نام درس تئوری پیشرفته مخابرات فصل 4: سیگنالینگ و مشخصه سازی

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1 نام درس تئوری پیشرفته مخابرات فصل 4: سیگنالینگ و مشخصه سازی کانال های محو 1

2 فصل 4 : سیگنالینگ و مشخصه سازی کانال های محو 1-4 مشخصه سازی کانال های چندمسیره محو 2-4 اثر مشخصات سیگنال روی انتخاب مدل کانال 3-4 کانال محو غیرفرکانس گزین کند 4-4 تکنیک های چندگانگی برای کانال های چندمسیره محو 5-4 سیگنالینگ روی کانال محو فرکانس گزین کند: دمدوالتور RAKE 2

3 نام مبحث آموزشی 1-4 مشخصه سازی کانال های چندمسیره محو 3

4 Characterization of fading multipath channels(1) The multipath fading channels with additive noise Time spread phenomenon of multipath channels (Unpredictable) Time-variant factors Delay Number of spreads Size of the receive pulses 4

5 Characterization of fading multipath channels(2) Transmitted signal S(t)= Re{s l (t)e i2πf ct } Received signal in absence of additive noise: r t = න c τ; t s t τ dτ = c τ; t Re{sl (t τ)e j2πf c(t τ) } dτ )} Re = c τ; t sl (t τ) e j2πf cτ dτ)e j2πf ct } =Re{(c τ; t e j2πf cτ s l (t))e j2πf ct } c l (τ; t)=c τ; t e j2πf cτ and c τ; t = c l (τ; t) 5

6 Characterization of fading multipath channels(3) Note that now it is not appropriate to write s l (t) c l (t) because t and τ are now specifically for time argument and convolution argument, respectively! We should perhaps write s l (t) c l (τ)and s l t c l τ; t which respectively denote: And =( τ ) s l (t) c l cl τ s l t τ dτ = s l t c l τ; t cl τ; t s l (t τ)dτ 6

7 Rayleigh and Rician The measurement suggests that c l τ; t is a 2-D Gaussian random process in t (not in τ), which can be supported by the central limit theorem (CT) because it is the sum effect of many paths. If zero mean, c l τ; t is Rayleigh distributed. The channel is said to be a Rayleigh fading channel. If nonzero mean, c l τ; t is Rician distributed. The channel is said to be a Rician fading channel. When diversity technique is used, c l Nakagami m-distribution. τ; t can be well modeled by 7

8 ҧ Channel correlation Function (1) Assumption(WSS) Assume c l τ; t is WSS in t. τ, τ; t R cl is only a function of time difference t. = E{c l ( τ; ҧ t + t)c l (τ; t)} Assumption (Uncorrelated scattering or US of a WSS channel) For τ ҧ τ, assume c l τ; ҧ t and c l τ; t are uncorrelated for any t 1, t 2. τ is the convolution argument and actually represents the delay for a certain path. Assumption (Math definition of US) R cl τ, ҧ τ; t =R cl τ; t δ( τ- ҧ τ) It may appear unnatural to define the autocorrelation function of a channel impulse response using the Dirac delta function. However, we have already learned that τ is the convolution argument, and δ(τ ) is the impulse response of the identity channel. 8

9 Channel correlation(2) This somehow hints that there is a connection between channel impulse response and Dirac delta function. Recall that a WSS white (noise) process z(τ ) is defined based on R z ( τ) = E{z( τ+τ ) z (τ)}= N 0 2 δ( τ) We can extensively view that the autocorrelation function of a 2-dimensional WSS white noise z(τ ; t) is defined as E{z( τ+τ ; t 1 ) z (τ ; t 2 )}= N 0(t 1,t 2 ) δ( τ) 2 US indicates that the accumulated power correlation from all other paths is essentially zero! Some researchers interpret US as zero-correlation scattering. So, from this, they don t interpret it as E{z( τ+τ ; t 1 ) z (τ ; t 2 )}=E{z( τ+τ ; t 1 )} E{z (τ ; t 2 )}=0 which requires zero-mean assumption but simply say E{z( τ+τ ; t 1 ) z (τ ; t 2 )}=0 when τ 0. 9

10 Multipath intensity profile of a US- WSS channel The multipath intensity profile or delay power spectrum for a US- WSS multipath fading channel is given by: R cl τ = R cl (τ; t = 0). It can be interpreted as the average signal power remained after delay τ. The multipath spread or delay spread of a US-WSS multipath fading channel The multipath spread is the range of τ over which R cl (τ) is essentially non-zero; it is usually denoted by T m. 10

11 Multipath intensity profile of a US-WSS channel Each τ corresponds to one path. No Tx power will remain at Rx for paths with delay τ >T m. 11

12 Transfer function of a multipath fading channel The transfer function of a channel impulse response c l (τ ; t) is the Fourier transform with respect to the convolutional argument τ : C l (f; t) = න Property: If c l (τ; t)is WSS, so is c l f; t. c l (τ; t)e j2πft dτ The autocorrelation function of WSS c l f; t is equal to: f, ҧ f; t = E{C l ( f; t+t) ҧ C l (f; t)} R cl 12

13 R cl Transfer function of a multipath fading channel (2) With an additional US assumption f, ҧ f; t = E{C l ( ҧ f; t+t) C l (f; t)} = } E cl ( τ; ҧ t + t)e j2πfതτ ҧ dτ cl (τ; t)e j2πfτ dτ} Rcl = τ; t δ(τ τ) e j2π(fτ fതτ) ҧ dτ dτҧ = Rcl τ; t e j2πτ( ҧ f f) dτ =R cl f; t where f= ҧ f f For a US-WSS multipath fading channel, R cl f; t = E{C l ( f + f; t+t) C l (f; t)} This is often called spaced-frequency, spaced-time correlation function of a US-WSS channel. 13

14 For the case of t = 0, we have Coherent bandwidth = R cl f; t Rcl τ; t e j2πτ( f) dτ = R cl f Rcl τ e j2πτ( f) dτ Recall that R cl τ = 0 outside [0, T m ). In freq domain, ( f) c = 1 T m is correspondingly called coherent bandwidth. R C f رابطه R c τ بین و 14

15 Example Given R cl τ = τ for 0 τ <100ns. Then R cl f = π f 2 e j2π10 7 f 1 1 2π f j 15

16 Coherent bandwidth since the channel output due to input s l is equal to: r l f =s l f C l f; t where we abuse the notations to denote the Fourier transforms of r l t and s l t respectively as r l f and s l f,we would say r l f =s l f C l f; t will have weak (power) affection on r l f+ f =s l f + f C l f + f; t when f >( f) c 16

17 If signal transmitted bandwidth B >( f) c, the channel is called frequency selective. If signal transmitted bandwidth B < ( f) c, the channel is called frequency non-selective Coherent bandwidth (2) For frequency selective channels, the signal shape is more severely distorted than that of frequency non-selective channels. Criterion for frequency selectivity: 1 B >( f) c > 1 T< T T T m. m 17

18 Time varying characterization: Doppler Doppler effect appears via the argument t. 18

19 Doppler power spectrum of a US- WSS channel (2) The Doppler power spectrum is =( λ ) S cl Rcl f = 0; t e j2πλ( f) t where λ is referred to as the Doppler frequency. B d = Doppler spread is the range such that S cl (λ)is essentially zero. ( t) c = 1 is called the coherence time. B d If symbol period T > ( t) c, the channel is classified as Fast Fading. i.e., channel statistics changes within one symbol! If symbol period T < ( t) c, the channel is classified as Slow Fading. 19

20 Doppler power spectrum of a US-WSS channel R C t S C رابطه λ بین و 20

21 Summary: Scattering function R cl (τ; t) Channel autocorrelation function 1-D FT : R c l f; t = F τ τ; t. S τ; λ = F t R cl τ; t Spaced-freq/Spaced-time 2-D FT:???=F τ, t R c l τ; t R cl ( f; t) Correlation-time/Scattering func??? = F 1-D FT: ቐ f R cl f; t S cl f; λ = F t R cl f; t Doppler power spectrum( f=0) 2-D FT: S τ; λ = F f, t R cl f; t 21

22 The scattering function can be used to identify delay spread and Doppler spread at the same time. Scattering function (2) 22

23 Example. Medium-range tropospheric scatter channel Scattering function of a medium-range tropospheric scatter channel. The taps delay increment is 0.1 μs. 3dB BW =1~ 30 Hz 23

24 Example study of delay spread The median delay spread is the 50% value, meaning that 50% of all channels has a delay spread that is lower than the median value. Clearly, the median value is not so interesting for designing a wireless link, because you want to guarantee that the link works for at least 90% or 99% of all channels. Therefore the second column gives the measured maximum delay spread values. The reason to use maximum delay spread instead of a 90% or 99% value is that many papers only mention the maximum value. From the papers that do present cumulative distribution functions of their measured delay spreads, it can be deduced that the 99% value is only a few percent smaller than the maximum delay spread. 24

25 Measured delay spread Measured delay spreads in frequency range of 800M to 1.5 GHz (surveyed by Richard van Nee) 25

26 Measured delay spread(2) Measured delay spreads in frequency range of 1.8 GHz to 2.4 GHz (surveyed by Richard van Nee) 26

27 Measured delay spread(3) Measured delay spreads in frequency range of 4 GHz to 6 GHz (surveyed by Richard van Nee) Conclusion by Richard van Nee: Measurements done at different frequencies show the multipath channel characteristics are almost the same from 1 to 5 GHz. 27

28 Jakes model: Example A widely used model for Doppler power spectrum is the so-called Jakes model (Jakes, 1974) R cl ( t)=j 0 2πf m t And S cl = 1 1 πf m 1 (λτf m ) 2 λ < f m 0 O. W. f m = vf c /c is the maximum Doppler shift v is the vehicle speed (m/s) c is the light speed (3 10^8 m/s) f c is the carrier frequency J 0 is the zero-order Bessel function of the first kind 28

29 Doppler shift Difference in path length = (sin θ ) 2 +(cos θ + v. t) 2 Phase change φ = 2π Τ c f c = v. t. cos θ + (v. t) 2 29

30 Estimated Doppler shift λ m = lim Δt 0 = 1 Τ c f c 1 2π lim Δt 0 φ t 1 2π Doppler shift (2) 2 +2.v. t.cos θ +(v. t) 2 = vf c c cos θ = f mcos θ Example. v = 108 km/hour, fc = 5 GHz and c = 3 10^8 m/s. λ m =500cos θ t This is ok because 500 Hz 5 GHz = 0.1 ppm. 30

31 Jakes model Here, a rough (and not so rigorous) derivation is provided for Jakes model. Just to give you a rough idea of how this model is obtained. Suppose τ (t) is the delay of some path. + τ τ t+ t τ(t) t = lim = lim c + c= lim c Δt 0 t Δt 0 t Δt 0 t = v c cos(θ) τ t v c cos(θ)+τ 0 (Assume for simplicity τ 0 = 0.) c = lim c t Δt 0 31

32 Jakes model (2) Assume that (i) c(τ; t) a δ (τ τ (t )) (a constant attenuation single path system); (ii) c (τ; t) a δ(τ τ (t )) WSS c l τ; t =c(τ; t) e j2πf cτ a δ (τ τ (t )) e j2πf cτ (t ) a δ (τ (v/c) cos(θ )t) e j2πf c(v/c) cos(θ )t =a δ (τ (f m /f c ) cos(θ )t) e j2πf m cos(θ )t R cl τ; t = න E{c l ( τ; ҧ t+t) c l (τ; t)} dτҧ = E{a δ (τ (fm /f c ) cos(θ )(t+ t)) e j2πf m cos(θ)(t+ t). a δ (τ (f m /f c ) cos(θ )t) e j2πfmcos(θ)t } dτҧ =a 2. E{e j2πf m cos(θ ) t } δ (τ (f m /f c ) cos(θ)t) Since Rcl τ; t is nothing to do with t, we take t = 0 for simplicity; thus 32

33 R cl τ; t =a 2. E{e j2πf m cos(θ ) t } δ (τ) Jakes model (3) R cl = f = 0; t Rcl τ; t e j2π fτ dτ = Rcl = dτ τ; t a 2. E{e j2πf m cos(θ ) t } δ (τ) dτ = a 2. E{e j2πf m cos(θ ) t } =J 0 (2πf m t) where the last step is valid if θ uniformly distributed over [ π, π), and a = 1. θ can be treated as uniformly distributed over [ π, π) and independent of attenuation α and delay path τ. 33

34 Channel model from IEEE Handbook A consistent channel model is required to allow comparison among different WAN systems. The IEEE Working Group adopted the following channel model as the baseline for predicting multipath for modulations used in IEEE a and IEEE b, which is ideal for software simulations. The phase is uniformly distributed. The magnitude is Rayleigh distributed with average power decaying exponentially. 34

35 Channel model from IEEE Handbook (2) where i max 1 c l τ; t = α i e jφ iδ(τ it s ) i=0 T s sampling period α i e jφ i N 0, σ 2 i /2 + jn 0, σ 2 i /2 σ 2 i = σ 2 0 e it S/τ max σ 2 0 = 1 e T S/τ max 35

36 Channel model from IEEE Handbook (3) R cl τ = න E{c l ( τ; ҧ t)c l (τ; t)} dτҧ i max 1 = σ i=0 i = σ max 1 i=0 E{αi 2 }δ(τ it s ) i = σ max 1 2 i=0 σ0 e it S/τ max δ(τ it s ) E {αi 2 δ(τ it s )δ( τҧ τ)}dτҧ By this example, we want to introduce the rms delay. By definition, the effective rms delay is T2 rms = τ 2 R cl τ dτ - τrcl τ dτ ) Rcl τ dτ )2 dτ Rcl τ = σ imax 1 i=0 (its ) 2 σ 2 0 e it S /τ rms imax 1 σ σ0 2 i=0 e it S /τ rms - ( σ imax 1 i=0 (its ) 2 σ 2 0 e it S /τ rms imax 1 σ σ0 2 i=0 e it S /τ rms ) 2 36

37 ǁ Channel model from IEEE Handbook (4) We wish to choose i max such that T rms τ rms. et τǁ rms = τ rms, T T rms = T rms and iǁ s T max = i max s τ rms We obtain T rms = e 1/τ rms (1 e 1/τ rms) 2 ǁ = ( τǁ 2 rms i max 2 e iǁ max (1 e ǁ 240τ rms 2 + ) ǁ Taking T rms ( τǁ 2 rms 1 ) and ǁ 12 1 i max) 2 2 τ rms i max 2 e iǁ max (1 e ǁ 2 i max e imax ǁ (1 e ǁ iǁ max = (or equivalently, i max =10 τ rms ) T s 1 i max) 2 2 τ rms i max) 2 1 yield

38 Channel model from IEEE Handbook (5) Typical multipath delay spread for indoor environment (Table 8-1 in IEEE Handbook) i max =10 τ rms T s = /( ) = 10 i max =10 τ rms = =20 T s 1/( ) i max =10 τ rms = =40 T s 1/( ) 38

39 Statistical models for fading channel (1) In addition to zero-mean Gaussian (Rayleigh), non-zero-mean Gaussian (Rice) and Nakagami-m distributions, there are other models for c l τ; t have been proposed in literature. Example. Channels with a direct path and a single multipath component, such as airplane-to-ground communications c l τ; t =αδ τ + β(t)δ(τ τ 0 (t)) where α controls the power in the direct path and is named specular component, and β(t) is modeled as zero-mean Gaussian 39

40 Example. Statistical models(2) Microwave OS radio channels used for long-distance voice and video transmission by telephone companies in the 6 GHz band (Rummler 1979) c l τ =α[δ τ βe i2πf 0τ δ τ τ 0 ] where α: overall attenuation parameter β: shape parameter due to multipath components τ 0 : time delay f 0 : frequency of the fade minimum, i.e.,f 0 = argmin c l f fεr 40

41 Rummler found that α β (independent) F(β) (1 β) 2.3 Statistical models(3) -log(α) Gaussian distributed (i.e., α lognormal distributed) τ 0 6.3ns Deep fading phenomenon: At f = f 0, the so-called deep fading occurs. 41

42 نام مبحث آموزشی اثر مشخصات سیگنال روی انتخاب مدل کانال

43 signal characteristics(1) Usually, we prefer slowly fading and frequency non-selectivity. So we wish to choose symbol time T and transmission bandwidth B such that T < ( t) c and B< ( f) c Hence, using BT = 1, we wish T B = B ( f) d T m < 1. c ( t) c The term B d T m is an essential channel parameter and is called spread factor. 43

44 Underspread versus overspread Underspread B d T m <1 Overspread B d T m >1 Multipath Spread, Doppler Spread, and Spread Factor for Several Time-Variant Multipath Channels 44

45 45 نام مبحث آموزشی 3-4 کانال محو غیرفرکانس گزین کند

46 Slowly fading channel(1) For a frequency-nonslective, slowly fading channel, i.e., the signal spectrum s l f hence, T m < 1 B = T < ( t) c is almost unchanged by C l (f; t); C l (f; t) C l (0; t) within the signal bandwidth and it is almost time-invariant; hence, C l (f; t) C l (0) within the signal bandwidth This gives r l t = c l (τ;t) s l t + z(t) = Cl (f; t) s l f e j2πft df + z(t) Cl (0) s l f e j2πft df + z(t) =C l (0)s l t + z(t) 46

47 Slowly fading channel(2) Assume that the phase of C l 0 = αe jφ can be perfectly estimated and compensated. The channel model becomes: r l t = αs l t + z(t). After demodulation, we obtain r l = αs l + n l. Question: What will the error probability be under random α? 47

48 Case 1: Equal-prior BPSK r = ±α Ԑ + n (passband vectorization with E n 2 = N 0 ) 2 2 r l,real = ±α 2Ԑ +n l,real (baseband vectorization with E[n l,real ] = N 0 ) The optimal decision is r 0, regardless of α (due to equal-prior). Thus, Pr error α = Q( 2γ b ) where γ b = γ b (α) = α 2 Ԑ/N 0 Given that α is Rayleigh distributed, γ b (α) is χ 2-distributed with two degrees of freedom; hence, 0 = e,bpsk P Pr error α f α dα 0 = Q( 2γb )f γ b dγ b 48

49 Case 1: Equal-prior BPSK (2) P e,bpsk = 0 Q( 2γb ) 1 γ b e γ b/γ b dγ b, where γ b =E[γ b ] = γ b 1+γ b (= 1 2(1+γ b + γ b 2 +γb ) 1 4γ b when γ b large) 49

50 Case 2: Equal-prior BFSK r = α Ԑ 0 or + n. 0 α Ԑ The optimal decision is r1 r2, regardless of α. P e,bfsk = 0 Pr error α f α dα = 0 Q( γb )f γ b dγ b = (= γ b 2+γ b 1 2+γ b + γ b 2 +2γb ) 1 2γ b when γ b large) 50

51 Case 3 & 4: BDPSK & Noncoherent BFSK P e,bdpsk = 0 Pr error α f α dα And 1 = 0 2 e γ bf γ b dγ b = 1 1 when γ 2(1+γ b ) 2γ b large b P e,noncoherent BFSK = න Pr error α f α dα 0 1 = 0 2 e γb/2 f γ b dγ b 1 (= 1 when γ 2+γ b + γ b large) b 51

52 Case 3 & 4: BDPSK & Noncoherent BFSK (2) 52

53 BPSK is 3dB better than BDPSK/BFSK; 6dB better than noncoherent BFSK. P e decreases inversely proportional with SNR under fading P e decreases exponentially with SNR when no fading. Comparison 53

54 To achieve P e =10 4, the system must provide an SNR higher than 35dB, which is not practically possible! So alternative solution should be used to compensate the fading such as the diversity technique. Comparison (2) 54

55 o If α Nakagami-m fading, Nakagami fading o Turin et al. (1972) and Suzuki (1977) have shown that the Nakagami-m distribution is the best-fit for urban radio multipath channels. f γ b = mm Γ(m) γ b m γ b m 1 e mγ b/γ b where γ b =E[α 2 ] Ԑ b /N 0. m < 1: Worse than Rayleigh fading in performance m = 1: Rayleigh fading m > 1: Better than Rayleigh fading in performance 55

56 PDF of Nakagami-m 56

57 BPSK performance under Nakagami-m fading P e,bpsk = න 0 Q( 2γ b ) m m m γ m 1 Γ(m) γ b e mγ b/γ b dγ b. b 57

58 BPSK performance under Nakagami-m fading(2) In some channel, the system performance may degrade even worse, such as Rummler s model, where deep fading occurs at some frequency. The lowest is equal to α(1 β), which is itself a random variable (which makes the performance worse as we have seen from Rayleigh example study) even if f0 is fixed! 58

59 نام مبحث آموزشی تکنیک های چندگانگی برای کانال های چندمسیره محو

60 Diversity techniques(1) Solutions to compensate deep fading Frequency diversity Separation of carriers ( f )c = 1/Tm to obtain uncorrelation in signal replicas. Time diversity Separation of time slots ( t) c = 1/B d to obtain uncorrelation in signal replicas. Space diversity (Multiple receiver antennas) Spaced sufficiently far apart to ensure received signals faded independently (usually, > 10 wavelengths) RAKE correlator or RAKE matched filter (Price and Green 1958) It is named wideband approach, since it is usually applied to situation where signal bandwidth is much greater than the coherent bandwidth ( f )c. 60

61 Diversity techniques(2) It is clear for the first three diversities, we will have identical replicas at the Rx. The idea is that as long as not all of them are deep-faded, the demodulation is ok. For the last one (i.e., RAKE), where B ( f )c, which results in a frequency selective channel, we have = B ( f )c 61

62 Assumption Binary signals (1) 1. identical and independent channels. 2. Each channel is frequency-nonselective and slowly fading with Rayleigh-distributed envelope. 3. Zero-mean additive white Gaussian background noise. 4. Assume the phase-shift can be perfectly compensated. 5. Assume the attenuation {α k } k=1 can be perfectly estimated at Rx. Hence, α k = α k s + n k k = 1,2,, How to combine these outputs when making decision? Maximal ratio combiner (Brennan 1959) r = k=1 α k r k = k=1 α k α k s + k=1 α k n k 62

63 Idea behind maximal ratio combiner Binary signals (2) Trust more on the strong signals and trust less on the weak signal. Advantage of maximal ratio combiner Theoretically tractable; so we can predict how good the system can achieve without performing simulations 63

64 Case 1: Equal-prior BPSK r = ±α Ԑ + n where α = α 2 k k=1 The optimal decision is r 0, regardless of α. Thus,(ቊ r = s 1 or s 2 + n n 0 mean Gaussian with E[nn ] = σ 2 I and n = k=1 α k n k P e=q( d2 12 ) 4σ 2 Pr {error {α k } k=1 }= Q( (2α 2 Ԑ) 2 4(α 2 (N 0 /2)) )=Q( 2γ b) where γ b =γ b (α)=α 2 Ԑ /N 0 Given that {α k } k=1 is i.i.d. Rayleigh distributed, γ b (α) is χ2-distributed with 2 degrees of freedom; hence, P e,bpsk = 0 0 Pr {error {αk } k=1 }f α 1,, α dα 1 dα = 0 Q( 2γb )f(γ b )dγ b 64

65 Case 1: Equal-prior BPSK (2) 0 = P e,bpsk Q( 2γb ) γ b 1 1!γ c e γ b/γ c dγ b Where ഥγ c = E[α 2 k ] Ԑ b /N 0 P e,bpsk =( 1 μ 2 ) σ 1 k=0 Where μ= γ c 1+γ c 1+k k P e,bpsk 2 1 ( 1 ) 4γ c Where we have 1 μ = 1 2 ( 1+μ 2 )k when ഥγ c large 2(1+γ c + γ c +γ c 2 1 4γ c and 1 μ

66 Case 2: Equal-prior BFSK r = α2 Ԑ 0 or 0 α 2 + n, where α= σ Ԑ k=1 α 2 k The optimal decision is r1 r2, regardless of α. P e,bfsk = 0 Q( γb )f γ b dγ b = ( 1 μ 2 ) σ 1 k=0 1+k k ( 1+μ 2 )k but μ= γ c 2+γ c P e,bfsk 2 1 ( 1 2γ c ) when ഥγ c large 66

67 With t being the time index, m = arg max 1 m 2 Case 3: BDPSK Re r l t 1 r l t e jθ m = arg max Re r l t 1 r l t (m = 1), Re r l t 1 r l t Now with previous receptions and current receptions, m = arg max (U l (m = 1), U l (m = 2)) Where U l = σ k=1 = σ k=1 Re r k,l t 1 r k,l t t 1 t 1 Re α k s l +nk,l which closely resembles maximal ratio combining. t t (α k s l +nk,l ) (m = 2) 67

68 Case 3: BDPSK (2) 68

69 Case 4: Noncoherent FSK Recall that the noncoherent M computes From Chapter 2 (course), Hence, 69

70 Case 4: Noncoherent FSK (2) Now we have k diversities/channels: r k,1,l r m,l = = α k s m,l + n k,l k=1,2,, r k,m,l Instead of maximal ratio combining, we do square-law combining: m = arg max 1 m M P e,noncoherent BFSK = ( 1 μ 2 ) σ 1 k=0 P e,noncoherent BFSK 1+k k k=1 r k,m,l ( 1 γ c ) when ഥγ c large ( 1+μ 2 )k but μ= γ c 2+γ c 70

71 Comparison Summary (what the theoretical results indicate?) With th order diversity, the BER of binary signals decreases inversely with th power of the SNR. 71

72 Comparison (2) Comparing the PDFs of γ b for 1- diversity (no diversity) Nakagami fading and -diversity Rayleigh fading,we found that f γ b = f γ b = 1 γ Γ(m) (γ b /m) m b m 1 e γ b/ (γ b /m) 1-divert Nakagami 1 γ Γ() (γ b /) b 1 e γ b/ (γ b /) -divert Rayleigh We can then conclude: -diversity in Rayleigh fading = 1-diversity in Nakagami- or further m-diversity in Rayleigh fading = -diversity in Nakagami-m 72

73 Multiphase signals(1) For M-ary phase signal over Rayleigh fading channels, the symbol error rate P e can be derived as (Appendix C) P e = μ 2 1 ൭ π 1! b 1 1 (b μ 2 ) π M M 1 μsin(π/m) μcos(π/m) ൩ൡ൱ b μ 2 cos 2 (π/m) cot 1 b μ 2 cos 2 (π/m) M 1 1 log 2 (M) sin 2 M ary PSK & =1 (π/m) 2Mγ b M 1 1 log 2 (M) sin 2 M ary DPSK & =1 (π/m) Mγ b b=1 73

74 Where Multiphase signals(2) ഥγ c 2 + ഥγ μ= c M ary PSK ഥγ c 2 + ഥγ c M ary DPSK and in this case, the system SNR ഥγ t = γ b log 2 M = ഥγ c. PSK is about 3dB better than DPSK for all M. 74

75 DPSK performance with diversity Bit error P b is calculated based on Gray coding. arger M, worse P b except for equal P b at M = 2, 4. Multiphase signals(3) 75

76 Orthogonal signals(1) Here, the derivation assumes that both passband and lowpass equivalent signals are orthogonal; hence, the frequency separation is 1/T rather than 1/(2T ). Based on lowpass (baseband) orthogonality, -diversity square-law combining gives M 1 ( 1) m+1 M 1 1 P e = 1! m (1 + m + mγ c ҧ ) m=1 where β k,m satisfies 1 U k k=0 k! m m( 1) k=0 = 1 + γ c ҧ β k,m 1 + k! ( ) k 1 + m + mγ c ҧ m( 1) k=0 β k,m U k 76

77 Orthogonal signals(2) M = 2 case: et ഥγ t = ഥγ c be the total system power. For fixed ഥγ t, there is an that minimizes P e. This hints that ഥγ c = 3 will give the best performance. 77

78 Orthogonal signals(3) M = 4 case: et ഥγ t = ഥγ c be the total system power. For fixed ഥγ t, there is an that minimizes P e. This hints that ഥγ c = 3 will give the best performance. 78

79 Discussions: arger M, better performance but larger bandwidth. arger, better performance. An increase in is more efficient in performance gain than an increase in M. 79

80 80 نام مبحث آموزشی 5-4 سیگنالینگ روی کانال محو فرکانس گزین کند: دمدوالتور RAKE

81 Tapped-delay line channel model (1) Assumption (Time-invariant channel) c l τ ; t = c l τ Assumption (Bandlimited signal) s l t is band-limited, i.e., s l f = 0 for f > W /2 In such case, we shall add a lowpass filter at the Rx. 1 f W/2 where f = ቊ 0 otherwise 81

82 Tapped-delay line channel model (2) Equivalent channel with Cl(f) random Equivalent channel with Cl(f) random and bandlimited, zw(t) bandlimited white noise. s l t = න s l f C l f e i2πft df + z W t 82

83 Tapped-delay line channel model (3) For a bandlimited C l c l t = C l f = න = n= C l n f,sampling theorem gives: W sinc W t n W c l (t)e j2πft dt 1 W n C l W n= e j2πfn/w f W/2 0 otherwise 83

84 r l t = න Tapped-delay line channel model (4) s l (f)c l (f)e j2πft dt + z W (t) = 1 σ n W n= c l W = 1 σ n W n= c l W 2/W W/2 sl (f) e j2π(t n W ) df + z W (t) s l(t n W ) + z W(t) = σ n= c n. s l (t n ) + z W W(t) where c n = 1 c W l W For a time-varying channel, we replace c l (τ)and C l (f)by c l (τ; t) and c l (f; t) and obtain n r l t = σ n= c n (t). s l (t n ) + z W W(t) where c n (t)= 1 c W l n W ; t 84

85 Tapped-delay line channel model (5) Statistically, c l τ = 0for τ > T m and τ < 0 with probability one. So, c l τ is assumed band-limited and is also statistically time-limited! Hence, c n t = 0 for n < 0 and n > T m W (since τ = n/w > T m ). T r l t = σ m W n= cn (t). s l (t n ) + z W W(t) For convenience, the text re-indexes the system as r l t = σ k=1 c k t. s l t k W +z W (t) 85

86 RAKE demodulator (1) Assumption (Gaussian and US (uncorrelated scattering)) {c k (t)} k=1 complex i.i.d. Gaussian and can be perfectly estimated by Rx. So the Rx can regard the transmitted signal as one of m = arg max 1 m M =arg max 1 m M Re r l. v m,l Re σ k=1 v 1,l t = c k t. s 1,l (t k W ) k=1 v M,l t = c k t. s M,l (t k W ) k=1 =arg max 1 m M Re 0 T rl T rl 0 t c k t s m,l (t k )dt W t v m,l dt 86

87 RAKE demodulator (1) Discussions on assumptions: We assume: s l (t) is band-limited to W. c l (τ) is causal and (statistically) time-limited to T M and, at the same time, band-limited to W. W ( f) c = 1 T m (i.e., WT m 1) as stated in page 879 in textbook. The definition of Um requires T T m (See page 871 in textbook) such that the longest delayed version s l (t /W)=s l (t WT m /W)=s l (t T m ) is still well-confined within the integration range [0,T ). As a result, the signal bandwidth is much larger than 1/T; RAKE is used in the demodulation of spread-spectrum signals! WT WT m 1 W 1/T 87

88 The receiver collects the signal energy from all received paths, which is somewhat analogous to the garden rake that is used to gather leaves, hays, etc. Consequently, the name RAKE receiver has been coined for this receiver structure by Price and Green (1958). (s m,l is used but the text uses s l,m.) M = 2 case 88

89 Alternative realization of RAKE receiver The previous structure requires M delay lines. We can reduce the number of the delay lines to one by the following derivation. et u=t-k/w U m,l = Re σ k=1 T rl 0 t c k (t)s m,l ((t k ))dt W = Re σ k=1 Re σ k=1 T K W r l u + k c W k (u + k ) s W m,l(u)du K W T rl 0 u + k c W k (u + k ) s W m,l(t)dt where the last approximation follows from K W W WT m W = T m T. 89

90 Optimum demodulator for wideband binary signals (delayed received signal configuration). RAKE is abbreviated as ck(t) in the above figure. 90

91 Performance of RAKE receiver Suppose c k t = c k and the signal corresponding to m = 1 is transmitted. Then, letting U m,l = 1 2ε s U m,l and sǁ m,l t k W = 1 2ε s s m,l t k W U m,l = Re න = Re න k=1 k=1 T 0 = Re k=1 n=1 T r l 0 ( n=1 c n c k න 0 +Re[σ k=1 c T k zw 0 t sǁ m,l t c k s m,l ((t k W ))dt (normalization), we have c n s 1,l t n W + z W(t)) c k sǁ m,l (t k W )dt T s 1,l t n W ǁ t k W s m,l dt] (t k W )dt 91

92 RAKE(2) The transmitted signal is orthogonal to the shifted counterparts of all signals, including itself. T z k = zw 0 t sǁ m,l because sǁ m,l t k W hence,with α k = c k U m,l = Re =σ k=1 k=1 t k W k=1 c k 2 න 0 dt complex Gaussian with E z k 2 = 2N 0 k=1 T orthonormal. s 1,l t k W sǁ m,l (t k W )dt + Re[ k=1 α 2 k Re s 1,l t k, s W m,l ((t k )) +σ W k=1 α k n k,l z k c k ] where n k,l = Re[e j c kz k ] k=1 i.i.d. Gaussian with E n 2 k,l = N 0 T Under T T m, 0 ( s1,l t k sǁ W m,l ((t k ))dt is almost independent of k; so, W s 1,l t k W, s m,l ((t k W )) s 1,l(t), s m,l (t). 92

93 RAKE(3) Therefore, the performance of RAKE is the same as the -diversity maximal ratio combiner if α k k=1 i.i.d. However, α k = c k k=1 may not be identically distributed. In such case, we can still obtain the pdf of γ b = σ k=1 γ k from characteristic function of γ k = Ψ k jυ = 1 1 jυγ k 1 characteristic function of γ b = γ k = Ψ k jυ = 1 jυγ k The pdf of γ is then given by the Fourier transform of characteristic function: f γ = σ k=1 Where π k = ς i=1,i k π k γ k e γ/γ k k=1 k=1 γ k γ k γ i, provided γ k ഥγ i for k i k=1 93

94 BPSK: P e = BFSK: U 1,l U 2,l k=1 U 1,l k=1 U 2,l 1 2 π k (1 k=1 1 2 π k (1 k=1 BPSK and BFSK α 2 k Re s 1,l t, s 1,l t + α k n k,l = k=1 α 2 k Re s 1,l t, s 2,l t + α k n k,l = k=1 k=1 Hint: E n k,l 2 = N 0 k=1 2 α k k=1 k=1 α 2 k Re s 1,l t, s 1,l t + α k n k,l = k=1 α 2 k Re s 1,l t, s 2,l t + α k n k,l = γ k ) γ k γ k ) γ k k=1 1 k=1 4γ k 1 k=1 2γ k α k 2 ( 2 α k k=1 k=1, BPSK, RAKE, FPSK, RAKE 2ε s + α k n k,l k=1 2ε s ) + α k n k,l k=1 2ε s + α k n k,l k=1 α 2 k (0) + α k n k,l k=1 94

95 Estimation of c k For orthogonal signaling, we can estimate c n via T න 0 =σ k=1 0 T z r l t + n W s 1,l t + + s M,l t dt c k 0 T sm,l t + n W k W s 1,l t + + s M,l t dt + t + n W s 1,l t + + s M,l t dt T =σ k=1 c k sm,l 0 t + n k s W W m,l t dt + 0 T z t + n W s 1,l t + + s M,l t dt (orthogonality) =c n 0 T sm,l (t) 2 dt + noise term (add-and-delay) 95

96 M=2 case 96

97 Decision-feedback estimator The previous estimator only works for orthogonal signaling. For, e.g., PAM signal with s l t = I. g t where I {±1, ±3,, ±(M 1)} We can estimate c n via T න 0 = න 0 r l t + n W g t dt T =σ k=1 ( k=1 c k. I. g t + n W k W + z t + n W )g t dt T n c k. I. 0 g t + k W W g t dt + noise term =c k. I 0 T g t 2 dt + noise term (add-and-delay) 97

98 Usually it requires ( t) c. T Final notes > 100 in order to have an accurate estimate of {c n } n=1 Note that for DPSK and FSK with square-law combiner, it is unnecessary to estimate {c n } n=1. So, they have no further performance loss (due to an inaccurate estimate of {c n } n=1 ). Statistical model of (US-WSS) (linear) multipath fading channels: c l τ; t = c τ; t e i2πf cτ and c τ; t = c l τ; t Multipath intensity profile or delay power spectrum R cl τ = R cl τ; t = 0. Multipath delay spread Tm vs coherent bandwidth ( f )c 98

99 Final notes (2) Frequency-selective vs frequency-nonselective Spaced-frequency, spaced-time correlation function R cl f; t = E{C l f + f; t + t C l f; t } Doppler power spectrum S cl λ = Rcl f = 0; t e i2πλ t d( t) Doppler spread Bd vs coherent time ( t) c Slow fading versus fast fading Scattering function (Good to know) Jakes model S cl τ; λ = F t {R cl τ; t } Rayleigh, Rice and Nakagami-m, Rummler s 3-path model 99

100 Deep fading phenomenon Final notes (3) B d T m spread factor: Underspread vs overspread Analysis of error rate under frequency-nonselective, slowly Rayleigh- and Nakagami-m-distributed fading channels ( diversity under Rayleigh) with M = 2. (Good to know) Analysis of the error rate with M > 2. Rake receiver under frequency-selective, slowly fading channels Assumption: Bandlimited signal with ideal lowpass filter and perfect channel estimator at the receiver This assumption results in a (finite-length) tapped-delay-line channel model under a finite delay spread Error analysis under add-and-delay assumption on the transmitted signals 100