## Introduction

This document describes the impact of amplitude and phase
mismatch in an I/Q (in-phase/quadrature-phase) signal pair. The result of a mismatch is an
unwanted spectral component at the negative signal frequency.

## Theory

An ideal sinusoidal I/Q signal pair with no amplitude or
phase mismatch can be represented by the following complex function:

Equation 1:

y(t) = A exp(i(ωt + φ)) =
A cos(ωt + φ) + i A sin(ωt + φ)

The real part (the first term in the second expression) of
y(t) is the I signal, and the imaginary part (the second term except the "i")
is the Q signal. ω, A, and φ are the angular frequency, amplitude and
phase, respectively.

If an amplitude and phase mismatch is introduced, the
function becomes:

Equation 2:

y(t) = A cos(ωt + φ) +
i α A sin(ωt + φ + ε)

Here, A and φ are the amplitude and phase of the I
signal, α is the amplitude mismatch (the ratio of Q amplitude to I
amplitude), and ε is the phase mismatch (the Q phase plus 90 degrees minus the I
phase). For α=0 dB (α=1) and ε=0 degrees there is no mismatch.

By rewriting Equation 2 we can split the mismatched I/Q signal into its spectral components:

Equation 3:

y(t) = 0.5 (1 + α exp(i ε)) A exp(i(ωt + φ))
+ 0.5 (1 - α exp(-i ε)) A exp(-i(ωt + φ))

The first term is the wanted signal (including both I and Q
parts) - a pure complex sinusoid at the frequency ω. The second term is an
unwanted signal at the negative frequency -ω. The wanted signal is simply
the ideal I/Q signal from Equation 1 multiplied by a complex constant, which modifies the amplitude and phase (here, the I and Q parts are modified equally).

From Equation 3 the ratio of the unwanted signal magnitude to the wanted signal magnitude can be found:

Equation 4:

## Conclusion

It has been shown that the introduction of an amplitude
and/or phase mismatch between the I and Q signals causes an unwanted spectral
component to appear at the mirror frequency. Also, the amplitude and phase of
the wanted spectral component will change. Equation 3 quantifies these effects.