# Quadrature Amplitude Modulation - Analog QAM

Analog QAM

When transmitting two signals by modulating them with QAM, the transmitted signal will be of the form:

\begin{align} s(t) &= \Re \left\{\left e^{j 2 \pi f_0 t}\right\} \\ &= I(t) \cos(2 \pi f_0 t) - Q(t) \sin(2 \pi f_0 t) \end{align}

where, and are the modulating signals and is the carrier frequency.

At the receiver, these two modulating signals can be demodulated using a coherent demodulator. Such a receiver multiplies the received signal separately with both a cosine and sine signal to produce the received estimates of and respectively. Because of the orthogonality property of the carrier signals, it is possible to detect the modulating signals independently.

In the ideal case is demodulated by multiplying the transmitted signal with a cosine signal:

\begin{align} r_i(t) &= s(t) \cos (2 \pi f_0 t) \\ &= I(t) \cos (2 \pi f_0 t)\cos (2 \pi f_0 t) - Q(t) \sin (2 \pi f_0 t)\cos (2 \pi f_0 t) \end{align}

Using standard trigonometric identities, we can write it as:

\begin{align} r_i(t) &= \frac{1}{2} I(t) \left - \frac{1}{2} Q(t) \sin (4 \pi f_0 t) \\ &= \frac{1}{2} I(t) + \frac{1}{2} \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 may multiply by a sine wave and then low-pass filter to extract .

The phase of the received signal is assumed to be known accurately at the receiver. If the demodulating phase is even a little off, it results in crosstalk between the modulated signals. This issue of carrier synchronization at the receiver must be handled somehow in QAM systems. The coherent demodulator needs to be exactly in phase with the received signal, or otherwise the modulated signals cannot be independently received. For example analog television systems transmit a burst of the transmitting colour subcarrier after each horizontal synchronization pulse for reference.

Analog QAM is used in NTSC and PAL television systems, where the I- and Q-signals carry the components of chroma (colour) information. "Compatible QAM" or C-QUAM is used in AM stereo radio to carry the stereo difference information.