In a QAM signal, one carrier lags the other by 90°, and its amplitude modulation is customarily referred to as the
in-phase component, denoted by The other modulating function is the
quadrature component, So the composite waveform is mathematically modeled as: :s_s(t) \triangleq \sin(2\pi f_c t) I(t)\ +\ \underbrace{\sin\left(2\pi f_c t + \tfrac{\pi}{2} \right)}_{\cos\left(2\pi f_c t\right)}\; Q(t),
or: {{NumBlk|:|s_c(t) \triangleq \cos(2\pi f_c t) I(t)\ +\ \underbrace{\cos\left(2\pi f_c t + \tfrac{\pi}{2} \right)}_{-\sin\left(2\pi f_c t\right)}\; Q(t),|}} where is the carrier frequency. At the receiver, a
coherent demodulator multiplies the received signal separately with both a
cosine and
sine signal to produce the received estimates of and . For example: :r(t) \triangleq s_c(t) \cos (2 \pi f_c t) = I(t) \cos (2 \pi f_c t) \cos (2 \pi f_c t) - Q(t) \sin (2 \pi f_c t) \cos (2 \pi f_c t). Using standard
trigonometric identities, we can write this as: :\begin{align} r(t) &= \tfrac{1}{2} I(t) \left[1 + \cos (4 \pi f_c t)\right] - \tfrac{1}{2} Q(t) \sin (4 \pi f_c t) \\ &= \tfrac{1}{2} I(t) + \tfrac{1}{2} \left[I(t) \cos (4 \pi f_c t) - Q(t) \sin (4 \pi f_c t)\right]. \end{align}
Low-pass filtering removes the high frequency terms (containing ), leaving only the term. This filtered signal is unaffected by showing that the in-phase component can be received independently of the quadrature component. Similarly, we can multiply by a sine wave and then low-pass filter to extract (dotted blue) functions are sinusoids of different phases. 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 (double-sideband) 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". Clock synchronization is typically achieved by transmitting a burst
subcarrier or a
pilot signal. The phase reference for
NTSC, for example, is included within its
color burst signal. Analog QAM is used in: •
NTSC and
PAL analog
color television systems, where the I- and Q-signals carry the components of chroma (colour) information. The QAM carrier phase is recovered from a special color burst waveformn transmitted at the beginning of each scan line. •
C-QUAM ("Compatible QAM") is used in
AM stereo radio to carry the stereo difference information. == Fourier analysis ==