Fading Channels: Capacity, BER and Diversity

Transcription

1 Fading Channel: Capacity, BER and Diverity Mater Univeritario en Ingeniería de Telecomunicación I. Santamaría Univeridad de Cantabria

2 Introduction Capacity BER Diverity Concluion Content Introduction Capacity BER Diverity Concluion Fading Channel: Capacity, BER and Diverity 0/50

3 Introduction Capacity BER Diverity Concluion Introduction We have een that the randomne of ignal attenuation (fading) i the main challenge of wirele communication ytem In thi lecture, we will dicu how fading affect 1. The capacity of the channel 2. The Bit Error Rate (BER) We will alo tudy how thi channel randomne can be ued or exploited to improve performance diverity Fading Channel: Capacity, BER and Diverity 1/50

4 Introduction Capacity BER Diverity Concluion General communication ytem model Source Channel Encoder b[n] Modulator [n] Channel ˆ [ n] Demod. Channel Decoder b ˆ[ n ] Noie The channel encoder (FEC, convolutional, turbo, LDPC...) add redundancy to protect the ource againt error introduced by the channel The capacity depend on the fading model of the channel (contant channel, ergodic/block fading), a well a on the channel tate information (CSI) available at the Tx/Rx Let u tart reviewing the Additive White Gauian Noie (AWGN) channel: no fading Fading Channel: Capacity, BER and Diverity 2/50

5 Introduction Capacity BER Diverity Concluion AWGN Channel Let u conider a dicrete-time AWGN channel y[n] = h[n] + r[n] where r[n] i the additive white Gauian noie, [n] i the tranmitted ignal and h = h e jθ i the complex channel The channel i aumed contant during the reception of the whole tranmitted equence The channel i known to the Rx (coherent detector) The noie i white and Gauian with power pectral denity N 0 /2 (with unit W/Hz or dbm/hz, for intance) We will mainly conider paband modulation (BPSK, QPSK, M-PSK, M-QAM), for which if the bandwith of the lowpa ignal i W the bandwidth of the paband ignal i 2W Fading Channel: Capacity, BER and Diverity 3/50

6 Introduction Capacity BER Diverity Concluion Signal-to-Noie-Ratio Tranmitted ignal power: P Received ignal power: P h 2 Total noie power: N = 2WN 0 /2 = WN 0 The received SNR i SNR = γ = P WN 0 h 2 In term of the energy per ymbol P = E /T We aume Nyquit pule with β = 1 (roll-off factor) o W = 1/T Under thee condition SNR = E T T N 0 h 2 = E N 0 h 2 Fading Channel: Capacity, BER and Diverity 4/50

7 Introduction Capacity BER Diverity Concluion For M-ary modulation, the energy per bit i E b = E log(m) SNR = E b N 0 log(m) h 2 To model the noie we generate zero-mean circular complex Gauian random variable with σ 2 = WN 0 r[n] CN(0, σ 2 ) or r[n] CN(0, WN 0 ) The real and imaginary part have variance σi 2 = σq 2 = σ2 2 = WN 0 2 Fading Channel: Capacity, BER and Diverity 5/50

8 Introduction Capacity BER Diverity Concluion Capacity AWGN Channel Let u conider a dicrete-time AWGN channel y[n] = h[n] + r[n] where r[n] i the additive white Gauian noie, [n] i the tranmitted ymbol and h = h e jθ i the complex channel The channel i aumed contant during the reception of the whole tranmitted equence The channel i known to the Rx (coherent detector) The capacity (in bit/eg or bp) i given by the well-known Shannon formula C = W log (1 + SNR) = W log (1 + γ) where W i the channel bandwidth, and SNR = P h 2 WN 0 ; with P being the tranmit power, h 2 the power channel gain and N 0 /2 the power pectral denity (PSD) of the noie Fading Channel: Capacity, BER and Diverity 6/50

9 Introduction Capacity BER Diverity Concluion Sometime, we will find ueful to expre C in bp/hz (or bit/channel ue) C = log (1 + SNR) A few thing to recall 1. Shannon coding theorem prove that a code exit that achieve data rate arbitrarily cloe to capacity with vanihingly mall probability of bit error The codeword might be very long (delay) Practical (delay-contrained) code only approach capacity 2. Shannon coding theorem aume Gauian codeword, but digital communication ytem ue dicrete modulation (PSK,16-QAM) Fading Channel: Capacity, BER and Diverity 7/50

10 Introduction Capacity BER Diverity Concluion Capacity (bp/hz) 10 Shaping gain (1.53 db) 8 6 Gauian input log(1+snr) 64-QAM 4 16-QAM 2 4-QAM (QPSK) 0 Pe (ymbol) = 10-6 (uncoded) SNR db For dicrete modulation, with increaing SNR we hould increae the contellation ize M (or ue a le powerful channel encoder) to increae the rate and hence approach capacity: Adaptive modulation and coding (more on thi later) Fading Channel: Capacity, BER and Diverity 8/50

11 Introduction Capacity BER Diverity Concluion AWGN v. Fading channel AWGN channel: h i a determinitic contant the SNR at the receiver i contant Intantaneou SNR = Average SNR = wlog we may take h 2 = 1 P WN 0 h 2 Fading channel: The intantaneou ignal-to-noie-ratio SNR = γ = γ h 2 i a random variable Intantaneou SNR = γ h 2 = Average SNR = γ = wlog we may take E[ h 2 ] = 1 γ = P WN 0 h 2 P WN 0 E[ h 2 ] P WN 0 Fading Channel: Capacity, BER and Diverity 9/50

12 Introduction Capacity BER Diverity Concluion Capacity fading channel Let u conider the following example 1 Fading channel Channel Encoder Random Switch AWGN C = 1 bp/hz AWGN C = 2 bp/hz The channel encoder i fixed The witch take both poition with equal probability We wih to tranmit a codeword formed by a long equence of coded bit, which are then mapped to ymbol What i the capacity for thi channel? 1 Taken from E. Biglieri, Coding for Wirele Communication, Springer, 2005 Fading Channel: Capacity, BER and Diverity 10/50

13 Introduction Capacity BER Diverity Concluion The anwer depend on the rate of change of the fading 1. Fat fading: If the witch change poition every ymbol period, the codeword experience both channel with equal probabilitie o we could ue a fixed channel encoder, and tranmit at a maximum rate C = 1 2 C C 2 = 1.5 bp/hz 2. Slow fading: If the witch remain fixed at the ame (unknown) poition during the tranmiion of the whole codeword, then we hould ue a fixed channel encoder, and tranmit at a maximum rate C = C 1 = 1 bp/hz or otherwie half of the codeword would be lot Fading Channel: Capacity, BER and Diverity 11/50

14 Introduction Capacity BER Diverity Concluion What would be the capacity if the witch chooe from a continuum of AWGN channel whoe SNR, γ = P h 2 WN 0, 0 γ <, follow an exponential ditribution (Rayleigh channel) C1 = log( 1+ γ 1) Channel Encoder Random Switch C2 = log( 1+ γ 2) C3 = log( 1+ γ 3) Fading Channel: Capacity, BER and Diverity 12/50

15 Introduction Capacity BER Diverity Concluion Fat fading (ergodic) channel Each ymbol uffer a different channel realization The channel encoder operate over L channel realization h1 h2 L hl h 1 h2 The channel capacity (aka ergodic capacity) i C = E [log(1 + γ)] = 0 L L h log(1 + γ)f (γ)dγ (1) where f (γ) i the pdf of the SNR, which in turn depend on the fading ditribution Fading Channel: Capacity, BER and Diverity 13/50

16 Introduction Capacity BER Diverity Concluion Remember that for a Rayleigh channel where γ i the average SNR f (γ) = 1 γ e γ/γ (2) The ergodic capacity of a Rayleigh channel i ( ) ( ) 1 1 C = log 2 (e) exp E 1 γ γ where E 1 (x) = x e t t dt i the Exponential integral function 2 2 y=expint(x) in Matlab Fading Channel: Capacity, BER and Diverity 14/50

17 Introduction Capacity BER Diverity Concluion The ergodic capacity i in general hard to compute in cloed form for other fading ditribution (we can alway reort to numerical integration) We can apply Jenen inequality to gain ome qualitative inight into the effect of fat fading on capacity Jenen inequality If f (x) i a convex function and X i a random variable E [f (X )] f (E [X ]) If f (x) i a concave function and X i a random variable E [f (X )] f (E [X ]) log( ) i a concave function, therefore C = E [log(1 + γ)] log (1 + E[γ]) = log (1 + γ) Fading Channel: Capacity, BER and Diverity 15/50

18 Fading Channel: Capacity, BER and Diverity 16/50 17 SISO Fading Channel: Ergodic Capacity Introduction Capacity BER Diverity Concluion Fading hurt: The capacity of a fading channel with receiver CSI me iid i le Rayleigh than the fading capacity with of σ an AWGN channel with the ame h 2 average SNR = E[ h 2 ] = 1, and σz 2 = Capacity (bp/hz) AWGN Channel Capacity Fading Channel Ergodic Capacity Average SNR (db)

19 Introduction Capacity BER Diverity Concluion Slow (block) fading channel The channel remain fixed during the tranmiion of a equence of L ymbol (a packet or a frame) For the next frame or packet, the channel take a new value (a new channel realization) The channel encoder operate over a packet: no coding over conecutive packet h 1 L L Block fading model: The channel realization are iid h2 L hn Fading Channel: Capacity, BER and Diverity 17/50

20 Introduction Capacity BER Diverity Concluion Slow (block) fading channel For a Rayleigh ditribution there i no nonzero rate at which codeword can be tranmitted with vanihingly mall probability or error Strictly, the capacity for the Rayleigh block fading channel would be zero In thi ituation, it i more ueful to define the outage capacity C out = r Pr(log(1 + γ) < r) = Pr(C(h) < r) = P out Suppoe we tranmit at a rate C out with probability of outage P out = 0.01 thi mean that with probability 0.99 the intantaneou capacity of the channel (a realization of a r.v.) will be larger than rate and the tranmiion will be ucceful; wherea with probability 0.01 the intantaneou capacity will be lower than the rate and the all bit in the codeword will be decoded incorrectly Fading Channel: Capacity, BER and Diverity 18/50

21 Introduction Capacity BER Diverity Concluion P out i the error probability ince data i only correctly received on 1 P out tranmiion What i the outage probability of a block fading Rayleigh channel with average SNR = γ for a tranmiion rate r? P out = Pr(log(1 + γ) < r) The rate i a monotonic increaing function with the SNR = γ; therefore, there i a γ min needed to achieve the rate log(1 + γ min ) = r γ min = 2 r 1 and P out can be obtained a P out =Pr(γ < γ min ) = =1 e γ min γ γmin 0 = 1 e 2r 1 γ f (γ)dγ = γmin 0 1 γ e γ/γ dγ = Fading Channel: Capacity, BER and Diverity 19/50

22 Introduction Capacity BER Diverity Concluion Rate (bp/hz) P out v C out (rate) for γ = 100 (10 db) P out Fading Channel: Capacity, BER and Diverity 20/50

23 Introduction Capacity BER Diverity Concluion Summary h 1 L AWGN L h1 h1 L C = log( 1+ γ ) h1 h2 L Ergodic (fat fading) h L h 1 h2 L h L C = E[log( 1+ γ )] Block fading h 1 L h2 L L hn C outage Fading Channel: Capacity, BER and Diverity 21/50

24 Introduction Capacity BER Diverity Concluion Final note All capacity reult een o far correpond to the cae of perfect CSI at the receiver ide (CSIR) If the tranmitter alo know the channel, we can perform power adaptation and the reult are different We ll ee more on thi later Now, let move to analyze the impact of fading on the Bit Error Rate (BER) Fading Channel: Capacity, BER and Diverity 22/50

25 Introduction Capacity BER Diverity Concluion BER analyi for the AWGN channel (a brief reminder) BPSK [n] { 1, +1} P = P b = Q 2E h 2 = Q N 0 ( ) 2SNR where SNR i the contant average ignal-to-noie ratio and Q(x) = 1 x 2π exp (x 2 /2) QPSK [n] { 1 j 2, 1+j 2, 1 j 2, 1+j 2 } ( ( )) 2 P = 1 1 Q SNR ( ) Nearet neighbor approximation P 2Q SNR With Gray encoding Pe = P /2 Fading Channel: Capacity, BER and Diverity 23/50

26 Introduction Capacity BER Diverity Concluion M-PAM contellation A i = (2i 1 M)d/2, i = 1,..., M Ditance between neighbor: d Average ymbol energy E = 1 3 (M2 1) ( ) 2 d = (M2 1)d Symbol Error Rate P = M 2 M 2Q = 2(M 1) M ( ) d + 2 ( ) d 2σ r M Q 2σ r Q 6 SNR M 2 1 With Gray encoding Pe P /M Fading Channel: Capacity, BER and Diverity 24/50

27 Introduction Capacity BER Diverity Concluion M-QAM: the contellation for the I and Q branche are A i = (2i 1 M)d/2, i = 1,..., M At the I and Q branche we have orthogonal M PAM ignal, each with half the SNR Symbol Error Rate P = 1 1 2( M 1) Q M 3 SNR M 1 Nearet neighbor approximation (4 nearet neighbor) P 4Q 3 SNR M 1 2 Fading Channel: Capacity, BER and Diverity 25/50

28 Introduction Capacity BER Diverity Concluion The impact of fading on the BER Let u conider for implicity a SISO channel with a BPSK ource ignal x[n] = h[n] + r[n] [n] {+1, 1}, r[n] CN(0, σ 2 ) i the noie (additive, white and Gauian) An AWGN channel or a fading channel behave alo differently in term of BER AWGN channel: h i contant Rayleigh fading channel: h ~ CN(0,1) h h Fading Channel: Capacity, BER and Diverity 26/50

29 Introduction Capacity BER Diverity Concluion The receiver know perfectly the channel coherent det. Since we are ending a BPSK ignal over a complex channel the optimal coherent detector i ( h ) x[n] +1 Re 0 (3) h 1 Auming that the tranmitted power i normalized to unity E[ [n] 2 ] = 1, the average SNR i SNR = 1 σ 2 and the intantaneou Signal-to-Noie-Ratio i SNR h 2 For an AWGN channel, SNR h 2 i a contant value and the BER i ( ) ( ) 2 h P e = Q = Q 2 h σ 2 SNR Fading Channel: Capacity, BER and Diverity 27/50

30 Introduction Capacity BER Diverity Concluion For a fading channel SNR h 2 i now a random variable ( ) We can therefore think of P e = Q 2 h 2 SNR a another random variable The BER for the fading channel would be the average over the channel ditribution E[P e ] Thi applie to both fat fading and low fading channel model BER for a Rayleigh channel h i a Rayleigh random variable and z = h 2 i exponential with pdf f (z) = exp( z), the BER i given by ( ) P e = Q 2zSNR e z dz = 1 SNR SNR 4SNR 0 Fading Channel: Capacity, BER and Diverity 28/50

31 Introduction Capacity BER Diverity Concluion BER AWGN Rayleigh SNR (db) At an error probability of P e = 10 3, a Rayleigh fading channel need 17 db more than an AWGN (contant) channel, even though we have perfect channel knowledge in both cae!! Fading Channel: Capacity, BER and Diverity 29/50

32 Introduction Capacity BER Diverity Concluion The main reaon of the poor performance i that there i a ignificant probability that the channel i in a deep fade, and not becaue of noie The intantaneou ignal-to-noie-ratio i h 2 SNR, and we may conider that the channel i in a deep fade when h 2 SNR < 1 In thi event, the eparation between contellation point will be of the order of magnitude of σ (noie tandard deviation) and the probability of error will be high For a Rayleigh channel, the prob. of being in a deep fade i Pr( h 2 SNR < 1) = ( = 1 1/SNR SNR + which i of the ame order a P e e z dz = 1 e 1/SNR = ( ) ) SNR 1 SNR Fading Channel: Capacity, BER and Diverity 30/50

33 Introduction Capacity BER Diverity Concluion Remark Although our example conidered a BPSK modulation, the ame reult i obtained for other modulation On the Rayleigh channel, all modulation cheme are equally bad and for all them the BER decreae a SNR 1 The gap between the AWGN and the Rayleigh channel in term of BER i much higher than in term of capacity Thi ugget that coding can be very beneficial to compenate fading Fading Channel: Capacity, BER and Diverity 31/50

34 Introduction Capacity BER Diverity Concluion Diverity The negative lope (at high SNR) of the BER curve with repect to the SNR i called the diverity gain or diverity order log(p e d = lim log(snr) SNR In other word, in a ytem with diverity order d, the P e at high SNR varie a P e = SNR d For a SISO Rayleigh channel the diverity order i 1 (remember that P e 1/(4SNR) SNR 1!! Fading Channel: Capacity, BER and Diverity 32/50

35 Introduction Capacity BER Diverity Concluion SIMO ytem Conider now that the Rx ha N antenna and that the reulting SIMO (ingle-input multiple-output) channel i patially uncorrelated [n] h 1 h 2 r 2 [ n] r 1 [ n] h1[ n] r1 [ n] h n] r [ 2 [ 2 n ] r h N [n] N hn [ n] rn [ n] If the channel h i are faded independently, the probability that the SIMO channel i in a deep fade will be approximately ( ) N Pr(( h 1 2 SNR < 1)&... &( h N 2 SNR < 1)) = Pr( h 2 1 SNR < 1) SNR N Fading Channel: Capacity, BER and Diverity 33/50

36 Introduction Capacity BER Diverity Concluion In conequence, the diverity order of a SIMO ytem with N uncorrelated receive antenna i N Notice the impact of antenna correlation: if all antenna are very cloe to each other o that the channel become fully correlated, then h 1 h 2 h N, and the diverity reduce to that of a SISO channel There are different way to proce the receive vector and extract all patial diverity of the SIMO channel Auming that the channel h = (h 1, h 2,..., h N ) T i known at the Rx, the optimal cheme called maximum ratio combining (MRC) Fading Channel: Capacity, BER and Diverity 34/50

37 Introduction Capacity BER Diverity Concluion Maximum Ratio Combining The MRC cheme linearly combine (with optimal weight w i ) the output of the ignal received at each antenna r 1 [ n] h [ ] [ ] h 1 n r1 n w 1 1 h r 2 [ n] 2 h2[ n] r2 [ n] w 2 h N r N [n] h [ n] r [ n] N N w N z[n] The ignal z[n] can be written in matrix form a h 1 r 1 [n] z[n] = ( ) h 2 w 1 w 2... w N. [n] + r 2 [n]. = wt (h[n] + r[n]) h N r N [n] Fading Channel: Capacity, BER and Diverity 35/50

38 Introduction Capacity BER Diverity Concluion A long a w = 1 the noie ditribution after combining doe not change: i.e., w T r[n] CN(0, σ 2 ) It i eay to how that the optimal weight are given by w = h h, which i jut the matched filter for thi problem! Thee are the optimal weight becaue they maximize the ignal-to-noie-ratio at the output of the combiner: SNR MRC = h 2 σ 2 = h 2 SNR For a BPSK tranmitted ignal, the optimal detector i Re(z[n]) = Re ( ) h H x[n] +1 0 h 1 where ( ) H denote Hermitian (complex conjugate and tranpoe). Notice the imilarity with the optimal coherent detector for the SISO cae, which wa given by (3) Fading Channel: Capacity, BER and Diverity 36/50

39 Introduction Capacity BER Diverity Concluion Similarly to the SISO cae, the P e can be derived exactly ( ) P e = Q 2 h 2 SNR, which i a random variable becaue h i random (fading channel) Under Rayleigh fading, each h i i i.i.d CN(0, 1) and x = h 2 = N h i 2 follow a Chi-quare ditribution with 2N degree of freedom f (x) = i=1 1 (N 1)! x N 1 exp( x), x 0 Note that the exponential ditribution (SISO cae with N = 1) i a Chi-quare with 2 degree of freedom The average error probability can be explicitly computed, here we only provide a high-snr approximation P e = 0 ( ( ) Q 2xSNR f (x)dx 2N 1 N ) 1 (4SNR) N Fading Channel: Capacity, BER and Diverity 37/50

40 Introduction Capacity BER Diverity Concluion The probability of being in a deep fade with MRC i Pr( h 2 SNR < 1) = Concluion (for x mall) 1/SNR 0 1/SNR 0 1 (N 1)! x N 1 exp( x)dx 1 (N 1)! x N 1 dx = 1 N!SNR N The P e i again dominated by the probability of being in a deep fade Both (the P e and the probability of being in a deep fade) decreae with the number of antenna roughly a (SNR) N MRC extract all patial diverity of the SIMO channel!! However, MRC i not the only multiantenna technique that extract all patial diverity of a SIMO channel Fading Channel: Capacity, BER and Diverity 38/50

41 Introduction Capacity BER Diverity Concluion Antenna election r 1 h 1 h 2 r 2 RF chain ADC x[ n] hmax[ n] r[ n] h N r N Selection of the bet link In antenna election the path with the highet SNR i elected and proceed: x[n] = h max[n] + r[n], where h max = max ( h 1, h 2,..., h N ) In comparion to MRC, only a ingle RF chain i needed!! Fading Channel: Capacity, BER and Diverity 39/50

42 Introduction Capacity BER Diverity Concluion Intuitively, the probability that the bet channel i in a deep fade i Pr( h max 2 SNR < 1) = N Pr( h i 2 SNR < 1) 1 i=1 SNR N and, therefore, the diverity of antenna election i alo N The ame concluion can be reached by tudying the P e (we would need the ditribution of h max 2 ) However, the output SNR of MRC i higher than that of antenna election (AS) SNR MRC = h 1 2 σ h N 2 σ 2 SNR AS = h max 2 σ 2 Fading Channel: Capacity, BER and Diverity 40/50

43 Introduction Capacity BER Diverity Concluion Array gain The SNR increae in a multiantenna ytem with repect to that of a SISO ytem i called array gain Wherea the diverity gain i reflected in lope of the BER v. SNR, the array gain provoke a hift to the left in the curve N=1 BER N= Array Gain SNR (db) Fading Channel: Capacity, BER and Diverity 41/50

44 Introduction Capacity BER Diverity Concluion For a receiver with N antenna, MRC i optimal becaue it achieve maximum patial diverity and maximum array gain MRC achieve the maximum array gain by coherently combining all ignal path On average, the output SNR of MRC i N time that of a SISO ytem, o it array gain i 10 log 10 N (db) For intance, uing 2 Rx antenna, MRC provide 3 extra db in comparion to a SISO ytem and in addition to the patial diverity gain Fading Channel: Capacity, BER and Diverity 42/50

45 Introduction Capacity BER Diverity Concluion 10 0 Antenna Selection 10 1 MRC SER N=3 N= SNR (db) Fading Channel: Capacity, BER and Diverity 43/50

46 Introduction Capacity BER Diverity Concluion Maximum Ratio Tranmiion Similar concept apply for a a MISO (multiple-input ingle-input) channel: the optimal cheme i now called Maximum Ratio Tranmiion (MRT) w 1 w 2 h2 [n] w N h N h 1 r[n] w T h [ n] r[ ] z[ n] n Again, the optimal tranmit beamformer that achieve full patial diverity (N) and full array gain (10 log 10 N) i w = h h, The main difference i that now the channel mut be known at the Tx (through a feedback channel) Fading Channel: Capacity, BER and Diverity 44/50

47 Introduction Capacity BER Diverity Concluion Spatial diverity of a MIMO channel An n R n T MIMO channel with i.i.d. entrie ha a patial diverity n R n T, which i the number of independent path offered to the ource ignal for going from the Tx to the Rx Some cheme extract the full patial diverity of the MIMO channel Optimal combining at both ide (MRT+MRC) Antenna election at both ide Antenna election at one ide and optimal combining at the other ide Repetition coding at the Tx ide (ame ymbol tranmitted through all Tx antenna) + optimal combining at the Rx ide But other might not: think of a ytem that tranmit independent data tream over different tranmit antenna (more on thi later) Fading Channel: Capacity, BER and Diverity 45/50

48 Introduction Capacity BER Diverity Concluion Time and frequency diverity We have mainly focued the dicuion on the patial diverity concept, but the ame idea can be applied to the time and frequency domain Time diverity: 1. The ame ymbol i tranmitted over different time intant, t 1 and t 2 2. For the channel to fade more or le independently: t 1 t 2 > T c = 1 D Frequency diverity: 1. The ame ymbol i tranmitted over different frequencie (ubcarrier in OFDM), f 1 and f 2 2. For the channel to fade more or le independently: f 1 f 2 > B c = 1 τ rm To achieve time or frequency diverity, interleaving i typically ued Fading Channel: Capacity, BER and Diverity 46/50

49 Introduction Capacity BER Diverity Concluion Interleaving The ymbol or coded bit are dipered over different coherence interval (in time or frequency) A typical interleaver conit of an P Q matrix: the input ignal are written rowwie into the matrix and then read columnwie and tranmitted (after modulation) Example: A 4 8 interleaving matrix from ource to channel Fading Channel: Capacity, BER and Diverity 47/50

50 Introduction Capacity BER Diverity Concluion burt of error Without interleaving h1 h2 h h With interleaving h1 h h 3 h Fading Channel: Capacity, BER and Diverity 48/50

51 Introduction Capacity BER Diverity Concluion Remark: Doppler pread in typical ytem range from 1 to 100 Hz, correponding roughly to coherence time from 0.01 to 1 ec. If tranmiion rate range from to bp, thi would imply that block of length L ranging from L = = 200 bit to L = = bit would be affected by approximately the ame fading gain Deep interleaving might be needed in ome cae But notice that interleaving involve a delay proportional to the ize of the interleaving matrix In delay-contrained ytem (tranmiion of real-time peech) thi might be difficult or even unfeaible Fading Channel: Capacity, BER and Diverity 49/50

52 Introduction Capacity BER Diverity Concluion Concluion We have analyzed the effect of fading from the point of view of capacity (only CSIR) and BER Depending of the channel model (ergodic or block fading) we ue ergodic capacity or outage capacity With CSIR only, the capacity of a fading channel i alway le than the capacity of an AWGN with the ame average SNR The effect of fading i more ignificant in term of BER than in term of capacity (power of coding gain + interleaving) Diverity: lope of the BER wrt SNR at high SNR Diverity technique with multiantenna ytem: MRT, MRC, antenna election, etc We can extract the time and frequency diverity of the channel through interleaving Fading Channel: Capacity, BER and Diverity 50/50