6.976 High Speed Communication Circuits and Systems Lecture 20 Performance Measures of Wireless Communication
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1 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
2 Recall Digital Modulation for Wireless Link Baseband Input Receiver Output i t (t) Lowpass ir (t) t 2cos(2πf 1 t) 2sin(2πf 1 t) 2cos(2πf 1 t) 2sin(2πf 1 t) t q t (t) Lowpass qr (t) t Decision Boundaries Sample Times t Send discrete-leveled values on I and Q channels Performance issues - Spectral efficiency (transmitter) - Bit error rate performance (receiver) Nonidealities - Intersymbol interference - Noise - Interferers
3 Impact of Receiver Noise Baseband Input i t (t) Input-referred Receiver Noise Lowpass ir (t) Receiver Output t 2cos(2πf 1 t) 2sin(2πf 1 t) 2cos(2πf 1 t) 2sin(2πf 1 t) t q t (t) Lowpass qr (t) t Decision Boundaries Performance impact - SNR is reduced, leading to possible bit errors Methods of increasing SNR - 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 Sample Times t receiver
4 View SNR Issue with Constellation Diagram Q Decision Boundaries I 10 Noise causes sampled I/Q values to vary about their nominal values Bit errors are created when sampled I/Q values cross decision boundaries Decision Boundaries
5 Mathematical Analysis of SNR versus Bit Error Rate (BER) Q Decision Boundaries Probability Density Function for Noise (I-Component) I Probability Density Function for Noise (Q-Component) 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
6 Key Parameters for SNR/BER Analysis Q Decision Boundaries d min d min Probability Density Function for Noise (I-Component) I Probability Density Function for Noise (Q-Component) 10 Decision Boundaries Bit error rate (BER) is a function of - Variance of noise - Distance between constellation points (d min ) Larger d min with a fixed noise variance leads to higher SNR and a lower bit error rate
7 Relationship Between Amplitude and Constellation Peak Amplitude Q Average Amplitude Probability Density Function for Noise (I-Component) d min d min I Probability Density Function for Noise (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)
8 Impact of Increased Signal Power At Receiver Peak Amplitude Q Average Amplitude d min Probability Density Function for Noise (I-Component) I Probability Density Function for Noise (Q-Component) d min Separation between constellation points, d min, 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!
9 Impact of Modulation Scheme on SNR/BER Peak Amplitude Q Average Amplitude Probability Density Function for Noise (I-Component) d min I d min Probability Density Function for Noise (Q-Component) Lowering the number of constellation points increases d min 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
10 Alternate View of Previous Constellation Peak Amplitude Q d min Average Amplitude Probability Density Function for Noise (-45 Degree Component) I d min Common modulation method is to transmit independent binary signals on I and Q channels The above constellation has the same d min as the one on the previous slide - Obtains the same SNR/BER performance given that the noise on I/Q channels is symmetric Probability Density Function for Noise (+45 Degree Component)
11 Impact of Noise Factor on Input-Referred SNR Source (Antenna) e R ns s SNR in Band Select LNA Image Reject Channel Amp SNR out IF out V in Rin Gain Synthesizer f 0 Channel Bandwidth = B Hz Overall Receiver Noise Factor = NF To achieve acceptable bit error rates (BER) Refer SNR requirement to input
12 Minimum Input Power to Achieve Acceptable SNR Source (Antenna) e ns SNR in Overall Receiver Noise Factor = NF V in Rin f 0 Channel Bandwidth = B Hz (1) Calculation of input SNR in terms of input power Combine (1) and (2) (2)
13 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:
14 Receiver Sensitivity Source (Antenna) e ns SNR in Overall Receiver Noise Factor = NF V in Rin f 0 Channel Bandwidth = B Hz Sensitivity of receiver is defined as minimum input power that achieves acceptable SNR (in units of dbm)
15 Example Calculation for Receiver Sensitivity Source (Antenna) e ns SNR in Overall Receiver Noise Factor = NF V in Rin f 0 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
16 Calculation of Noise Figure of Cascaded Stages Source Input-referred Noise for Stage 1 Input-referred Noise for Stage 2 Note: noise of R L not included in Noise Factor calculation e ns e n1 i n1 V in1 R in1 Stage 1 (Noiseless) Unloaded voltage gain = A v1 R out1 V out1 e n2 i n2 V in2 R in2 Stage 2 (Noiseless) Unloaded voltage gain = A v2 R out2 V out2 R L 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 noiseless voltage controlled voltage source with an unloaded gain equal to A v - Ignore noise of final load resistor (or could input refer to previous stage)
17 Method: Refer All Noise to Input of First Stage Source Input-referred Noise for Stage 1 Input-referred Noise for Stage 2 Note: noise of R L not included in Noise Factor calculation e ns e n1 i n1 V in1 R in1 Stage 1 (Noiseless) Unloaded voltage gain = A v1 R out1 V out1 e n2 i n2 V in2 R in2 Stage 2 (Noiseless) Unloaded voltage gain = A v2 R out2 V out2 R L Model for referring stage 2 noise to input of stage 1 Source resistance for Stage 1 Gain Stage 1 (Noiseless) R out1 Input impedance and input-referred noise for Gain Stage 2 e n2 v s v in1 R in1 A v1 v in1 i n2 R in2
18 Step 1: Create an Equivalent Noise Voltage Source R out1 e n2 v s v in1 R in1 A v1 v in1 i n2 R in2 e n2 R out1 v s v in1 R in1 A v1 v in1 i n2 R in2 e n2 +R out1 i n2 R out1 v s v in1 R in1 A v1 v in1 R in2
19 Step 2: Input Refer Voltage Noise Source e n2 +R out1 i n2 R out1 v s v in1 R in1 A v1 v in1 R in2 e n2 +R out1 i n2 α 1 R out1 v s v in1 R in1 A v1 v in1 R in2 Scaling factor α 1 is a function of unloaded gain, A v, and input voltage divider
20 Input Referral of Noise to First Stage Source Input-referred Noise for Stage 1 Input-referred Noise for Stage 2 Note: noise of R L not included in Noise Factor calculation e ns e n1 i n1 V in1 R in1 Stage 1 (Noiseless) R out1 V out1 e n2 i n2 V in2 R in2 Stage 2 (Noiseless) Unloaded = A v1 = A v2 voltage gain Unloaded voltage gain R out2 V out2 R L e ns e n2 +R out1 i n2 α 1 Stage 2 Noise e n1 i n1 V in1 R in1 Stage 1 (Noiseless) R out1 V out1 V in2 R in2 Stage 2 (Noiseless) Unloaded = A v1 = A v2 voltage gain Unloaded voltage gain R out2 V out2 R L Where
21 Noise Factor Calculation Source Input-referred Noise for Stage 1 Input-referred Noise for Stage 2 Note: noise of R L not included in Noise Factor calculation e ns e n1 i n1 V in1 R in1 Stage 1 (Noiseless) Unloaded voltage gain = A v1 R out1 V out1 e n2 i n2 V in2 R in2 Stage 2 (Noiseless) Unloaded voltage gain = A v2 R out2 V out2 R L e ns e n2 +R out1 i n2 α 1 e n1 Stage 2 Noise i n1 i sc
22 Alternate Noise Factor Expression
23 Define Available Power Gain Available Source Power Source Source Available Power at Output Gain Stage 1 R out1 v s v in1 R in1 = v s v in1 R in1 A v1 v in1 R L =R out1 Available power gain for stage 1 defined as Available power at output (matched load) A p1 = Available source power (matched load) - Available power at output - Available source power
24 Available Power Gain Versus Loaded Voltage Gain Available Source Power Source Source Available Power at Output Gain Stage 1 R out1 v s v in1 R in1 = v s v in1 R in1 A v1 v in1 R L =R out1 Available power gain for stage 1 If R in1 = R out1 = - Where A v1_l is defined as the loaded gain of stage 1
25 Final Expressions for Cascaded Noise Factor Calculation Source Note: noise of R L not included in Noise Factor calculation e ns NF 1 NF 2 Stage 1 Stage 2 R in1 Unloaded voltage gain = A v1 R out1 R in2 Unloaded voltage gain = A v2 R out2 R ink Stage k NF k Unloaded voltage gain = A vk R outk V out R L Overall Noise Factor (general expression) Overall Noise Factor when all input and output impedances equal :
26 Calculation of Noise Factor for Lossy Passive Networks Source e ns Passive Network with Loss L R e nloss loss Note: noise of R L not included in Noise Factor calculation V in C V out R L R in 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 R out - See pages of Razavi book
27 Define Available Power Gain For Passive Networks Available Power at Output Available Source Power R out V in R in A v V in R L =R out R in = V out V in Available power at output Available source power
28 Equivalent Noise Model and Resulting NF Calculation Equivalent Model for Computing Total Noise Equivalent Model for Computing Source Noise Contribution R out e ns R out e nout V out R L V in R in A v V in V out R L Total noise at output Noise due to source (referred to output) Noise factor
29 Example: Impact of Cascading Passive with LNA Source e ns Note: noise of R L not included in Noise Factor calculation Passive LNA V out R L Insertion Loss = 2 db Noise Factor calculation Noise Figure = 1.5 db Noise Figure
30 Example: Noise Factor Calculation for RF Receiver Source (Antenna) 50 Ω e ns Band Select Image Reject Channel A B C D E F G LNA Amp IF out 50 Ω A p1 = -2.5 db A p3 = -4 db A p5 = -3 db A v2_l = 17 db A v4_l = 15 db A v6_l = 15 db NF 2 = 1.5 db A p4_l = 5 db NF 6 = 9 db NF 4 = 13 db LO 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
31 Methodology for Cascaded NF Calculation Source (Antenna) 50 Ω e ns Band Select Image Reject Channel A B C D E F G LNA Amp IF out 50 Ω A p1 = -2.5 db A p3 = -4 db A p5 = -3 db A v2_l = 17 db A v4_l = 15 db A v6_l = 15 db NF 2 = 1.5 db LO A p4_l = 5 db NF 4 = 13 db NF 6 = 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
32 Cumulative Noise Factor Calculations Source (Antenna) 50 Ω e ns Band Select Image Reject Channel A B C D E F G LNA Amp IF out 50 Ω A p1 = -2.5 db A p3 = -4 db A p5 = -3 db A v2_l = 17 db A v4_l = 15 db A v6_l = 15 db NF 2 = 1.5 db LO A p4_l = 5 db NF 4 = 13 db NF 6 = 9 db
33 Level Diagram for Gain, NF Calculation Source (Antenna) 50 Ω e ns Band Select Image Reject Channel A B C D E F G LNA Amp IF out 50 Ω A p1 = -2.5 db A p3 = -4 db A p5 = -3 db A v2_l = 17 db A v4_l = 15 db A v6_l = 15 db NF 2 = 1.5 db A p4_l = 5 db NF 6 = 9 db NF 4 = 13 db LO Stage Gain (db) Voltage (loaded) Power Cumulative Voltage Gain (db) A B C D E F G Stage NF (db) Cumulative NF (db)
34 The Issue of Receiver Nonlinearity Source (Antenna) Band Select Image Reject Channel e ns V in LNA Amp IF out V in (f) Rin f 0 f 1 f 2 Two-Tone Test Synthesizer Gain f 0 Channel Bandwidth = B Hz Overall Receiver IIP3 = P IIP3 dbm 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 IM3 product 0 P f f 3 f 4 2f 3 -f 4 2f 3 -f 4 performance of receiver
35 Receiver Dynamic Range Source (Antenna) Band Select Image Reject Channel e ns V in LNA Amp IF out V in (f) Rin f 0 f 1 f 2 Two-Tone Test Synthesizer Gain f 0 Channel Bandwidth = B Hz Overall Receiver IIP3 = P IIP3 dbm Overall Receiver Noise Factor = NF Noise Floor 0 B Hz P f 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
36 A Key IIP3 Expression 0 Fundamental IM3 product f 3 f 4 P 2f 3 -f 4 2f 3 -f 4 f Output Power (dbm) P OIP3 P Fundamental Power 1 P in 3 x PIIP3 Input Power (dbm) IM3 Product Power By inspection of the right figure Combining the above expressions:
37 Refer All Signals to Input in Previous IIP3 Expression 0 Fundamental IM3 product f 3 f 4 P 2f 3 -f 4 2f 3 -f 4 f Output Power (dbm) P OIP3 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 P Fundamental Power 1 P in 3 x PIIP3 Input Power (dbm) IM3 Product Power Applying algebra:
38 Calculation of Spurious Free Dynamic Range (SFDR) Key expressions: - Minimum P in (dbm) set by SNR min 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 P in (in db)
39 Calculation of Overall IIP3 for Cascaded Stages Memoryless Nonlinearity Memoryless Nonlinearity x y z Assume nonlinearity of each stage characterized as Multiply nonlinearity expressions and focus on first and third order terms Resulting IIP3 expression
40 Alternate Expression for Overall IIP3 Memoryless Nonlinearity Memoryless Nonlinearity x y z Stage 1 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
41 A Closer Look at Impact of Second Order Nonlinearity Memoryless Nonlinearity Memoryless Nonlinearity X(w) Two tone test x y z W 0 w 1 w 2 α 1, α 2, α 3 β 1, β 2, β 3 α 2, α 3 Y(w) α 2 α 3 α 3 w w 2 -w 1 w 2 2w 1 2w 2 3w 1 3w 2 W 0 1 2w 1 -w 2 2w 2 -w 1 w 1 +w 2 2w 1 +w 2 2w 2 +w 1 Z(w) w 1 w 2 2w 1 -w 2 2w 2 -w 1 W Influence of α 2 of Stage 1 produces tones that are at frequencies far away from two tone input
42 Impact of Having Narrowband Amplification X(w) Two tone test x Memoryless Nonlinearity Bandpass ing Action of Amplifier y Memoryless Nonlinearity z Y(w) W 0 w 1 w 2 α 3 α 3 α 1, α 2, α 3 β 1, β 2, β 3 w 1 w 2 2w 1 -w 2 2w 2 -w 1 W Z(w) w 1 w 2 2w 1 -w 2 2w 2 -w 1 W Removal of outside frequencies dramatically simplifies overall IIP3 calculation
43 Cascaded IIP3 Calculation with Narrowband Stages Memoryless Nonlinearity Bandpass ing Action of Amplifier Memoryless Nonlinearity Bandpass ing Action of Amplifier Memoryless Nonlinearity x y z r α 1, α 2, α 3 β 1, β 2, β 3 γ 1, γ 2, γ 3 Stage 1 Stage 2 Stage 3 Note that α 1 and β 1 correspond to the loaded voltage gain values for Stage 1 and 2, respectively
44 Example: IIP3 Calculation for RF Receiver Source (Antenna) 50 Ω e ns Band Select Image Reject Channel A B C D E F G LNA Amp IF out 50 Ω A p1 = -2.5 db IP3 1 = very large A p3 = -4 db IP3 3 = very large A p5 = -3 db IP3 5 = very large A v2_l = 17 db A v4_l = 15 db A v6_l = 15 db IP3 2 = 11 dbm IP3 4 = 8 dbm IP3 6 = 800 mv rms LO 30 db rejection of interfers 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
45 Key Formulas for IIP3 Calculation Source (Antenna) 50 Ω e ns Band Select Image Reject Channel A B C D E F G LNA Amp IF out 50 Ω A p1 = -2.5 db IP3 1 = very large A p3 = -4 db IP3 3 = very large A p5 = -3 db IP3 5 = very large A v2_l = 17 db A v4_l = 15 db A v6_l = 15 db IP3 2 = 11 dbm IP3 4 = 8 dbm IP3 6 = 800 mv rms LO 30 db rejection of interfers 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
46 Cumulative IIP3 Calculations Source (Antenna) 50 Ω e ns Band Select Image Reject Channel A B C D E F G LNA Amp IF out 50 Ω A p1 = -2.5 db IP3 1 = very large A p3 = -4 db IP3 3 = very large A p5 = -3 db IP3 5 = very large A v2_l = 17 db A v4_l = 15 db A v6_l = 15 db IP3 2 = 11 dbm IP3 4 = 8 dbm IP3 6 = 800 mv rms IP3 2 = mv rms IP3 4 = mv rms LO 30 db rejection of interfers
47 Level Diagram for Gain, IIP3 Calculations Source (Antenna) 50 Ω e ns Band Select Image Reject Channel A B C D E F G LNA Amp IF out 50 Ω A p1 = -2.5 db IP3 1 = very large A p3 = -4 db IP3 3 = very large A p5 = -3 db IP3 5 = very large A v2_l = 17 db A v4_l = 15 db A v6_l = 15 db IP3 2 = 11 dbm IP3 4 = 8 dbm IP3 6 = 800 mv rms IP3 2 = mv rms IP3 4 = mv rms LO 30 db rejection of interfers Stage Gain (db) Voltage (loaded) Power Cumulative Voltage Gain (db) Stage IIP3 Power (dbm) Voltage (mv rms ) A B C D E F G Very large 11 Very large 8 Very large 1.07 Very large Very large Very large 800 Cumulative IIP3 Voltage (mv rms ) e3 800 Power (dbm)
48 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) - ing 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
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