ECE6604 PERSONAL & MOBILE COMMUNICATIONS. Week 4. Envelope Correlation Space-time Correlation

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1 ECE6604 PERSONAL & MOBILE COMMUNICATIONS Week 4 Envelope Correlation Space-time Correlation 1

2 Autocorrelation of a Bandpass Random Process Consider again the received band-pass random process r(t) = g I (t)cos2πf c t g Q (t)sin2πf c t where g I (t) = N n=1 g Q (t) = N n=1 C n cosφ n (t) C n sinφ n (t) Assuming that r(t) is wide-sense stationary, the autocorrelation of r(t) is φ rr (τ) = E[r(t)r(t+τ)] = E[g I (t)g I (t+τ)]cos2πf c τ +E[g Q (t)g I (t+τ)]sin2πf c τ = φ gi g I (τ)cos2πf c τ φ gi g Q (τ)sin2πf c τ where E[ ] is the ensemble average operator, and φ gi g I (τ) = E[g I (t)g I (t+τ)] φ gi g Q (τ) = E[g I (t)g Q (t+τ)]. Note that the wide-sense stationarity of r(t) imposes the condition φ gi g I (τ) = φ gq g Q (τ) φ gi g Q (τ) = φ gq g I (τ). 2

3 Auto- and Cross-correlation of Quadrature Components The phases φ n (t) are statistically independent random variables at any time t, uniformly distributed over the interval [ π, π). The azimuth angles of arrival, θ n are all independent due to the random placement of scatterers. Also, in the limit N, the discrete azimuth angles of arrival θ n can be replaced by a continuous random variable θ having the probability density function p(θ). By using the above properties, the auto- and cross-correlation functions can be obtained as follows: φ gi g I (τ) = φ gq g Q (τ) = lim N E τ,θ,φ [g I(t)g I (t+τ)] = Ω p 2 E θ[cos(2πf m τ cosθ)] φ gi g Q (τ) = φ gq g I (τ) = lim N E τ,θ,φ [g I(t)g Q (t+τ)] = Ω p 2 E θ[sin(2πf m τ cosθ)] τ = (τ 1,τ 2,...,τ N ) θ = (θ 1,θ 2,...,θ N ) φ = (φ 1,φ 2,...,φ N ) and Ω p is the total received envelope power. Ω p = E[g 2 I (t)]+e[g2 Q (t)] = N Cn 2 n=1 3

4 2-D Isotropic Scattering Evaluation of the expectations for the auto- and cross-correlation functions requires the azimuth distribution of arriving plane waves p(θ), and the receiver antenna gain pattern G(θ), as a function of the azimuth angle θ. With 2-D isotropic scattering, the plane waves are confined to the x y plane and arrive uniformly distributed angle of incidence, i.e., p(θ) = 1 2π, π θ π With 2-D isotropic scattering and an isotropic receiver antenna with gain G(θ) = 1,θ [ π, π), the auto- and cross-correlation functions become φ gi g I (τ) = = Ω p 2 J 0(2πf m τ) φ gi g Q (τ) = 0 where J 0 (x) = 1 π π cos(xcosθ)dθ 0 is the zero-order Bessel function of the first kind. 4

5 Autocorrelation, φ ri r I (τ)/(ω p /2) Time Delay, f m τ Normalized autocorrelation function of the quadrature components of the received complex envelope with 2-D isotropic scattering and an isotropic receiver antenna. 5

6 Doppler Spectrum The autocorrelation function and power spectral density (psd) are Fourier transform pairs. S gg (f) = φ gg(τ)e j2πfτ dτ φ gg (τ) = S gg(f)e j2πfτ df The autocorrelation of the received complex envelope g(t) = g I (t)+jg Q (t) is φ gg (τ) = 1 2 E[g (t)g(t+τ)] = φ gi g I (τ)+jφ gi g Q (τ) The Fourier transform of φ gg (τ) gives the Doppler psd S gg (f) = S gi g I (f)+js gi g Q (f). Sometimes S gg (f) is just called the Doppler spectrum. 6

7 Bandpass Doppler Spectrum We can also relate the power spectrum of the complex envelope g(t) to that of the band-pass process r(t). We have φ rr (τ) = Re [ φ gg (τ)e j2πf cτ ]. By using the identity Re[z] = z +z 2 and the property φ gg (τ) = φ gg( τ), it follows that the band-pass Doppler psd is S rr (f) = 1 2 [S gg(f f c )+S gg ( f f c )]. Since φ gg (τ) = φ gg( τ), the Doppler spectrum S gg (f) is always a real-valued function of frequency, but not necessarily even. However, the band-pass Doppler spectrum S rr (f) is always real-valued and even. 7

8 Isotropic Scattering For 2-D isotropic scattering, the psd and cross psd of g I (t) and g Q (t) are S gi g I (f) = F[φ gi g I (τ)] = S gi g Q (f) = 0 Ω p 1 2πf m 1 ( fm f )2, f f m 0, otherwise 10.0 Power Spectrum, S gi g I (f)/(ω p /2πf m ) (db) Normalized Doppler Frequency, f/f m Normalized psd of the quadrature components of the received complex envelope with 2-D isotropic scattering and an isotropic receiver antenna. Sometimes this is called the CLASSICAL Doppler power spectrum. 8

9 Non-isotropic Scattering Rician Fading Suppose that the propagation environment consisting of a strong specular component plus a scatter component. The azimuth distribution p(θ) might have the form p(θ) = 1 K +1ˆp(θ)+ K K +1 δ(θ θ 0) where ˆp(θ) is the continuous AoA distribution of the scatter component, θ 0 is the AoA of the specular component, and K is the ratio of the received specular to scattered power. One such scattering environment, assumes that the scatter component exhibits 2-D isotropic scattering, i.e., ˆp(θ) = 1/(2π),θ [ π,π). The correlation functions φ gi g I (τ) and φ gi g Q (τ) are φ gi g I (τ)= 1 Ω p K +1 φ gi g Q (τ)= K K +1 2 J 0(2πf m τ)+ K K +1 Ω p 2 sin(2πf mτ cosθ 0 ). Ω p 2 cos(2πf mτ cosθ 0 ) 9

10 p(θ) K=1 θ 0 = π/ π π Arrival Angle, θ Plot of p(θ) vs. θ with 2-D isotropic scattering plus a LoS or specular component arriving at angle θ 0 = π/2. 10

11 The azimuth distribution p(θ) = 1 K +1ˆp(θ)+ K K +1 δ(θ θ 0) yields a complex envelope having a Doppler spectrum of the form S gg (f) = 1 K +1 Sc gg(f)+ K K +1 Sd gg(f) (1) where Sgg d (f) is the discrete portion of the Doppler spectrum due to the specular component and Sgg(f) c is the continuous portion of the Doppler spectrum due to the scatter component. For the case when ˆp(θ) = 1/(2π),θ [ π,π], the power spectrum of g(t) = g I (t)+jg Q (t) is S gg (f) = 1 K+1 Ω p 2πf m 1 + K K+1 1 (f/fm ) 2 Ω p 2 δ(f f mcosθ 0 ) 0 f f m 0 otherwise Note the discrete tone at frequency f c +f m cosθ 0 due to the line-of-sight or specular component arriving from angle θ

12 Non-isotropic scattering Other Cases Sometimes the azimuth distribution p(θ) may not be uniform, a condition commonly called non-isotropic scattering. Several distributions have been suggested to model non-isotropic scattering. Once possibility is the Gaussian distribution p(θ) = 1 exp 2πσS (θ µ)2 2σ 2 S where µ is the mean AoA, and σ S is the rms AoA spread. Another possibility is the von Mises distribution 1 p(θ) = 2πI 0 (k) exp[kcos(θ µ)], where θ [ π,π), I 0 ( ) is the zeroth-order modified Bessel function of the first kind, µ [ π,π) is the mean AoA, and k controls the spread of scatterers around the mean. 12

13 k=2 k=5 k= p(θ) π π 4 Arrival Angle, θ Plot of p(θ) vs. θ for the von Mises distribution with a mean angle-to-arrival µ = π/2. 13

14 Calculating the Doppler Spectrum The Doppler spectrum can be derived by using a different approach that is sometimes very useful because it can avoid the need to evaluate integrals. The Doppler spectrum can be expressed as S gg (f) df = Ω p (G(θ)p(θ) + G( θ)p( θ)) dθ. 2 The Doppler frequency associated with the incident plane wave arriving at angle θ is f = f m cos(θ), and, hence, Therefore, where df = f m sin(θ)dθ = f 2 m f 2 dθ. S gg (f) = Ω p/2 f 2 m f2(g(θ)p(θ) + G( θ)p( θ)), θ = cos 1 (f/f m ). Hence, if p(θ) and G(θ) are known, the Doppler spectrum can be easily calculated. For example, with 2-D isotropic scattering and an isotropic antenna G(θ)p(θ) = 1/(2π). 14

15 Space-time Correlation Functions Many mobile radio systems use antenna diversity, where spatially separated receiver antennas provide multiple faded replicas of the same information bearing signal. The spatial decorrelation of the channel tell us the required spatial separation between antenna elements so that they will be sufficiently decorrelated. Sometimes it is desirable to simultaneously characterize both the spatial and temporal correlation characteristics of the channel, e,g, when using space-time coding. This can be described by the space-time correlation function. To obtain the spatial or space-time correlation functions, we must specify some kind of radio scattering geometry. 15

16 Spatial Correlation at the Mobile Station ( n) S M n n2 n1 ( n ) M (1) A M M M v O B (2) A M O M M D R Single-ring scattering model for NLoS propagation on the forward link of a cellular system. The MS is surrounded by a scattering ring of radius R and is at distance D from the BS, where R D. 16

17 Model Parameters O B : base station location O M : mobile station location D: LoS distance from base station to mobile station R: scattering radius γ M : mobile station moving direction w.r.t x-axis v: mobile station speed θ M mobile station array orientation w.r.t. x-axis A (i) M: location of ith mobile station antenna element δ M : distance between mobile station antenna elements S (n) M : location of nth scatterer. α (n) M : angle of arrival from the nth scatterer. ǫ n : distance O B S (n) M. ǫ ni : distance S (n) M A (i) M. 17

18 Received Complex Envelope The channel from O B to A (q) M has the complex envelope g q (t) = N n=1 C n e jφ n j2π(ǫ n +ǫ nq )/λ c e j2πf mtcos(α (n) M γ M), q = 1,2 where ǫ n and ǫ nq denote the distances O B S (n) M and S (n) M A (q) M, q = 1,2, respectively, and φ n is a uniform random phase on the interval [ π,π). From the Law of Cosines, the distances ǫ n and ǫ nq can be expressed as a function of the angle-of-arrival α (n) M as follows: ǫ 2 n = D 2 +R 2 +2DRcosα (n) M Note sign change since the angle is π α (n) M ǫ 2 nq = [(1.5 q)δ M ] 2 +R 2 2(1.5 q)δ M Rcos(α (n) M θ M ),q = 1,2. Assuming that R/D 1 (local scattering), δ M R and 1±x 1±x/2 for small x, we have Hence, ǫ n D +Rcosα (n) M ǫ nq R (1.5 q)δ M cos(α (n) M θ M ),q = 1,2. g q (t) = N n=1 ( C n e jφ n j2π D+Rcosα (n) M +R (1.5 q)δ M cos(α (n) ) M θ M) /λ c e j2πf mtcos(α (n) M γ M),q = 1,2. 18

19 Space-time Correlation Function Thespace-timecorrelationfunctionbetween thetwo complex fadedenvelopes g 1 (t)andg 2 (t) is φ g1,g 2 (δ M,τ) = 1 2 E[g 1(t)g 2 (t+τ)] The space-time correlation function between g 1 (t) and g 2 (t) can be written as φ g1,g 2 (δ M,τ) = Ω p 2N N n=1 E [ e j2π(δ M/λ c )cos(α (n) M θ M) e j2πf mτ cos(α (n) M γ M) ]. Since the number of scatters is infinite, the discrete angles-of-arrival α (n) M can be replaced with a continuous random variable α M with probability density function p(α M ). Hence, the space-time correlation function becomes φ g1,g 2 (δ M,τ) = Ω p 2 2π where a = 2πf m τ and b = 2πδ M /λ c. 0 e jbcos(α M θ M ) e jacos(α M γ M ) p(α M )dα M. 19

20 2-D Isotropic Scattering Forthecaseof2-Disotropicscatteringwithanisotropicreceiveantennas,p(α M ) = 1/(2π), π α M π, and the space-time correlation function becomes φ g1,g 2 (δ M,τ) = Ω ( ) p 2 J 0 a 2 +b 2 2abcos(θ M γ M ) Thespatialandtemporalcorrelationfunctionscanbeobtainedbysettingτ = 0andδ M = 0, respectively. This gives Finally, we note that φ g1,g 2 (δ M ) = φ g1,g 2 (δ M,0) = Ω p 2 J 0(2πδ M /λ c ) φ gg (τ) = φ g1,g 2 (0,τ) = Ω p 2 J 0(2πf m τ) f m τ = v τ = δ M λ c λ c For this scattering environment, the normalized time f m τ is equivalent to the normalized distance δ M /λ c. The antenna branches are uncorrelated if they are separated by δ M 0.5λ c.. 20

21 Autocorrelation, φ ri r I (τ)/(ω p /2) Time Delay, f m τ Temporal and spatial correlation functions at the MS with 2-D isotropic scattering and an isotropic receiver antenna. Note that f m τ = δ M /λ c. 21

22 Spatial Correlation at the Base Station v ( m) S M m1 (1) A B ( m) M M m ( m) B B O M m2 O B B (2) A B R D Single-ring scattering model for NLoS propagation on the reverse link of a cellular system. The MS is surrounded by a scattering ring of radius R and is at distance D from the BS, where R D. 22

23 Model Parameters O B : base station location O M : mobile station location D: LoS distance from base station to mobile station R: scattering radius γ M : mobile station moving direction w.r.t x-axis v: mobile station speed θ B base station array orientation w.r.t. x-axis A (i) B : location of ith base station antenna element δ B : distance between mobile station antenna elements S (m) M : location of mth scatterer. α (m) M : angle of departure to the nth scatterer. ǫ m : distance S (m) M O B. ǫ mi : distance S (m) M A (i) B. 23

24 Received Complex Envelope The channel from O M to A (q) B has the complex envelope g q (t) = N m=1 C m e jφ m j2π(r+ǫ mq )/λ c e j2πf mtcos(α (m) M γ M), q = 1,2 (2) where ǫ mq denote the distance S (m) M A (q) B, q = 1,2, and φ m is a uniform random phase on ( π,π]. To proceed further, we need to express ǫ mq as a function of α (m) ApplyingtheLawofCosinestothetriangle S (m) M O B A (q) B, thedistanceǫ mq canbeexpressed as a function of the angle θ (m) B θ B as follows: ǫ 2 mq = [(1.5 q)δ B ] 2 +ǫ 2 m 2(1.5 q)δ B ǫ m cos(θ (m) B θ B ),q = 1,2. (3) where ǫ m is the distance S (m) M O B. By applying the Law of Sines to the triangle O M S (m) M O B we obtain following identity ǫ m sinα (m) M = sin ( R π θ (m) B D ) = ( sin π α (m) ( M π θ (m) B M. )). 24

25 Received Complex Envelope Since the angle π θ (m) B is small, we can apply the small angle approximations sinx x and cosx 1 for small x, to the second equality in the above identity. This gives or R (π θ (m) B ) sin It follows that the cosine term in (2) becomes ( D π α (m) M (π θ (m) B ) (R/D)sin(π α (m) M ). cos(θ (m) B θ B ) = cos(π θ B (π θ (m) B )) = cos(π θ B )cos(π θ (m) B )+sin(π θ B )sin(π θ (m) B ) cos(π θ B )+sin(π θ B )(R/D)sin(π α (m) M ) = cos(θ B )+(R/D)sin(θ B )sin(α (m) M ) (4) Using the approximation in (4) in (2), along with δ B /ǫ m 1, gives ǫ 2 mq ǫ2 m 1 2(1.5 q) δ B ǫ m ) [ (R/D)sin(θ B )sin(α (m) M ) cos(θ B )]. 25

26 Received Complex Envelope Applying the approximation 1±x 1±x/2 for small x, we have [ ǫ mq ǫ m (1.5 q)δ B (R/D)sin(θ B )sin(α (m) M ) cos(θ B ) ]. (5) Applying the Law of Cosines to the triangle O M S (m) M O B we have ǫ 2 m = D2 +R 2 2DRcos(α (m) M ) ] D 2 [ 1 2(R/D)cos(α (m) M ) and again using the approximation 1±x 1±x/2 for small x, we have ǫ m D Rcos(α (m) M ) (6), Finally, using (5) in (4) gives ǫ mq D Rcos(α (m) M ) (1.5 q)δ B [ (R/D)sin(θ B )sin(α (m) M ) cos(θ B ) ]. (7) Substituting (7) into (2) gives the result g q (t) = N C m e jφ m+j2πf m tcos(α (m) M γ M) m=1 ( e j2π R+D Rcos(α (m) M ) (1.5 q)δ B [ (R/D)sin(θ B )sin(α (m) ]) M ) cos(θ B) /λ c, which no longer depends on the ǫ mq and is a function of the angle of departure α (m) M. (8) 26

27 Space-time Correlation Function Thespace-timecorrelationfunctionbetween thetwo complex fadedenvelopes g 1 (t)andg 2 (t) at the BS is once again given by φ g1,g 2 (δ B,τ) = 1 2 E[g 1(t)g 2 (t+τ)] Using (8), the space-time correlation function between g 1 (t) and g 2 (t) can be written as φ g1,g 2 (δ B,τ) = Ω p 2N N m=1 E [ [ e j2π(δ B/λ c ) (R/D)sin(θ B )sin(α (m) ] M ) cos(θ B) e j2πf mτ cos(α (m) M γ M) ]. Since the number of scatters around the MS is infinite, the discrete angles-of-departure α (m) M can be replaced with a continuous random variable α M with probability density function p(α M ). Hence, the space-time correlation function becomes. φ g1,g 2 (δ B,τ) = Ω p 2 π where a = 2πf m τ and b = 2πδ B /λ c. π e jacos(α M γ M ) e jb[(r/d)sin(θ B)sin(α M ) cos(θ B )] p(α M )dα M, 27

28 2-D Isotropic Scattering For the case of 2-D isotropic scattering with an isotropic MS transmit antenna, p(α M ) = 1/(2π), π α M π, and the space-time correlation function becomes φ g1,g 2 (δ B,τ) = Ω p 2 e jbcos(θ B) J 0 ( a 2 +b 2 (R/D) 2 sin 2 (θ B ) 2ab(R/D)sin(θ B )sin(γ M ) ). The spatial and temporal correlation functions can be obtained by setting τ = 0 and and δ B = 0, respectively. The temporal correlation function φ gg (τ) = φ g1,g 2 (0,τ) = Ω p 2 J 0(2πf m τ) which matches our result for the received signal at a mobile station. The spatial correlation function is φ g1,g 2 (δ B ) = φ g1,g 2 (δ B,0) = Ω p 2 e jbcos(θb) J 0 (b(r/d)sin(θ B )) = Ω p 2 e jbcos(θb) ( ) J 0 (2πδ B /λ c )(R/D)sin(θ B ) Observe that a much greater spatial separation is required to achieve a given degree of envelope decorrelation at the BS as compared to the MS. This can be readily seen by the term R/D 1 in the argument of the Bessel function. 28

29 Normalized Spatial Crosscovariane, φ g1 g 2 (δ B ) φ B =π/2 φ B =π/3 φ B =π/ BS Antenna Separation, δ B /λ Envelope crosscorrelation magnitude at the base station for different base station antenna orientation angles, θ B ; D = 3000 m, R = 60 m. Broadside base station antennas have the lowest crosscorrelation. 29

30 Normalized Spatial Crosscovariance, φ g1,g 2 (δ B ) R=30 R=60 R= BS Antenna Separation, δ B /λ c Envelope crosscorrelation magnitude at the base station for θ B = π/3 and various scattering radii, R; D = 3000 m. Smaller scattering radii will result in larger a crosscorrelations. 30