How do I predict the frequencies and levels of intermodulation products in a multi-carrier system?

In a multi-carrier system (multiple signals passing through a shared nonlinear component), intermodulation products are generated at frequencies determined by the carrier frequencies and at levels determined by the carrier powers and the component nonlinearity: (1) IM product frequencies: for N carriers at frequencies f1, f2, ..., fN: second-order products: fi ± fj (all pairs). Third-order products: 2fi - fj, fi + fj - fk (all combinations). The number of IM3 products grows rapidly with N: 2 carriers: 2 IM3 products (2f1-f2, 2f2-f1). 3 carriers: 12 IM3 products. 4 carriers: 30 IM3 products. N carriers: N(N-1)(N-2)/2 + N(N-1) IM3 products. Some of these products fall on or near the carrier frequencies (in-band IM3) and cannot be filtered. (2) IM product levels: for equal-power carriers (each at power P, with IP3 of the device known): two-tone IM3 at 2fi - fj: P_IM3 = 3P - 2×IIP3 (same as two-tone test). Three-tone IM3 at fi + fj - fk: P_IM3 = 3P - 2×IIP3 + 3.5 dB. The three-tone products are approximately 3.5 dB higher than the two-tone products at the same individual carrier power. (3) Multi-carrier IM3 stacking: when multiple IM3 products fall at the same frequency (or within the same channel bandwidth): the products add. At the center of the carrier group: the IM3 products from all carrier combinations accumulate. The total IM3 power at the center frequency can be 6-15 dB higher than the single-pair IM3 level. (4) Prediction tools: for a small number of carriers (2-10): use a spreadsheet to enumerate all IM3 products, compute their frequencies and levels, and identify which products fall in-band. For a large number of carriers (OFDM, multi-carrier cellular): use NPR (noise power ratio) to characterize the total intermodulation noise across the band.