Editing Quadrature amplitude modulation (section)

== Demodulation ==
{{unsourced|section|date=December 2018}}
[[File:PAL_Vector.png|200px|right|thumb|Analog QAM: PAL color bar signal on a [[vectorscope]]]]

In a QAM signal, one carrier lags the other by 90°, and its amplitude modulation is customarily referred to as the [[In-phase_and_quadrature_components|in-phase component]], denoted by {{math|''I''(''t'').}} The other modulating function is the [[quadrature component]], {{math|''Q''(''t'').}} So the composite waveform is mathematically modeled as:

:<math>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),</math> &nbsp; &nbsp; '''or:'''
{{NumBlk|:|<math>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),</math>|{{EquationRef|Eq.1}}}}

where {{math|''f''{{sub|c}}}} is the carrier frequency.&nbsp; At the receiver, a [[product detector|coherent demodulator]] multiplies the received signal separately with both a [[cosine]] and [[sine]] signal to produce the received estimates of {{math|''I''(''t'')}} and {{math|''Q''(''t'')}}. For example:

:<math>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).</math>

Using standard [[list of trigonometric identities#Product-to-sum and sum-to-product identities|trigonometric identities]], we can write this as:

:<math>\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}</math>

[[Low-pass filter]]ing {{math|''r''(''t'')}} removes the high frequency terms (containing {{math|4π''f''{{sub|c}}''t''}}), leaving only the {{math|''I''(''t'')}} term. This filtered signal is unaffected by {{math|''Q''(''t''),}} showing that the in-phase component can be received independently of the quadrature component.&nbsp; Similarly, we can multiply {{math|''s''{{sub|c}}(''t'')}} by a sine wave and then low-pass filter to extract {{math|''Q''(''t'').}}

[[File:Sine and Cosine.svg|thumb|180px|right|The graphs of the sine (solid red) and [[cosine]] (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 [[colorburst]] 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 colorburst transmitted at the beginning of each scan line.
* [[C-QUAM]] ("Compatible QAM") is used in [[AM stereo]] radio to carry the stereo difference information.