Physics:Quadrature modulation

Quadrature modulation is the general technique of combining two amplitude-modulated (DSB) carrier signals in such a way that the original amplitude modulations are separable, by coherent demodulation, at the receiver. 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

With typical constraints on the two amplitude modulation waveforms, quadrature modulation can result in a single constant-envelope, phase-modulated carrier described mathematically by the trigonometric identity:

[math]\displaystyle{ \cos[2\pi f_c t + \phi(t)]\ \equiv \ \cos(2\pi f_c t)\cdot I(t)\ +\ \underbrace{\cos\left(2\pi f_c t + \tfrac{\pi}{2} \right)}_{-\sin(2\pi f_c t)} \cdot Q(t), }[/math]

and equivalently:

[math]\displaystyle{ \sin[2\pi f_c t + \phi(t)]\ \equiv \ \sin(2\pi f_c t)\cdot I(t)\ +\ \underbrace{\sin\left(2\pi f_c t + \tfrac{\pi}{2} \right)}_{\cos(2\pi f_c t)} \cdot Q(t), }[/math]

where:

[math]\displaystyle{ \phi(t) \ \stackrel{\text{def}}{=}\ \tan^{-1}\left(Q(t)/I(t)\right). }[/math]

Both carrier frequencies are [math]\displaystyle{ f_c }[/math] Hz, and the first one lags the second one by π/2 radians (90 degrees). It is in-phase with the composite waveform, and its amplitude modulation is designated by [math]\displaystyle{ I(t) }[/math]. The other carrier is said to be in-quadrature with the first, and its amplitude modulation is designated by [math]\displaystyle{ Q(t) }[/math]. The 90° phase difference between the original carriers makes them orthogonal, which is key to the subsequent separability of the modulations. Another key is that the modulations are low-frequency/low-bandwidth waveforms compared to [math]\displaystyle{ f_c, }[/math]  which is known as the narrowband assumption.

Demodulation

The addition of two sinusoids is a linear operation that creates no new frequency components. So the bandwidth of the composite signal is comparable to the bandwidth of the DSB components. Effectively, the spectral redundancy of DSB enables a doubling of the information capacity using this technique. This comes at the expense of demodulation complexity. In particular, a DSB signal has zero-crossings at a regular frequency, which makes it easy to recover the phase of the carrier sinusoid. It is said to be self-clocking. But the sender and receiver of a quadrature-modulated signal must share a clock or otherwise send a clock signal. If the clock phases drift apart, the demodulated I and Q signals 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.

See also

• Polar modulation
• Quadrature amplitude modulation
• In-phase and quadrature components
• Matrix decoder