Baseband Equivalent Modeling

The model operates on the complex envelope of the signal, not the RF carrier. A bandpass signal x(t) = A(t)cos(ωct + φ(t)) is represented by its complex baseband equivalent x̃(t) = I(t) + jQ(t), where I(t) is the in-phase component and Q(t) is the quadrature component. This eliminates the carrier frequency ωc from all computations.

• Captures both amplitude A(t) and phase φ(t) information
• Reduces a high-frequency problem to a low-frequency one
• The original RF signal is recoverable via x(t) = Re{x̃(t)ejωct}

A critical property is that the baseband equivalent accurately captures the nonlinear behavior of the RF system. The nonlinear transfer function is transformed so that only in-band and adjacent-band distortion products are modeled. Out-of-band harmonics at multiples of the carrier are intentionally discarded, as they are typically filtered out in real systems.

• Models AM/AM (amplitude-to-amplitude) and AM/PM (amplitude-to-phase) distortion
• Captures intermodulation products that fall near the carrier
• Preserves memory effects through baseband Volterra kernels

By eliminating the carrier frequency, the required simulation sampling rate drops dramatically. A 2 GHz carrier modulated by a 100 MHz signal requires a ~4 GHz Nyquist rate for RF simulation. The baseband equivalent only needs to sample at a rate sufficient for the modulation bandwidth (e.g., 200-300 MHz), reducing computational load by over an order of magnitude.

• Sampling rate determined by signal bandwidth, not carrier frequency
• Enables practical simulation of long data sequences
• Critical for iterative DPD coefficient extraction algorithms

Every component in the RF chain—filters, amplifiers, mixers—has a low-pass equivalent transfer function. A bandpass filter centered at ωc with response H(ω) is modeled by its low-pass equivalent H̃(ω) shifted to DC. This allows the entire transmitter or receiver chain to be simulated as a cascaded low-pass system.

• Simplifies frequency-selective component modeling
• Enables unified simulation of linear and nonlinear blocks
• Standard technique in tools like MATLAB RF Toolbox and Keysight SystemVue

Memory effects—where the PA output depends on past input values—are modeled using baseband Volterra series or memory polynomials. The discrete-time baseband equivalent is ỹ(n) = Σk Σm hk(m) x̃(n-m) |x̃(n-m)|^(k-1), where hk(m) are complex baseband kernels. This formulation captures both short-term (electrical) and long-term (thermal) memory.

• Odd-order kernels dominate due to bandpass filtering of even-order products
• Memory depth M determines how many past samples influence the output
• Directly compatible with indirect learning architecture for DPD

For static nonlinearities, the baseband model simplifies to a complex gain function G(|x̃|) that depends only on the instantaneous envelope magnitude. The output is ỹ = G(|x̃|) · x̃, where G(·) is a complex-valued function representing both gain compression (magnitude) and phase shift (angle). This is the foundation of AM/AM and AM/PM characterization.

• Measured directly using a vector network analyzer
• Basis for look-up table (LUT) predistorters
• Valid for narrowband signals where memory effects are negligible