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Gain and phase I/Q mismatch

Wednesday, January 1, 2020

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Understanding the equation describing I/Q mismatch is critical in complex transceivers. Insufficient image rejection can cause smearing or rotation of the signal constellation. An ideal arbitrary complex signal can be given as
$$y\left(t\right)=A\cdot e^{j\left(\omega t+\phi \right)}=\cos \left(\omega t+\phi \right)+j\sin \left(\omega t+\phi \right)$$
Equation (1)
where ω is the angular frequency (rad/s) and φ is the angular phase (in rad). It is assumed that the I channel signal is represented by the cosine comonent of (1) and the Q channel signal is represented by the sine component of (1). A received signal, with gain and phase errors added to the Q channel can be represented as
$$y"\left(t\right)=A\cdot e^{j\left(\omega t+\varphi \right)}=\cos \left(\omega t+\varphi \right)+j\sin \left(\omega t+\varphi +\theta \right)$$
Equation (2)
where α is the gain error (i.e. a 1% gain error results in α=1.01) and ε is the phase error in radians. Note that the gain and phase error terms are added to the Q channel. Using Euler's formula, (2) can be rewritten (to separate the desired from undesired components) as
$$y"\left(t\right)=\frac{1}{2}\left\{1+\alpha \cdot {}^{j\theta }\right\}A\cdot e^{j\left(\omega t+\varphi \right)}+\frac{1}{2}\left\{1-\alpha \cdot {}^{-j\theta }\right\}A\cdot e^{-j\left(\omega t+\varphi \right)}$$
Equation (3)
The I/Q mismatch can now be written the magnitude of the ratio of the undesired to desired signals which computes to
$${\mathrm{IQ}}_{\mathrm{mismatch}}=\sqrt{\frac{1+{\alpha }^2-2\alpha \cos \left(\theta \right)}{1+{\alpha }^2+2\alpha \cos \left(\theta \right)}}$$
Equation (4)
Equation (4) can also be approximated as
$${\mathrm{IQ}}_{\mathrm{mismatch}}=\sqrt{{\left(\frac{\Delta \theta }{2}\right)}^2+{\left(\frac{\Delta A}{2}\right)}^2}$$
Equation (5)
To illustrate these equations, two examples are given: Example 1. If there is no gain mismatch and only a 1° of phase mismatch, this results in 41.12dB of I/Q mismatch. Example 2. If there is 1% gain mismatch and no phase mismatch, this results in 46dB of I/Q mismatch.

Table 1. Comparison of I/Q mismatch calculation methods

Case	Example 1	Example 2
Equation (4)	41.12dB	46dB
Equation (5)	41.19dB	46dB

References

Keywords: I/Q mismatch, RFIC, wireless transceivers, quadrature transceivers

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