Lecture 6: Modeling of MIMO Channels Theoretical Foundations of Wireless Communications 1. Overview. CommTh/EES/KTH

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1 : Theoretical Foundations of Wireless Communications 1 Wednesday, May 11, :00-12:00, Conference Room SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 1 Overview Lecture 5: Spatial Diversity, MIMO Capacity SIMO, MISO, MIMO Degrees of freedom MIMO capacity : MIMO Channel Modeling 2 / 1

2 Overview Motivation: How does the multiplexing capability of MIMO channels depend on the physical environment? When can we gain (much from MIMO? How do we have to design the system? 3 / 1 Physical Modeling Line-of-Sight : SIMO Free space without scattering and reflections. Antenna separation r λ c, with carrier wavelength λ c and the normalized antenna separation r ; n r receive antennas. Distance between transmitter and i-th receive antenna: d i Continuous-time impulse between transmitter and i-th receive antenna: h i (τ = a δ(τ d i /c Base-band model (assuming d i /c 1/W, signal BW W : ( h i = a exp j 2πfcd ( i = a exp j 2πd i c λ c SIMO model: y = h x + w, with w CN (0, N 0I h: signal direction, spatial signature. 4 / 1

3 Physical Modeling Line-of-Sight : SIMO Paths are approx. parallel, i.e., d i d + (i 1 r λ c cos(φ Directional cosine Ω = cos(φ Spatial signature can be expressed as ( h = a exp j 2πd λ c Phased-array antenna. SIMO capacity (with MRC C = log (1 + P h 2 N 0 1 exp( j2π r Ω exp( j2π2 r Ω. exp( j2π(n r 1 r Ω ( = log 1 + Pa2 n r N 0 Only power gain, no degree-of-freedom gain. 5 / 1 Physical Modeling Line-of-Sight : MISO Similar to the SIMO case: t, λ c, d i, φ, Ω,... MISO channel model: y = h x + w, with w CN (0, N 0. Channel vector ( h = a exp j 2πd λ c 1 exp( j2π tω exp( j2π2 tω. exp( j2π(n t 1 tω Unit spatial signature in the directional cosine Ω: e(ω = 1/ n [1, exp( j2π Ω,..., exp( j2π(n 1 Ω] T e t(ω t and e r(ω r with n t, t and n r, r, respectively. 6 / 1

4 Physical Modeling Line-of-Sight : MIMO Linear transmit and receive array with n t, t and n r, r. Gain between transmit antenna k and receive antenna i h ik = a exp ( j2πd ik /λ c Distance between transmit antenna k and receive antenna i d ik = d + (i 1 r λ c cos(φ r (k 1 tλ c cos(φ t MIMO channel matrix (with Ω t = cos(φ t and Ω r = cos(φ r H = a ( n tn r exp j 2πd e r(ω r e t(ω t λ c H is a rank-1 matrix with singular value λ 1 = a n tn r Compare with SVD decomposition in Lecture 5: H = k λ i u i vi i=1 7 / 1 Physical Modeling Line-of-Sight : MIMO MIMO capacity ( C = log 1 + Pa2 n r n t N 0 Only power gain, no degree-of-freedom gain. n t = 1: power gain equals n r receive beamforming. n r = 1: power gain equals n t transmit beamforming. General n t, n r : power gain equals n r n t Transmit and receive beamforming. Conclusion: In LOS environment, MIMO provides only a power gain but no degree-of-freedom gain. 8 / 1

5 Physical Modeling Geographically Separated Antennas at the Transmitter Example/special case (D. Tse and P. Viswanath, Fundamentals of Wireless Communications. 2 distributed transmit antennas, attenuations a 1, a 2, angles of incidence φ r 1, φr 2, negligible delay spread. Spatial signature (n r receive antennas ( h k = a k nr exp j 2πd 1k e r(ω rk λ c Channel matrix H = [h 1, h 2] H has independent columns as long as (Ω r i = cos(φ r i Ω r = Ω r 2 Ω r 1 0 mod 1 r Two non-zero singular values λ 2 1, λ 2 2; i.e., two degrees of freedom. But H can still be ill-conditioned! 9 / 1 Physical Modeling Geographically Separated Antennas at the Transmitter Conditioning of H is determined by how the spatial signatures are aligned (with L r = n r r : cos(θ = f r (Ω r2 Ω r1 = e r(ω r1 e r(ω r2 = sin(πl r Ω r }{{} n r sin(πl r Ω r /n r =f r (Ω r2 Ω r1 Example (a 1 = a 2 = a λ 2 1 = a 2 n r (1 + cos(θ λ 2 2 = a 2 n r (1 cos(θ λ 1 λ 2 = 1 + cos(θ 1 cos(θ f r (Ω r is periodic with n r /L r. Maximum at Ω r = 0; f r (0 = 1. f r (Ω r = 0 at Ω r = k/l r with k = 1,..., n r 1. Resolvability 1/L r, if Ω r 1/L r, then the signals from the two antennas cannot be resolved. 10 / 1

6 Physical Modeling Geographically Separated Antennas at the Transmitter Beamforming pattern Assumption: signal arrives with angle φ 0; receive beamforming vector e r(cos(φ 0. A signal form any other direction φ will be attenuated by a factor (D. Tse and P. Viswanath, Fundamentals of Wireless Communications. e r(cos(φ 0 e r(cos(φ = f r (cos(φ cos(φ 0 Beamforming pattern ( φ, f r (cos(φ cos(φ 0 Main lobes around φ 0 and any angle φ for which cos(φ = cos(φ 0. In a similar way, separated receive antennas can be treated. 11 / 1 Physical Modeling LOS Plus One Reflected Path Direct path: φ t1, Ω r1, d (1, and a 1. Reflected path: φ t2, Ω r2, d (2, and a 2. Channel model follows from signal superposition with H = a b 1e r(ω r1e t(ω t1 + a b 2e r(ω r2e t(ω t2, a b i H has rank 2 as long as = a i ntn r exp ( j (i 2πd. λ c Ω t1 Ω t2 mod 1 t and Ω r1 Ω r2 mod 1 r. H is well conditioned if the angular separations Ω t, Ω r at the transmit/receive array are of the same order or larger than 1/L t,r. 12 / 1

7 Physical Modeling LOS Plus One Reflected Path Direct path: φ t1, Ω r1, d (1, and a 1. Reflected path: φ t2, Ω r2, d (2, and a 2. H can be rewritten as H = H H, with H = [a b 1e r(ω r1, a b 2e r(ω r2] and H = [ et (Ω t1 e t (Ω t2 Two imaginary receivers at points A and B (virtual relays. Since the points A and B are geographically widely separated, H and H have rank 2 and hence H has rank 2 as well. Furthermore, if H and H are well-conditioned, H will be well-conditioned as well. Multipath fading can be viewed as an advantage which can be exploited! ] 13 / 1 Physical Modeling LOS Plus One Reflected Path Significant angular separation is required at both the transmitter and the receiver to obtain a well-conditioned matrix H. If the reflectors are close to the receiver (downlink, we have a small angular separation not very well-conditioned matrix H. Similar, if the reflectors are close to the transmitter (uplink. Size of an antenna array at a base station will have to be many wavelengths to be able to exploit the spatial multiplexing effect. 14 / 1

8 Physical Modeling Summary 15 / 1 Fading General Concept Antenna lengths L t, L r limit the resolvability 2 of the transmit and receive antenna in the angular domain. Sample the angular domain at fixed angular spacings of 1/L t at the transmitter and 1/L r at the receiver. Represent the channel (the multiple paths in terms of these input and output coordinates. The (k, l-th channel gain follows as the aggregation of all paths whose transmit and receive directional cosines lie in a (1/L t 1/L r bin around the point (l/l t, k/l r. 2 Note, if Ω r,t 1/L r,t, the paths cannot be separated. 16 / 1

9 Fading Angular Domain Representation (ADR Orthonormal basis for the received signal space (n r basis vectors { S r = e r(0, e 1 r(,..., e r( nr 1 } L r L r Orthogonality follows directly from the properties of f r (Ω. Orthonormal basis for the transmitted signal space (n t basis vectors { S t = e t(0, e 1 t(,..., e t( nt 1 } L t L t Orthogonality follows directly from the properties of f t(ω. Orthonormal bases provide a very simple (but approximate decomposition of the total received/transmitted signal up to a resolution 1/L r, 1/L t. 17 / 1 Fading Angular Domain Representation Examples: Receive beamform patterns of the angular basis vectors in S r (a Critically spaced ( r = 1/2, each basis vector has a single pair of main lobes. (b Sparsely spaced ( r > 1/2, some of the basis vectors have more than one pair of main lobes. (c Densely spaced ( r < 1/2, some of the basis vectors have no pair of main lobes. 18 / 1

10 Fading ADR of MIMO (Assumption: critically spaced antennas Observation: The vectors in S t and S r form unitary matrices U t and U r with dimensions (n t n t and (n r n r, respectively. ( IDFT matrices! With 3 x a = U t x and y a = U r y we get y a = U r HU tx a + U r w = H a x a + w a, with w a CN (0, N 0I nr. Furthermore, with H = i ab i e r(ω ri e t(ω ti, we get h a kl = e r(k/l r H e t(l/l t = i ai b [e r(k/l r e r(ω ri ] [e t(ω ti e t(l/l t] }{{}}{{} (1 (2 The terms (1 and (2 are significant for the i th path if Ω ri k < 1 and L r L Ω ti k < 1. r L t L t ( Projections on the basis vectors in S r, S t. 3 The superscript a denotes angular domain quantities. 19 / 1 Fading Statistical Modeling in the Angular Domain Let T l and R k be the sets of physical paths which have most energy in directions of e t(l/l t and e r(k/l r. hkl a corresponds to the aggregated gains ai b of paths which lie in R k T l. Independence and time variation Gains of the physical paths ai b [m] are independent. The path gains hkl a [m] are independent across m. The angles {φ ri [m]} m and {φ ti [m]} m evolve slower than a b i [m]. The physical paths do not move from one angular bin to another. The path gains h kl [m] are independent across k and l. If there are many paths in an angular bin Central Limit Theorem h a kl[m] can be modeled as complex circular symmetric Gaussian. If there are no paths in an angular bin h a kl[m] 0. Since U t and U r are unitary matrices, the matrix H has the same i.i.d. Gaussian distribution as H a. 20 / 1

11 Fading Statistical Modeling in the Angular Domain Example for H a (a Small angular spread at the transmitter. (b Small angular spread at the receiver. (c Small angular spread at both transmitter and receiver. (d Full angular spread at both transmitter and receiver. 21 / 1 Fading Degrees of Freedom and Diversity Degrees of freedom Based on the derived statistical model, we get the following result: with probability 1, the rank of the random matrix H a is given by rank(h a = min{ number of non-zero rows, number of non-zero columns }. The number of non-zero rows and columns depends on two factors: Amount of scattering and reflection; the more scattering and reflection, the larger the number of non-zero entries in H a. Lengths L r and L t; for small L r, L t many physical paths are mapped into the same angular bin; with higher resolution, more paths can be represented. Diversity: The diversity is given by the number of non-zero entries in H a. Example: Same number of degrees of freedom but different diversity. 22 / 1

12 Fading Antenna Spacing So far: critically spaced antennas with r = 1/2. One-to-one correspondence between the angular windows and the resolvable bins. Setup 1: vary the number of antennas for a fixed array length L r,t. Sparsely spaced case ( r > 0.5 Beamforming patterns of some basis vectors have multiple main lobes. Different paths with different directions are mapped onto the same basis vector. Resolution of the antenna array, number of degrees of freedom, and diversity are reduced. Densely spaced case ( r < 0.5 There are basis vectors with no main lobes which do not contribute to the resolvability. Adds zero rows and columns to H a and creates correlation in H. Setup 2: vary the antenna separation for a fixed number of antennas. Rich scattering: number of non-zero rows in H a is already n r ; i.e., no improvement possible. Clustered scattering: scattered signal can be received in more bins; i.e., increasing number of degrees of freedom. 23 / 1