Electronic Noise Dynamic range in the analog domain Resistor noise Amplifier noise Maximum signal levels Tow Thomas Biquad noise example Implications on power dissipation A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 1 Analog Dynamic Range Finite precision effects in digital filters are rapidly becoming negligible Floating point digital filters with huge mantissas will be reduced to negligible cost The only fixed point numbers will come from ADCs But we will always have thermal noise A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 2 Analog Dynamic Range Let s say you ve selected the poles and zeroes of your analog filter transfer function Of the infinitely many ways to build a filter with a given transfer function each of those ways has a different output noise A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 3 Analog Dynamic Range The job of a high performance analog filter designer is to get reasonably close to the optimal noise for a given transfer function Not the absolute minimum noise just close The job of a mixed signal chip architect is to appreciate filter noise and to be able to model filters well enough to know that a given dynamic range objective is feasible A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 4 Analog Dynamic Range We ll begin our adventure in analog filter implementation by looking at the noise in resistors and simple RC filters A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 5 Resistor Noise Capacitors are noiseless Resistors have thermal noise This noise is uniformly distributed from dc to infinity Frequencyindependent noise is called white noise A D DSP EECS 247 Lecture 4 Dynamic Range vIN R vOUT C 2002 B Boser 6 Resistor Noise Resistor noise has A mean value of zero A mean squared value vn2 4k BTr R f vIN R ohms vOUT C measurement bandwidth Hz Volts2 absolute temperature K Boltzmann s constant 1 38e 23 J K A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 7 Resistor Noise Resistor rms noise voltage in a 10Hz band centered at 1kHz is the same as resistor rms noise in a 10Hz band centered at 1GHz Resistor noise spectral density N0 is the rms noise per Hz of bandwidth N0 A D DSP vIN R vOUT C vn2 4k BTr R f EECS 247 Lecture 4 Dynamic Range 2002 B Boser 8 Resistor Noise Don t bother to remember Boltzmann s constant Instead remember forever that N0 for a 1k resistor at room temperature is 4nV Hz R vIN vOUT C Scaling R A 10M resistor gives 400nV Hz A 50 resistor gives 0 9nV Hz Or remember that kBTr 4x10 21 J Tr 17 oC A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 9 Resistor Noise Resistor noise gives our filter a non zero output when vIN 0 In this simple example both the input signal and the resistor noise obviously have the same transfer functions to the output Since noise has random phase we can use any polarity convention for a noise source but we have to use it consistently A D DSP EECS 247 Lecture 4 Dynamic Range vIN R vOUT e C 2002 B Boser 10 Resistor Noise What is the thermal noise of the RC filter Let s ask SPICE Netlist vIN Noise from RC LPF vin vin 0 ac 1V r1 vin vout 8kOhm c1 vout 0 1nF ac dec 100 10Hz 1GHz noise V vout vin end A D DSP e R 8k vOUT C 1nF EECS 247 Lecture 4 Dynamic Range 2002 B Boser 11 LPF1 Output Noise Density Noise Spectral Density nV Hz 100 20 kHz corner 10 N 0 4k BTr R 1 0 1 0 01 101 A D DSP nV Hz nV 11 3 Hz 8 4 103 EECS 247 Lecture 4 Dynamic Range 105 107 109 Hz 2002 B Boser 12 Total Noise Suppose we want to know the value of vo now what s the standard deviation error E g on the display of a volt meter connected to vo Answer v 4k BTR H 2 jf df 2 o 2 0 A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 13 Total Noise Note that noise is integrated in the meansquared domain because noise in a bandwidth df around frequency f1 is uncorrelated with noise in a bandwidth df around frequency f2 Powers of uncorrelated random variables add Squared transfer functions appear in the meansquared integral A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 14 Total Noise v 4k BTR H 2 jf df 2 2 o 0 2 1 df 4k BTR 1 2 jfRC 0 A D DSP k BT C EECS 247 Lecture 4 Dynamic Range 2002 B Boser 15 Total Noise vo2 k BT C This interesting and somewhat counterintuitive result means that even though resistors provide the noise sources capacitors set the total noise For a given capacitance as resistance goes up the increase in noise density is balanced by a decrease in noise bandwidth A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 16 kT C Noise The rms noise voltage of the simplest possible first order filter is kBT C For 1pF kBT C 64 V rms at 298 K 1000pF gives 2 V rms The noise of a more complex filter is K x kBT C K depends on implementation and features such as filter order A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 17 LPF1 Output Noise Integrated Noise Vrms Noise Spectral Density nV Hz 100 10 2 Vrms 1 0 1 0 01 101 A D DSP 103 EECS 247 Lecture 4 Dynamic Range 105 107 109 Hz 2002 B Boser 18 LPF1 Output Noise Note that the integrated noise essentially stops growing above 100kHz for this 20kHz lowpass filter Beware of faulty intuition which might tempt you to believe that an 80 1000pF filter has lower integrated noise than our 8000 1000pF filter A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 19 LPF1 Output Noise Integrated Noise Vrms Noise Spectral Density nV Hz 100 80 1000pF 10 1 0 1 0 01 101 A D DSP 103 EECS 247 Lecture 4 Dynamic Range 105 107 109 Hz 2002 B Boser 20 LPF1 Output Noise Of course an 80 100 000pF filter has both the same bandwidth AND lower integrated noise than our 8000 1000pF filter In the analog filter dynamic range game the highest capacitance wins A D DSP EECS 247 Lecture 4 Dynamic Range 2002 B Boser 21 LPF1 Output Noise Integrated Noise Vrms Noise Spectral Density nV Hz 100 80 100000pF 10 1 0 1 0 01 101 A D DSP 103 EECS 247 Lecture 4 Dynamic Range 105 107 109 Hz 2002 B Boser 22 Analog Circuit Dynamic Range The biggest signal we can ever expect at the output of a circuit is limited by the supply voltage VDD hence for sinusoids Vmax rms …
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