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Quadrature Multiplexing

 

Quadrature Multiplexing

Quadrature multiplexing (often called quadrature modulation / QAM/QPSK) is a technique in which two independent signals are transmitted simultaneously over the same carrier frequency by placing them on orthogonal carriers.

How it works

Two carriers of the same frequency are used:

  • In-phase carrier (I): cos(2Ï€fct)
  • Quadrature carrier (Q): sin(2Ï€fct)

These carriers are 90° out of phase (orthogonal).

If:

  • m1(t) → In-phase signal
  • m2(t) → Quadrature signal

The transmitted signal is:

s(t) = m1(t) cos(2Ï€fct) + m2(t) sin(2Ï€fct)

Why it works (key idea)

∫ cos(2Ï€fct) sin(2Ï€fct) dt = 0

So the two signals do not interfere with each other and can be separated at the receiver.

Advantages

  • Doubles data rate without increasing bandwidth
  • Efficient use of spectrum
  • Basis of QPSK, QAM, OFDM

Where it is used

  • Quadrature Phase Shift Keying (QPSK)
  • Quadrature Amplitude Modulation (QAM)
  • Modern wireless systems (Wi-Fi, LTE, 5G)
  • Cable and satellite communication

Quadrature multiplexing is a technique in which two independent signals are transmitted over the same carrier frequency using orthogonal carriers that are 90° out of phase.

Orthogonality & Integration Interval for Sine & Cosine 


Key identity

cos(2Ï€fct) sin(2Ï€fct) = (1/2) sin(4Ï€fct)

Correct integration interval

The integral is zero over any integer number of carrier periods.

One carrier period

Carrier period:

Tc = 1 / fc
∫0Tc cos(2Ï€fct) sin(2Ï€fct) dt = (1/2) ∫0Tc sin(4Ï€fct) dt = 0

General case

For any integer n:

∫0nTc cos(2Ï€fct) sin(2Ï€fct) dt = 0

Why this matters

  • The symbol duration is chosen as an integer multiple of Tc
  • This guarantees orthogonality
  • Allows perfect separation of I and Q channels at the receiver


The integral of cos(2Ï€fct) sin(2Ï€fct) is zero over any interval equal to an integer multiple of the carrier period Tc = 1/fc.

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