1. f( )xy = f(x) Any f(x) can be represented as a Taylor series expansion: a 0 represents a DC offset a 1 represents the linear gain a 2 represents the 2 nd order response a 3 represents the 3 rd order response etc.... General Non-linearity We model devices as linear, but no practical devices are perfectly linear, so the Taylor coefficients a i 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:

2. 3 1 3 2 + 2 1 + 2 + 1 + 2 2 X = 2 1 - 2 The Culprit Then X = 2 1 –( 1 + 1 - 1 2 X Let 2 = 1 +. The amplitude of Third Order Responses are proportional to the third power of the amplitudes of the generating signals. A 1 dB increase in the amplitude of the generating signals creates a 3 dB increase in the third order response.

3. P N,i P N,o P SF P s,i P s,o P d,o P d,i P IP,i P IP,o Linear Response to Desired Signal 3 rd order Response to Interference SNR IMDR P o (dBm) P i (dBm) Spurious Free 1 1 1 3 3 rd Order Intercept Point Intermodulation Characteristics

4. Example An amplifier has a gain of 22 dB and a 3 rd order output intercept point of 27 dBm. Assume the effective noise input power is P N,i = -130 dBm. Determine Spurious Free Range P N,i + G = P N,o = P d,o = 3(P SF + G) – 2P IP,o P SF = (P N,i + 2P IP,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. P d,i = (P s,i + 2(P IP,o -G)– SNR )/3 = (– 80 dBm + 2(27 dBm – 22 dB)) – 15 dB )/3 = – 28.3 dBm IMDR = P d,i - P s,i = – 28.3dBm – (– 80 dBm) = 51.7 dB

5. P d,o P N,i P N,o P s,o P IP,o P SF IMDR P s,i P d,i S/N