General Non-linearity x f( ) y = f(x) Any f(x) can be represented as a Taylor series expansion: a0 represents a DC offset a1 represents the linear gain a2 represents the 2nd order response a3 represents the 3rd order response etc. . . . We model devices as linear, but no practical devices are perfectly linear, so the Taylor coefficients ai are small, but not zero for i ≠ 1 . We are going to examine the third order response when the input, x, is the sum of two sinusoids having equal amplitudes but different frequencies:
Then wX = 2w1 –(w1 + Dw) = w1 - Dw The amplitude of Third Order Responses are proportional to the third power of the amplitudes of the generating signals. The “Culprit” The Other “Culprit” wX = 2w1 - w2 Let w2 = w1 + Dw. Then wX = 2w1 –(w1 + Dw) = w1 - Dw A 1 dB increase in the amplitude of the interfering signals creates a 3 dB increase in the third order interference. Dw Dw Dw wX w1 w2 wY
Intermodulation Characteristics Po(dBm) 3rd Order Intercept Point PIP,o = PIP,i + G PIP,o Linear Response to Desired Signal 3rd order Response to Interference 1 3 1 Ps,o SNR 1 Pd,o IMDR = Pd,i – Ps,i PN,o Pi(dBm) PN,i Ps,i PSF Pd,i PIP,i “Spurious Free”
Example An amplifier has a gain of 22 dB and a 3rd order output intercept point of 27 dBm. Assume the effective noise input power is PN,i = -130 dBm. Determine Spurious Free Range PN,i + G = PN,o = Pd,o = 3(PSF + G) – 2PIP,o PSF = (PN,i + 2PIP,o -2G)/3 = (– 130 dBm + 54 dB – 44 dB)/3 = – 40 dBm Determine IMDR for an input signal level of -80 dBm and SNR = 15 dB. Pd,i = (Ps,i + 2(PIP,o -G)– SNR )/3 = (– 80 dBm + 2(27 dBm – 22 dB)) – 15 dB )/3 = – 28.3 dBm IMDR = Pd,i - Ps,i = – 28.3dBm – (– 80 dBm) = 51.7 dB
PIP,o Ps,o S/N Pd,o PN,o IMDR Ps,i PN,i PSF Pd,i