Phase modulation is the basis for many modern mobile telecommunications formats, including IS-136 and GSM. Using in-phase (I) and quadrature (Q) signal components, modulation is commonly generated by means of an analog I/Q modulator. Since variations in components and manufacturing processes can easily introduce imbalance in I/Q modulator arms, resulting in poor transmission quality, I/Q imbalance measurements are critical to maintaining good system performance. This report will examine such measurements and highlight why traditional measurements can be misleading, offering an improved, alternative approach which can be applied to existing test processes.

Figure 1 shows a model for data transfer in a digital wireless-communication system. The particular modulation format is not critical here; the important assumption is that the data modulator creates complex signals with I and Q components. The I and Q components can be treated as a complex number, with I being the real part and Q being the imaginary part of the complex number. As a convention, complex quantities will be underlined throughout for clarity.

Since a complex signal cannot be transmitted, it must be converted to a real value and modulated onto a carrier signal. The I/Q modulator provides the means for this. An I/Q modulator consists of a local oscillator (LO) feeding two (I and Q) mixers. The LO signal that feeds the Q mixer is shifted by +p/2 rad. in phase relative to the LO signal that feeds the I mixer. The mixers' outputs are combined to form the complex signal.

Due to differential additional a_I and a_Q phase shifts along the path leading to the mixer, the phase shift might not be exactly +p/2 rad., however. Due to different gains, G_I and G_Q , the two mixers may also receive LO signals at different levels, further complicating matters. These errors can be expressed by the complex gains of Eqs. 1 and 2.

Due to leakage, some portion of the LO signal may also appear at the output of the mixers. Such leakage can stem from various sources, which are not discussed here, but it can be represented by additive offsets, O_I and O_Q, to the in-phase and quadrature-phase input signals, respectively. These errors are traditionally characterized by the measures of Eqs. 3-5. Ideally, Eqs. 3 and 4 should be 0 dB.

In Eq. 5, O is called the "complex origin offset" and is shorthand for the expression in Eq. 6.

In Eq. 5, P_s is the average power of the baseband input signal s(t) . Ideally, this measure should be as low as possible. In practice, this is -30 dB or lower.

The RF channel incorporates all the RF components between the I/Q modulator and I/Q demodulator. As a simple but usable model (especially in manufacturing where the mobile units are connected to test equipment by means of a cable), this signal path is considered to be free of distortion. The signal path is assumed without loss of generality to have 0 dB gain and 0 group delay. As part of the RF signal chain, the I/Q demodulator, which translates a received signal back to complex baseband signals, is also considered to be ideal for modelling purposes. Additional assumptions are that the LO for the I/Q modulator has unit amplitude while the LO for the I/Q demodulator has an amplitude of two units, and that both LOs have a starting phase of 0 when time, t, is 0.

In this model, the received complex baseband signal, r(t) , can be described by Eq. 7. The Re{s(t)} and Im{s(t)} operators return the real (I) and imaginary (Q) components of the baseband input signal, s(t), respectively.

Generally, the aim of an I/Q modulator characterization is to measure gain imbalance, quadrature skew, and LO leakage since these parameters are used in I/Q modulator data sheets, and because manufacturers often provide some adjustable devices (e.g., potentiometers) that are dedicated to reduce these errors. Equation 7 provides a way to determine the imbalances via estimation of the G_I , G_Q , and O intermediate parameters. Once G_I , G_Q , and O are obtained, the gain imbalance, quadrature skew, and LO leakage can be calculated with Eqs. 3, 4, and 5.

Equation 7 contains the three unknowns G_I , G_Q , and O, so three independent equations are needed to solve them. This can be accomplished by capturing three samples of the transmitted complex baseband signal, s(t), and the corresponding three samples of the received complex baseband signal, r(t).

The received complex baseband signal, r(t), is determined directly by sampling the received signal from a device under test (DUT) at the output of the I/Q demodulator. However, the transmitted signal, s(t) , is not readily available. It is synthesized in the measuring device from the received digital data, which can be recovered by demodulating r(t). This synthesized signal is denoted as s'(t).

Unfortunately, all digital modulation formats are designed so that the absolute phase is always irrelevant. This is necessary because the phase at the receiving end is always random. Hence, there is an infinite number of s'(t) that carry the same digital content as the actually transmitted s(t), differing only in phase. They can be rotated into the actually transmitted signal, s(t), by a phase shift, d, shown in Eq. 8.

As a result, the system of equations that is solved is Eq. 9 rather than Eq. 7. The question is whether this phase ambiguity, d, affects the measurement results or not. Substituting Eq. 8 into Eq. 7 and expanding yields Eq. 10.

Comparing this with Eq. 9 reveals the relationship between G'_I, G'_Q, O' calculated using the synthesized transmitted signal, s'(t) , and the actual G_I, G_Q, O and shown in Eqs. 11-13.

Based on these results, the relationships between the perceived GainImbalance', QuadratureSkew', LOLeakage', and the actual GainImbalance, QuadratureSkew, LOLeakage are shown in Eqs. 14-16.

Equations 14 and 15 show that the calculated gain imbalance and quadrature skew do differ from their actual values if d != 0, that is, if the synthesized transmitted signal, s'(t), differs in phase from the actual transmitted signal, s(t).