EECS 242 Analysis of Memoryless Weakly Non Lineary Systems Review of Linear Systems Linear Linear Complete description of a general time varying linear system Note output cannot have a DC offset UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Time invariant Linear Systems Time invariant Linear Systems has h t h t Relative function of time rather than absolute The transfer function is stationary convolution in time is product in frequency UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Stable Systems x Linear time invariant LTI system cannot generate frequency content not present in input x x x Poles of H are strictly in the left hand plane LHP x x UC Berkeley EECS 242 Copyright Prof Ali M Niknejad if Memoryless Linear System No DC No Delay If function is continuous at xo then we can do a Taylor Series expansion about xo yo xo UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Taylor Series Expansion This expansion has a certain radius of convergence If we truncate the series we can compute a bound on the error Let s assume Maximum excursion must be less than radius of convergence Certainly the max Ak has to be smaller than the radius of convergence UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Sinusoidal Exciation m times UC Berkeley EECS 242 Copyright Prof Ali M Niknejad General Mixing Product We have frequency components where kp ranges over 2N values Terms in summation Example Take m 3 N 2 64 Terms in summation HD3 64 Terms IM3 gain expression or compression UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Vector Frequency Notation Define 2N vector where kj denotes the number of times a particular frequency appears in a give summation No DC terms 2 1 0 1 2 Sum order of non linearity UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Multinomial Coefficient For a fixed vector how many different sum vectors are there m frequencies can be summed m different ways but order is immaterial Each coefficient kj can be ordered kj ways Therefore we have Multinomial coefficient UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Game of Cards example 3 Cards 3 or six ways to order cards ways to order UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Since R1 R2 Reds not distinguished Making Conjugate Pairs Usually we only care about a particular frequency mix generated by certain order non linearity Since our signal is real each term has a complex conjugate Hence there is another reverse order Taking the complex conjugates in pairs UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Amplitude of Mix Thus the amplitude of any particular frequency component is Ex IM3 product generated by the cubic term IM3 2 1 1 0 m 3 N 2 Amplitude of IM3 relative to fundamental UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Gain Compression Expansion How much gain compression occurs due to cubic and pentic x5 terms cubic m 3 N 1 amp of fund App Gain appear anywhere This to appear twice anywhere Gain depends on signal amplitude pentic m 5 N 1 1 1 App Gain UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Who wins Pentic or Cubic Gain Reduction due to Cubic R Gain Reduction due to Pentic Take an exponential transfer function and consider gain compression UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Compression for Exp BJT When R 1 pentic non linearity contributes equally to gain compression R 1 UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Summary of Distortion x t f x y t Due to non linearity y t has frequency components not present in input For sinusoidal excitation by N tones we M tones in output m Order of highest term in non linearity Taylor exp UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Amplitude of Frequency Mix Particular frequency mix has frequency The amplitude of any particular frequency mix amplitude UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Harmonic Distortion dB HD2 HD3 Signal amplitude For an input frequency j each order non linearity power produces a jth order harmonic in output Signal amplitude 2 2 dB increase for 1 dB signal increase UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Intermodulation For a two tone input to a memoryless non linearity output contains due to cubic power and due to second order power Power dB RF band or channel IM2 IM3 terms UC Berkeley EECS 242 Copyright Prof Ali M Niknejad IM2 Filtering Intermodulation IM2 important direct conv receiver RF DC AMP IM3 important LO RF AC coupled IM2 products fall at much lower DC and higher frequencies 2 o These signals appear as interference to others but can be attenuated by filtering IM3 products cannot be filtered for close tones In a direct conversion receiver IM2 is important due to DC UC Berkeley EECS 242 Copyright Prof Ali M Niknejad IM Harmonic Relations Signal level Signal level 2 UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Triple Beat Triple Beat Apply three sine waves and observe effect of cubic non linearity 3 2 1 1 2 3 UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Intercept Point Intercept Point Apply a two tone input and plot output power and IM powers The intercept point in the extrapolated signal power level which causes the distortion power to equal the fundamental power UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Intercept IM Calculations Say an amplifier has an IIP3 10 dBm What is the amplifier signal distortion IM3 ratio if we drive it with 25 dBm Note IM3 0 dB at Pin 10 dBm If we back off by 15 dB the IM3 improves at a rate of 2 1 For Pin 25 dBm 15 dB back off we have therefore IM3 30dBc intercept signal level UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Gain Compression and Expansion To regenerate the fundamental for the N th power we need to sum k positive frequencies with k 1 negative frequencies so N 2k 1 N must be an odd power k GP k 1 1dB P 1dB UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Pin P1dB Compression Point An important specification for an amplifier is the 1dB compression point or the input power required to lower the gain by 1dB Assume a3 a1 0 About 9 6dB lower than IIP3 UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Dynamic Range P 1dB is a convenient maximum signal level which sets the upper bound on the amplifier linear regime Note that at this power the IM3 20 dBc The lower bound is set by the amplifier noise figure UC Berkeley EECS 242 Copyright Prof Ali M Niknejad Blocking or Jamming Blocker Any large interfering signal PBL Blocking level Interfering signal level in dBm which causes a 3dB drop in gain