How do I calculate the level of second and third order intermodulation products?
Intermodulation Product Calculation
Intermodulation products arise from the nonlinear transfer function of active devices. Any nonlinearity can be expressed as a power series: Vout = a1·Vin + a2·Vin² + a3·Vin³ + ..., where the higher-order coefficients (a2, a3) represent the nonlinearity. When two tones at frequencies f1 and f2 are applied, the squaring term (a2) produces products at f1±f2, 2f1, and 2f2. The cubing term (a3) produces products at 2f1±f2, 2f2±f1, 3f1, and 3f2.
Third-order products at 2f1-f2 and 2f2-f1 are the most problematic because they fall close to the original signals and within the system bandwidth. If f1 = 1000 MHz and f2 = 1001 MHz, the IM3 products appear at 999 MHz and 1002 MHz, just 1 MHz from the fundamentals and impossible to filter without also filtering the desired signals.
The IP3 figure of merit allows prediction of IM3 levels at any operating power. The formula IM3 (dBm) = 3·Pin - 2·OIP3 gives the absolute IM3 power, while IM3 (dBc) = 2×(Pin - OIP3) gives the level relative to the fundamental. These formulas are valid only in the linear region, well below P1dB. Near compression, the actual IM3 levels deviate from these predictions.
IM2 product frequencies: f₁+f₂ and f₁-f₂
IM3 level (dBm) = 3·Pin - 2·OIP3
IM3 level (dBc) = 2×(Pin - OIP3)
Example: Pin = -10 dBm, OIP3 = +30 dBm
IM3 = 3(-10) - 2(30) = -90 dBm = -80 dBc
Frequently Asked Questions
What about higher-order products?
Fifth-order products (3f1-2f2) grow at 5 dB/dB and are at 4×(Pin-OIP5) dBc. Seventh-order products grow at 7 dB/dB. Higher orders are usually negligible except in highly nonlinear systems or when many carriers are present, creating a dense spectral forest of products.
Are IM2 or IM3 more important?
IM3 dominates in narrowband systems because the products fall in-band. IM2 dominates in wideband or direct-conversion receivers where the f1-f2 product can fall at baseband. Direct-conversion receivers require very high IIP2 (typically +50 to +70 dBm) to prevent IM2 from corrupting the desired signal.
How many IM products exist with N tones?
With N tones, the number of third-order IM products grows as N²(N-1)/2. For 2 tones: 2 products. For 10 tones: 450 products. For 100 tones: 495,000 products. This is why multi-carrier systems like OFDM are extremely sensitive to IM distortion.