MIMO I: Spatial Diversity

COS 463: Wireless Networks Lecture 16 Kyle Jamieson

[Parts adapted from D. Halperin et al., T. Rappaport] What is MIMO, and why? • Multiple-Input, Multiple-Output (MIMO) communications – Sends and receive more than one signal on different transmit and receive antennas

• We’ve already seen frequency, time, spatial multiplexing in 463:

– MIMO is a more powerful way to multiplex wireless medium in space

– Transforms multipath propagation from an impediment to an advantage

2 Many Uses of MIMO • At least three different ways to leverage space:

1. Spatial diversity: Send or receive redundant streams of information in parallel along multiple spatial paths – Increases reliability and range (unlikely that all paths will be degraded simultaneously)

2. Spatial multiplexing: Send independent streams of information in parallel along multiple spatial paths – Increases rate, if we can avoid interference

3. Interference alignment: “Align” two streams of interference at a remote receiver, resulting in the impact of just one interference stream MIMO-OFDM

(5) QPSK Modulated f subcarriers Symbols (6) Mapped onto Subcarriers as OFDM Symbol

Figure 1: A graphical• Multipath view of fading: the OFDM different encoding effects on process different for frequencies the 18 Mbps rate (QPSK, 3/4)of802.11a. The data bits (0) are encoded– OFDM: by Orthogonal a rate-1/2 convolutionalFrequency Domain code Multiplexing (1) and then optionally punctured by dropping certain bits for higher coding rates (here, 3/4) that send fewer redundant bits (2). The remaining bits are in- terleaved (3) to spread– theDifferent redundancy subcarriers across are subcarriers independent and protectof each other against frequency-selective fades. These bits are grouped into symbols (4) based on the modulation (QPSK encodes 2 bits per symbol), modulated (5), and ﬁnally mapped• ontoChannel the dimodel↵erent for subcarriers OFDM: y = toh∙x form+ w an OFDM symbol (6). – A single complex number h captures the effect of the channel on data h11 h h11 iny1 a = particular h11 x subcarrier 11 y = (+h11 h21 )x y1 = h11x1+h21x2 x x x1 h12 h21 Tx•h For MIMO: Rx Think about each subcarrier, independentTxof other subcarriers Rx 12 Txh21 Rx

x x2 h22

y2 = h12 x y y2 = h12x1 +h22x2 (a) Receive diversity (b) Transmit diversity (c) Spatial multiplexing

Figure 2: Using some of the transmit/receive antennas in an example 2x2 system to exploit diversity and multiplexing gain. xi and yi represent transmitted and received signals. The channel gain hij is a complex number indicating a signal’s attenuated amplitude and phase shift over the channel between the ith transmit antenna and the jth receive antenna. The received signals yi will additionally include thermal RF noise. modulation sending more bits per symbol and being used volutional code of rate 1/2 adds redundant information. It when there is a higher SNR. There are minor di↵erences be- is then punctured [3] by removing bits as needed to support tween 802.11a/g and 802.11n. In 802.11a/g there are 48 data coding rates of 2/3 and 3/4, plus 5/6 for 802.11n. At a rate subcarriers, 4 pilot tones for control, and 6 unused guard of 3/4, for example, a quarter of the data on the subcarriers subcarriers at each edge of the channel. In 802.11n, there is redundant. An alternative LDPC (Low-Density Parity- are only 4 guard subcarriers at each edge of the channel, and Check) code with slightly better performance can also be two adjacent 20 MHz channels can be merged into a single used for 802.11n. Figure 1 presents a pictorial overview of 40 MHz channel. the OFDM encoding process. The beauty of OFDM is that it divides the channel in a The net e↵ect of OFDM plus coding is to provide consis- way that is both computationally and spectrally ecient. tently good 802.11 performance despite signiﬁcant variabil- High aggregate data rates can be achieved, while the en- ity in the wireless signal due to multi-path fading. coding and decoding on di↵erent subcarriers can use shared hardware components. More relevant to our point here, however, is that OFDM transforms a single large channel into 3. SPATIAL DIVERSITY many relatively independently faded channels. This is be- In this section we look at spatial diversity techniques that cause multi-path fading is frequency selective, so the di↵er- can be applied at the receiver and at the transmitter. Adding ent subcarriers will experience di↵erent fades. Some adja- multiple antennas to an 802.11n receiver or transmitter pro- cent subcarriers may be faded in a similar way, but the fading vides a new set of independently faded paths, even if the for more distant subcarriers is often uncorrelated. Dividing antennas are separated by only a few centimeters. This the channel also increases the symbol time per channel, since adds spatial diversity to the system, which can be exploited many slow symbols will be sent in parallel instead of many to improve resilience to fades. There is also a power gain fast symbols in sequence. This adds time diversity because from multiple receive antennas because, everything else be- the channel is more likely to average out fades over a longer ing equal, two receive antennas will receive twice the signal. period of time. These factors combine to improve performance at a given 802.11 makes use of the frequency diversity provided by distance, and hence increase range. OFDM by coding across the data carried on the subcarriers. This uses a fraction of them for redundant information that 3.1 Receive Diversity Techniques can later be used to correct errors that occur when fading Consider the arrangement in Figure 2(a). One transmit reduces the SNR on some of the subcarriers. First, a con- antenna at a node is sending to two receive antennas at a Plan 1. Today: Diversity in Space – Receive Diversity – Transmit Diversity

2. Next time: Multiplexing in Space

3. Next time: Interference Alignment

5 Path Diversity: Motivation

1. Multi-Antenna Access Points (APs), especially 802.11n,ac:

2. Multiple APs cooperating with each other:

Wired backhaul 3. Distributed Antenna systems, separating antenna from AP: Antenna 2 Antenna 1

AP Coaxial / Fiber backhaul 6 Review: Fast Fading

• Typical outdoor multipath propagation environment, channel h

• On one link each subcarrier’s power level experiences Rayleigh fading:

! "

7 Uncorrelated Rayleigh Fading

• Suppose two antennas, separated by distance d12

• Channels from each to a distant third antenna (h13, h23) can be uncorrelated – Fading happens at different times with no bias for a simultaneous fade

# # !" , !#

8 When is Fading Uncorrelated, and Why?

≫ , !"#

• Channels from each antenna (h13, h23) to a third antenna – Channels are uncorrelated when !"# > ≈ &. () – Channels correlated, fade together when !"# < ≈ )

• This correlation distance depends on the radio environment around the pair of antennas – Increases, e.g., atop cellular phone tower

9 Plan 1. Today: Diversity in Space – Receive Diversity • Selection Diversity • Maximal Ratio Combining – Transmit Diversity

2. Next time: Multiplexing in Space

3. Next time: Interference Alignment

10 Channel Model for Receive Diversity • One transmit antenna sends a symbol to two receive antennas – Receive diversity, or Single-Input, Multi-Output (SIMO)

y =3ei3π/4x 9/p13 h 1 1 Receive antenna 1 rotate, scale by 3/p13 xx n 1 p13 h2 4/p13 Receive antenna 2 rotate, scale by 2/p13 n expected 1 n2 -iπ/6 y2=2e x

h11 y = h x h11 y =0(+h h )x h11 y = h x +h x x second node. This is known1 11 as a 1x2 system. Realx systems 11 21 x 1 11 1 21 2 may have more than two receive antennas, but two will suf- 1 h fice for our explanation. With this setup, each receive an- 12 -5 h21 Txtenna receives a copy of Rx the transmitted signal modified by Tx Rx the channelh12 between the transmitter andTx itself. The chan-h Rx 21 -10 nel gains hij are complex numbers that represent both the x x2 amplitude attenuation over the channel as well as the path- h22 dependent phase shifty (see = h Figure x 3 for a graphical example). y-15 y = h x +h x The receiver measures2 the12 channel gains based on training A 2 12 1 22 2 C fields in the packet preamble. Note that the gains di↵er for -20 B and SEL (a)each Receive subcarrier diversity (in frequency-selective fading)(b) as well Transmit as for diversity (c) SpatialAB (MRC) multiplexing each antenna. The question now is how to combine the two ABC (MRC) Normalized power (dB) Figure 2: Usingreceived some signals to of make the best transmit/receive use of them. antennas in-25 an example 2x2 system to exploit diversity and -20 -10 0 10 20 multiplexing gain.We considerx twoand diversityy represent techniques to show transmitted the extremes. and received signals. The channel gain h is a complex The simplest methodi is toi use the antenna with the strongest Subcarrier index ij number indicatingsignal (hence a signal’s the largest SNR) attenuated to receive the amplitude packet and ig- and phase shift over the channel between the ith transmit Figure 4: Frequency-selective fading over testbed antenna andnore the thej others.th receive We will callantenna. this method The SEL, received for selection signals yi will additionally include thermal RF noise. combining. This is essentially what is done by 802.11a/g APs links: the figure shows, for an example 5.2 GHz link, with multiple antennas. It helps with reliability, because the received power measured on each subcarrier for modulation sendingboth signals more are unlikely bits per to be symbol bad, but and it wastes being perfectly used individualvolutional antennas code and of rate under 1/2 SEL adds and MRC redundant diver- information. It when there isgood a higher received SNR. power There at the antennas are minor that di are↵erences not chosen. be- sity,is thennormalized punctured to the [3] strongest by removing subcarrier bits power. as needed to support tween 802.11a/gThe and better 802.11n. method In is 802.11a/g to add the signals there arefrom 48 the data two coding rates of 2/3 and 3/4, plus 5/6 for 802.11n. At a rate antennas together. However, this cannot be done by simply tudes of 3 and 2. With expected noise power 1, these gains subcarriers, 4superimposing pilot tones their for signals, control, or we and will 6 have unused just recreated guard correspondof 3/4, to for SNRs example, of 9 and 4, a given quarter that aof signal’s the data power on is the subcarriers subcarriers atthe each e↵ects edge of multi-path of the channel. fading. Rather, In 802.11n, the signals should there theis square redundant. of its magnitude. An alternative The MRC receiver LDPC scales (Low-Density each Parity- are only 4 guardeach subcarriers be delayed until at they each are edge in the of same the phase; channel, then, and the antenna’sCheck) signal code by itswith magnitude, slightly normalized better to performance the total; can also be power in the signals will combine coherently. To do this, delays the signals to a common phase reference; and then two adjacent 20the receiverMHz channels needs a dedicated can be RF merged chain for each into antenna a single to addsused them. for The 802.11n. result has magnitudeFigure 1p presents13, and the a normal- pictorial overview of 40 MHz channel.process the signals. This increases the hardware complexity izedthe weighted OFDM sum encoding of noise still process. has expected power 1. The The beautyand of power OFDM consumption, is that butit divides yields better the performance. channel in a combinedThe signal net thuse↵ect has of a resulting OFDM sum plus SNR coding of 13. is to provide consis- As a twist in the above, the signals are also weighted by As an example of how MRC and SEL work in 802.11, con- way that is boththeir SNRs. computationally This gives less and weight spectrally to a signal that ecient. has a sidertently Figure good 4. This 802.11 figure shows performance the wireless signal despite strength significant variabil- High aggregatelarger data fraction rates of noise, can so be that achieved, the e↵ects of while the noise the are en-not of eachity in subcarrier the wireless using three signal antennas due for to a real multi-path 802.11n link fading. coding and decodingamplified. on The di result↵erent is maximal-ratio subcarriers combining can use, or shared MRC. in our indoor wireless testbed. The subcarrier strengths are MRC is known to be optimal (it maximizes SIMO capacity), measured in decibels normalized to the strongest subcarrier hardware components.and produces More an SNR relevant that is to the our sum point of the here, component how- strength. This figure gives a much more detailed view than ever, is that OFDMSNRs. Note transforms that in frequency-selective a single large fading, channel this process into metrics3. such SPATIAL as the RSSI DIVERSITY (Received Signal Strength Indi- many relativelyis performed independently di↵erently faded for each channels. subcarrier according This is to be- its cation)In for this a link, section which we gives look only at the spatial sum of diversity the signal techniques that specific channel gains. strengthcan be over applied all subcarriers. at the receiver and at the transmitter. Adding cause multi-pathFigure fading 3 depicts is frequency MRC operation selective, graphically so the for di a↵ 1x2er- For each antenna labeled A, B, or C, the signal varies ent subcarrierschannel. will experience In this example, di the↵erent two channel fades. gains Some have magni- adja- overmultiple the channel, antennas changing to gradually an 802.11n from onereceiver subcarrier or transmitter pro- cent subcarriers may be faded in a similar way, but the fading vides a new set of independently faded paths, even if the for more distant subcarriers is often uncorrelated. Dividing antennas are separated by only a few centimeters. This the channel also increases the symbol time per channel, since adds spatial diversity to the system, which can be exploited many slow symbols will be sent in parallel instead of many to improve resilience to fades. There is also a power gain fast symbols in sequence. This adds time diversity because from multiple receive antennas because, everything else be- the channel is more likely to average out fades over a longer ing equal, two receive antennas will receive twice the signal. period of time. These factors combine to improve performance at a given 802.11 makes use of the frequency diversity provided by distance, and hence increase range. OFDM by coding across the data carried on the subcarriers. This uses a fraction of them for redundant information that 3.1 Receive Diversity Techniques can later be used to correct errors that occur when fading Consider the arrangement in Figure 2(a). One transmit reduces the SNR on some of the subcarriers. First, a con- antenna at a node is sending to two receive antennas at a Selection Diversity

y =3ei3π/4x 9/p13 h 1 1 Rx 1 rotate, scale by 3/p13 xx n Select stronger 1 p13 h2 Radio 4/p13 Rx 2 rotate, scale by 2/p13 n expected 1 n2 -iπ/6 y2=2e x

i3⇡/4 i⇡/6 Figure 3: MRC operation on a sample channel. The channel gains are ~h = 3e , 2e ,withGaussian Figure 1: A graphical view of the OFDM encoding process for the 18h Mbps ratei (QPSK, 3/4)of802.11a. noise ~n = n1,n2 of expected power 1. The antennas have respective SNRs of 9 and 4. To implement MRC, • Twoh receivei antennas share one receiving radio The data bitsthe (0) receiver are encoded multiplies the by received a rate- signal1/2 convolutional~y = ~hx + ~n by the codeunit vector (1) and~h⇤/ ~h then, where optionally~h⇤ denotes the punctured complex by dropping || || certain bitsconjugate for higher of ~h coding. This operation rates (here, scales each3/4) antenna’s that send signal fewer by its redundant magnitude, and bits rotates (2). The the signals remaining into bits are in- terleaved (3)the to• same spreadChooses phase the reference redundancy the beforeantenna adding across with them. subcarriers stronger (For graphical andsignal, clarity, protect wesends depict against thethat common frequency-selectiveto the phase radio vertically, fades. These bits are groupedrather into than symbolsat 0). The resulting(4) based sum on has the magnitude modulationp13, and (QPSK expected encodes noise power2 bits1 because per symbol), the scaling is modulated (5), normalized.– Helps Thus, by reliability coherently combining (both unlikely received bad) signals from di↵erent antennas, the MRC output has and finally mappedthe expected onto SNR the of 13 di.↵ Inerent systems subcarriers with OFDM, to MRC form is performed an OFDM separately symbol for each (6). subcarrier. – Wastes received signal from other antenna(s) h11 y = h x h11 y =0(+h h )x h11 y = h x +h x x second node. This is known1 11 as a 1x2 system. Realx systems 11 21 x 1 11 1 21 2 may have more than two receive antennas, but two will suf- 1 h fice for our explanation. With this setup, each receive an- 12 -5 h21 Txtenna receives a copy of Rx the transmitted signal modified by Tx Rx the channelh12 between the transmitter andTx itself. The chan-h Rx 21 -10 nel gains hij are complex numbers that represent both the x x2 amplitude attenuation over the channel as well as the path- h22 dependent phase shifty (see = h Figure x 3 for a graphical example). y-15 y = h x +h x The receiver measures2 the12 channel gains based on training A 2 12 1 22 2 C fields in the packet preamble. Note that the gains di↵er for -20 B and SEL (a)each Receive subcarrier diversity (in frequency-selective fading)(b) as well Transmit as for diversity (c) SpatialAB (MRC) multiplexing each antenna. The question now is how to combine the two ABC (MRC) Normalized power (dB) Figure 2: Usingreceived some signals to of make the best transmit/receive use of them. antennas in-25 an example 2x2 system to exploit diversity and -20 -10 0 10 20 multiplexing gain.We considerx twoand diversityy represent techniques to show transmitted the extremes. and received signals. The channel gain h is a complex The simplest methodi is toi use the antenna with the strongest Subcarrier index ij number indicatingsignal (hence a signal’s the largest SNR) attenuated to receive the amplitude packet and ig- and phase shift over the channel between the ith transmit Figure 4: Frequency-selective fading over testbed antenna andnore the thej others.th receive We will callantenna. this method The SEL, received for selection signals yi will additionally include thermal RF noise. combining. This is essentially what is done by 802.11a/g APs links: the figure shows, for an example 5.2 GHz link, with multiple antennas. It helps with reliability, because the received power measured on each subcarrier for modulation sendingboth signals more are unlikely bits per to be symbol bad, but and it wastes being perfectly used individualvolutional antennas code and of rate under 1/2 SEL adds and MRC redundant diver- information. It when there isgood a higher received SNR. power There at the antennas are minor that di are↵erences not chosen. be- sity,is thennormalized punctured to the [3] strongest by removing subcarrier bits power. as needed to support tween 802.11a/gThe and better 802.11n. method In is 802.11a/g to add the signals there arefrom 48 the data two coding rates of 2/3 and 3/4, plus 5/6 for 802.11n. At a rate antennas together. However, this cannot be done by simply tudes of 3 and 2. With expected noise power 1, these gains subcarriers, 4superimposing pilot tones their for signals, control, or we and will 6 have unused just recreated guard correspondof 3/4, to for SNRs example, of 9 and 4, a given quarter that aof signal’s the data power on is the subcarriers subcarriers atthe each e↵ects edge of multi-path of the channel. fading. Rather, In 802.11n, the signals should there theis square redundant. of its magnitude. An alternative The MRC receiver LDPC scales (Low-Density each Parity- are only 4 guardeach subcarriers be delayed until at they each are edge in the of same the phase; channel, then, and the antenna’sCheck) signal code by itswith magnitude, slightly normalized better to performance the total; can also be power in the signals will combine coherently. To do this, delays the signals to a common phase reference; and then two adjacent 20the receiverMHz channels needs a dedicated can be RF merged chain for each into antenna a single to addsused them. for The 802.11n. result has magnitudeFigure 1p presents13, and the a normal- pictorial overview of 40 MHz channel.process the signals. This increases the hardware complexity izedthe weighted OFDM sum encoding of noise still process. has expected power 1. The The beautyand of power OFDM consumption, is that butit divides yields better the performance. channel in a combinedThe signal net thuse↵ect has of a resulting OFDM sum plus SNR coding of 13. is to provide consis- As a twist in the above, the signals are also weighted by As an example of how MRC and SEL work in 802.11, con- way that is boththeir SNRs. computationally This gives less and weight spectrally to a signal that ecient. has a sidertently Figure good 4. This 802.11 figure shows performance the wireless signal despite strength significant variabil- High aggregatelarger data fraction rates of noise, can so be that achieved, the e↵ects of while the noise the are en-not of eachity in subcarrier the wireless using three signal antennas due for to a real multi-path 802.11n link fading. coding and decodingamplified. on The di result↵erent is maximal-ratio subcarriers combining can use, or shared MRC. in our indoor wireless testbed. The subcarrier strengths are MRC is known to be optimal (it maximizes SIMO capacity), measured in decibels normalized to the strongest subcarrier hardware components.and produces More an SNR relevant that is to the our sum point of the here, component how- strength. This figure gives a much more detailed view than ever, is that OFDMSNRs. Note transforms that in frequency-selective a single large fading, channel this process into metrics3. such SPATIAL as the RSSI DIVERSITY (Received Signal Strength Indi- many relativelyis performed independently di↵erently faded for each channels. subcarrier according This is to be- its cation)In for this a link, section which we gives look only at the spatial sum of diversity the signal techniques that specific channel gains. strengthcan be over applied all subcarriers. at the receiver and at the transmitter. Adding cause multi-pathFigure fading 3 depicts is frequency MRC operation selective, graphically so the for di a↵ 1x2er- For each antenna labeled A, B, or C, the signal varies ent subcarrierschannel. will experience In this example, di the↵erent two channel fades. gains Some have magni- adja- overmultiple the channel, antennas changing to gradually an 802.11n from onereceiver subcarrier or transmitter pro- cent subcarriers may be faded in a similar way, but the fading vides a new set of independently faded paths, even if the for more distant subcarriers is often uncorrelated. Dividing antennas are separated by only a few centimeters. This the channel also increases the symbol time per channel, since adds spatial diversity to the system, which can be exploited many slow symbols will be sent in parallel instead of many to improve resilience to fades. There is also a power gain fast symbols in sequence. This adds time diversity because from multiple receive antennas because, everything else be- the channel is more likely to average out fades over a longer ing equal, two receive antennas will receive twice the signal. period of time. These factors combine to improve performance at a given 802.11 makes use of the frequency diversity provided by distance, and hence increase range. OFDM by coding across the data carried on the subcarriers. This uses a fraction of them for redundant information that 3.1 Receive Diversity Techniques can later be used to correct errors that occur when fading Consider the arrangement in Figure 2(a). One transmit reduces the SNR on some of the subcarriers. First, a con- antenna at a node is sending to two receive antennas at a ECE 5325/6325 Spring 2010 76

Selection Diversity: Performance Improvement

• In general, might have M receive Probability (%) that Selected antennas (average SNR Γ) Antenna’s SNR Exceeds Threshold γ th – !": SNR of the i receive antenna

• Probability selected SNR is less than some threshold γ: ( – Pr !%, ⋯ , !( ≤ γ = Pr !" ≤ γ

• One more “9” of reliability per additional selection branch

Higher probability (better) ↓

ß lower threshold SNR 13

Figure 32: Rappaport Figure 7.11, the impact of selection combining. Leveraging All Receive Antennas

i3π/4 p y1=3e x 9/ 13 h1 Rx 1 rotate, scale by 3/p13 xx n 1 p13 h2 4/p13 Rx 2 rotate, scale by 2/p13 n expected 1 n2 -iπ/6 y2=2e x

i3⇡/4 i⇡/6 Figure 3: MRC operation on a sample channel. The channel gains are ~h = 3e , 2e ,withGaussian Figure 1: A graphical• Want to view just of add the the OFDM two received encoding signals process together for the 18h Mbps ratei (QPSK, 3/4)of802.11a. noise ~n = n1,n2 of expected power 1. The antennas have respective SNRs of 9 and 4. To implement MRC, The data bits (0) areh encodedi by a rate-1/2 convolutional code (1) and then optionally punctured by dropping the receiver– But multiplies if we the did received the signals signal ~y = ~hxwould+ ~n by theoften unit vectorcancel~h⇤/ out~h , where ~h⇤ denotes the complex || || certain bitsconjugate for higher of ~h coding. This operation rates (here, scales each3/4) antenna’s that send signal fewer by its redundant magnitude, and bits rotates (2). The the signals remaining into bits are in- terleaved (3)the to same spread phase the reference redundancy before adding across them. subcarriers (For graphical and clarity, protect we depict against the common frequency-selective phase vertically, fades. These bits are groupedrather• into thanSolution: symbolsat 0). The Receive resulting(4) based sumM onradios, has the magnitude modulationalignp13 signal, and (QPSK expected phases, encodes noise then power2 bits 1addbecause per symbol), the scaling is modulated (5), normalized. Thus, by coherently combining received signals from di↵erent antennas, the MRC output has and ﬁnally mappedthe expected– ontoRequires SNR the of 13 di. ↵ InMerent systemsreceive subcarriers with radios, OFDM, to MRCin form general is performed an OFDM separately symbol for each (6). subcarrier.

i3π/4 p y1=3e x 9/ 13 h1 Rx 1 rotate, scale by 3/p13 xx n 1 y p13 h2 4/p13 Rx 2 rotate, scale by 2/p13 n expected 1 n2 -iπ/6 y2=2e x

i3⇡/4 i⇡/6 Figure 3: MRC operation on a sample channel. The channelth gains are ~h = 3e , 2e ,withGaussian Figure 1: A graphical• Suppose view phase of theof incoming OFDM signal encoding on the process i branch for the is !"18h Mbps ratei (QPSK, 3/4)of802.11a. noise ~n = n1,n2 of expected power 1. The antennas have respective SNRs of 9 and 4. To implement MRC, h i The data bitsthe (0) receiver are encoded multiplies the by received a rate- signal1/2 convolutional~y = ~hx + ~n by the codeunit vector (1) and~h⇤/ ~h then, where optionally~h⇤ denotes the punctured complex by dropping || || certain bitsconjugate for higher of ~h coding. This operation rates (here, scales each3/4) antenna’s that send signal fewer by its redundant magnitude,( and bits+,- rotates. (2). The the signals remaining into bits are in- • To align { yi } in phase, let the combiner output # = ∑"&' )"* terleaved (3)the to same spread– phaseHow the reference to redundancy choose before amplitudes adding across them. subcarriers a (For? graphical and clarity, protect we depict against the common frequency-selective phase vertically, fades. These bits are groupedrather into than symbolsat 0). The resulting(4) based sum on has the magnitude modulationi p13, and (QPSK expected encodes noise power2 bits1 because per symbol), the scaling is modulated (5), normalized. Thus, by coherently combining received signals from di↵erent antennas, the MRC output has and finally mappedthe expected onto SNR the of 13 di.↵ Inerent systems subcarriers with OFDM, to MRC form is performed an OFDM separately symbol for each (6). subcarrier. • Idea: Put more weight into branches with high SNR: Let /0 = 10 h–11 This isy called= h x Maximal Ratio Combiningh11 y = (MRC)0(+h h )x h11 y = h x +h x x second node. This is known1 11 as a 1x2 system. Realx systems 11 21 x 1 11 1 21 2 may have more than two receive antennas, but two will suf- 1 h fice for our explanation. With this setup, each receive an- 12 -5 h21 Txtenna receives a copy of Rx the transmitted signal modified by Tx Rx the channelh12 between the transmitter andTx itself. The chan-h Rx 21 -10 nel gains hij are complex numbers that represent both the x x2 amplitude attenuation over the channel as well as the path- h22 dependent phase shifty (see = h Figure x 3 for a graphical example). y-15 y = h x +h x The receiver measures2 the12 channel gains based on training A 2 12 1 22 2 C fields in the packet preamble. Note that the gains di↵er for -20 B and SEL (a)each Receive subcarrier diversity (in frequency-selective fading)(b) as well Transmit as for diversity (c) SpatialAB (MRC) multiplexing each antenna. The question now is how to combine the two ABC (MRC) Normalized power (dB) Figure 2: Usingreceived some signals to of make the best transmit/receive use of them. antennas in-25 an example 2x2 system to exploit diversity and -20 -10 0 10 20 multiplexing gain.We considerx twoand diversityy represent techniques to show transmitted the extremes. and received signals. The channel gain h is a complex The simplest methodi is toi use the antenna with the strongest Subcarrier index ij number indicatingsignal (hence a signal’s the largest SNR) attenuated to receive the amplitude packet and ig- and phase shift over the channel between the ith transmit Figure 4: Frequency-selective fading over testbed antenna andnore the thej others.th receive We will callantenna. this method The SEL, received for selection signals yi will additionally include thermal RF noise. combining. This is essentially what is done by 802.11a/g APs links: the figure shows, for an example 5.2 GHz link, with multiple antennas. It helps with reliability, because the received power measured on each subcarrier for modulation sendingboth signals more are unlikely bits per to be symbol bad, but and it wastes being perfectly used individualvolutional antennas code and of rate under 1/2 SEL adds and MRC redundant diver- information. It when there isgood a higher received SNR. power There at the antennas are minor that di are↵erences not chosen. be- sity,is thennormalized punctured to the [3] strongest by removing subcarrier bits power. as needed to support tween 802.11a/gThe and better 802.11n. method In is 802.11a/g to add the signals there arefrom 48 the data two coding rates of 2/3 and 3/4, plus 5/6 for 802.11n. At a rate antennas together. However, this cannot be done by simply tudes of 3 and 2. With expected noise power 1, these gains subcarriers, 4superimposing pilot tones their for signals, control, or we and will 6 have unused just recreated guard correspondof 3/4, to for SNRs example, of 9 and 4, a given quarter that aof signal’s the data power on is the subcarriers subcarriers atthe each e↵ects edge of multi-path of the channel. fading. Rather, In 802.11n, the signals should there theis square redundant. of its magnitude. An alternative The MRC receiver LDPC scales (Low-Density each Parity- are only 4 guardeach subcarriers be delayed until at they each are edge in the of same the phase; channel, then, and the antenna’sCheck) signal code by itswith magnitude, slightly normalized better to performance the total; can also be power in the signals will combine coherently. To do this, delays the signals to a common phase reference; and then two adjacent 20the receiverMHz channels needs a dedicated can be RF merged chain for each into antenna a single to addsused them. for The 802.11n. result has magnitudeFigure 1p presents13, and the a normal- pictorial overview of 40 MHz channel.process the signals. This increases the hardware complexity izedthe weighted OFDM sum encoding of noise still process. has expected power 1. The The beautyand of power OFDM consumption, is that butit divides yields better the performance. channel in a combinedThe signal net thuse↵ect has of a resulting OFDM sum plus SNR coding of 13. is to provide consis- As a twist in the above, the signals are also weighted by As an example of how MRC and SEL work in 802.11, con- way that is boththeir SNRs. computationally This gives less and weight spectrally to a signal that ecient. has a sidertently Figure good 4. This 802.11 figure shows performance the wireless signal despite strength significant variabil- High aggregatelarger data fraction rates of noise, can so be that achieved, the e↵ects of while the noise the are en-not of eachity in subcarrier the wireless using three signal antennas due for to a real multi-path 802.11n link fading. coding and decodingamplified. on The di result↵erent is maximal-ratio subcarriers combining can use, or shared MRC. in our indoor wireless testbed. The subcarrier strengths are MRC is known to be optimal (it maximizes SIMO capacity), measured in decibels normalized to the strongest subcarrier hardware components.and produces More an SNR relevant that is to the our sum point of the here, component how- strength. This figure gives a much more detailed view than ever, is that OFDMSNRs. Note transforms that in frequency-selective a single large fading, channel this process into metrics3. such SPATIAL as the RSSI DIVERSITY (Received Signal Strength Indi- many relativelyis performed independently di↵erently faded for each channels. subcarrier according This is to be- its cation)In for this a link, section which we gives look only at the spatial sum of diversity the signal techniques that specific channel gains. strengthcan be over applied all subcarriers. at the receiver and at the transmitter. Adding cause multi-pathFigure fading 3 depicts is frequency MRC operation selective, graphically so the for di a↵ 1x2er- For each antenna labeled A, B, or C, the signal varies ent subcarrierschannel. will experience In this example, di the↵erent two channel fades. gains Some have magni- adja- overmultiple the channel, antennas changing to gradually an 802.11n from onereceiver subcarrier or transmitter pro- cent subcarriers may be faded in a similar way, but the fading vides a new set of independently faded paths, even if the for more distant subcarriers is often uncorrelated. Dividing antennas are separated by only a few centimeters. This the channel also increases the symbol time per channel, since adds spatial diversity to the system, which can be exploited many slow symbols will be sent in parallel instead of many to improve resilience to fades. There is also a power gain fast symbols in sequence. This adds time diversity because from multiple receive antennas because, everything else be- the channel is more likely to average out fades over a longer ing equal, two receive antennas will receive twice the signal. period of time. These factors combine to improve performance at a given 802.11 makes use of the frequency diversity provided by distance, and hence increase range. OFDM by coding across the data carried on the subcarriers. This uses a fraction of them for redundant information that 3.1 Receive Diversity Techniques can later be used to correct errors that occur when fading Consider the arrangement in Figure 2(a). One transmit reduces the SNR on some of the subcarriers. First, a con- antenna at a node is sending to two receive antennas at a MRC: Performance Improvement

Probability that MRC’s SNR is Under Threshold γ0 0 10

−1 10

M = 1

−2

out 10 P

M = 2

M = 3

−3 10 M = 4

Lower probability M = 10

(better) ↓ M = 20 −4 10 −10 −5 0 5 10 15 20 25 30 35 40 10log ( / ) 10 γ γ0 10log'( )/+( Lower threshold SNR à

Figure 7.5: Pout for MRC with i.i.d. Rayleigh fading. • Two “9”s of reliability improvement between one (i.e., no MRC) and two Rayleigh fading, where p (γ)isgivenby(7.16),itcanbeshownthat[4,Chapter6.3] MRC branches γΣ M M 1 m ∞ 1 Γ − M 1+m 1+Γ P = Q( 2γ)pγ (γ)dγ = − − , (7.18) b Σ 2 m 2 16 0 # $ m=0 & ' # $ ! " % where Γ = γ/(1 + γ). This equation is plotted in Figure 7.6. Comparing the outage probability for MRC in" Figure 7.5 with that of SC in Figure 7.2 or the average probability of error for MRC in Figure 7.6 with that of SC in Figure 7.3 indicates that MRC has significantly better performance than SC. In Section 7.7 we will use a different analysis based on MGFs to compute average error probability under MRC, which can be applied to any modulation type, any number of diversity branches, and any fading distribution on the different branches.

7.6 Equal-Gain Combining

MRC requires knowledge of the time-varying SNR on each branch, which can be very diﬃcult to mea- sure. A simpler technique is equal-gain combining, which co-phases the signals on each branch and then θ combines them with equal weighting, αi = e− i . The SNR of the combiner output, assuming equal noise power N in each branch, is then given by

2 1 M γΣ = ri . (7.19) NM & ' %i=1 212 i3π/4 p y1=3e x 9/ 13

rotate, scale by 3/p13 x n 1 p13 4/p13 rotate, scale by 2/p13 n expected 1 n2 -iπ/6 y2=2e x

i3⇡/4 i⇡/6 Figure 3: MRC operation on a sample channel. The channel gains are ~h = 3e , 2e ,withGaussian h i noise ~n = n1,n2 of expected power 1. The antennas have respective SNRs of 9 and 4. To implement MRC, h i the receiver multiplies the received signal ~y = ~hx + ~n by the unit vector ~h⇤/ ~h , where ~h⇤ denotes the complex || || conjugate of ~h. This operation scales each antenna’s signal by its magnitude, and rotates the signals into the same phase reference before adding them. (For graphical clarity, we depict the common phase vertically, rather than at 0). The resulting sum has magnitude p13, and expected noise power 1 because the scaling is normalized. Thus, by coherently combiningSelection received signals from diDiversity,↵erent antennas, the MRC in output Frequency has the expected SNR of 13. In systems with OFDM, MRC is performed separately for each subcarrier. second node. This is known as a 1x2 system. Real systems 0 may have more than two receive antennas, but two will suf- fice for our explanation. With this setup, each receive an- -5 tenna receives a copy of the transmitted signal modified by the channel between the transmitter and itself. The chan- -10 nel gains hij are complex numbers that represent both the amplitude attenuation over the channel as well as the path- dependent phase shift (see Figure 3 for a graphical example). -15 The receiver measures the channel gains based on training A C fields in the packet preamble. Note that the gains di↵er for -20 B and SEL each subcarrier (in frequency-selective fading) as well as for AB (MRC) each antenna. The question now is how to combine the two ABC (MRC) received signals to make best use of them. Normalized power (dB) -25 We consider two diversity techniques to show the extremes. -20 -10 0 10 20 The simplest method is to use the antenna with the strongest Subcarrier index signal (hence the largest SNR) to receive the packet and ig- nore the others. We will call this method• Antennas SEL, for selection A and FigureC experience 4: Frequency-selective different fadingfades over on testbeddifferent subcarriers combining. This is essentially what is done by 802.11a/g APs links: the figure shows, for an example 5.2 GHz link, with multiple antennas. It helps with reliability, because the received power measured on each subcarrier for both signals are unlikely to be bad, but it wastes perfectly individual antennas and under SEL and MRC diver- good received power at the antennas• thatSelection are not chosen. Combiningsity, (“SEL”) normalized improves to the strongestbut certain subcarrier subcarriers power. still experience fading The better method is to add the signals from the two antennas together. However, this cannot be done by simply tudes of 3 and 2. With expected noise power 1, these gains superimposing their signals, or we• willMRC have just increases recreated correspond power toand SNRs flattens of 9 and 4, given nulls, that aleading signal’s power to is fewer bit errors the e↵ects of multi-path fading. Rather, the signals should the square of its magnitude. The MRC receiver scales each each be delayed until they are in the same phase; then, the antenna’s signal by its magnitude, normalized to the total; power in the signals will combine coherently. To do this, delays the signals to a common phase reference; and then the receiver needs a dedicated RF chain for each antenna to adds them. The result has magnitude p13, and the normal- process the signals. This increases the hardware complexity ized weighted sum of noise still has expected power 1. The and power consumption, but yields better performance. combined signal thus has a resulting sum SNR of 13. As a twist in the above, the signals are also weighted by As an example of how MRC and SEL work in 802.11, con- their SNRs. This gives less weight to a signal that has a sider Figure 4. This figure shows the wireless signal strength larger fraction of noise, so that the e↵ects of the noise are not of each subcarrier using three antennas for a real 802.11n link amplified. The result is maximal-ratio combining, or MRC. in our indoor wireless testbed. The subcarrier strengths are MRC is known to be optimal (it maximizes SIMO capacity), measured in decibels normalized to the strongest subcarrier and produces an SNR that is the sum of the component strength. This figure gives a much more detailed view than SNRs. Note that in frequency-selective fading, this process metrics such as the RSSI (Received Signal Strength Indi- is performed di↵erently for each subcarrier according to its cation) for a link, which gives only the sum of the signal specific channel gains. strength over all subcarriers. Figure 3 depicts MRC operation graphically for a 1x2 For each antenna labeled A, B, or C, the signal varies channel. In this example, the two channel gains have magni- over the channel, changing gradually from one subcarrier MRC’s Capacity Increase • MRC with M branches increases SNR – Increased Shannon capacity

• Sub-linear (logarithmic) capacity increase in M: – !"#$ = &' ( log 1 + . ( /01 bits/second/Hz Plan 1. Today: Diversity in Space – Receive Diversity – Transmit Diversity • Channel reciprocity • Transmit beamforming • Introduction to Space-Time Coding: Alamouti’s Scheme

2. Next time: Multiplexing in Space

3. Next time: Interference Alignment

19 An Aside: Radio Channels are “Reciprocal”

a2,d2,τ2

Transmitter T Receiver R a1,d1,τ1

()*+,/. ()*+0/. • Forward channel (T to R) is ℎ"# = %& ' + %)'

• Switch T and R roles, changing nothing else: ()*+,/. ()*+0/. – Reverse channel (R to T) is ℎ#" = %& ' + %)' = ℎ"# – The reverse radio channel is “reciprocal”

• Practical radio receiver circuitry induces differences between ℎ"#, ℎ#"

20 Transmit Diversity: Motivation • More space, power, processing capability available at the transmitter? – Yes, likely! e.g. Cellular base station, Wi-Fi AP transmitting downlink traffic to mobile

• But, a (possible) requirement: May need to know the radio channel at the transmitter before the transmission commences – cf. receive diversity: channel from preamble reception

• Then, a tension: Separate transmit antennas for path diversity – Antenna 1, Antenna 2, transmit radio non co-located • Then, harder to move transmit signals, radio channel measurements i.e. channel state information (CSI) between the three locations

21 Transmit Beamforming: Motivation

“receiver”

• Suppose the transmitter knows the CSI to receivers

• Transmitters align their signals so that constructive interference occurs at the single receive antenna – Align before transmission, not after reception (receive beamforming)

22 Transmit Beamforming • Leverage channel reciprocity, receive beamforming “in reverse”

• Send one data symbol x from two antennas

,'( % &i3π/4 ) p y1=3e# x ℎ 9/ 13 # = % # & '( rotate,Tx scale1 by 3/p13 ) x n 1 p13 '(+ Receive p & 4=/ %13# ℎ* rotate,Tx scale2 by 2/p13 n expected 1 n2 y =2e-iπ/6x 2 ,'(+ %*&

i3⇡/4 i⇡/6 Figure 3: MRC operation on a sample channel. The channel gains are ~h = 3e , 2e ,withGaussian h i noise ~n = n1,n2 of expected power 1. The antennas have respective SNRs of 9 and 4. To implement MRC, h i the receiver multiplies the received signal ~y = ~hx + ~n by the unit vector ~h⇤/ ~h , where ~h⇤ denotes the complex • Multiply (pre-code) x by the complex|| conjugate|| of each channel conjugate of ~h. This operation scales each antenna’s signal by its magnitude, and rotates the signals into the same phase reference before adding them. (For graphical clarity, we depict the common phase vertically, rather than at 0). The resulting sum has magnitude p13, and expected noise power 1 because the scaling is 23 normalized. Thus, by coherently combining received signals from di↵erent antennas, the MRC output has the expected SNR of 13. In systems with OFDM, MRC is performed separately for each subcarrier. second node. This is known as a 1x2 system. Real systems 0 may have more than two receive antennas, but two will suf- fice for our explanation. With this setup, each receive an- -5 tenna receives a copy of the transmitted signal modified by the channel between the transmitter and itself. The chan- -10 nel gains hij are complex numbers that represent both the amplitude attenuation over the channel as well as the path- dependent phase shift (see Figure 3 for a graphical example). -15 The receiver measures the channel gains based on training A C fields in the packet preamble. Note that the gains di↵er for -20 B and SEL each subcarrier (in frequency-selective fading) as well as for AB (MRC) each antenna. The question now is how to combine the two ABC (MRC) received signals to make best use of them. Normalized power (dB) -25 We consider two diversity techniques to show the extremes. -20 -10 0 10 20 The simplest method is to use the antenna with the strongest Subcarrier index signal (hence the largest SNR) to receive the packet and ig- nore the others. We will call this method SEL, for selection Figure 4: Frequency-selective fading over testbed combining. This is essentially what is done by 802.11a/g APs links: the figure shows, for an example 5.2 GHz link, with multiple antennas. It helps with reliability, because the received power measured on each subcarrier for both signals are unlikely to be bad, but it wastes perfectly individual antennas and under SEL and MRC diver- good received power at the antennas that are not chosen. sity, normalized to the strongest subcarrier power. The better method is to add the signals from the two antennas together. However, this cannot be done by simply tudes of 3 and 2. With expected noise power 1, these gains superimposing their signals, or we will have just recreated correspond to SNRs of 9 and 4, given that a signal’s power is the e↵ects of multi-path fading. Rather, the signals should the square of its magnitude. The MRC receiver scales each each be delayed until they are in the same phase; then, the antenna’s signal by its magnitude, normalized to the total; power in the signals will combine coherently. To do this, delays the signals to a common phase reference; and then the receiver needs a dedicated RF chain for each antenna to adds them. The result has magnitude p13, and the normal- process the signals. This increases the hardware complexity ized weighted sum of noise still has expected power 1. The and power consumption, but yields better performance. combined signal thus has a resulting sum SNR of 13. As a twist in the above, the signals are also weighted by As an example of how MRC and SEL work in 802.11, con- their SNRs. This gives less weight to a signal that has a sider Figure 4. This figure shows the wireless signal strength larger fraction of noise, so that the e↵ects of the noise are not of each subcarrier using three antennas for a real 802.11n link amplified. The result is maximal-ratio combining, or MRC. in our indoor wireless testbed. The subcarrier strengths are MRC is known to be optimal (it maximizes SIMO capacity), measured in decibels normalized to the strongest subcarrier and produces an SNR that is the sum of the component strength. This figure gives a much more detailed view than SNRs. Note that in frequency-selective fading, this process metrics such as the RSSI (Received Signal Strength Indi- is performed di↵erently for each subcarrier according to its cation) for a link, which gives only the sum of the signal specific channel gains. strength over all subcarriers. Figure 3 depicts MRC operation graphically for a 1x2 For each antenna labeled A, B, or C, the signal varies channel. In this example, the two channel gains have magni- over the channel, changing gradually from one subcarrier Plan 1. Today: Diversity in Space – Receive Diversity – Transmit Diversity • Channel reciprocity • Transmit beamforming • Introduction to Space-Time Coding: Alamouti’s Scheme

2. Next time: Multiplexing in Space

3. Next time: Interference Alignment

24 Alamouti Scheme: Motivation • Suppose transmitters don’t know CSI information to receiver: what to do?

1. Naïve beamforming (just send same signals) – Signals would often cancel out

2. Repetition – Each antenna takes turns transmitting same symbol • Receiver combines coherently

– Use M symbol times • Increases diversity (“SNR” term in Shannon capacity) • Cuts Shannon rate by 1/M factor

25 Alamouti Scheme • Scope: A two-antenna transmit diversity system (M = 2)

• Sends two symbols, s1 and s2, in two symbol time periods:

Symbol Time Period 1 2 ∗ Antenna 1: Send !" Send −!$ ∗ Antenna 2: Send !$ Send !"

• Then, by superposition the receiver hears:

Symbol Time Period 1 2 ∗ ∗ Receiver hears: ℎ"!" + ℎ$!$ −ℎ"!$ + ℎ$!"

26 Alamouti Receiver Processing

Symbol Time Period 1 2 ∗ ∗ Receiver hears: y[1] = ℎ'(' + ℎ*(* +[2] = −ℎ'(* + ℎ*(' ∗ ∗ ∗ + 2 = ℎ*(' − ℎ'(*

+[1] ℎ' ℎ* (' ∗ = ∗ ∗ + [2] ℎ* −ℎ' (*

• Rewrite into two equations in two unknowns (s1 and s2): – (Receiver has CSI information)

∗ (' ℎ' ℎ* +[1] ∝ ∗ ∗ (* ℎ* −ℎ' + [2]

• But, what’s happening in terms of the physical wireless channel?

27 Intuition for Alamouti Receiver Processing • Start with the inverted channel matrix:

∗ !" &" &( ,[1] ∝ ∗ ∗ #$ ℎ$ −ℎ+ , [2]

• Consider the computation for s1: – Rotate ,[1] by −1" ∗ – Rotate , [2] by 1( – Sum the result

28 Alamouti: Impact of Phase Rotations

• Consider the computation for s1: – Rotate ![1] by −&' ∗ – Rotate ! [2] by &* – Sum the result

Symbol Time Period 1 2 ∗ ∗ ∗ Receiver hears: y[1] = ℎ./. + ℎ1/1 ! 2 = ℎ1/. − ℎ./1 ↓ ↓ ↓ ↓ Phase after rotation: 2 &* − &' 2 &* − &'

29 Alamouti: Receiver-Side Picture

Symbol Time Period 1 2 ∗ ∗ ∗ Receiver hears: y[1] = ℎ'(' + ℎ*(* + 2 = ℎ*(' − ℎ'(* ↓ ↓ ↓ ↓ Phase after rotation: / 01 − 02 / 01 − 02

• Receiver then sums all terms above:

Received signal: Q

∡4* − 4' s2

s2 s1 s1 I

30 Alamouti: Interpretation

∗ !" ℎ" ℎ# ([1] ∝ ∗ ∗ !# ℎ# −ℎ" ( [2]

• Two new signal dimensions:

1. Multiply two received symbols by the top column of H ∗ - – Name this dimension ℎ" ℎ# – s1 arrives along this dimension (only!)

2. Multiply two received symbols by the lower column of H ∗ - – Name this dimension ℎ# −ℎ" – s2 arrives along this dimension (only!)

31 Alamouti: Performance

∗ % ∗ % • Two dimensions: ℎ" ℎ$ , ℎ$ −ℎ" Received Q signal:

s1 s1 I • Send half power on each antenna |,|./|,|. – For both symbols, '() = - . $0.

• Rate gain from enhanced SNR, and maintains one symbol per symbol time

32 Multi-Antenna Diversity: Summary • Leverage path diversity – Decrease probability of “falling into” to deep Rayleigh fade on a single link

• Defined new “dimensions” of independent communication channels based on space

33 Thursday Topic: MIMO II: Spatial Multiplexing

Friday Precept: Exploiting Doppler