6.976 High Speed Communication Circuits and Systems Lecture 20
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6.976 High Speed Communication Circuits and Systems Lecture 20 Performance Measures of Wireless Communication Michael Perrott Massachusetts Institute of Technology Copyright © 2003 by Michael H. Perrott Recall Digital Modulation for Wireless Link Receiver Output Baseband Input it(t) t 2cos(2πf1t) 2sin(2πf1t) qt(t) t Lowpass 2cos(2πf1t) 2sin(2πf1t) Lowpass ir(t) t qr(t) Decision Boundaries t Sample Times Send discrete-leveled values on I and Q channels Performance issues - Spectral efficiency (transmitter) - Bit error rate performance (receiver) Nonidealities - Intersymbol interference - Noise Interferers M.H. Perrott MIT OCW Impact of Receiver Noise Baseband Input it(t) t Input-referred Receiver Noise 2cos(2πf1t) 2sin(2πf1t) qt(t) t Performance impact Methods of increasing SNR Receiver Output Lowpass 2cos(2πf1t) 2sin(2πf1t) Lowpass ir(t) t qr(t) Decision Boundaries t Sample Times - SNR is reduced, leading to possible bit errors - Decrease bandwidth of receiver lowpass SNR is traded off for intersymbol interference - Increase input power into receiver Increase transmit power and/or shorten its distance from receiver M.H. Perrott MIT OCW View SNR Issue with Constellation Diagram Q 00 01 11 10 00 01 Decision Boundaries I 11 10 Decision Boundaries Noise causes sampled I/Q values to vary about their nominal values Bit errors are created when sampled I/Q values cross decision boundaries M.H. Perrott MIT OCW Mathematical Analysis of SNR versus Bit Error Rate (BER) Q 00 01 11 10 00 01 Decision Boundaries Probability Density Function for Noise (I-Component) I 11 Probability Density Function for Noise (Q-Component) 10 Decision Boundaries Model noise impacting I and Q channels according to a probability density function (PDF) - Gaussian shape is often assumed Receiver bit error rate can be computed by calculating probability of tail regions of PDF curves M.H. Perrott MIT OCW Key Parameters for SNR/BER Analysis Q 00 01 11 10 00 Decision Boundaries 01 dmin Probability Density Function for Noise (I-Component) I Probability Density Function for Noise dmin (Q-Component) 11 10 Decision Boundaries Bit error rate (BER) is a function of - Variance of noise - Distance between constellation points (d min) Larger dmin with a fixed noise variance leads to higher SNR and a lower bit error rate M.H. Perrott MIT OCW Relationship Between Amplitude and Constellation Q Peak Amplitude Average Amplitude Probability Density Function for Noise (I-Component) dmin I Probability Density Function for Noise dmin (Q-Component) Distance of I/Q constellation point from origin corresponds to instantaneous amplitude of input signal at that sample time Amplitude is measured at receiver and a function of - Transmit power Distance between transmitter and receiver (and channel) M.H. Perrott MIT OCW Impact of Increased Signal Power At Receiver Peak Amplitude Q Average Amplitude Probability Density Function for Noise (I-Component) dmin I Probability Density Function for Noise (Q-Component) dmin Separation between constellation points, dmin, increases as received power increases Noise variance remains roughly constant as input signal power is increased - Noise variance primarily determined by receiver circuits Bit error rate improves with increased signal power! M.H. Perrott MIT OCW Impact of Modulation Scheme on SNR/BER Q Peak Amplitude Average Amplitude Probability Density Function for Noise (I-Component) dmin I dmin Probability Density Function for Noise (Q-Component) Lowering the number of constellation points increases dmin for a fixed input signal amplitude - SNR is increased (given a fixed noise variance) - Bit error is reduced Actual situation is more complicated when coding is used M.H. Perrott MIT OCW Alternate View of Previous Constellation Q Average Amplitude Peak Amplitude dmin Probability Density Function for Noise (-45 Degree Component) I dmin Probability Density Function for Noise (+45 Degree Component) Common modulation method is to transmit independent binary signals on I and Q channels The above constellation has the same dmin as the one on the previous slide - Obtains the same SNR/BER performance given that the M.H. Perrott noise on I/Q channels is symmetric MIT OCW Impact of Noise Factor on Input-Referred SNR Source (Antenna) Rs SNRin Band Select Filter ens Image Reject Filter Channel Filter Amp LNA Vin Vs SNRout IF out |Gain| Rin Synthesizer 0 f Channel Bandwidth = B Hz Overall Receiver Noise Factor = NF To achieve acceptable bit error rates (BER) Refer SNR requirement to input M.H. Perrott MIT OCW Minimum Input Power to Achieve Acceptable SNR Source (Antenna) Rs SNRin Overall Receiver Noise Factor = NF ens Vin Vs Rin 0 f (1) Channel Bandwidth = B Hz Calculation of input SNR in terms of input power Combine (1) and (2) M.H. Perrott (2) MIT OCW Simplified Expression for Minimum Input Power Assume that the receiver input impedance is matched to the source (i.e., antenna, etc.) Resulting expression - At room temperature: M.H. Perrott MIT OCW Receiver Sensitivity Source (Antenna) Rs SNRin Overall Receiver Noise Factor = NF ens Vin Vs Rin 0 f Channel Bandwidth = B Hz Sensitivity of receiver is defined as minimum input power that achieves acceptable SNR (in units of dBm) M.H. Perrott MIT OCW Example Calculation for Receiver Sensitivity Source (Antenna) Rs SNRin Overall Receiver Noise Factor = NF ens Vin Vs Rin 0 f Channel Bandwidth = B Hz Suppose that a receiver has a noise figure of 8 dB, channel bandwidth is 1 MHz, and the minimum SNR at the receiver output is 12 dB to achieve a BER of 1e-3 - Receiver sensitivity (for BER of 1e-3) is M.H. Perrott MIT OCW Calculation of Noise Figure of Cascaded Stages Input-referred Noise for Stage 1 Source ens Rs Vs Note: noise of RL not included in Noise Factor calculation en2 en1 in1 Vin1 Rin1 Input-referred Noise for Stage 2 Stage 1 (Noiseless) Unloaded voltage gain = Av1 Vout1 Rout1 in2 Vin2 Rin2 Stage 2 (Noiseless) Unloaded voltage gain = Av2 Vout2 RL Rout2 Want to calculate overall noise figure of above system Assumptions - Input refer the noise sources of each stage - Model amplification (or attenuation) of each stage as a - M.H. Perrott noiseless voltage controlled voltage source with an unloaded gain equal to Av Ignore noise of final load resistor (or could input refer to previous stage) MIT OCW Method: Refer All Noise to Input of First Stage Input-referred Noise for Stage 1 Source Rs in1 Note: noise of RL not included in Noise Factor calculation en2 en1 ens Vs Stage 1 (Noiseless) Vin1 Rin1 Input-referred Noise for Stage 2 Unloaded voltage gain = Av1 Vout1 in2 Vin2 Rin2 Rout1 Stage 2 (Noiseless) Unloaded voltage gain = Av2 Vout2 RL Rout2 Model for referring stage 2 noise to input of stage 1 Source resistance for Stage 1 Gain Stage 1 (Noiseless) Rs vs M.H. Perrott Rout1 vin1 Rin1 Av1vin1 Input impedance and input-referred noise for Gain Stage 2 en2 in2 Rin2 MIT OCW Step 1: Create an Equivalent Noise Voltage Source Rs vs Rout1 vin1 Rin1 en2 Rs vs vin1 Rin1 M.H. Perrott vin1 Rin1 in2 Rin2 Rout1 Av1vin1 en2+Rout1in2 Rs vs Av1vin1 en2 Av1vin1 in2 Rin2 Rout1 Rin2 MIT OCW Step 2: Input Refer Voltage Noise Source en2+Rout1in2 Rs vs vin1 Rin1 Rout1 Av1vin1 Rin2 en2+Rout1in2 α1 vs Rs Rout1 vin1 Rin1 Av1vin1 Rin2 Scaling factor α1 is a function of unloaded gain, Av, and input voltage divider M.H. Perrott MIT OCW Input Referral of Noise to First Stage Input-referred Noise for Stage 1 Source Rs Input-referred Noise for Stage 2 en2 en1 ens Vs in1 Note: noise of RL not included in Noise Factor calculation Vin1 Rin1 Stage 1 (Noiseless) Unloaded voltage gain = Av1 Vout1 in2 Stage 2 (Noiseless) Vin2 Rin2 Rout1 Unloaded voltage gain = Av2 Vout2 RL Rout2 en2+Rout1in2 Rs Vs α1 ens Stage 2 Noise en1 in1 Vin1 Rin1 Stage 1 (Noiseless) Unloaded voltage gain = Av1 Vout1 Vin2 Rout1 Rin2 Stage 2 (Noiseless) Unloaded voltage gain = Av2 Vout2 RL Rout2 Where M.H. Perrott MIT OCW Noise Factor Calculation Input-referred Noise for Stage 1 Source Rs ens Vs Input-referred Noise for Stage 2 en2 en1 in1 Note: noise of RL not included in Noise Factor calculation Vin1 Rin1 Stage 1 (Noiseless) Unloaded voltage gain = Av1 Vout1 Rout1 in2 Vin2 Rin2 Stage 2 (Noiseless) Unloaded voltage gain = Av2 Vout2 RL Rout2 en2+Rout1in2 Rs Vs M.H. Perrott ens α1 Stage 2 Noise en1 in1 isc MIT OCW Alternate Noise Factor Expression M.H. Perrott MIT OCW Define “Available Power Gain” Available Source Power Available Power at Output Source Rs vs Gain Stage 1 Source Rs vin1 Rin1=Rs vs Rout1 vin1 Rin1 Av1vin1 RL=Rout1 Available power gain for stage 1 defined as Ap1 = - Available power at output (matched load) Available source power (matched load) Available power at output - Available source power M.H. Perrott MIT OCW Available Power Gain Versus Loaded Voltage Gain Available Source Power Available Power at Output Source Rs vs Gain Stage 1 Source Rs vin1 Rin1=Rs vs Rout1 vin1 Available power gain for stage 1 If Rin1 = Rout1 = Rs - Where A v1_l M.H. Perrott Rin1 Av1vin1 RL=Rout1 is defined as the “loaded gain” of stage 1 MIT OCW Final Expressions for Cascaded Noise Factor Calculation Note: noise of RL not included in Noise Factor calculation Source Rs ens Vs Rin1 Stage 1 Stage 2 Stage k NF1 NF2 NFk Unloaded voltage gain = Av1 Unloaded voltage gain = Av2 Unloaded voltage gain = Avk Rout1 Rin2 Rout2 Rink Overall Noise Factor (general expression) Overall Noise Factor when all input and output impedances equal Rs: M.H. Perrott Vout RL Routk MIT OCW Calculation of Noise Factor for Lossy Passive Networks Source Rs Vs ens L Rloss enloss Vin C Rin Note: noise of RL not included in Noise Factor calculation Passive Network with Loss Vout RL Rout RF systems often employ passive filters for band select and channel select operations - Achieve high dynamic range and excellent selectivity Practical filters have loss - Can model as resistance in equivalent RLC network - Such resistance adds thermal noise, thereby lowering noise factor of receiver We would like to calculate noise factor contribution of lossy passive networks in a straightforward manner - See pages 46-48 of Razavi book M.H. Perrott MIT OCW Define “Available Power Gain” For Passive Networks Available Power at Output Rs Vs Rs Rout Vin Rin AvVin Vout RL=Rout Available power at output Available source power M.H. Perrott Available Source Power Vs Vin Rin=Rs MIT OCW Equivalent Noise Model and Resulting NF Calculation Equivalent Model for Computing Total Noise Equivalent Model for Computing Source Noise Contribution Rout enout Rs Vout RL Vs ens Rout Vin Rin Total noise at output Noise due to source (referred to output) Noise factor M.H. Perrott AvVin Vout RL MIT OCW Example: Impact of Cascading Passive Filter with LNA Source Rs Note: noise of RL not included in Noise Factor calculation ens Passive Filter LNA Insertion Loss = 2 dB Noise Figure = 1.5 dB Vs Noise Factor calculation Noise Figure M.H. Perrott Vout RL MIT OCW Example: Noise Factor Calculation for RF Receiver Source (Antenna) 50 Ω Vs ens Band Select Filter A B 50 Ω LNA Ap1 = -2.5 dB Image Reject Filter C D Ap3 = -4 dB Av2_l = 17 dB NF2 = 1.5 dB LO Channel Filter E F Amp G IF out Ap5 = -3 dB Av4_l = 15 dB Ap4_l = 5 dB NF4 = 13 dB Av6_l = 15 dB NF6 = 9 dB Ports A, B, C, and D are conjugate-matched for an impedance of 50 Ohms - Noise figure of LNA and mixer are specified for source impedances of 50 Ohms Ports E and F and conjugate-matched for an impedance of 500 Ohms - Noise figure of rightmost amplifier is specified for a source impedance of 500 Ohms M.H. Perrott MIT OCW Methodology for Cascaded NF Calculation Source (Antenna) 50 Ω Vs ens Band Select Filter A B 50 Ω Ap1 = -2.5 dB LNA Image Reject Filter C D LO Ap3 = -4 dB Av2_l = 17 dB NF2 = 1.5 dB Channel Filter E F Amp G IF out Ap5 = -3 dB Av4_l = 15 dB Ap4_l = 5 dB NF4 = 13 dB Av6_l = 15 dB NF6 = 9 dB Perform Noise Figure calculations from right to left Calculation of cumulative Noise Factor at node k - If source and load impedances are equal True for all blocks except mixer above M.H. Perrott MIT OCW Cumulative Noise Factor Calculations Source (Antenna) 50 Ω Vs M.H. Perrott ens Band Select Filter A B 50 Ω Ap1 = -2.5 dB LNA Image Reject Filter C D Ap3 = -4 dB Av2_l = 17 dB NF2 = 1.5 dB LO Channel Filter E F Amp G IF out Ap5 = -3 dB Av4_l = 15 dB Ap4_l = 5 dB NF4 = 13 dB Av6_l = 15 dB NF6 = 9 dB MIT OCW “Level Diagram” for Gain, NF Calculation Source (Antenna) 50 Ω Band Select Filter A B ens Vs Av2_l = 17 dB NF2 = 1.5 dB A Stage Gain (dB) Voltage (loaded) Power B -2.5 -2.5 Cumulative Voltage Gain (dB) Stage NF (dB) M.H. Perrott -2.5 6.7 4.2 E D 14.5 Amp G IF out 10.5 F -3 -3 25.5 13 13.9 Av6_l = 15 dB NF6 = 9 dB E 15 5 4 17.9 F Ap5 = -3 dB -4 -4 1.5 Channel Filter Av4_l = 15 dB Ap4_l = 5 dB NF4 = 13 dB C 17 17 2.5 LO Ap3 = -4 dB Ap1 = -2.5 dB 50 Ω Cumulative NF (dB) LNA Image Reject Filter C D 15 15 22.5 3 12 G 37.5 9 9 MIT OCW The Issue of Receiver Nonlinearity Source (Antenna) ens Rs Band Select Filter Vin Rin Vin(f) 0 f1 f2 IF out Amp LNA Synthesizer 0 IM3 product ∆P f Channel Bandwidth = B Hz f Two-Tone Test Channel Filter |Gain| Vs Image Reject Filter Overall Receiver IIP3 = PIIP3 dBm 0 f3 f4 2f3-f4 2f3-f4 f Lower limit of input power into receiver is limited by sensitivity (i.e., required SNR, Noise Figure, etc.) Upper limit of input power into receiver is determined by nonlinear characteristics of receiver - High input power will lead to distortion that reduces SNR (even in the absence of blockers) - Nonlinear behavior often characterized by IIP3 M.H. Perrott performance of receiver MIT OCW Receiver Dynamic Range Source (Antenna) ens Rs Band Select Filter Vin Channel Filter Rin Vin(f) 0 f1 f2 f Two-Tone Test IF out Amp LNA |Gain| Vs Image Reject Filter Synthesizer 0 Noise Floor ∆P f Channel Bandwidth = B Hz 0 f B Hz Overall Receiver IIP3 = PIIP3 dBm Overall Receiver Noise Factor = NF Defined as difference (in dB) between max and min input power levels to receiver - Min input power level set by receiver sensitivity - Max input power set by nonlinear characteristics of receiver Often defined as max input power for which third order IM products do not exceed the noise floor in a two tone test M.H. Perrott MIT OCW Fundamental IM3 product 0 ∆P f3 f4 2f3-f4 2f3-f4 f Output Power (dBm) A Key IIP3 Expression Fundamental Power POIP3 ∆P 1 IM3 Product Power x Pin PIIP3 Input Power (dBm) By inspection of the right figure Combining the above expressions: M.H. Perrott 3 MIT OCW Fundamental IM3 product 0 ∆P f3 f4 2f3-f4 2f3-f4 f Output Power (dBm) Refer All Signals to Input in Previous IIP3 Expression POIP3 ∆P Fundamental Power 1 3 IM3 Product Power x Pin PIIP3 Input Power (dBm) Difference between fundamental and IM3 products, ∆P, is the same (in dB) when referred to input of amplifier - Both are scaled by the inverse of the amplifier gain Applying algebra: M.H. Perrott MIT OCW Calculation of Spurious Free Dynamic Range (SFDR) Key expressions: - Minimum P in (dBm) set by SNRmin and noise floor Where F is the input referred noise floor of the receiver - Max P in (dBm) occurs when IM3 products = noise floor Dynamic range: subtract min from max Pin (in dB) M.H. Perrott MIT OCW Calculation of Overall IIP3 for Cascaded Stages Memoryless Nonlinearity x Memoryless Nonlinearity y z Assume nonlinearity of each stage characterized as Multiply nonlinearity expressions and focus on first and third order terms Resulting IIP3 expression M.H. Perrott MIT OCW Alternate Expression for Overall IIP3 Memoryless Nonlinearity x Memoryless Nonlinearity y Stage 1 z Stage 2 Worst case IIP3 estimate – take absolute values of terms Square and invert the above expression Express formulation in terms of IIP3 of stage 1 and stage 2 M.H. Perrott MIT OCW A Closer Look at Impact of Second Order Nonlinearity Memoryless Nonlinearity X(w) Two tone test Y(w) α2 α3 y z W w 1 w2 0 x Memoryless Nonlinearity α3 α2, α3 α1, α2, α3 β1, β2, β3 W w1 w2 2w1 2w2 3w1 3w2 0 w2-w1 2w1-w2 2w2-w1 w1+w2 2w1+w2 2w2+w1 Z(w) w1 w2 2w1-w2 2w2-w1 W Influence of α2 of Stage 1 produces tones that are at frequencies far away from two tone input M.H. Perrott MIT OCW Impact of Having Narrowband Amplification Memoryless Nonlinearity X(w) Two tone test Y(w) α3 Memoryless Nonlinearity y z W w 1 w2 0 x Bandpass Filtering Action of Amplifier α3 w1 w2 2w1-w2 2w2-w1 α1, α2, α3 β1, β2, β3 W Z(w) w1 w2 2w1-w2 2w2-w1 W Removal of outside frequencies dramatically simplifies overall IIP3 calculation M.H. Perrott MIT OCW Cascaded IIP3 Calculation with Narrowband Stages Memoryless Nonlinearity Bandpass Filtering Action of Amplifier x y α1, α2, α3 Stage 1 Bandpass Filtering Action of Amplifier Memoryless Nonlinearity Memoryless Nonlinearity z β1, β2, β3 Stage 2 r γ1, γ2, γ3 Stage 3 Note that α1 and β1 correspond to the loaded voltage gain values for Stage 1 and 2, respectively M.H. Perrott MIT OCW Example: IIP3 Calculation for RF Receiver Source (Antenna) 50 Ω ens Band Select Filter A B LNA Image Reject Filter C D LO 30 dB rejection of interfers Channel Filter E F Amp G IF out Ap1 = -2.5 dB 50 Ω IP3 = very large Vs 1 Ap3 = -4 dB Ap5 = -3 dB IP33 = very large IP35 = very large Av6_l = 15 dB Av2_l = 17 dB Av4_l = 15 dB IP36 = 800 mVrms IP32 = 11 dBm IP34 = 8 dBm Ports A, B, C, and D are conjugate-matched for an impedance of 50 Ohms - IIP3 of LNA and mixer are specified for source impedances of 50 Ohms Ports E and F and conjugate-matched for an impedance of 500 Ohms - IIP3 of rightmost amplifier is specified for a source impedance of 500 Ohms M.H. Perrott MIT OCW Key Formulas for IIP3 Calculation Source (Antenna) 50 Ω ens Band Select Filter A B LNA Image Reject Filter C D LO 30 dB rejection of interfers Channel Filter E F Amp G IF out Ap1 = -2.5 dB 50 Ω IP3 = very large Vs 1 Ap3 = -4 dB Ap5 = -3 dB IP33 = very large IP35 = very large Av6_l = 15 dB Av2_l = 17 dB Av4_l = 15 dB IP36 = 800 mVrms IP32 = 11 dBm IP34 = 8 dBm Perform IIP3 calculations from right to left Calculation of cumulative IIP3 at node k (IIP3 in units of rms voltage) - Conversion from rms voltage to dBm M.H. Perrott MIT OCW Cumulative IIP3 Calculations Source (Antenna) 50 Ω Vs ens Band Select Filter A B LO 30 dB rejection of interfers Channel Filter E F Amp G IF out Ap1 = -2.5 dB 50 Ω IP3 = very large 1 M.H. Perrott LNA Image Reject Filter C D Ap3 = -4 dB Ap5 = -3 dB IP33 = very large IP35 = very large Av6_l = 15 dB Av2_l = 17 dB Av4_l = 15 dB IP36 = 800 mVrms IP32 = 11 dBm IP34 = 8 dBm IP32 = 793.4 mVrms IP34 = 561.7 mVrms MIT OCW “Level Diagram” for Gain, IIP3 Calculations Source (Antenna) 50 Ω Band Select Filter A B ens LO Image Reject Filter C D LNA 30 dB rejection of interfers Channel Filter E F Amp G IF out Ap1 = -2.5 dB 50 Ω IP3 = very large Vs Ap3 = -4 dB Ap5 = -3 dB IP33 = very large IP35 = very large Av6_l = 15 dB Av2_l = 17 dB Av4_l = 15 dB IP36 = 800 mVrms IP32 = 11 dBm IP34 = 8 dBm IP32 = 793.4 mVrms IP34 = 561.7 mVrms 1 A B Stage Gain (dB) Voltage (loaded) Power -2.5 -2.5 Cumulative Voltage Gain (dB) Stage IIP3 Power (dBm) Voltage (mVrms) 17 17 Very large Very large 14.5 11 793.4 123.3 -5.17 D E -4 -4 -2.5 Cumulative IIP3 Voltage (mVrms) 164.4 Power (dBm) -2.67 M.H. Perrott C 15 5 883.4 11.93 -3 -3 10.5 Very large Very large 25.5 8 561.7 557.4 7.93 F 15 15 22.5 Very large Very large 25.3e3 91 G 800 1.07 37.5 1.07 800 MIT OCW Final Comments on IIP3 and Dynamic Range Calculations we have presented assume - Narrowband stages Influence of second order nonlinearity removed - IM3 products are the most important in determining maximum input power Practical issues - Narrowband operation cannot always be assumed - Direct conversion architectures are also sensitive to IM2 products (i.e., second order distortion) - Filtering action of channel filter will not reduce in-band IM3 components of blockers (as assumed in the previous example in node E calculation) Must perform simulations to accurately characterize IIP3 (and IIP2) and dynamic range of RF receiver M.H. Perrott MIT OCW
