Complex Baseband

The signal is mathematically modeled as z(t) = I(t) + jQ(t) , where I(t) is the in-phase component and Q(t) is the quadrature component. This single complex stream captures both the instantaneous amplitude (envelope) and instantaneous phase of the original bandpass signal without loss of information. The complex representation is not a physical signal but a mathematical convenience that simplifies modulation, filtering, and detection algorithms.

The defining characteristic is that the signal's spectrum is centered at 0 Hz (DC) , not at the original carrier frequency. This is achieved through digital down conversion (DDC) , which mixes the signal with a local oscillator and applies a low-pass filter. The resulting spectrum is asymmetric around zero, containing all the information from the original positive-frequency passband signal. This shift to baseband dramatically reduces the required sampling rate to just above the signal's bandwidth, satisfying the Nyquist criterion efficiently.

The I and Q channels are orthogonal to each other, meaning they occupy the same bandwidth but do not interfere. This is achieved by modulating them with carrier signals that are 90 degrees out of phase. Orthogonality effectively doubles the spectral efficiency, as two independent data streams can be transmitted simultaneously over the same frequency channel. In an ideal system, a receiver can perfectly separate the I and Q components. Hardware impairments like IQ imbalance destroy this orthogonality, causing cross-talk between the channels.

At any given instant, a complex baseband sample can be visualized as a phasor on the complex plane. The distance from the origin represents the instantaneous amplitude, A(t) = sqrt(I² + Q²) , and the angle from the positive I-axis represents the instantaneous phase, φ(t) = arctan(Q/I) . This geometric view is the basis for the IQ constellation diagram, where each valid symbol is a discrete point on this plane. Modulation schemes are defined by the specific set of allowed phasor positions and the trajectories between them.

Unlike real-valued signals, which have Hermitian symmetric spectra, a complex baseband signal can have independent content at positive and negative frequencies. The spectrum is not constrained to be conjugate-symmetric. This property is critical for representing signals with asymmetric sidebands or for modeling the effects of IQ imbalance, which creates a mirror-image interference at the negative of the desired signal's frequency. The ability to process negative frequencies independently is a key advantage of complex signal processing.

A complex random signal is termed proper or circular if it is uncorrelated with its own complex conjugate, meaning its probability distribution is rotationally invariant. For a proper signal, the complementary autocorrelation function is zero. Many communication signals, like high-order QAM, are designed to be proper. However, signals corrupted by IQ imbalance or certain modulated formats like BPSK become improper. Detecting and exploiting this impropriety through widely linear filtering can yield significant performance gains in interference suppression.