Digital Modulation and Detection

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(1)Digital Modulation and Detection. Wha Sook Jeon Mobile Computing & Communication Lab..

(2) Signal Space Analysis. 2.

(3) Signal and System Model (1)    . System sends K = log 2 M bits every T seconds Each bit sequence of length K comprises a message mi ={b1, … , bK} The message i has the probability pi of being selected for transmission ( ∑iM=1 pi = 1 ) Each message is mapped to a unique analog signal si (t), is transmitted over the channel during the time interval [0, T], and has energy T.   . ESi = ∫ si2 (t ) dt , i = 1,  , M 0. The transmitted signal is sent through an AWGN channel (a white Gaussian noise process n(t) of power spectral density N0/2) Received signal r(t)=s(t)+n(t) The receiver should determine the best estimate of the transmitted signal and outputs the best estimate of the transmitted message mˆ = {bˆ1 ,  , bˆK } 3.

(4) Signal and System Model (2)  Goal of the receiver design −. minimizing the probability of message estimation error M. Pe = ∑ p (mˆ ≠ mi | mi sent) p (mi sent) i =1. Communication system model. 4.

(5) Geometric Representation of Signals (1)  By representing the signals geometrically, we can solve for the optimal receiver design in AWGN based on a minimum distance criterion.  Basis function representation −. Any set of M real energy signals S = {s1 (t ), , sM (t )} can be represented as a liner combination of N (≤ M) real orthonormal basis functions {φ1 (t ),, φ N (t )} according to Gram-Schmidt orthogonalization procedure N : Modulation si (t ) = ∑ sijφ j (t ), 0≤t <T j =1. −. T Orthonormal basis function: ∫ φi (t )φ j (t )dt = 1 if i = j 0. −. Real coefficient representing the projection of si(t) onto φ j (t ). 0 if i ≠ j. T. sij = ∫ si (t )φ j (t )dt. : Demodulation. 0. 5.

(6) Geometric Representation of Signals (2)  Basis set of linear pathband modulation technique − φ (t ) = 2 cos(2πf t ), 1 c T. −. . With fcT >> 1 for orthonormality. Transmitted signal 2 2 cos(2πf c t ) + si 2 sin( 2πf c t ) T T. si (t ) = si1. . 2 sin( 2πf c t ) T. φ2 (t ) =. The basis set can include a bandpass pulse-shaping filter g(t) to improve the spectral characteristics of the transmitted signal si (t ) = si1 g (t ) cos(2πf c t ) + si 2 g (t ) sin(2πf c t ). . The pulse shape g(t) must maintain the orthonormal property of basis functions. ∫. T. 0. g (t ) cos (2πf c t )dt = 1, 2. 2. ∫. T. 0. g 2 (t ) cos(2πf c t ) sin( 2πf c t )dt = 0 6.

(7) Geometric Representation of Signals (3)  Signal space representation −. Signal constellation point of the signal si(t) ■ ■. ■. −. s i = ( si1 ,  siN ) ∈ R N the vector of coefficients in the basis representation of si(t), N that is, si (t ) = ∑ sijφ j (t ) j =1 One-to-one correspondence between si(t) and si. The distance between two signal constellation points si and sk si − sk = =. N. ∑ (s j =1. ∫. T. 0. ij. − skj ) 2. ( si (t ) − sk (t )) 2 dt. Signal space of MQAM or MPSK is two-dimensional 7.

(8) Receiver Structure and Sufficient Statistics (1) . Given the channel output r(t)=si(t)+n(t), 0 ≤ t < T, the receiver determines which constellation point (or message) is sent over time interval [0, T ). 8.

(9) Receiver Structure and Sufficient Statistics (2)  In this receiver structure − r (t ) = si (t ) + n(t ). = ∑ j =1 sijφ j (t ) + ∑ j =1 n jφ j (t ) + nr (t ) N. N. = ∑ j =1 ( sij + n j )φ j (t ) + nr (t ) = ∑ j =1 rjφ j (t ) + nr (t ) N. N. T. T. − sij = ∫0 si (t )φ j (t )dt , n j = ∫0 n(t )φ j (t )dt − nr(t) is the “remainder” noise which is orthogonal to signal space. 9.

(10) Receiver Structure and Sufficient Statistics (3). (r1 , r2 , nr ). r = (r1 , r2 ) [r (t ) = r1φ1 (t ) + r1φ1 (t ) + nrφnr (t )] 10.

(11) Receiver Structure and Sufficient Statistics (3)  Goal of receiver design is to minimize the error probability Pe = p (mˆ ≠ mi | r (t )) = 1 − p (mˆ = mi | r (t )).  Maximizing p ( si sent | r (t )) = p (( si1 ,  , siN ) sent | (ri1 ,  , riN ) , nr (t )) p (( si1 ,  , siN ) sent, (ri1 ,  , riN ) , nr (t )) = p ((ri1 ,  , riN ) , nr (t )) p (( si1 ,  , siN ) sent, (ri1 ,  , riN )) p (nr (t )) = p ((ri1 ,  , riN )) p (nr (t )) = p (( si1 ,  , siN ) sent | (ri1 ,  , riN )).  r =(r1, …, rN) is a sufficient statistic for r(t) in optimal detection of transmitted message 11.

(12) Decision Region  . Optimal receiver selects m ˆ = mi corresponding to constellation si that satisfies p( si sent | r ) ≥ p( s j sent | r ) for all j ≠ i Design region: Z i = {r : p( si sent | r ) > p( s j sent | r ) ∀ j ≠ i}. 12.

(13) Maximum Likelihood Decision Criteria (1) . ˆ = mi corresponding to constellation si Optimal receiver selects m that maximizes p (r | si ) p ( si ) p ( si | r ) = p(r ) p (r | si ) p ( si ) arg max = arg max p (r | si ) p ( si ) si si p(r ) = arg max p (r | si ) si.  . Let p(si)=1/M. Likelihood function: L( si ) = p (r | si ) ˆ = micorresponding to A maximum likelihood receiver outputs m constellation si that maximizes L(si). 13.

(14) Maximum Likelihood Decision Criteria (2)  Conditional distribution of r − −. Since n(t) is a Gaussian random process, r(t)=si(t)+n(t) is also a Gaussian random process and n(t) has a zero mean.. rj = sij + n j µ r |s = E[rj | sij ] = E[ sij + n j | sij ] = sij j. i. σ 2 r |s = E[(rj − µ r |s ) 2 ] = E[( sij + n j − sij ) 2 | sij ] = E[n j 2 ] = N 0 2 j. i. j. ij. cov[rj rk | si ] = E[(rj − µ r j )(rk − µ rk ) | si ] = E[n j nk ] N 2 = 0  0. −. j=k j≠k. rj is a Gaussian random variable that is independent of rk ( j ≠ k ) with mean sij and variance N0/2. 14.

(15) Maximum Likelihood Decision Criteria (3).  Likelihood function L(si): conditional distribution of r  1 1 exp − p (r | si ) = ∏ p (rj | sij ) = N /2 ( ) N π j =1 0  N0 N.  (rj − sij )  ∑ j =1  N. 2.  Log likelihood function 1 l ( si ) = − N0. N. 1 (rj − sij ) = − r − si ∑ N0 j =1 2. 2. minimizing. 15.

(16) Maximum Likelihood Decision Criteria (4)  A maximum likelihood receiver outputs mˆ = mi corresponding to constellation si that satisfies N. arg min ∑ (rj − sij ) 2 = arg min r − si si. j =1. 2. si.  Decision Region Z i = {r : r − si < r − s j ∀ j = 1,  , M , j ≠ i}.  Constellation point si is determined from the decision Zi that contains r. 16.

(17) Union Bound of Error Probability (1)   . Aik: the event that r − sk < r − si given that the constellation point si was sent If the event Aik occurs, the constellation will be decoded in error.. M  M  Pe (mi sent) = p  Aik  ≤ ∑ p (Aik )  k =1  k =1  k ≠i  k ≠i a one-dimensional Gaussian r.v.. sk − si. p ( Aik ) = p ( r − sk < r − si | si sent) = p ( ( si + n) − sk < ( si + n) − si ) = p ( n − ( sk − si ) < n ) The probability that n is closer to the vector sk-si than to the origin 17.

(18) Union Bound of Error Probability (2)  . . The event Aik occurs if n > d ik 2 , where d ik = si − sk  d ik   − v2  d ik  ∞ 1   exp  p ( Aik ) = p n > dv = Q  = ∫d / 2   2N  ik 2  πN 0   N0  0   M  d ik   Pe (mi sent) ≤ ∑ Q  2N  k =1 0   k ≠i M. 1 M Pe = ∑ p (mi ) Pe (mi sent) ≤ ∑ M i =1 i =1.  d ik   Q ∑  2N  k =1 0   k ≠i M. 18.

(19) Approximation of Error Probability . The minimum distance of constellation: dmin.  d min   − Pe ≤ ( M − 1)Q  2 N 0  . . The number of neighbors at the minimum distance: M d min. − −. . : looser bound.  d min   Pe ≈ M d min Q  2N  0  . (. In case of binary modulation (M=2): Pb = Q d min. 2N 0. ). Gray code: mistaking a constellation point for one of its nearest neighbors results in a single bit error − P ≈ Pe b log 2 M 19.

(20) Amplitude and Phase Modulation. 20.

(21) Amplitude and Phase Modulation (1) . Over the time interval Ts, K (log2M) bits are encoded in the amplitude and/or phase of the transmitted signal − Signal: s(t ) = sI (t ) cos(2π f ct ) − sQ (t ) sin(2π f ct ) − In signal space: s (t ) = si1 (t )φ1 (t ) + si 2 (t )φ2 (t ) φ1 (t ) = g (t ) cos(2π f c t + φ0 ) φ2 (t ) = − g (t ) sin(2π f c t + φ0 ). . There are three main types of amplitude/phase modulation: − Pulse Amplitude Modulation (MPAM) : Uses amplitude only − Phase Shift Keying (MPSK) : Uses phase only − Quadrature Amplitude Modulation (MQAM). 21.

(22) Amplitude and Phase Modulation (2). Amplitude/Phase Modulator. 22.

(23) Amplitude and Phase Modulation (3) Amplitude/Phase Demodulator.  Coherent detection (φ = φ0 ).  if φ − φ0 = ∆φ ≠ 0, r1 = si1 cos(∆φ ) + si 2 sin( ∆φ ) + n1 and r2 = − si1 sin( ∆φ ) + si 2 cos(∆φ ) + n2 => performance degradation  Synchronization or timing recovery:. the sampling function is synchronized to the start of every symbol period. 23.

(24) Amplitude and Phase Modulation (4).     . Pulse Amplitude Modulation Phase-Shift Keying Quadrature Amplitude Modulation Differential Modulation Modulator with Quadrature Offset. 24.

(25) Pulse Amplitude Modulation (MPAM) (1)   . . Encodes all of the information into the signal amplitude (Ai) Transmitted signal over time:. −. {. }. s (t ) = Re Ai g (t )e j 2π fct = Ai g (t ) cos(2π f c t ), 0 ≤ t ≤ Ts ⟩⟩ 1 / f c. Signal constellation: { Ai = (2i − 1 − M )d , i = 1, 2, ..., M }. − − −. Parameterization distance d is typically a function of a signal energy Minimum distance : d min = min i , j Ai − A j = 2d Constellation mapping is usually done by Gray encoding. Amplitude of each transmitted signal has M different values Gray. −. Each pulse conveys K = log2M bits per symbol time Ts Coding. 25.

(26) Pulse Amplitude Modulation (MPAM) (2) . Over each symbol period, the MPAM signal associated with the ith constellation has energy Es = T si2 (t )dt = T Ai2 g 2 (t ) cos 2 (2π f c t )dt = Ai2 i. −. . 1 Average energy: Es = M. ∫. ∫. s. 0. s. 0. M. ∑A i =1. 2 i. Decision region associated with signal amplitude Ai = (2i − 1 − M )d. i = 1, (− ∞, Ai + d )  Z i = [ Ai − d , Ai + d ) 2 ≤ i ≤ M − 1, [ A − d , ∞ ) i=M  i. 26.

(27) Pulse Amplitude Modulation (MPAM) (3) Coherent MPAM demodulator. 27.

(28) Phase Shift Keying (MPSK) (1)  . Encodes information in the phase of the transmitted signal Transmitted signal over one symbol time:. {. si (t ) = Re A g (t )e j 2π (i −1) / M e j 2πfct. }. 2π (i − 1)  = A g (t ) cos 2π f c t + M    2π (i − 1)  2π (i − 1) = A g (t ) cos  − cos 2 π f t A g ( t ) sin c  M  sin 2π f c t  M . . Constellation points: ( si1 , si 2 ). si1 = A cos[2π (i − 1) / M ], si 2 = A sin[2π (i − 1) / M ] for i = 1,..., M.   . Minimum distance:d min = 2 A sin(π / M ), where A is typically function of signal energy All possible signals have equal energy A2 Constellation mapping usually uses Grady encoding 28.

(29) Phase Shift Keying (MPSK) (2) Gray Coding for MPSK. 29.

(30) Phase Shift Keying (MPSK) (3)  Decision Regions for MPSK. {. Z i = re jθ : 2π (i − 3 2 ) M < θ < 2π (i − 1 2 ) M. }. 30.

(31) Phase Shift Keying (MPSK) (4) Coherent Demodulator for BPSK. 31.

(32) Quadrature Amplitude Modulation (MQAM) (1)   . Information bits are encoded in both amplitude and phase of the transmitted signal MPSK and MPAM have only one degree-of-freedom, but MQAM has two degree-of-freedom. Thus, MQAM is more spectral-efficient Transmitted signal:. {. S i (t ) = Re Ai e jθi g (t )e j 2π fct. }. = Ai cos(θ i ) g (t ) cos(2π f c t ) − Ai sin(θ i ) g (t ) sin (2π f c t ), 0 ≤ t ≤ Ts.  . Ts. Signal energy in si(t): Esi = ∫ si2 (t )dt = Ai2 0 Distance between constellation points: d ij = si − s j =. (s. i1 − s j1 ) + (si 2 − s j 2 ) 2. 2. 32.

(33) Quadrature Amplitude Modulation (MQAM) (2) . . For square signal constellation, − Values on (2i-1-L)d for i = 1, …, L − dmin= 2d Good constellation mapping can be hard to find for QAM signal. 4-QAM and 16-QAM constellations 33.

(34) Quadrature Amplitude Modulation (MQAM) (3) Decision Regions for 16-QAM. 34.

(35) Differential Modulation (1) . . Information in MPSK and MQAM signals is carried in the signal phase. − − −. MPSK and MQAM require coherent detection Phase recovery mechanism required in receiver Coherent demodulation ■ makes receiver complex ■ is hard in rapidly fading channel ■ is more susceptible to phase drift of the carrier. The principle of differential modulation is to use the previous symbol as a phase reference for current symbol for avoiding the need for a coherent phase modulation. − −. Differential BPSK (DPSK) ■ 0-bit: no change in phase, 1-bit: a phase change of π Differential QPSK (DQPSK) ■ 00: no change in phase, 01: a phase change of π/2 ■ 10: a phase change of - π/2, 11: a phase change of π. 35.

(36) Differential Modulation (2) . Phase Comparator − Transmitted signal: s(k ) = Ae j (θ ( k )+φ0 ) − Received signal at time k: r (k ) = r1 (k ) + jr2 (k ) = Ae j (θ ( k )+φ −φ ) + n(k ) − Received signal at time (k-1): 0. −. r (k − 1) = r1 (k − 1) + jr2 (k − 1) = Ae j (θ ( k −1) +φ0 −φ ) + n(k − 1). Phase difference r (k ) r * (k − 1) = A2 e j (θ ( k ) −θ ( k −1) ) + Ae j (θ ( k ) +φ0 −φ ) n* (k − 1). + A e − j (θ ( k −1) +φ0 −φ )n(k ) + n(k ) n* (k − 1) Phase difference in the absence of noise (n(k) = n(k-1) = 0).   . modulation with memory less sensitive to random drift in the carrier phase With non-zero Doppler frequency, previous symbol is not good for phase reference 36.

(37) Differential Modulation (3) DPSK demodulator. 37.

(38) Modulation with Quadrature Offset (1) . . Linearly modulated signal may cause transition to symbol which makes phase change up to π, and signal amplitude to cross zero point − Abrupt phase transition and amplitude variations can be distorted by non-linear filters and amplifiers To avoid the above problem − Offsetting the quadrature branch pulse g(t) half a symbol period − Phase can change maximum π/2. 38.

(39) Modulation with Quadrature Offset (2) Modulator with quadrature offset. 39.

(40) Pulse Shaping (1)  .  . Bandwidth of the baseband and passband modulated signal is a function of the bandwidth of the pulse shape g(t). The effective received pulse: p(t ) = g (t ) ∗ c(t ) ∗ g * (−t ) − c(t): the channel impulse response − g*(-t): the matched filter − In AWGN channel (c(t)=δ(t)), p(t ) = g (t ) ∗ g * (−t ) To avoid ISI between the received pulses, p(t) must satisfy Nyquist criterion, which requires the pulse to equal zero at the ideal sampling point associated with past or future symbols. Pulse shapes that satisfy the Nyquist criterion. − − −. Rectangular pulse Cosine pulse Raised Cosine pulse. 40.

(41) Pulse Shaping (2) . Raised Cosine Pulse − These pulses are designed in the frequency domain   Ts  P( f ) =   Ts 1 − sin π Ts  2 β  . 0≤ f ≤  1   f −  2Ts  . 1− β 2Ts. 1− β 1+ β ≤ f ≤ 2Ts 2Ts. where β is a rolloff factor. −. The pulse p(t) in the time domain: p (t ) =. sin π t / Ts cos βπ t / Ts × 1 − 4 β 2t 2 / Ts2 π t / Ts. 41.

(42) Pulse Shaping (3). Raised Cosine Pulse in frequency domain. P(f). 42.

(43) Pulse Shaping (4). Raised Cosine Pulse in time domain p(t). 43.

(44) Error Probability of Digital Modulation over AWGN Channel.  BPSK and QPSK  MPSK  MPAM and MQAM. 44.

(45) Signal-to-Noise Power Ratio (SNR)  In an AWGN channel − − −. j 2πf t. Modulated (transmitted) signal: s (t ) = Re{u (t )e c } Received signal: r (t ) = s (t ) + n(t ) n(t): a white Gaussian random process with mean zero and power spectral density N0/2.  SNR − − −. Ratio of the received signal power Pr to the power of the noise within the bandwidth of the transmitted signal Es Eb P SNR = r = = N 0 B N 0 BTs N 0 BTb In system with interference Pr ■ SINR = N 0 B + PI. 45.

(46) Bit/Symbol Errors  For pulse shaping with Ts=1/B (e.g., raised cosine pulse with β=1), SNR=Es/N0  For general pulse, Ts= k/B and SNR = Es N 0 ×1 k  Define − SNR per symbol: γ s = Es N 0 − SNR per bit: γ b = Eb N 0  We are interested in bit error probability Pb as a function of γ b  Approach − First, compute the symbol error probability Ps as a function of γ s − Then, obtain bit error probability as a function of SNR per bit using assumptions. ■. −. ■. The symbol energy is divided equally among all bits, Gray encoding is used. These assumptions for M-ary signaling lead to the approximations. γb ≈. γs. log 2 M. and. Pb ≈. Ps log 2 M. 46.

(47) Error Probability for BPSK  For binary modulation (M=2), −   d   Pb = Q. −. min.  2N  0  . d min = s1 − s0 = 2 A, Tb. Tb. Eb = ∫ s (t )dt = ∫ A2 g 2 (t ) cos 2 (2πf c t )dt =A2 0. . 2 1.  2 Eb Pb = Ps = Q  N0. 0. (.   = Q 2γ b  . ). 47.

(48) Error probability for QPSK  The QPSK system is equivalent to the system consisting of BPSK modulation on . both the in-phase and quadrature components of the signal. The bit error probability on each component is the same as for BPSK. − Pb = Q( 2γ b ). [. ].  The symbol error probability is Ps = 1 − 1 − Q( 2γ b ). 2.  Since the transmitted symbol energy Es is split between each branch, the signal . energy per branch is ( Es 2) The symbol error probability is the probability that either branch has a error. −. P = 1 − [1 − Q ( γ )]. 2. s. s.  For the same Eb N 0 and therefore the same average probability of bit error, QPSK system transmits data at twice the bit rate of a BPSK system for the same channel bandwidth.. [ ( )]. − Ps = 1 − 1 − Q γ s. 2. ( ). (. ≈ 2Q γ s = 2Q 2γ b. ). ( ). Ps ≈ 2Q γ s. 48.

(49) Error Probability for MPSK  When using the union bound of.  Signal-space diagram. error probability and the nearest neighbor approximation,. for 8PSK. (. φ2. Ps ≈ M d min Q d min. Es.  Es. 2N0. ). For MPSK,. d min = 2 Es sin(π M ) ,. φ1. M d min = 2. . (. Ps = 2Q 2γ s sin(π M ). ) 49.

(50) Error Probability for MPAM  The constellation for MPAM is Ai = (2i − 1 − M )d , i = 1,2,..., M  Since each of the M-2 inner constellation points has two nearest neighbor at distance 2d, Ps ( si ) = p ( n > d ),. i = 2,..., M − 1.  For outer constellation points, there is only one nearest neighbor. M 1  The average energy per symbol for MPAM is Es = ∑ Ai 2 = 1 ( M 2 − 1)d 2 M. i =1. 3. 2 × Q 2d M. 2N0.  The symbol error probability Ps in terms of the average energy as 1 Ps = M. M. ∑ Ps (si ) = i =1. (. M −2 × 2 × Q 2d M. Ps =. 2( M − 1) M. ). 2N0 +. (. ).  6γ  s  Q 2  M −1   . 50.

(51) Error Probability for MQAM  MQAM system can be viewed as two MPAM systems with signal constellations of size L = M transmitted over the in-phase and quadrature signal components, each with half the energy of the original MQAM system..  The constellation points in the in-phase and quadrature branches take values Ai = (2i − 1 − L)d ,. i = 1,2,..., L.  The symbol error probability for each branch is Ps ,branch. 2( M − 1)  3γ s  = Q  M −1  M  .  The probability of symbol error the for MQAM system is Ps = 1 − (1 − Ps ,branch ) 2  If we take a conservative approach and set the number of nearest neighbors to be four,.  3γ  s  Ps ≈ 4 × Q  M −1    51.

(52) Summary in Error Probability for Coherent Modulation (1)  Many of the exact or approximation values for Ps derived for coherent modulation are in the following form:. (. Ps (γ s ) ≈ α M Q β M γ s. − −. ). αM : the number of nearest neighbors at the minimum distance dmin βM : a constant that relates the minimum distance to average symbol energy.  Performance specifications are generally most concerned with the bit error rate as a function of the bit energy. Pb (γ b ) ≈ αˆ M Q βˆM γ b   . −. With Gray coding and high SNR, αˆ M = α M log 2 M and βˆM = β M × log 2 M. 52.

(53) Summary in Error Probability for Coherent Modulation (2) Ps (γ s ). Modulation. Pb (γ b ). (. BPSK QPSK MPAM. MPSK. MQAM. Pb = Q 2γ b. ( ). (. Pb ≈ Q 2γ b. Ps ≈ 2Q γ s Ps =. 6γ s  2(M − 1)   Q 2  M  M −1 .  π Ps ≈ 2Q 2γ s sin  M   3γ s   Ps ≈ 4Q  M − 1  . ).    . ). Pb ≈. 2(M − 1)  6γ b log 2 M Q M log 2 M  M 2 −1. Pb ≈.  2  π  Q 2γ b log 2 M sin    log 2 M   M . Pb ≈.  3γ b log 2 M 4 Q M −1 log 2 M .    .    . 53.

(54) Flat Fading Channel  Outage Probability  Average Probability of Error. 54.

(55) Performance Criteria (1)  In a fading environment, the received signal power varies randomly over distance or time due to shadowing and/or multipath fading.. −. In fading γ s is a random variable with distribution f γ s (γ ) , and therefore Ps (γ s ) is also random..  Performance criteria − −. The outage probability, Pout , defined as the probability that γ s falls below a given value corresponding to the maximum allowable Ps The average error probability, Ps , averaged over the distribution of γ s. 55.

(56) Performance Criteria (2) . . When the fading coherence time is on the order of a symbol time (Ts ≈ Tc ) − The signal fading level is roughly constant over a symbol period − The error correction coding techniques can recover from a few bit errors − An average error probability is a reasonably good figure When the signal fading is changing slowly (Ts << Tc ) − A deep fade affects simultaneously many symbols − Large error bursts that cannot be corrected for with coding of reasonable. − −. . complexity Outage probability When the channel is modeled as a combination of fast and slow fading (e.g., log-normal shadowing with fast Rayleigh fading), outage and average error probability is often combined. When Ts >> Tc , the fading will be averaged out by the matched filter in the demodulator − For very fast fading, performance is the same as in AWGN. 56.

(57) Outage Probability (1) Outage. γs. Ts. γ0.  Probability that γ s is below a target γ 0, which is the minimum SNR required for acceptable performance. − P = p(γ < γ ) = γ p (γ )dγ out. s. 0. ∫. 0. 0. γs. 57.

(58) Outage Probability (2)  In Rayleigh fading with mean zero and variance σ 2 (dB) − −. −. The received signal power is exponentially distributed with average 2σ 2 The received SNR γ s also has an exponential distribution with average γ s Es 2 σ 2 Ts ■ γs = = N0 N0 1 ■. The probability density function of γ s :. Outage probability. Pout =. γ0. 1. ∫γ 0. −. pγ s (γ ) =. γs. e −γ / γ s. e − γ / γ s dγ s = 1 − e − γ 0 / γ s. s. Average SNR ■. γs =. −γ0 ln(1 − Pout ). 58.

(59) Average Error Probability  The averaged probability of error is computed by integrating the error probability in AWGN over the fading distributions. ∞. Ps = ∫ Ps (γ ) pγ s (γ )dγ. −. 0. An error probability in AWGN with SNR γ :.  In Rayleigh fading, ∞. (. )γ. Ps = ∫ α M Q β M γ ⋅ 0. −. 1. e. s. −γ / γ s. α M . (. Ps (γ ) ≈ α M Q β M γ s. 0.5β M γ s dγ = 1− 2  1 + 0.5β M γ s . ).   ≈ αM  2β M γ s . BPSK: Pb (γ b ) = Q( 2γ b ). γb 1  Pb = 1 − 2 1+ γ b .    . 59.

(60) Average Pb for MQAM in Rayleigh Fading and AWGN. 60.

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