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Section 3 2 PerformanceDMoverFading
Introduction AWGN channels Fading Channels Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels Introduction AWGN channels Fading Channels Outline of the lecture notes Introduction AWGN channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Fading Channels Introduction Outage probability Average probability of error Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels Introduction AWGN channels Fading Channels Introduction We now consider the performance of the digital modulation techniques discussed in the previous chapter when used over AWGN channels and channels with flat-fading There are two performance criteria of interest: the probability of error, defined relative to either symbol or bit errors, and the outage probability, defined as the probability that the instantaneous signal-to-noise ratio falls below a given threshold Wireless channels may also exhibit frequency selective fading and Doppler shift Frequency-selective fading gives rise to intersymbol interference (ISI), which causes an irreducible error floor in the received signal Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels Introduction AWGN channels Fading Channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Signal-to-Noise power ratio and bit/symbol energy In this section we define the signal-to-noise power ratio (SNR) and its relation to energy-per-bit (𝐸𝑏 ) and energy-per-symbol (𝐸𝑠 ) We then examine the error probability on AWGN channels for different modulation techniques as parameterized by these energy metrics Our analysis uses the signal space concepts of previous section ] [ In an AWGN channel, the modulated signal 𝑠(𝑡) = Re 𝑢(𝑡)𝑒𝑗2𝜋𝑓𝑐 𝑡 has receiver noise 𝑛(𝑡) added to it prior to reception The noise 𝑛(𝑡) is a white Gaussian random process with zero-mean and power spectral density 𝑁0 /2 The received signal is thus 𝑟(𝑡) = 𝑠(𝑡) + 𝑛(𝑡) Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels Introduction AWGN channels Fading Channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Signal-to-Noise power ratio and bit/symbol energy (cont.) We define the received signal-to-noise power ratio (SNR) as the ratio of the received signal power 𝑃𝑟 to the power of the noise within the bandwidth of the transmitted signal 𝑠(𝑡) The received power 𝑃𝑟 is determined by the transmitted power and the path loss and multipath fading The noise power is determined by the bandwidth of the transmitted signal and the spectral properties of 𝑛(𝑡) Specifically, if the bandwidth of the complex envelope 𝑢(𝑡) of 𝑠(𝑡) is 𝐵 then the bandwidth of the transmitted signal 𝑠(𝑡) is 2𝐵 Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels Introduction AWGN channels Fading Channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Signal-to-Noise ratio and bit/symbol energy (cont.) Since the noise 𝑛(𝑡) has uniform power spectral density 𝑁0 /2, the total noise power within the bandwidth 2𝐵 is 𝑃𝑛 = 𝑁0 /2 × 2𝐵 = 𝑁0 𝐵 So, the received SNR is given by 𝑆𝑁 𝑅 = 𝑃𝑟 𝑁0 𝐵 (1) In systems with interference, we often use the received signal-to-interference-plus-noise power ratio (SINR) in place of SNR for calculating error probability If the interference statistics approximate those of Gaussian noise then this is a reasonable approximation The received SINR is given by 𝑆𝑁 𝑅 = 𝑃𝑟 𝑁0 𝐵 + 𝑃𝐼 (2) where 𝑃𝐼 is the average power of the interference Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels Introduction AWGN channels Fading Channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Signal-to-Noise ratio and bit/symbol energy (cont.) The SNR is often expressed in terms of the signal energy per bit 𝐸𝑏 or per symbol 𝐸𝑠 as 𝑆𝑁 𝑅 = 𝑃𝑟 𝐸𝑠 𝐸𝑏 = = 𝑁0 𝐵 𝑁0 𝐵𝑇𝑠 𝑁0 𝐵𝑇𝑏 (3) where 𝑇𝑠 and 𝑇𝑏 are the symbol and bit durations, respectively For binary modulation (e.g., BPSK), 𝑇𝑠 = 𝑇𝑏 and 𝐸𝑠 = 𝐸𝑏 For data shaping pulses with 𝑇𝑠 = 1/𝐵 (e.g., raised cosine pulses with 𝛽 = 1), one will have SNR = 𝐸𝑠 /𝑁0 for multilevel signaling and SNR = 𝐸𝑏 /𝑁0 for binary signaling For general pulses, 𝑇𝑠 = 𝑘/𝐵 for some constant 𝑘, we have 𝑘 × SNR = 𝐸𝑠 /𝑁0 The quantities 𝛾𝑠 = 𝐸𝑠 /𝑁0 and 𝛾𝑏 = 𝐸𝑏 /𝑁0 are sometimes called the SNR per symbol and the SNR per bit, respectively Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels Introduction AWGN channels Fading Channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Signal-to-Noise ratio and bit/symbol energy (cont.) For performance specification, we are interested in the bit error probability 𝑃𝑏 as a function of 𝛾𝑏 However, for M-array signaling (e.g., MPAM and MPSK), the bit error probability depends on both the symbol error probability and the mapping of bits to symbols Thus, we typically compute the symbol error probability 𝑃𝑠 as a function of 𝛾𝑠 based on the signal space concepts of previous section and then obtain 𝑃𝑏 as a function of 𝛾𝑏 using an exact or approximate conversion The approximate conversion typically assumes that the symbol energy is divided equally among all bits, and that Gray encoding is used so that at reasonable SNRs, one symbol error corresponds to exactly one bit error These assumptions for M-array signaling lead to the approximations: 𝛾𝑏 ≈ 𝛾𝑠 𝑃𝑠 and 𝑃𝑏 ≈ log2 𝑀 log2 𝑀 Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels (4) Introduction AWGN channels Fading Channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Error probability for BPSK and QPSK Consider BPSK modulation with coherent detection and perfect recovery of the carrier frequency and phase With binary modulation each symbol corresponds to one bit, so the symbol and bit error rates are the same The transmitted signal is 𝑠1 (𝑡) = 𝐴𝑔(𝑡)𝑐𝑜𝑠(2𝜋𝑓𝑐 𝑡) to send a bit and 𝑠2 (𝑡) = −𝐴𝑔(𝑡)𝑐𝑜𝑠(2𝜋𝑓𝑐 𝑡) to send a bit Note that for binary modulation where 𝑀 = 2, there is only one way to make an error and 𝑑𝑚𝑖𝑛 is the distance between the two signal constellation points, so the probability of error is also the bound: ) ( 𝑑𝑚𝑖𝑛 (5) 𝑃𝑏 = 𝑄 √ 2𝑁0 In previous chapter, we have 𝑑𝑚𝑖𝑛 =∥ s1 − s2 ∥=∥ 𝐴 − (−𝐴) ∥= 2𝐴 The energy-per-bit can be determined by ∫ 𝑇𝑏 ∫ 𝑇𝑏 ∫ 𝑇𝑏 𝐸𝑏 = 𝑠21 (𝑡)𝑑𝑡 = 𝑠22 (𝑡)𝑑𝑡 = 𝐴2 𝑔 (𝑡) cos2 (2𝜋𝑓𝑐 𝑡)𝑑𝑡 = 𝐴2 0 Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels (6) Introduction AWGN channels Fading Channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Error probability for BPSK and QPSK (cont.) Thus, the signal √ constellation for √ BPSK in terms of energy-per-bit is given by s0 = 𝐸𝑏 and s√1 = − 𝐸𝑏 This yields the minimum distance 𝑑𝑚𝑖𝑛 = 2𝐴 = 𝐸𝑏 Substituting this into (5) yields (√ ) ( √ ) (√ ) 2𝐸𝑏 𝐸𝑏 2𝛾𝑏 (7) 𝑃𝑏 = 𝑄 √ =𝑄 =𝑄 𝑁0 2𝑁0 QPSK modulation consists of BPSK modulation on both the in-phase and quadrature components of the signal With perfect phase and carrier recovery, the received signal components corresponding to each of these branches are orthogonal Therefore, the bit error probability (√ ) on each branch is the same as for BPSK:𝑃𝑏 = 𝑄 2𝛾𝑏 Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels 10 Introduction AWGN channels Fading Channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Error probability for BPSK and QPSK (cont.) The symbol error probability equals the probability that either branch has a bit error: [ (√ )]2 2𝛾𝑏 𝑃𝑠 = − − 𝑄 (8) Example: Find the bit error probability 𝑃𝑏 and symbol error probability 𝑃𝑠 of QPSK assuming 𝛾(𝑏√= )dB Solution: We have 𝛾𝑏 = 107/10 = 5.012, then 𝑃𝑏 = 𝑄 2𝛾𝑏 = 7.726 × 10−4 and [ (√ )]2 𝑃𝑠 = − − 𝑄 2𝛾𝑏 = 1.545 × 10−3 Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels 11 Introduction AWGN channels Fading Channels Signal-to-Noise power ratio and bit/symbol energy Error probability for BPSK and QPSK Approximate symbol and bit error probabilities for typical modulations Approximate symbol and bit error probabilities for typical modulations Many of the approximations or exact values for 𝑃𝑠 derived above for coherent modulation are in the following form: (√ ) 𝑃𝑠 (𝛾𝑠 ) ≈ 𝛼𝑀 𝑄 𝛽𝑀 𝛾𝑠 (9) where 𝛼𝑀 and 𝛽𝑀 depend on the type of approximation and the modulation type In the below table, we summarize the specific values of 𝛼𝑀 and 𝛽𝑀 for common 𝑃𝑠 expressions for PSK, QAM, and FSK modulations based on the derivations in the prior sections Modulation BFSK: BPSK: QPSK,4QAM: MPAM: MPSK: Rectangular MQAM: Nonrectangular MQAM: Ps (γs ) √ Ps ≈ Q γs q 6γ s Ps ≈ 2(MM−1) Q M −1 √ Ps ≈ 2Q 2γs sin(π/M ) q √ 3γ s Ps ≈ 4( √MM−1) Q M −1 q 3γ s Ps ≈ 4Q M −1 Mobile communications - Chapter 3: Physical-layer transmission techniques Pb (γb ) √ Pb = Q γb √ Pb = Q 2γb √ Pb ≈ Q 2γb q 6γ b log2 M 2(M −1) Pb ≈ M log2 M Q (M −1) p Pb ≈ log2 M Q 2γb log2 M sin(π/M ) √ q 3γ b log2 M −1) Pb ≈ √4(M M Q (M −1) log2 M q 3γ b log2 M Pb ≈ log M Q (M −1) Section 3.2: Performance analysis over fading channels 12 Introduction AWGN channels Fading Channels Introduction Outage probability Average probability of error Introduction In AWGN the probability of symbol error depends on the received SNR 𝛾𝑠 In a fading channel, the received signal power varies randomly over distance or time due to shadowing and/or multipath fading Thus, in fading 𝛾𝑠 is a random variables with distribution 𝑝𝛾𝑠 (𝛾), and therefore 𝑃𝑠 (𝛾𝑠 ) is also random The performance metric when 𝛾𝑠 is random depends on the rate of change of the fading There are three different performance criteria that can be used to characterize the random variable 𝑃𝑠 : The outage probability, 𝑃𝑜𝑢𝑡 , defined as the probability that 𝛾𝑠 falls below a given value corresponding to the maximum allowable 𝑃𝑠 The average error probability, 𝑃𝑠 , averaged over the distribution of 𝛾𝑠 Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels 13 Introduction AWGN channels Fading Channels Introduction Outage probability Average probability of error Introduction (cont.) If the power of the received signal (with fading) is changing slowly (slow-fading), then a deep fade will affect many simultaneous symbols Thus, fading may lead to large error bursts, which cannot be corrected for with coding of reasonable complexity Therefore, these error bursts can seriously degrade end-to-end performance In this case acceptable performance cannot be guaranteed over all time or, equivalently, throughout a cell, without drastically increasing transmit power Under these circumstances, an outage probability is specified so that the channel is deemed unusable for some fraction of time or space Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels 14 Introduction AWGN channels Fading Channels Introduction Outage probability Average probability of error Outage probability The outage probability relative to 𝑃𝑜𝑢𝑡 is defined as ∫ 𝛾0 𝑃𝑜𝑢𝑡 = 𝑝 (𝛾𝑠 < 𝛾0 ) = 𝑝𝛾𝑠 (𝛾)𝑑𝛾 (10) where 𝛾0 typically specifies the minimum SNR required for acceptable performance For example, if we consider digitized voice, 𝑃𝑏 = 10−3 is an acceptable error rate since it generally cannot be detected by the human ear Thus, for a BPSK signal in Rayleigh fading, 𝛾𝑏 < dB would be declared an outage, so we set 𝛾0 = dB In Rayleigh fading with 𝑝𝛾𝑠 (𝛾) = 𝛾1 𝑒−𝛾𝑠 /𝛾 𝑠 , one will have 𝑠 𝑃𝑜𝑢𝑡 = ∫ 𝛾0 −𝛾𝑠 /𝛾 𝑠 𝑒 𝑑𝛾 = − 𝑒−𝛾0 /𝛾 𝑠 𝛾𝑠 Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels (11) 15 Introduction AWGN channels Fading Channels Introduction Outage probability Average probability of error Outage probability (cont.) Example: Determine the required 𝛾 𝑏 for BPSK modulation in slow Rayleigh fading such as 50% of the time (or in space), 𝑃𝑏 (𝛾𝑏 ) < 10−4 Solution: For BPSK modulation in AWGN, √ the target BER is obtained at 8.5 dB (i.e., for 𝑃 (𝛾 ) = 𝑄( 2𝛾𝑏 ), one have 𝑏 𝑏 ( ) 𝑃𝑏 100.85 = 10−4 ) Thus, 𝛾0 = 8.5 dB, since we want 𝑃𝑜𝑢𝑡 = 𝑝(𝛾𝑏 < 𝛾0 ), we have 𝛾𝑏 = 𝛾0 10.85 = = 21.4 dB − ln(1 − 𝑃𝑜𝑢𝑡 ) − ln(1 − 05) Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels (12) 16 Introduction AWGN channels Fading Channels Introduction Outage probability Average probability of error Average probability of error The average probability of error is used as a performance metric when 𝛾𝑠 is roughly constant over a symbol time Then the averaged probability of error is computed by integrating the error probability in AWGN over the fading distribution: ∫ ∞ 𝑃𝑠 = 𝑃𝑠 (𝛾)𝑝𝛾𝑠 (𝛾)𝑑𝛾 (13) where 𝑃𝑠 (𝛾) is the probability of symbol error in AWGN channels with SNR 𝛾, which can be approximated by the expressions in the aforementioned table For a given distribution of the fading amplitude 𝑟 (i.e., Rayleigh, Rician, log-normal, etc.), we compute 𝑝𝛾𝑠 (𝛾) by making the change of variable 𝑝𝛾𝑠 (𝛾)𝑑𝛾 = 𝑝(𝑟)𝑑𝑟 Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels (14) 17 Introduction AWGN channels Fading Channels Introduction Outage probability Average probability of error Average probability of error (cont.) For instance, in Rayleigh fading (refer to Zheng’s paper on Modified Jake channel model), the received signal amplitude r has the Rayleigh distribution 𝑝(𝑟) = 𝑟 −𝑟2 /(2𝜎2 ) 𝑒 , 𝑟 ≥ 𝜎2 (15) The SNR per symbol for a given amplitude 𝑟 is 𝛾= 𝑟 𝑇𝑠 2𝜎𝑛2 (16) where 𝜎𝑛2 = 𝑁0 /2 is the PSD of the noise in the in-phase and quadrature branches Differentiating both sides of this expression yields 𝑑𝛾 = Mobile communications - Chapter 3: Physical-layer transmission techniques 𝑟𝑇𝑠 𝑑𝑟 𝜎𝑛2 Section 3.2: Performance analysis over fading channels (17) 18 Introduction AWGN channels Fading Channels Introduction Outage probability Average probability of error Average probability of error (cont.) Substituting (16) and (17) into (15) and then (14) yields 𝑝𝛾𝑠 (𝛾) = 𝜎𝑛2 −𝛾𝜎𝑛2 /(𝜎2 𝑇𝑠 ) 𝑒 𝜎 𝑇𝑠 (18) Since the average SNR per symbol 𝛾 𝑠 is just 𝜎 𝑇𝑠 /𝜎𝑛2 , one can rewrite (18) as 𝛾/𝛾𝑠 𝑝𝛾𝑠 (𝛾) = 𝑒 , (19) 𝛾𝑠 which is exponential For binary signaling, this reduces to 𝑝𝛾𝑏 (𝛾) = Mobile communications - Chapter 3: Physical-layer transmission techniques 𝛾/𝛾𝑏 𝑒 𝛾𝑏 Section 3.2: Performance analysis over fading channels (20) 19 Introduction AWGN channels Fading Channels Introduction Outage probability Average probability of error Average probability of error (cont.) Integrating the error probability of BPSK in AWGN over the distribution (20) yields the following average probability of error for BPSK in Rayleigh fading: √ ( ) 𝛾𝑏 𝑃𝑏 = 1− ≈ (21) + 𝛾𝑏 4𝛾 𝑏 where the approximation holds for large 𝛾 𝑏 (√ ) If we use the general approximation 𝑃𝑠 ≈ 𝛼𝑀 𝑄 𝛽𝑀 𝛾𝑠 then the average probability of symbol error in Rayleigh fading can be approximated as √ ( ) ∫ ∞ (√ ) 5𝛽𝑀 𝛾 𝑠 𝛼𝑚 −𝛾/𝛾 𝑠 𝑃𝑠 ≈ 𝛼𝑀 𝑄 𝛽𝑀 𝛾 𝑒 𝑑𝛾𝑠 = 1− 𝛾𝑠 + 5𝛽𝑀 𝛾 𝑠 𝛼𝑀 ≈ 2𝛽𝑀 𝛾 𝑠 Mobile communications - Chapter 3: Physical-layer transmission techniques Section 3.2: Performance analysis over fading channels 20