Principles of Digital Communications Part 2

Transcription

1 Principles of Digital Communications Part Jocelyn Fiorina

2 Course content Spread Spectrum Signaling Fading channel and diversity

3 Course content Spread Spectrum Signaling Direct Sequence Spread Spectrum (DS SS) -Performance in broad band and narrow band interference -Processing gain, Interference Margin, Coding Gain -Case of Uncoded DS SS -Applications of DS SS Frequency Hopped Spread Spectrum (FH SS) -Performance of slow FH and fast FH -Coding Gain -Partial Band Interference Synchronization 3

4 Course content Fading Channel Rayleigh Fading Channel Model Performance Diversity Capacity of Rayleigh Fading Channels -Ergodic Capacity -Outage Capacity 4

5 Spread Spectrum Signaling 5

6 Spread Spectrum Signaling Use a bandwidth W much greater than the information rate R. Bandwidth expansion factor Be=W/R Large redundancy overcome severe levels of interference. Coding is an important element in the design of spread spectrum (coded waveforms are characterized by a large expansion factor, redundancy and coding are linked) Pseudorandomness is another element in the design of spread spectrum signals. It makes appear the signal similar to random noise and difficult to demodulate by other receivers. 6

7 Spread Spectrum Signaling Combats interference due to jamming, interference from other users, self-interference due to multipath propagation. Hides the signal by transmitting at low power spectral density (LPI: low probability of intercept). Achieves message privacy in the presence of other listeners. Due to large bandwidth it allows also accurate ranging. 7

8 Spread Spectrum Signaling PSK (needs coherent detection) + PN (pseudonoise or pseudo random pattern) : DS (Direct Sequence Spread Spectrum) FSK (non-coherent detection)+pn : FH (Frequency Hopped Spread Spectrum) 8

9 Direct Sequence Spread Spectrum Direct Sequence Spread Spectrum 9

10 Direct Sequence Spread Spectrum Modulation : BPSK. In order to utilize the entire bandwidth, the phase of the carrier is shifted pseudorandomly according to the pattern from the PN at a rate W times/s. The pulse (or chip) is the basic element of DS-SS. The chip duration is Tc=1/W The bandwidth expansion is Be=W/R=Tb/Tc=Lc Lc is the number of chips per information bit. 10

11 The PN and data signals (a) and the QPSK modulator (b) for a DS spread spectrum system. Direct Sequence Spread Spectrum (for a BPSK modulator, just the I-channel is used) - 11

12 Direct Sequence Spread Spectrum For an encoder that takes k information bits and generated a binary linear (n,k) block code, there are k.tb seconds to transmit the n code elements. There are k.lc chips in that time, so the block length is n=k.lc. The coded bits are altered by a modulo- addition with the PN sequence. bi is the i-th bit of the PN sequence, ci is the corresponding bit from the encoder. We have: a i b i c i 1

13 Direct Sequence Spread Spectrum The sequence a i is mapped into a BPSK signal of the form according to the convention g i t s t Re[ ( ) e j f c t g t ] g( t i. Tc ) g( t i. Tc ) where g(t) represents a pulse of duration Tc and arbitrary shape. a i a i

14 Direct Sequence Spread Spectrum The modulo- addition may also be represented as a multiplication of the two waveforms: g i t p ( t) c ( t) i i ( bi 1)(ci 1) g( t itc ) 14

15 Direct Sequence Spread Spectrum The received equivalent low-pass signal for the i-th code element is: r t p ( t) c ( t) z( t) (b 1)(c 1) g( t it ) z( t) i i with z(t) the low-pass equivalent noise and interference. i i c When the sampled output from the matched filter is multibplied by ( b i 1), the effect of the PN sequence on the received coded bits is removed ( ). ( b i 1) 1 15

16 Direct Sequence Spread Spectrum Possible demodulator structres for DS spread spectrum signals. 16

17 Performances of Direct Sequence Spread Spectrum 17

18 The decoder computes the metrics CM i j1 1) y Where cij is the j-th bit in the i-th code word. We assume that the all zero code word is transmitted. The metric corresponding to the all zero code word is: CM With vj the noise and interference n (c ij j k i 1,,..., n n 1 nec (c1 j 1)(b j 1) v j nec (b j 1) j1 j1 * v j Re g ( t) z( t ( j 0 Tc 1) Tc ) dt The metric corresponding to the code word cm having weight wm is: n CM m DS SS: Performance ne c (1 w n m ) j1 (c mj 1)(b j 1) v j v j 18

19 DS SS: Performance There is an error if CMm>CM1. The difference is: D CM n 1 CM m 4Ecwm cmj (b j 1) j1 v j If the bandwidth expansion is greater than 10, we can evoke the central limit theorem and the sum in D becomes Gaussian. The variance of D is: m 4 n j1 n i1 c mi c mj E[(b j 1)(b i 1)] E[ v j v i ] 19

20 The PN is uncorrelated so: DS SS: Performance E[(b j 1)(b i 1)] ij and m 4w m E[ v j ] S zz 1 E[ v j ] G( f ) Szz ( f ) df ( f ) is the power spectral density of z(t). If the interference is spectrally flat within the bandwidth: 1 S zz ( f ) J 0, f W E[ v j ] EcJ 0 m 8EcJ 0 wm 0

21 Performance with broadband interference So, in the case of interference spectrally flat within the bandwidth, the probability that D<0 is: P ( m) Q( E J c w m 0 We have Ec=k/n Eb = Rc Eb so: P ( m) Q( E J b c m 0 The code word error probability may be upper-bounded by the union bound as: E M b k PM Q( Rcwm ), M m J 0 R w ) ) 1

22 Performance with narrow band interference Now we consider narrow band interference with total power Jav: 1 J av, f 1 ( ) 1 1 W Szz f W 0 f W1 We can note Jo the equivalent wideband interference J J W, J0 av 0 Jav / Taking the former computation: E[ v j ] it depends on the spectral characteristics of g(t) 1 J W av 1 W / 1 W 1 / W G( f ) df

23 Performance with narrow band interference At the limit as W1 becomes zero, the interference is a pure frequency tone and it is usually called a continuous wave (CW) interfering signal. S zz ( f ) J ( f av ) If g(t) is rectangular: Thus: m 1 4wm J av G(0) 4w m E T c c J av 3

24 Performance with narrow band interference So, for a CW interference and a rectangular pulse shape, we have: M P Q( M m 4Ec J T av c w m ) Knowing that J av / W J and T c 1/ W M P Q( M 0 m E J 0 b R c w m ) as in a broadband interference. It is as if the interfering power was spread. 4

25 5 Without taking the limit (W1 not zero) and keeping a rectangular pulse shape, we have: with the integral is bounded so / / ) ( 1 4 ] [ 4 W W av m i m m df f G W J w v E w / / 1 ) sin( 4 df x x W J E w av c m W 1 T c 1 4 W J E w av c m m Performance with narrow band interference

26 Processing gain, Interference Margin, Coding gain 6

27 We can write again the error probability in function of the signal power: P E / T av Processing Gain b b M P Q( M m 4P J av av L c R c w m ) Tb W Lc Be Tc R is the bandwidth expansion factor, or equivalently the number of chips per information bit. It is also called the processing gain of the DS SS system. 7

28 Interference Margin We can write again the error probability in function of the signal power: P E / T av b b M P Q( M m 4P J av av L c R c w m ) J av Pav is called the interference margin. It is the maximum ratio interference on signal power that can be taken to meet a specified performance for a given bandwidth expansion factor. 8

29 We can write again the error probability in function of the signal power: P E / T av Coding Gain b b M P Q( M m 4P J av av L c R c w 4Pav ( M 1) Q( Lc Rcd min J av m ) ) R c d min is the coding gain. 9

30 Classical case: Uncoded DS SS 30

31 We can now consider a trivial (n,k) code: the binary repetition code. For this case, k=1 and the weight of the non-zero code is w=n. Thus Rc.w=1. The probability of error is: P ( m) Uncoded DS SS Q( Q( E J There is no coding gain. The processing gain is W/R 0 b ) 4W / R J av / P av ) 31

32 Uncoded DS SS Another way of seeing the processing gain in the demodulation process for uncoded DS SS: (a) Before demodulation (b) after demodulation 3

33 1 Before spreading, P. with the Power Spectral Density t c Re ct t jt c b j N 1 n k0 t c b Rect t nt kt j After spreading, the spectrum is spread on W=1/Tc s c T Nnk n 1 The power is '. with being the new PSD. Tc As the spreading does not change the total power, P so ' /L c Uncoded DS SS ' T b T b b c 1 1 '.. Tc Tb 33

34 Uncoded DS SS For a big Lc, the psd of the spread spectrum signal is very low, it is hidden below the noise. For instance, if the signal to noise ration is 0dB before spreading. By using a processing gain (or spreading factor) of 30dB, the psd of the spread signal becomes 10dB lower than the noise. 34

35 After despreading, x t c Rect( t nt ) z( t) b Re ct t nt kt n n Uncoded DS SS The spectrum of the first term is between (-1/T b, +1/T b ) while the spectrum of the second term is between (-1/T c, +1/T c ). The interference spectrum has been spread. b If the signal to interference ratio on the channel is The power of the interference before the detection, in the bandwidth (-1/T b, +1/T b ), is Pi/Lc, while the power of the signal is still Ps, so the signal to interference ratio before the detection is: P' S PS S / I det ection N N ( S / I) channel P' P n N 1 k0 Nnk S / I PS / PI channel I I T c b c 35

36 Uncoded DS SS Conclusion:with restpect to narrow band interference, the spread spectrum technique gives a gain of Lc, or 10 log(lc) in db, in term of S/I ratio. 36

37 Applications of DS SS Signals: Low detectability signal transmission 37

38 We consider a communication in additive noise of spectral density No/. The noise power in the bandwidth is W.No. We want to hide the signal in the noise so we want Pav<<W.No=Nav We can take the former formula with Jo=No P M MQ MQ E N b ( Rcd min 0 W R P LPI av ( Rcd min Nav So even if Pav/Nav<<1, we can recover the information thanks to the processing gain and the coding gain. ) ) 38

39 LPI Moreover, any other receiver without the PN knowledge, is unable to take advantage of the processing gain and coding gain. It is a LPI signal. 39

40 Applications of DS SS Signals: Code division multiple access 40

41 CDMA The enhancement brought by processing gain and coding gain can be used to enable many DS SS signals to occupy the same band with distinct PN sequence. The other users are seen as broadband interference. We want to determine the number of simultaneous users that can be supported. If each user has a power Pav, the signal to interference ratio is: Pav Pav 1 J / ( Nu 1) P Nu 1 av av 41

42 CDMA So we can express the probability of error: Pe ( M 1) Q( Lc Rcd min ( Nu 1) ) 4

43 CDMA For instance, if an error probability of 10^-6 is targeted, then ( Nu 1) L c R d c min 40 so Nu L R d min / 0 1 c c If W/R=100 and Rc.dmin=4, the maximum number of users is Nu=1. If W/R=1000 and Rc.dmin=4, the maximum number of users is Nu=01. 43

44 CDMA IS-95 Uplink 44

45 CDMA CDMA downlink Can make downlink signals for different users orthogonal at the transmitter (Walsh-Hadamard codes) (power control less crucial, only for inter-cell interferences). Still because of multipaths, different users signals are not completely orthogonal at the receiver. Less interference averaging: interference come from a few high-power base stations as opposed to many low-power mobiles. 45

46 CDMA IS-95 Downlink 46

47 47

48 Frequency-Hopped (FH) Spread Spectrum 48

49 Frequency Hopped Spread Spectrum The channel bandwidth is subdivided into many frequency slots. The signal occupies the available slots according to a pseudorandom selection of the slots through a PN sequence. An example of a frequency-hopped (FH) pattern. 49

50 Frequency Hopped Spread Spectrum We consider here binary FSK modulation, and non coherent detection. PSK modulation gives better performance than FSK but it requires coherent detection (it is difficult to maintain phase coherence). Block diagram of an FH spread spectrum system. 50

51 Frequency Hopped Spread Spectrum Block hopping: There are M information-bearing tones that are separates in frequency by 1/Tc, Tc is the signaling interval. Independent tone hopping: the M possible tones are widely dispersed frequency slots. It is less vulnerable to jamming. If we have multiple hops per symbol, we have a fast-hopped signal. It also helps in preventing jammers (follower jammers have no time to intercept the frequency and retransmit interfering signal). Fast-hopped signals create penalty, because the energy is divided and combined noncoherently (noncoherent combining loss). 51

52 Frequency Hopped Spread Spectrum Timing synchronization is less stringent in FH SS than in DS SS. In DS SS timing synchronization must be established in a fraction of the chip interval Tc=~1/W. In FH SS the chip interval spent to transmit in a slot of bandwidth B<<W is approximately 1/B. 5

53 Frequency-Hopped (FH) Spread Spectrum: performance 53

54 (slow) FH SS Performance For slow FH SS using binary orthogonal FSK with noncoherent detection in AWGN of power spectral density No/, the probability of error is: with b E b N 0 P 1 b e / 54

55 55 If the bit interval is subdivided into L subintervals (fast frequency hopping with L hops per bit), the non coherent detection perform a square-law combining of the output signals of each L matched filters. The probability of error becomes: with and i b L i L K i e L P b ) / ( 1 ) ( 1 0 / 1 0 N 0 E L L N E c c b b i L r i r L i K 1 0 1! 1 Fast FH SS Performance

56 Fast FH SS Performance E b For the same SNR per bit b N 0, the error rate with fast FH signal (L subintervals) is worst than with slow frequency hopping (1 hop/bit). The difference, expressed in SNR, for a given error rate and a given L is called the noncoherent combining loss. 56

57 Frequency-Hopped (FH) Spread Spectrum: Performance with coding gain 57

58 FH SS, Coding Gain Suppose we use a linear binary (n,k) block code, binary FSK modulation with one hop per coded bit, soft-decision decoding of the square-law demodulated FSK signal. The probability of a codeword error is: P e M m P ( m ) where P(m) is the error probability in deciding between the m-th codeword and the all-zero codeword when the latter has been transmitted. 58

59 FH SS, Coding Gain P(m) has the same expression as the probability of error of fast FH without coding and L subdivisions, with L being replaced by and replaced by w m The product. R c d min b R c w m R b c w m represents the coding gain. It is not less than 59

60 FH SS, Coding Gain If we use fast FH with n hops per coded bits, the n hops may be interpreted as a repetition code, combined with a non-trivial (n1,k) binary linear code having weight distribution wm. It yields to a (n1n,k) binary linear code having weight distribution nwm. Hence, the error probability P(m) has the form given for fast FH, with L being replaced by n w m and replaced by, brcn wm where. R c k / n n R n 1 w k n w Note that b c m b m, which is the coding gain of only the (n1,k) code. 1 So, the use of the repetition code does not increase the coding gain, it only increase the noncoherent combining loss. b 60

61 FH SS, Coding Gain With hard-decision decoding and slow frequency hopping, the probability of a coded bit error with non coherent detection is p 1 b Rc e / The codeword error probability is upper bounded, by use of the Chernov bound: P e M m wm / 4 p(1 p) 61

62 FH SS, Coding Gain With hard-decision decoding and fast frequency hopping with n hops per coded bits, where the square-law-detected outputs corresponding to the matched filter output from the n hops are added as in soft-decision decoding, the probability of a coded bit error p is the same as P(L) of fast FH SS without coding, but replacing L by n and by, with the rate of the (n1,k) code. b R b c R c So, fast FH system is degraded relative to the slow FH system by an amount equal to the noncoherent combining loss of the signals received from the n hops. 6

63 FH SS, Coding Gain In both hard decoding and soft decoding, the use of repetition code in fast FH yields no coding gain with noncoherent combining. 63

64 In broadband interference, the signal to noise ratio per bit may be expressed as: E P b av / 0 J / R J av W with b R the information rate in bits per second E J b 0 W / R J / P av av We recognize W/R as the processing gain and Jav/Pav as the interference margin. 64

65 Frequency-Hopped (FH) Spread Spectrum: Performance in Partial-Band Interference 65

66 FH SS with partial band interference The partial-band interference is modeled as a zero-mean Gaussian random process with flat power spectral density over a fraction of the total bandwidth W and zero elsewhere. In the region(s) where the PSD is non zero, its value is: R zz ( f ) J0 /,

67 FH SS with partial band interference A slow FH system will be jammed with probability not jammed with a probability 1 and If the errors comes only from the jammer: when the signal is jammed, the probability of error is 1 / J e and when it is not jammed the demodulation is error free. So, the probability of error is P 1 ( ). E b e E b 0 / J 0 67

68 FH SS with partial band interference Performance of binary FSK with partial-band interference. 68

69 FH SS with partial band interference The jammer s optimum strategy is to select the value of alpha that maximizes the error probability. By differentiating P(alpha), we find that the most harmful jamming is obtained for: max E b 1 / J 1 0 E b E b / / J J 0 0 The corresponding error probability for the worst-case partialband jammer is: P 1 e E / J b 0 69

70 FH SS with partial band interference The corresponding error probability for the worst-case partialband jammer is: P 1 e E / J b 0 So, whereas the error probability decreases exponentially for fullband jamming, we find that the error probability decreases only inversely with Eb/Jo for the worst case partial-band jamming. (As in rayleigh fading channel, as we will see later) 70

71 Frequency-Hopped (FH) Spread Spectrum: Benefits of diversity in Partial-Band Interference 71

72 FH SS with partial band interference We use fast FH with L hops per bit and binary FSK. The signaling interval is divided into L subintervals. For each subinterval the signal is demodulated by passing through a pair of matched filter ( FSK) whose outputs are square-law-detected. The square-law-detected signals corresponding to the L frequency hops are weighted and summed to form the two decision variables U1 and U: U U L 1 k Ec k 1 L k Nk k 1 k N jk N 1k are the weighting coefficients represent the additive Gaussian noise 7

73 FH SS with partial band interference The coefficients are optimally selected. Ideally, is elected to be equal to the inverse of the variance of the corresponding noise term So, we need to have knowledge of jammer state at the decoder. N k k In the presence of the partial-band interference, the variance of the real and imaginary parts of the noise terms N1k and Nk are: 1 N E N E. N 1 E J 1k k c 0 0 k 73

74 FH SS with partial band interference So, in presence of the interferer we select: k 1/ And in the absence of interferer: k k N E c 0 E c. N k 0 J 0 1 N 1 E c 0 k is a random variable as the k-th frequency is pseudorandomly chosen. 74

75 An error occurs if U>U1. The Chernov upper bound is: P FH SS with partial band interference P( U U1 0) E exp ( U 1) U 0 Eexp where is a variable that has to be optimized to yield the tightest possible bound. The average is performed with respect to the noise and beta. L k 1 ( E k c N' 1 k N' k ) 75

76 76 By first averaging on the gaussian noise: L k k k c k N N E E P 1 1 ) ' ' ( exp ) ( 1 1 ' exp ' exp k k c k L k N E N E E 1 4 exp k c L k E FH SS with partial band interference

77 77 We now average on the weighting beta. Since the FSK tones are interfered with probability : L k c c N E J N E P exp / exp 4 1 L L k c c N E J N E exp / exp 4 1 FH SS with partial band interference

78 FH SS with partial band interference We have to find the optimal value of to get a tight bound. This can be done assuming No negligeable. In this case, for the worst jamming case and most precise bound we find : 3J 0 1 E c 1 4 P P 4 L 1,47 ( c b L) e 3 c c c J0 LJ0 L E E 78

79 FH SS with partial band interference We may express the bound as: P ( L ) exp ( ) h b c with h( ) c 1 4 ln 1 c c There is an optimum SNR per chip 10log c 6dB 79

80 FH SS with partial band interference So, at the best SNR per chip (choosing the best L for a given SNR per bit): P P opt) ( b L e / 4 In comparison with the broad band interference without fast hopping, we see that the worst-case partial-band interference and the noncoherent combining loss result in a 3dB loss. For a given SNR per bit, if the order of diversity L is not optimized, the loss will be greater. 80

81 FH SS with partial band interference Coding provides a mean for improving the performance of the FH system corrupted by partial band interference. Suppose we use a block orthogonal code with M=^k codewords and L-th order diversity per codeword, the probability of a codeword error is: P e k ( 1) P ( L ) ( And the equivalent bit error probability: k 1,47 1) c L ( k 1) 1,47 k. / L b L P b 1,47 L k 1 k. b / L 81

82 FH SS with partial band interference Performance of binary and octal FSK with L-order diversity for a channel with worst-case partial-band interference. 8

83 FH SS with partial band interference With an optimum choice of diversity, we have: P b k exp k b exp k b 4 4 ln Thus, we have an improvement in performance by an amount equal to 10log k(1,77 / b For instance, if b 10 and k 3 then the gain is 3.4 db, while if k=5 the gain is 5.6 db Additional gains can be achieved by employing concatenated codes with soft-decision decoding. ) 83

84 Other Types of Spread Spectrum Signals 84

85 Synchronization of Spread Spectrum Systems 85

86 Synchronization Time synchronization is divided into two phases: acquisition and tracking after the signal has been initially acquired. In DS SS, the PN code must be synchronized to within a smal fraction of the chip interval Tc=~1/W. The transmitter send a known pseudorandom data sequence. The receiver is continuously in a search mode for this sequence in order to establish initial synchronization. 86

87 Synchronization In principle, matched filter or cross correlation are optimum methods for establishing initial synchronization. The known reference sequence in the crosscorrelation is advanced of 1/Tc seconds untill a threshold is exceeded. 87

88 Synchronization A similar process may also be used for FH signals. In this case, the problem is to synchronize the PN code that controls the hopped frequency pattern. 88

89 Synchronization Alternative system for acquisition of an FH signal. 89

90 Synchronization We may have parallel or serial search. Serial search is generally time consuming. During the search mode, there may be false alarms that occur at the designed false alarm rate of the system. To handle false alarms, it is necessary to have an additional method that checks that the correlator output remains above the threshold. 90

91 Synchronization Sequential search methods are more efficient: the average search time is minimized. There is a correlator with a variable integration period and the output is compared to threshold. -If the upper threshold is exceeded, initial synchronization is established. -If the correlator output falls below the lower threshold, the signal is declared absent at that delay and search resumes. -If the correlator output falls between the two threshold, the integration time is increased by one chip and the output is compared with the two threshold again. 91

92 Synchronization There is not only time uncertainty, but also frequency uncertainty due to Doppler effect. It has to be searched also, by shifting the frequencies of the known pattern. 9

93 Synchronization Initial search for Doppler frequency offset in a DS system. 93

94 Synchronization Once the signal is acquired, tracking begins. The commonly used tracking loop for a DS SS is the delay locked loop (DLL) 94

95 Synchronization Autocorrelation function and tracking error signal for DLL 95

96 Synchronization Tracking method for FH signals. [From Pickholtz et al. (198). 198 IEEE.] 96

97 97

98 Fading Channels 98

99 Fading Channel Wireless Mulipath Channel Channel varies at two spatial scales: large scale fading small scale fading 99

100 Channel Model Physical Models Wireless channels can be modeled as linear time-varying systems: where a i (t) and i (t) are the gain and delay of path i. The time-varying impulse response is: H( f ; t) a i ( t) e jf ( t) i Consider first the special case when the channel is timeinvariant: 100

101 Channel Model Passband to Baseband Conversion Communication takes place at Processing takes place at baseband 101

102 Channel Model 10

103 Channel Model 103

104 Channel Model Complex Baseband Equivalent Channel The frequency response of the system is shifted from the passband to the baseband. Each path is associated with a delay and a complex gain. 104

105 Channel Model Discrete-time baseband model 105

106 Channel Model 106

107 Channel Model In the case where gains and delays are time-invariant we have: It can be interpreted as the sampled version of the channel response hb(τ) convolved with sinc(w τ). 107

108 Statistical Models Design and performance analysis based on statistical ensemble of channels rather than specific physical channel. jf i ( m/ W ) h [ m] a ( m / W ) e sin c[ l ( m / W ) W l i i Fading Channel c ] Rayleigh fading model: many small scattered paths i h [ m] ~ l CN(0, ) l Complex circular Gaussian. Squared magnitude is exponentially distributed. Rician model: 1 line-of-sight (specular path) plus scattered paths h [ m] l l e 1 j 1 CN(0, l 1 ) 108

109 Fading Channel Rayleigh fading model h [ m] ~ CN(0, ) l h l [m] Then the magnitude random variable with pdf: x x exp l l and the squared magnitude distributed with pdf: l, [m] h l 1 x exp, x l l of the lth path is a Raleigh x 0 is exponentially 0 109

110 Fading Channel Rayleigh flat fading model: y[ m] h[ m] x[ m] w[ m] w[ m] ~ CN(0, N0) h[ m] ~ CN(0,1) 110

111 Performance over Rayleigh Fading Channel Rayleigh flat fading model, non coherent detection: 111

112 Performance over Rayleigh Fading Channel 11

113 Performance over Rayleigh Fading Channel exponentially distributed : 113

114 Performance over Rayleigh Fading Channel BPSK: Coherent detection. Conditional on h, where SNR=a^/No Averaged over h, at high SNR. 114

115 Performance over Rayleigh Fading Channel Rayleigh vs AWGN 115

116 Diversity 116

117 Diversity Time Diversity Time diversity can be obtained by interleaving and coding over symbols across different coherent time periods. Coding alone is not sufficient!

118 Diversity Example:GSM Amount of time diversity limited by delay constraint and how fast channel varies. In GSM, delay constraint is 40ms (voice). To get full diversity of 8, needs v > 30 km/hr at f c = 900Mhz. 118

119 Diversity Simplest Code: Repetition After interleaving over L coherence time periods, Repetition coding: for all where and This is classic vector detection in white Gaussian noise. 119

120 Diversity 10

121 Diversity 11

122 Diversity Performance 1

123 Antenna Diversity Receive Transmit Both 13

124 Diversity Receive Diversity Same as repetition coding in time diversity, except that there is a further power gain. Optimal reception is via match filtering (receive beamforming). 14

125 y T h x Diversity Transmit Diversity If transmitter knows the channel, send: w x x h h * maximizes the received SNR by in-phase addition of signals at the receiver (transmit beamforming). Reduce to scalar channel: same as receive beamforming. What happens if transmitter does not know the channel? 15

126 Diversity Other schemes of receive or transmit diversity: Selection Combining (SC) Threshold Combining (TC) Equal Gain Combining (EGQ) 16

127 Diversity Transmit Diversity What happens if transmitter does not know the channel? 17

128 Diversity Space-time Codes Transmitting the same symbol simultaneously at the antennas doesn t work. Using the antennas one at a time and sending the same symbol over the different antennas is like repetition coding. More generally, can use any time-diversity code by turning on one antenna at a time. Space-time codes are designed specifically for the transmit diversity scenario. 18

129 Over two symbol times: Diversity Alamouti Scheme Projecting onto the two columns of the H matrix yields: double the symbol rate of repetition coding. 3dB loss of received SNR compared to transmit beamforming. 19

130 Capacity of fading channels 130

131 For a power-constrained discrete-time AWGN channel, the capacity can be expressed as: In bits per channel use per real dimension. C Capacity 1 log(1 P N ) For a complex-input complex-output channel with circular complex Gaussian noise with variance No, or No/ per real and imaginary components, the capacity is: C log(1 bits per complex dimension. P N 0 ) 131

132 The capacity of an ideal band-limited, power-limited additive white Gaussian waveform channel is : in bits per second. C Capacity W log(1 N P 0 W P is the signal power and No/ the noise PSD. ) For an infinite bandwidth channel with P/NoW tending to zero C 1 ln P N P N 0 13

133 Capacity The capacity in bits/sec/hz (ot bits per channel use) which determines the highest achievable spectral bit rate is: C log( 1 SNR) with the SNR defined as SNR P N W 0 133

134 The capacity of a non-ideal channel in which the channel frequency response is C(f) and the Gaussian noise has a PSD Sn(f): C The input PSD is selected such that and K is selected such as Capacity 1 P f C f ( ) ( ) log(1 ) df S ( f ) Sn( f ) P( f ) K x max0, x C( f ) P( f ) df P It is the principle of water-filling. n 134

135 Capacity 135

136 Capacity 136

137 Ergodic Capacity vs. Outage Capacity 137

138 Ergodic Capacity of the Rayleigh Fading Channel Ergodic capacity of the Rayleigh fading channel. Assumption: the channel coherence time and the delay restrictions are such that perfect interleaving is possible: model with independent Rayleigh channel coefficients: y i R x i i n i where xi and yi are the complex input and output, Ri is a complex iid random variable with Rayleigh distributed magnitude and uniforme phase. The PDF of the magnitude of Ri is: r p( r) e 0 r r r

139 Ergodic Capacity of the Rayleigh Fading Channel E[ R ] 1 As, we may assume for simplicity so the received power and the transmit power are the same. (The extension to the general case is straightforward). If no channel state information is available at the receiver (No CSI): -the capacity is plotted on the next slide. -for P/No tending to 0, we have as in AWGN channel: C 1 ln P N P N 0 139

140 Ergodic Capacity of the Rayleigh Fading Channel Ergodic capacity of a Rayleigh fading channel with no CSI 140

141 Ergodic Capacity of the Rayleigh Fading Channel If channel state information is available at the receiver (with CSI): the receiver can compensate the phase of the fading process, so without loss of generality the channel maybe modeled by a multiplicative real coefficient R with Rayleigh distribution. The effect on the power is a multiplicative coefficient R To find the ergodic capacity, we have to find the expected value of: C log(1 P ) N 0 141

142 Ergodic Capacity of the Rayleigh Fading Channel If channel state information is available at the receiver (with CSI): C P Elog(1 ) N 0 And by using Jensen s inequality on concave function: C log(1 log(1 E[ ] P N 0 ) P N 0 ) 14

143 Ergodic Capacity of the Rayleigh Fading Channel If channel state information is available at the receiver (with CSI), we can compute the ergodic capacity for SNR tending to zero or infinity, we find: C 1, 44SNR for low SNR, as in AWGN channels. C log SNR for high SNR, it lags the AWGN channel by 0.83 bits per channel use. Or equivalently, at high SNR the asymptotic difference is.5 db. 143

144 Ergodic Capacity of the Rayleigh Fading Channel Ergodic capacity of a Rayleigh fading channel with CSI at the decoder. 144

145 Ergodic Capacity of the Rayleigh Fading Channel If channel state information is available at the receiver and also at the transmitter, we have CSI at both sides. The transmitter may adjust its power level to the fading level similar to the water-filling approach: C P( ) log(1 ) e 0 N 0 d P( ) 1 1 N

146 Ergodic Capacity of the Rayleigh Fading Channel Ergodic capacity of a Rayleigh fading channel with CSI at both sides. 146

147 Ergodic Capacity of the Rayleigh Fading Channel Ergodic capacity of a Rayleigh fading channel. 147

148 Outage Capacity of Rayleigh Fading Channels 148

149 Outage Capacity of Rayleigh Fading Channels The outage capacity is considered when due to strict delay restrictions, ideal interleaving is impossible: the channel capacity cannot be expressed as the average of the capacities for all possible realizations. In this case the capacity is a ransom variable. Errors occur when the rate exceeds capacity (otherwise we use perfect coding) i.e. when the channel is in outage. 149

150 Outage Capacity of Rayleigh Fading Channels The outage capacity is : C max R : P( C R) max R : P out ( R) F 1 C ( ) F C 1 (.) Where is the cumulative distribution function (CDF) of the random variable representing the channel capacity. 150

151 Outage Capacity of Rayleigh Fading Channels For a Rayleigh fading channel with normalized channel gain: P out C log( 1 SNR) ( R) P( C R) P 1 e R 1 SNR R 1 SNR At high SNR: C R log 1 SNRln(1 Pout log 1 SNRln(1 ) ) 151

152 Outage Capacity of Rayleigh Fading Channels For a Rayleigh fading channel with normalized channel gain: C log( 1 SNR) P out ( R) P( C R) P 1 e R 1 SNR R 1 SNR At low SNR: C SNR ln 1 ln 1 for small : C C AWGN 15

153 Outage Capacity of Rayleigh Fading Channels At high SNR: C 1 log SNRln( ) 1 1 log( SNR) log(ln ) 1 At high SNR: C AWGN log(snr) So there is a difference of for small the difference is 1 log(ln ) 1 log bits per channel use, 153

154 Outage Capacity of Rayleigh Fading Channels The outage capacity of a Rayleigh fading channel. 154

155 Outage Capacity of Rayleigh Fading Channels Effect of diversity on Outage Capacity: With diversity of L-order, the channel maybe modeled with a Chisquares PDF with L degrees of freedom. By taking the cumulative distribution function we find the outage capacity. 155

156 Outage Capacity of Rayleigh Fading Channels The outage capacity of a Rayleigh fading channel with different diversity order and with epsilon=