex representation is used for harmonic signals is that the maths is trivial rather than tricky - to combine two signals with complex amplitudes you add the amplitudes.  The phase information is encoded in the complex amplitude and the sums all just work with complex addition instead of dealing with trigonometrical equations.  I'd rather do addition than solve a trigonometric identity the hard way - easier and little risk of mistakes.

In a deeper sense complex numbers are the essence of trigonometry - everything is straightforward using them.  And ultimately the universe is driven by complex numbers as quantum mechanics is entirely based on them.
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The phase of a sinusoid only means anything when it is compared to something else or a reference.
So for example the phase of a wave coming through a capacitor is 90 degrees out of phase with a signal that doesn't, providing they started in phase to begin with. Similarly an inductor gives a 90 degree phase shift in the other direction. This is important because when you try and mix signals of different phases you get a reduction or increase in the final signal. For example two equal amplitude sin waves with a phase shift of 180 degrees will produce no output. The waves cancel. When they are in phase the waves add up and you get twice as much.
When you start to drive signals through circuits consisting of capacitors and inductors the situation is complicated but by using complex numbers to describe the waves the net result can be easily calculated. This is useful in designing things like filters.

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And ultimately the universe is driven by complex numbers as quantum mechanics is entirely based on them.
Wrong way round, complex  numbers describe what is happening in the universe, they don't drive it.
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Well both ways of looking at it are anthropomorphic anyway! Its like the difference between believing we invented complex numbers as opposed to discovering them...  (my take is neither BTW, complex numbers invented _us_ as emergent behaviour.  Reommended reading: David Deutsch)
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http://xkcd.com/435/

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Yes very true.

I always say that Physics is the sub branch of mathematics that happens to deal with reality.

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Its like the difference between believing we invented complex numbers as opposed to discovering them.
No we actually invented them, they do not exist, they were not discovered. It turns out that they are amazingly powerful at describing a lot of things but they are a construct of the human mind. You can imagine some alien civilization inventing a way of describing the universe in terms of smell in a rigorous logically consistent way that would be similar to mathematics but would have no numbers or even the concept of numbers.     
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