# Quadrature modulation

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Quadrature modulation is the general technique of modulating two carriers.

Examples include Quadrature amplitude modulation, Phase-shift keying, and Minimum-shift keying.

Constellation diagrams are used to examine the modulation in the 2-D signal space.

## Explanation

Sending a signal by amplitude modulation consists of sending the function

$y(t) = I(t) \cdot \cos(\omega_c t)$

where $I(t)$ is the signal to encode and $\cos(\omega_c t)$ is the carrier wave, $\omega_c$ 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 to encode the signal, as in:

$z(t) = I(t) \cdot \cos(\omega_c t) - Q(t) \cdot \sin(\omega_c t)$

this 90° (the angle of a rectangle, or a 1/4 turn) is why it is called "quadrature" modulation, and the symbols $I(t)$ and $Q(t)$ indicate the "in-phase" signal and "quadrature" signal.

In terms of Euler's formula, $e^{it} = \cos t + i \sin t,$ 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 – 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 $(I,Q)$ 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 article: 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.