I/Q Modulation

I/Q is a common way to represent message signals in modern communication systems.

There are two fundamental ways (degrees of freedom) to produce a modulated wave from a carrier wave: you can perturb either its amplitude or its phase. I/Q signaling allows you to simultaneously change both amplitude and phase. Using both degrees of freedom when modulating a carrier allows greater spectral efficiency with regard to bits per hertz of channel bandwidth (for a given bits per second transmission rate) compared to using only one degree of freedom.

To simultaneously perturb both the amplitude and phase of a carrier, two canonical forms of frequency-translating transmitters use the two degrees of freedom in modulation simultaneously. These canonical forms are referred to as canonical polar modulation and canonical rectangular modulation.

Canonical Polar Modulation

Figure 1. Canonical Polar Modulation


In polar modulation, the message signal is split into two components.

  • R(t) controls the carrier wave amplitude changes
  • φ(t) drives the carrier wave phase changes

Canonical Rectangular Modulation

Figure 1. Canonical Rectangular Modulation


In rectangular modulation, the message signal is split into two components.

  • I(t) controls the in-phase carrier wave changes
  • Q(t) controls the quadrature-phase carrier wave changes

Mathematical Relationship between Canonical Modulation Forms

The modulated outputs of the two canonical forms are mathematically and physically equivalent. To show their equality, you can compare and translate between the polar and rectangular representations.

Begin with the following equation:

R(t)cos(ωc(t) + φ(t))

Plug in the following trigonometric identity:

cos(α + β) = cos α cos β - sin α sin β

to yield the following equation:

R(t)[cosωc(t)cosφ(t) - sinωc(t)sinφ(t)]

Simplify the preceding equation to arrive at the following relationship:

I(t)cos(ωc(t)) - Q(t)sin(ωc(t))

where

  • I(t) = R(t)cos(φ(t))
  • Q(t) = R(t)sin(φ(t))
  • ω c = 2 πfc
  • f c is the carrier frequency in Hz

The rectangular form of modulation, often called I/Q modulation, has become popular for certain technical reasons.

In the rectangular modulation figure, the real baseband signals of I(t) and Q(t) are created (in some way) to contain all the information of message input m(t).[1]1 The polar R(t) and φ(t) also contain the information of m(t). Because the carriers cos(ωc(t)) and sin(ωc(t)) are orthogonal functions, we use the terminology of quadrature modulation, where the signal applied to the cosine mixer is called the In-phase component (I), and the signal applied to the sine mixer is call the Quadrature-phase component (Q). The I and Q designations are useful because the I(t) baseband signal is applied to the cosine mixer, and the Q(t) baseband signal is applied to the sine mixer.

The analytical relationship between the polar form (R(t) and φ(t)) of the baseband signal and Cartesian form (I(t) and Q(t)) of the baseband signal is shown in the following figure.

Figure 1. Relationship Between R(t) and φ(t)


The diagram is a "snapshot in time" of the complex (analytical) baseband envelope g(t). Because the diagram is for an arbitrary instant in time, the independent time variable t is dropped. The amplitude of Q is projected onto the imaginary (j) axis and the amplitude of I is projected onto the real axis.

I/Q Receiver

An I/Q receiver recovers a signal containing both amplitude and phase modulation.

For the incoming signal, p(t), the normalized outputs of the cosine and sine mixers are respectively:

I(t) + I(t)cos(2ωct) + Q(t)sin(2ωct)

Q(t) + Q(t)sin(2ωct) + I(t)cos(2ωct)

After the mixers, the lowpass filters remove all the high-frequency 2ωct terms, leaving the recovered I(t) and Q(t) signals.

The diagrams and discussion in this topic are ideal. They imply that the original message signal m(t) is recovered along the transmitter-path-receiver chain. This implication is useful when focusing purely upon the signaling method. In reality, however, the receiver returns a version of m(t) corrupted by noise and distortion impairments.

The vector signal analyzer measures the unit under test (UUT) noise and distortion impairments and determines if the performance of the UUT is sufficient for the desired transmission scheme. Because the I/Q signal representation is relatively straightforward (and also has practical and physical application in communication systems), it is conceptually used to evaluate the noise and distortion performance of these communication systems.

1 The polar R(t) and φ(t) also contain the information of m(t).