Quadrature amplitude modulation (QAM) is the name of a family of signal modulation methods widely used in modern telecommunications to transmit information. At its core, it conveys two independent analog signals by changing (modulating) the amplitudes of two differently phased versions of a single carrier wave using amplitude modulation. These paired analog signal channels may then be used either directly or to encode digital bit streams using joint amplitude-shift keying across the synchronized channels. The two carrier waves are of the same frequency and are out of phase with each other by 90°, a condition known as orthogonality or quadrature. The transmitted signal is created by adding the two carrier waves together. At the receiver, the two waves can be coherently separated (demodulated) because of their orthogonality. Another key property is that the modulations are low-frequency/low-bandwidth waveforms compared to the carrier frequency, which is known as the narrowband assumption. In M-ary transmission amplitude-shift keying the phase is the same but with different amplitudes, while phase-shift keying (PSK) has the same amplitude but different phases. Combining these concepts leads to QAM, where both amplitude and phase are modulated, or two binary PSK signals are combined with orthogonal carriers. QAM is used extensively as a modulation scheme for digital communications systems, such as in 802.11 Wi-Fi standards. Arbitrarily high spectral efficiencies can be achieved with QAM by setting a suitable constellation size, limited only by the noise level and linearity of the communications channel. QAM is being used in optical fiber systems as bit rates increase; QAM16 and QAM64 can be optically emulated with a three-path interferometer.
Demodulation
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 I(t). The other modulating function is the quadrature component, Q(t). So the composite waveform is mathematically modeled as
s s ( t ) ≜ sin ( 2 π f c t ) I ( t ) + sin ( 2 π f c t + π 2 ) ⏟ cos ( 2 π f c t ) Q ( t ) , {\displaystyle 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
where fc 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 I(t) and Q(t). For example:
r ( t ) ≜ s c ( t ) cos ( 2 π f c t ) = I ( t ) cos ( 2 π f c t ) cos ( 2 π f c t ) − Q ( t ) sin ( 2 π f c t ) cos ( 2 π f c t ) . {\displaystyle 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
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