MIMO II: Physical Channel Modeling, Spatial Multiplexing COS 463: Wireless Networks Lecture 17 Kyle Jamieson
Today 1. Graphical intuition in the I-Q plane 2. Physical modeling of the SIMO channel 3. Physical modeling of the MIMO channel 2
The problem of wireless interference User A I-Q plot: Channel Access Point (AP) A send AP receive Channel AP can estimate the channel, so can decode User A s signal ( ) 3
The problem of wireless interference User B I-Q plot: B send Channel AP receive AP can estimate the channel, so can decode User B s signal ( ) 4
The problem of wireless interference User B User A I-Q plot: AP receive from A alone AP receive from B alone AP receive (A + B) One received signal ( ), two sent (, ), so AP can t decode 5
Leveraging Multiple Antennas Now, the AP hears two received signals, one on each antenna: Antenna 1 2 User A User A Access Point Send Antenna 1 6
Leveraging Multiple Antennas User B Antenna 1 User A User A A2 A1 User B A2 + = A1 2 Mixture of A and B Antenna 1 7
Intuition: Zero-Forcing Receiver MIMO zero-forcing receiver 1. Rotate one antenna s signal ( ) 2. Sum the two antennas signals together ( + ) User A User B Sum A2 Sum A2 Rotate A1 Rotate A1 Zero-forcing cancels B, revealing A Can re-run to cancel A, revealing B 8
Spatial Multiplexing: More Streams Send multiple streams of information over each of the spatial paths between sender and receiver This is called spatial multiplexing Potential for increased capacity by a factor of N (minimum number of send or receive antennas): C = BN log 1+ SNR ( )bits/s/hz Potential for a multiplicative rate speed-up 9
Today 1. Graphical intuition in the I-Q plane 2. Physical modeling of the SIMO channel 3. Physical modeling of the MIMO channel 10
Physical Modeling of Multi-Antenna Channels Gain intuition as to how the RF channel (ambient environment) impacts capacity Many physical antenna arrangement geometries possible Limit discussion today to linear antenna arrays, halfwavelength antenna spacing Details vary with more sophisticated antenna arrangements, but concepts do not 11
Line-of-Sight SIMO Channel: A Second Look Send x 8 3 5 5/2 antenna 1 separation 2 3 Receive y 1, y 2, y 3 Vector notation for the system:! "! #! $ =! = h( + * Wireless channel is now a three-tuple vector: h = +, -#./ 0 +, -#./ 1 +, -#./ 2 12
Line-of-Sight SIMO Channel: A Second Look Send x 7 / 3 3/2 antenna 1 separation 2 3 Wireless channel is now a three-tuple vector: h = Antenna separations: Assume / 0 = / / ' / + 0 ' 3 cos 7 / 8 / + 3 cos 7 Wireless channel: Receive y 1, y 2, y 3 1 h = $% &'()/9 % % $% &'() */, $% &'() -/, $% &'()./, &( ;<= > &'( ;<= > 13
Line-of-Sight SIMO Channel: The Spatial Signature Send x 5 1 3 3/2 antenna 1 separation 2 3 The wireless channel decomposes into two components: 1 h = $% &'()/+ % % Path component &( -./ 0 &'( -./ 0 Spatial Signature Receive y 1, y 2, y 3 The angle of arrival of the sender s signal at the receive array determines the spatial signature 14
Line-of-Sight SIMO Channel: Maximal Ratio Combining (Review) Send x ' ), 1 2 3 )/2 antenna separation Maximal ratio combining projects the received signals " onto the receive spatial signature: #" = h " Receive y 1, y 2, y 3 Reverses the phases in the spatial signature to align each antenna s component of the above sum SNR improvement but no multiplexing 15
Today 1. Graphical intuition in the I-Q plane 2. Physical modeling of the SIMO channel 3. Physical modeling of the MIMO channel Line-of-Sight MIMO Channel Geographically-Separated Transmit Antennas Geographically-Separated Receive Antennas MIMO Link in Multipath 16
The Line-of-Sight MIMO Channel //2 Send x 1, x 2, x 3 3 2 1 3 - / 2 1 2 3 //2 antenna separation Receive y 1, y 2, y 3 Want to transmit three symbols per symbol time: " = " $ " % " & ' () : channel between k th receive and l th transmit antenna * = H ", where H = h $$ h $% h $$ h %$ h %% h $$ is the MIMO channel matrix h &$ h &% h $$ 17
The Line-of-Sight MIMO Channel: Channel Matrix H/2 Send x 1, x 2, x 3 3 2 1 3 $ H 1 1 2 3 H/2 antenna separation! "# : channel between k th receive and l th transmit antenna Suppose as before, $ %% = $ Then $ '( = $ + % + 1 cos 1 + % 2 1 cos 3 * * Tx 1: Tx 2: Tx 3: 78 *@AB? Channel matrix 4 = 56 7*89/; 1 6 78 <=>? 6 6 78 @AB C 6 78 (@AB CE@AB?) 6 6 7*8 @AB C 6 78 (*@AB CE@AB?) 6 Receive y 1, y 2, y 3 78 (@AB CE*@AB?) 78 (*@AB CE*@AB?) 18
The Line-of-Sight MIMO Channel: Identical Spatial Signatures </2 Send x 1, x 2, x 3 3 2 1? : < > 1 2 3 </2 antenna separation Tx 1: Tx 2: Tx 3: %' &012 / Channel matrix! = #$ %&'(/* 1 $ %',-. / $ $ %' 012 3 $ %' (012 35012 /) $ $ %&' 012 3 $ %' (&012 35012 /) $ Receive y 1, y 2, y 3 %' (012 35&012 /) %' (&012 35&012 /) Transmit antenna 2 s channel and spatial signature: h 8& h && h 9& = #$ %&' ( 1 * 5012 / %' 012 3 $ $ %&' 012 3 19
The Line-of-Sight MIMO Channel: Takeaways Spatial signature tells us how to phase-shift the received signals in order to align them Spatial signature of Transmit antenna 1 Equals spatial signature of Transmit antenna 2 Equals spatial signature of Transmit antenna 3 So any receiver attempt to align signal from Transmit antenna 1 Also aligns transmit antennas 2 and 3 Spatial mux fail Result is interference between x 1, x 2, x 3 Can send same single symbol x on all transmit antennas Results in same power gain as MRC 20
Today 1. Graphical intuition in the I-Q plane 2. Physical modeling of the SIMO channel 3. Physical modeling of the MIMO channel Line-of-Sight MIMO Channel Geographically-Separated Transmit Antennas Geographically-Separated Receive Antennas MIMO Link in Multipath 21
Geographically-Separated Transmit Antennas Send x 1, x 2 1 2 2 2 4 2 4 6 & 6 9 1 2 4/2 antenna separation Tx 1: Tx 2: Channel matrix! = #$ %&'(/* 1 1 $ %',-. / 0 $ %',-. / 1 Sig. 1 Sig. 2 Different spatial signatures for Transmit Antenna 1, 2 Receive y 1, y 2 22
Spatial Signature = Series of Phase Differences Tx 1: Tx 2: Channel matrix! = #$ %&'(/* 1 1 $ %',-. / 0 $ %',-. / 1 Sig. 1, 2 34 Sig. 2, 2 35 At Receive Antenna 1 At Receive From Transmit Antenna 1: + From Transmit : Sig. 1 Sig. 2 At Receive At Receive Antenna 1 23
The Zero-Forcing Receiver (via Spatial Signatures) Suppose want to receive from Transmit Antenna 1 (Recall:) Rotate Receive s signal so that Signature 2 cancels itself At Receive Antenna 1 At Receive From Transmit Antenna 1: + From Transmit : Sig. 1 Sig. 2 Cancelled! At Receive At Receive Antenna 1 24
The Zero-Forcing Receiver (via Spatial Signatures) One spatial signature = One direction " #$ & Zero forcing is projection Onto subspace to " #$ " #% At Receive Antenna 1 At Receive From Transmit Antenna 1: + From Transmit : Sig. 1 Sig. 2 Cancelled! At Receive At Receive Antenna 1 25
MIMO Separability: Discussion Transmit antenna separation à Spatial signature separation à Better projection, Better performance! "$ %! "# MIMO antenna array without multipath No transmit antenna separation No spatial signature separation Cancel Tx Ant 2: cancels Tx Ant 1 No spatial multiplexing! "#! "$ % 26
Today 1. Graphical intuition in the I-Q plane 2. Physical modeling of the SIMO channel 3. Physical modeling of the MIMO channel Line-of-Sight MIMO Channel Geographically-Separated Transmit Antennas Geographically-Separated Receive Antennas MIMO Link in Multipath 27
Geographically-Separated Receive Antennas Transmitter d 1 Receiver 1 Antenna 1 θ 2 θ 1 Δ d 2 Receiver 2 Different spatial signatures for Receive Antennas 1, 2 Rows, instead of columns in the MIMO matrix 28
Today 1. Graphical intuition in the I-Q plane 2. Physical modeling of the SIMO channel 3. Physical modeling of the MIMO channel Line-of-Sight MIMO Channel Geographically-Separated Transmit Antennas Geographically-Separated Receive Antennas MIMO Link in Multipath 29
MIMO in Multipath Antenna 1 Transmitter θ 1 θ 2 φ 1 Antenna 1 φ 2 d 1 Receiver d 2 H = # $ % &'() */, + # ' % &'()./, # $ % &'( ) *, /012 3 * + # ' % &'( )., /012 3. # $ % &'( ) *, /012 4 * + # ' % &'( )., /012 4. #$ % &'( ) *, /012 3 * /012 4 * + # ' % &'( )., /012 3. /012 4. Channel matrix H has two different transmitter spatial signatures 30
Different Spatial Signatures: Intuition Antenna 1 Transmitter θ 1 θ 2 A B φ 1 Antenna 1 φ 2 d 1 Receiver d 2 Channel matrix H has two different spatial signatures Imagine perfect signal relays A, B This H is the product of: Geographically-separated receive antenna channel Geographically-separated transmit antenna channel 31
Poorly-Conditioned MIMO channels Receiver! 1! 2 Antenna 1 Antenna 1 Antenna 1 Transmitter Transmitter " 1 " 2 " 1 " 2 Only reflectors near receiver: ϕ 1 ϕ 2 Only reflectors near transmitter: θ 1 θ 2! 2! 1 Antenna 1 Receiver h 1 h 2 When channel is poorly conditioned, spatial signatures are closer aligned 32
Friday Precept: Exploiting Doppler Tuesday Topic: MIMO III: MIMO Channel Capacity, Interference Alignment 33