Notes Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH Lecture 6: Modeling of MIMO Channels Theoretical Foundations of Wireless Communications 1 Lars Kildehøj CommTh/EES/KTH Wednesday, May 11, 2016 9:00-12:00, Conference Room SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 1 Overview Notes Lecture 6 Modeling of MIMO Channels Lecture 5: Spatial Diversity, MIMO Capacity Lars Kildehøj • SIMO, MISO, MIMO CommTh/EES/KTH • Degrees of freedom • MIMO capacity Lecture 6: MIMO Channel Modeling 2 / 1
Overview Notes Lecture 6 Modeling of MIMO Channels Lars Kildehøj Motivation: CommTh/EES/KTH • 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 Notes – Line-of-Sight Channels: SIMO • Free space without scattering and reflections. Lecture 6 • Antenna separation ∆ r λ c , with Modeling of MIMO Channels carrier wavelength λ c and the Lars Kildehøj normalized antenna separation CommTh/EES/KTH ∆ r ; n r receive antennas. (D. Tse and P. Viswanath, Fundamentals of Wireless Communi- • Distance between transmitter cations .) 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 ): � � � � − j 2 π f c d i − j 2 π d i h i = a · exp = a · exp λ c c • SIMO model: y = h · x + w , with w ∼ CN (0 , N 0 I ) → h : signal direction, spatial signature. 4 / 1
Physical Modeling Notes – Line-of-Sight Channels: SIMO • Paths are approx. parallel, i.e., Lecture 6 d i ≈ d + ( i − 1)∆ r λ c cos( φ ) Modeling of MIMO Channels • Directional cosine Lars Kildehøj CommTh/EES/KTH Ω = cos( φ ) (D. Tse and P. Viswanath, Fundamentals of Wireless Communi- cations .) • Spatial signature can be expressed as 1 exp( − j 2 π ∆ r Ω) � � − j 2 π d exp( − j 2 π 2∆ r Ω) h = a · exp λ c . . . exp( − j 2 π ( n r − 1)∆ r Ω) → Phased-array antenna. • SIMO capacity (with MRC) � � � � 1 + P � h � 2 1 + Pa 2 n r C = log = log N 0 N 0 → Only power gain, no degree-of-freedom gain. 5 / 1 Physical Modeling Notes – Line-of-Sight Channels: MISO • Similar to the SIMO case: ∆ t , λ c , d i , φ , Ω,... Lecture 6 Modeling of MIMO • MISO channel model: Channels Lars Kildehøj y = h ∗ x + w , CommTh/EES/KTH with w ∼ CN (0 , N 0 ). (D. Tse and P. Viswanath, Fundamentals of Wireless Communi- cations .) • Channel vector 1 exp( − j 2 π ∆ t Ω) � � j 2 π d exp( − j 2 π 2∆ t Ω) h = a · exp λ c . . . exp( − j 2 π ( n t − 1)∆ t Ω) • Unit spatial signature in the directional cosine Ω: e (Ω) = 1 / √ n · [1 , exp( − j 2 π ∆Ω) , . . . , exp( − j 2 π ( n − 1)∆Ω)] T → e t (Ω t ) and e r (Ω r ) with n t , ∆ t and n r , ∆ r , respectively. 6 / 1
Physical Modeling Notes – Line-of-Sight Channels: MIMO • Linear transmit and receive array with n t , ∆ t and n r , ∆ r . Lecture 6 • Gain between transmit antenna k and receive antenna i Modeling of MIMO Channels Lars Kildehøj h ik = a · exp ( − j 2 π d ik /λ c ) CommTh/EES/KTH • Distance between transmit antenna k and receive antenna i d ik = d + ( i − 1)∆ r λ c cos( φ r ) − ( k − 1)∆ t λ c cos( φ t ) • MIMO channel matrix (with Ω t = cos( φ t ) and Ω r = cos( φ r )) � � H = a √ n t n r exp − j 2 π d e r (Ω r ) e t (Ω t ) ∗ λ c → H is a rank-1 matrix with singular value λ 1 = a √ n t n r → Compare with SVD decomposition in Lecture 5: k � λ i u i v ∗ H = i i =1 7 / 1 Physical Modeling Notes – Line-of-Sight Channels: MIMO Lecture 6 Modeling of MIMO Channels • MIMO capacity � � 1 + Pa 2 n r n t Lars Kildehøj CommTh/EES/KTH C = log 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
Physical Modeling Notes – Geographically Separated Antennas at the Transmitter Example/special case Lecture 6 • 2 distributed transmit antennas, Modeling of MIMO Channels attenuations a 1 , a 2 , angles of incidence Lars Kildehøj φ r 1 , φ r 2 , negligible delay spread. CommTh/EES/KTH • Spatial signature ( n r receive antennas) � � √ n r exp − j 2 π d 1 k h k = a k e r (Ω rk ) λ c (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) • Channel matrix H = [ h 1 , h 2 ] • H has independent columns as long as (Ω r i = cos( φ r i )) 1 Ω r = Ω r 2 − Ω r 1 � = 0 mod ∆ 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 Notes – Geographically Separated Antennas at the Transmitter • Conditioning of H is determined by how the spatial signatures are aligned (with L r = n r ∆ r ): Lecture 6 Modeling of MIMO Channels � � sin( π L r Ω r ) � � | cos( θ ) | = | f r (Ω r 2 − Ω r 1 ) | = | e r (Ω r 1 ) ∗ e r (Ω r 2 ) Lars Kildehøj | = � � CommTh/EES/KTH n r sin( π L r Ω r / n r ) � � � �� � = f r (Ω r 2 − Ω r 1 ) • Example ( a 1 = a 2 = a ) � λ 2 a 2 n r (1 + | cos( θ ) | ) λ 1 1 + | cos( θ ) | = 1 ⇒ = λ 2 a 2 n r (1 − | cos( θ ) | ) = λ 2 1 − | cos( θ ) | 2 • 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 (D. Tse and P. Viswanath, Fundamentals of Wireless Commu- be resolved. nications .) 10 / 1
Physical Modeling Notes – Geographically Separated Antennas at the Transmitter Beamforming pattern Lecture 6 Modeling of MIMO • Assumption: signal arrives with Channels angle φ 0 ; receive beamforming Lars Kildehøj CommTh/EES/KTH vector e r (cos( φ 0 )). • A signal form any other direction φ will be attenuated by a factor | e r (cos( φ 0 )) ∗ e r (cos( φ )) | = | f r (cos( φ ) − cos( φ 0 )) | • Beamforming pattern (D. Tse and P. Viswanath, Fundamentals of Wireless ( φ, | f r (cos( φ ) − cos( φ 0 )) | ) Communications .) • 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 Notes – LOS Plus One Reflected Path • Direct path: Lecture 6 φ t 1 , Ω r 1 , d (1) , and a 1 . Modeling of MIMO Channels Lars Kildehøj • Reflected path: CommTh/EES/KTH φ t 2 , Ω r 2 , d (2) , and a 2 . (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) • Channel model follows from signal superposition 1 e r (Ω r 1 ) e t (Ω t 1 ) ∗ + a b H = a b 2 e r (Ω r 2 ) e t (Ω t 2 ) ∗ , with � � − j 2 π d ( i ) √ n t n r exp a b i = a i . λ c → H has rank 2 as long as 1 1 Ω t 1 � = Ω t 2 mod and Ω r 1 � = Ω r 2 mod ∆ r . ∆ t → 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
Physical Modeling Notes – LOS Plus One Reflected Path • Direct path: Lecture 6 φ t 1 , Ω r 1 , d (1) , and a 1 . Modeling of MIMO Channels Lars Kildehøj • Reflected path: CommTh/EES/KTH φ t 2 , Ω r 2 , d (2) , and a 2 . (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) • H can be rewritten as H = H ′′ H ′ , with � e t � ∗ (Ω t 1 ) H ′′ = [ a b H ′ = 1 e r (Ω r 1 ) , a b 2 e r (Ω r 2 )] and ∗ (Ω t 2 ) e t → 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 Notes – LOS Plus One Reflected Path Lecture 6 Modeling of MIMO Channels Lars Kildehøj CommTh/EES/KTH (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) • 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
Recommend
More recommend
