Lecture 5: Antenna Diversity and MIMO Capacity Theoretical - PowerPoint PPT Presentation

Notes Lecture 5 Antenna Diversity, MIMO Capacity Lars Kildehøj CommTh/EES/KTH Lecture 5: Antenna Diversity and MIMO Capacity Theoretical Foundations of Wireless Communications 1 Lars Kildehøj CommTh/EES/KTH Wednesday, May 4, 2016 9:00-12:00, Conference Room SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 1 Overview Notes Lecture 5 Antenna Diversity, Lecture 1-4: Channel capacity MIMO Capacity • Gaussian channels Lars Kildehøj CommTh/EES/KTH • Fading Gaussian channels • Multiuser Gaussian channels • Multiuser diversity Lecture 5: Antenna diversity and MIMO capacity 2 / 1

Diversity Notes Multiuser diversity (lecture 4) Lecture 5 Antenna Diversity, • Transmissions over independent fading channels. MIMO Capacity Lars Kildehøj • Sum capacity increases with the number of users. CommTh/EES/KTH → High probability that at least one user will have a strong channel. Fading channels (point-to-point links) • Use diversity to mitigate the effect of (deep) fading. • Diversity: let symbols pass through multiple paths. • Time diversity: interleaving and coding, repetition coding. • Frequency diversity: for example OFDM. • Antenna Diversity. 3 / 1 Antenna/Spatial Diversity Notes Motivation: For narrowband channels with large coherence time or delay constraints, time diversity and frequency diversity cannot be exploited! Lecture 5 Antenna Diversity, MIMO Capacity Lars Kildehøj CommTh/EES/KTH (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) Antenna diversity • Multiple transmit/receive antennas with sufficiently large spacing: • Mobiles: rich scattering → 1 / 2 . . . 1 carrier wavelength. • Base stations on high towers: tens of carrier wavelength. • Receive diversity: multiple receive antennas, → single-input/multiple-output (SIMO) systems. • Transmit diversity: multiple transmit antennas, → multiple-input/single-output (MISO) systems. • Multiple transmit and receive antennas, → multiple-input/multiple-output (MIMO) systems. 4 / 1

Antenna/Spatial Diversity Notes – Receive Diversity (SIMO) • Channel model: flat fading channel, 1 transmit antenna, L receive antennas: Lecture 5 Antenna Diversity, y [ m ] = h [ m ] · x [ m ] + w [ m ] MIMO Capacity Lars Kildehøj y l [ m ] = h l [ m ] · x [ m ] + w l [ m ] , l = 1 , . . . , L CommTh/EES/KTH with • additive noise w l [ m ] ∼ CN (0 , N 0 ), independent across antennas, • Rayleigh fading coefficients h l [ m ]. • Optimal diversity combining: maximum-ratio combining (MRC) r [ m ] = h [ m ] ∗ · y [ m ] = � h [ m ] � 2 · x [ m ] + h ∗ [ m ] w [ m ] • Error probability for BPSK (conditioned on h ) � Pr( x [ m ] � = sign( r [ m ])) = Q ( 2 � h � 2 SNR) with the (instantaneous) SNR γ = � h � 2 SNR = � h � 2 E {| x | 2 } / N 0 = L SNR · 1 L � h � 2 L � h � 2 and power/array gain L SNR. → Diversity gain due to 1 → 3 dB gain by doubling the number of antennas. 5 / 1 Antenna/Spatial Diversity Notes – Transmit Diversity (MISO), Space-Time Coding Channel model Lecture 5 Flat fading channel, L transmit antennas, 1 receive antenna: Antenna Diversity, MIMO Capacity h T [ m ] · x [ m ] + w [ m ] , y [ m ] = with Lars Kildehøj CommTh/EES/KTH • additive noise w [ m ] ∼ CN (0 , N 0 ), • vector h [ m ] of Rayleigh fading coefficients h l [ m ]. Alamouti scheme • Rate-1 space-time block code (STBC) for transmitting two data symbols u 1 , u 2 over two symbol times with L = 2 transmit antennas. • Transmitted symbols: x [1] = [ u 1 , u 2 ] T and x [2] = [ − u ∗ 1 ] T . 2 , u ∗ • Channel observations at the receiver (with channel coefficients h 1 , h 2 ): � u 1 � − u ∗ 2 [ y [1] , y [2]] = [ h 1 , h 2 ] + [ w [1] , w [2]] . u ∗ u 2 1 6 / 1

Antenna/Spatial Diversity Notes – Transmit Diversity (MISO), Space-Time Coding • Alternative formulation � y [1] � h 1 � � u 1 � w [1] � � � h 2 Lecture 5 = + Antenna Diversity, y [2] ∗ h ∗ − h ∗ w [2] ∗ u 2 MIMO Capacity 2 1 � �� � Lars Kildehøj = y CommTh/EES/KTH � h 1 � w [1] � � � � h 2 = u 1 + u 2 + h ∗ − h ∗ w [2] ∗ 2 1 � �� � � �� � = v 1 = v 2 → v 1 and v 2 are orthogonal; i.e., the AS spreads the information onto two dimensions of the received signal space. • Matched-filter receiver 2 : correlate with v 1 and v 2 H y = � h � 2 u i + ˜ r i = v i w i , for i = 1 , 2 , w i ∼ CN (0 , � h � 2 N 0 ). with independent ˜ • SNR (under power constraint E {� x � 2 } = P 0 ): SNR = � h � 2 P 0 → diversity gain of 2! 2 N 0 2 The textbook uses a projection on the orthonormal basis v 1 / � v 1 � , v 2 / � v 2 � . 7 / 1 Antenna/Spatial Diversity Notes – Transmit Diversity (MISO), Space-Time Coding Determinant criterion for space-time code design • Model: codewords of a space-time code with L transmit antennas Lecture 5 and N time slots: X i , ( L × N ) matrix. Antenna Diversity, MIMO Capacity  y T Lars Kildehøj = [ y [1] , . . . , y [ N ] ] , CommTh/EES/KTH  y T = h ∗ X i + w T with h ∗ = [ h 1 , . . . , h L ] ,  w T = [ w 1 , . . . , w L ] . Example: Alamouti scheme: Repetition coding: � u 1 � u � � − u ∗ 0 2 X i = X i = u ∗ u 2 0 u 1 • Pairwise error probability of confusing X A with X B given h �� � � h ∗ ( X A − X B ) � 2 Pr( X A → X B | h ) = Q 2 N 0 �� � SNR h ∗ ( X A − X B )( X A − X B ) ∗ h = Q 2 (Normalization: unit energy per symbol → SNR = 1 / N 0 ) 8 / 1

Antenna/Spatial Diversity Notes – Transmit Diversity (MISO), Space-Time Coding • Average pairwise error probability Lecture 5 Antenna Diversity, MIMO Capacity Pr( X A → X B ) = E { Pr( X A → X B | h ) } Lars Kildehøj CommTh/EES/KTH • Some useful facts... • ( X A − X B )( X A − X B ) ∗ is Hermitian (i.e., Z ∗ = Z ). • ( X A − X B )( X A − X B ) ∗ can be diagonalized by an unitary transform, ( X A − X B )( X A − X B ) ∗ = UΛU ∗ , where U is unitary (i.e., U ∗ U = UU ∗ = I ) and Λ = diag { λ 2 1 , . . . , λ 2 L } , with the singular values λ l of X A − X B . • And we get (with ˜ h = U ∗ h )   �   SNR � L l =1 | ˜  h l | 2 λ 2  l Pr( X A → X B ) = E  Q  ,   2 L 1 � ≤ 1 + SNR λ 2 l / 4 l =1 9 / 1 Antenna/Spatial Diversity Notes – Transmit Diversity (MISO), Space-Time Coding Lecture 5 Antenna Diversity, MIMO Capacity • If all λ 2 l > 0 (only possible if N ≥ L ), we get Lars Kildehøj CommTh/EES/KTH L 4 L 1 � Pr( X A → X B ) ≤ l / 4 ≤ 1 + SNR λ 2 SNR L � L l =1 λ 2 l =1 l 4 L 1 SNR L · = det[( X A − X B )( X A − X B ) ∗ ] → Diversity gain of L is achieved. → Coding gain is determined by the determinant det[( X A − X B )( X A − X B ) ∗ ] (determinant criterion) . 10 / 1

Antenna/Spatial Diversity Notes – 2 × 2 MIMO Example Channel Model Lecture 5 • 2 transmit antennas, 2 receive antennas: Antenna Diversity, MIMO Capacity � y 1 � h 11 � x 1 � w 1 � � � � h 12 Lars Kildehøj = · + CommTh/EES/KTH y 2 h 21 h 22 x 2 w 2 � �� � � �� � � �� � � �� � y H x w • Rayleigh distributed channel gains h ij from transmit antenna j to receive antenna i . • Additive white complex Gaussian noise w i ∼ CN (0 , N 0 ). → 4 independently faded signal paths, maximum diversity gain of 4. H 11 H 12 H 11 H 12 H = H 21 H 21 H 22 H 22 11 / 1 Antenna/Spatial Diversity Notes – 2 × 2 MIMO Example Degrees of freedom • Number of dimensions of the received signal space. • MISO: one degree of freedom for every symbol time. Lecture 5 Antenna Diversity, → Repetition coding ( L = 2): 1 dimension over 2 time slots. MIMO Capacity → Alamouti scheme ( L = 2): 2 dimension over 2 time slots. Lars Kildehøj CommTh/EES/KTH • SIMO: one degree of freedom for every symbol time. → Only one vector is used to transmit the data, y = h x + w . • MIMO: potentially two degrees of freedom for every symbol time. → Two degrees of freedom if h 1 and h 2 are linearly independent. y = h 1 x 1 + h 2 x 2 + w . (D. Tse and P. Viswanath, Fundamentals of Wireless Communications .) 12 / 1