Final value theorem: Difference between revisions

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'''Quadrature modulation''' is the general technique of modulating two carriers.
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Examples include [[Quadrature amplitude modulation]], [[Phase-shift keying]], and [[Minimum-shift keying]].
 
[[Constellation diagram]]s are used to examine the modulation in the 2-D signal space.
 
== Explanation ==
Sending a signal by [[amplitude modulation]] consists of sending the function
:<math>y(t) = I(t) \cdot \cos(\omega_c t)</math>
where <math>I(t)</math> is the signal to encode and <math>\cos(\omega_c t)</math> is the carrier wave, <math>\omega_c</math> is the carrier frequency – one is changing the amplitude of a carrier wave to encode the signal, hence amplitude modulation.
 
In general one could also change the phase of the carrier wave, as in [[phase modulation]] – there is a dimension of phase that is not being used. In fact, one can encode another signal that is 90° out of phase by using a sine wave, as in:
:<math>z(t) = I(t) \cdot \cos(\omega_c t) - Q(t) \cdot \sin(\omega_c t)</math>
this 90° (the angle of a rectangle, or a 1/4 turn) is why it is called "quadrature" modulation, and the symbols <math>I(t)</math> and <math>Q(t)</math> indicate the "in-phase" signal and "quadrature" signal.
 
In terms of [[Euler's formula]], <math>e^{it} = \cos t + i \sin t,</math> amplitude modulation encodes a 1-dimensional ''real'' signal, while quadrature modulation encodes a 2-dimensional ''complex'' signal. This viewpoint, that a wave of a given frequency can encode 2 dimensions of data, is elaborated in [[Fourier analysis]], and is the principle that quadrature modulation exploits.
 
=== Clocking ===
The added channel capacity is not costless, however.
 
An amplitude-modulated signal is [[self-clocking signal|self-clocking]] – it has zero-crossings at a regular frequency as a [[clock pulse]]. A quadrature-modulated signal, by contrast, has no such pulse, and thus sender and receiver must share a clock or otherwise send a clock signal – if the clocks drift by phase φ, which corresponds to rotation by φ in the <math>(I,Q)</math> plane, then the ''I'' and ''Q'' signal bleed into each other, yielding [[crosstalk]].
In this context, the clock signal is called a "phase reference" – in [[NTSC]], which uses [[quadrature amplitude modulation]], this is conveyed by the [[color burst]], a synchronization signal.
 
By contrast, in polar modulation, clock drift simply degrades the phase-modulated signal.
 
== Polar modulation ==
{{Main|Polar modulation}}
 
Quadrature modulates two signals by changing the in-phase and quadrature phase components, corresponding to [[Cartesian coordinates]]. By contrast, one can instead consider this to be changing the amplitude and phase of a wave, which corresponds to [[polar coordinates]]. The corresponding modulation is called [[polar modulation]], and was developed earlier, in the 1874 [[quadruplex telegraph]] by [[Thomas Edison]].
 
== See also ==
* [[Matrix decoder]]
* [[Polar modulation]]
 
{{DEFAULTSORT:Quadrature Modulation}}
[[Category:Radio modulation modes]]
 
 
{{telecomm-stub}}

Latest revision as of 16:42, 19 November 2014

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