MegaMIMO and Full Duplex Radios
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* Examples of physical layer wireless technologies
* Two different approaches to improving wireless throughput at the physical layer
* Being commercialized in some form (MegaMIMO Inc. and Kumu Networks)
* Relevant to emerging wireless standards (e.g., 5G and recent versions of WiFi)
* However, both papers implement their techniques in the WiFi context.

What problems are they trying to solve?
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* MegaMIMO (or MIMO in general): Harnessing multiple antennas to increase throughput.
  --> So far we have looked at time and frequency as two sets of resources for communication.
  --> MIMO introduces another axis: space (or more precisely independent antennas)
  --> Prior work: Much work on MIMO using transmitters and receivers each with multiple antennae.
  --> MegaMIMO's innovation: Making MIMO work in a distributed context with *independent* transmitters and receivers.
      Essentially treat a collection of APs and clients as one large MIMO sender+one large MIMO receiver by synchronizing them.
* Full Duplex: Transmitting and receiving at the same time. Another "axis" for resources.
  --> Taken for granted on Ethernet: typically have separate wires.
  --> Historically impossible for wireless media: transmitted signal typically overwhelms receiver
  --> Prior work: Use multiple antennae to subtract out transmitted (or self-interfering) signal.
  --> Bharadia et al.'s innovation: Making it work with a single antenna.
* What these papers show
  --> We still don't fully understand how much capacity is available in a wireless network.
  --> There is still more work to do (e.g., combining MIMO and Full Duplex, discovering other axes)
* Great examples of research papers
  --> Problems are simple to state ("how to harness multiple antennae" and "how to simultaneously transmit and receive")
  --> Rich theory that guides system design
  --> But considerable ingenuity required to make things work. The papers are quite dense and a bit hard to fully digest.
  --> And important (any capacity increase in the physical layer is a huge deal given our insatiable appetite for bandwidth).

Wireless communication basics:
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* Sender transmits symbols.
* Think of the symbol as a voltage level where the voltage level represents a bit pattern.
  --> E.g., 5 V could represent the bit 1. 0 V could represent the bit 0.
      Here 2 voltage levels represent 1 bit.
  --> Could divide it up more finely:
      0 V represents bits 00. 2 V represents bits 01. 3 V represents bits 11. 4 V represents bits 10
      Now 4 voltage levels represent 2 bits.
* Think of the voltage level abstractly as a real number standing for a certain bit pattern.
  --> The number of bits in the bit pattern is log_2(# of voltage levels)
* The arrangement of voltage levels is called a constellation.
  --> How tightly voltage levels can be packed is governed by the amount of noise (or signal-to-noise ratio).
* In practice, can send two voltages at once on orthogonal waves (sine and cos).
* So you can think of the symbol as a complex number with a real voltage level and an imaginary voltage level.
  --> There is nothing complex about the voltages itself. It's just a representation.

Wireless channels:
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* Can think of a wireless channel as a transformation from input symbols to output symbols.
* The channel is typically denoted h.
* The channel can attenuate the symbols in strength and/or delay them.
* Wireless systems have some way to estimate the channel (using known pilot symbols)
  so that they can "undo" the channel when receiving.

MIMO
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* Let's start with MIMO itself.
* Simple setup: One transmitter and one receiver. Each has 2 antennas.
* This already shows us something very surprising.
* Let's say we want to transmit symbols x1 and x2 to client 1 and 2.
* Simplest way to do this: First transmit x1 and then x2 (this is what traditional WiFi does)
* Can we do better?
* Try transmitting them at the same time.
* The problem is that the signals get mixed up at the receiver.
  --> Can't untangle them.
  --> But can if we can ensure that each receiver receives only a modified version of their own signal.
  --> This is what beamforming does.

Beamforming
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* Multiply the transmitted signals with some complex numbers (precoding)
* Choose these numbers so that you invert the channel matrix.
* This way each receiver only gets a scaled copy of their own signal.
* Works great when all transmitters are synchronized to the same clock.
* Otherwise the channel is continuously changing due to differences in clock frequencies between the transmitters.
* Performing beamforming with the inverse of the original channel matrix doesn't work.

MegaMIMO
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* Correct for phase and frequency offsets between different transmitters
* Can rewrite channel matrix into time dependent and time independent parts
* Basic idea: Time dependent precoding can be applied independently at each transmitter
* Still doesn't work. Can't measure phase exactly at each AP locally.
* Some more linear algebra:
  --> Decomposes matrix into part that standard WiFi can handle using standard carrier frequency offset techniques ...
      (exp(j.omega_t1.t)*R(t))
  --> ... and a time dependent precoding matrix that only needs us to estimate relative phase and frequency offsets.

MegaMIMO details
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* Track phase difference at the beginning of each packet accurately.
* Then track frequency difference to extrapolate phase difference through the duration of the packet.
  Doesn't accumulate too much error within the lifetime of one packet.

Diversity vs. multiplexing gains
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* Multiplexing is what the paper devotes most of its attention to. This is what beamforming does
  --> Essentially deliver packets to different clients concurrently through precoding.
* Diversity is another benefit of MIMO.
  --> Deliver the same data across multiple independent paths so that you can combine them to get better throughput.
  --> Basic idea:
      If the received signals were corrupted on one path, they can be recovered from the received signal on another path.

MegaMIMO evaluation
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* Figure 9 illustrates the benefit of MIMO in different regimes.
* Figure 10 shows this benefit is uniform across all clients.
* Figure 11 shows the diversity gains of MIMO.

Full duplex paper
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* Goes even deeper down the rabbit hole.
* While MegaMIMO still operates within the domain of discrete time, the full duplex paper also relies on analog electronics.
* What's the problem it's trying to solve?
* Making wireless a full duplex medium: Transmit and receive simultaneously
* The problem: self interference from transmitted signal.
  --> Transmitted signal overwhelms the receiver antenna.
  --> Need to cancel this out so that you can receive the actual signal and not your own "echo".
* Figure 1 and 2 illustrate this self interference problem.
* Need to cancel out considerable amount of self interference (110 dB)
* What is 110 dB?
  --> It's a ratio between two powers. The ratio is about 10**11!
  --> Essentially the transmitted signal is about 10**11 larger than what you want it to be.

Full duplex ideas
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* Analog cancelation:
  --> To remove transmit noise.
  --> To reduce power of the signal before it hits the receiver circuitry.
      --> Receiver circuitry can't handle more than a certain power level. This is what is called saturation.
* Digital cancelation of both linear and non-linear functions of the original transmitted symbols.
* Difference between analog and digital cancelation:
  --> Analog cancelation doesn't really assume anything about how the signal is transformed from input (x) to output (y).
      --> This is required because the noise term is anyway random and can't be modeled.
      --> And the noise term is significant (50 dB)
  --> Digital cancelation does assume a model about how the signal is transformed from the input (x) to output (y)
      --> Either linear transformation or non-linear transformation.

Analog cancelation:
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* Prior work: Used antenna separation. For instance, use two separate transmit antennae that destructively interfere at receiver.
* This work: Carry out analog cancelation without separate antennas.
  --> Use a set of delay lines with fixed delays but programmable attenuations.
  --> Solve an optimization problem online to minimize error between reconstructed and original signal.
  --> Do all this without assuming anything about the nature of the signal.

Full duplex results:
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* Both analog and digital cancelation are required. Problem is too hard without using both.
* In combination achieves the required 110 dB cancelation.
* Median gain of 87% using full duplex (ideally get 100%).

Comparing the two schemes:
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* Full duplex: Don't need more antennas, but can't scale beyond a factor of 2.
* MIMO: Can scale indefinitely, but need more antennas.
* MIMO: applicable in most crowded WiFi scenarios (e.g., malls)
* Full duplex: A bit rarer to require simultaneous transmit and receive.
  --> One application is wireless relays.
* Both technologies seem to be gaining traction in emerging wireless standards.