Wallace
In the paper 'Automatic Secret Keys From Reciprocal MIMO Wireless Channels: Measurement and Analysis' Jon W. Wallace and Rajesh K. Sharma describe two methods of quantizing a channel. These methods are called channel quantization with guardband (CQG) and channel quantization alternating (CQA) with CQA being based on CQG which itself is a generalization of the simple channel quantization proposed by A. Sayeed and A. Perrig [1] as well as S. Marthur et al. [2]
Use Case
Both CQG and CQA are to be used if two parties Alice and Bob receive a similar signal and want to receive the same quantization of said signal. A common use case is Alice and Bob calculating a shared cryptographic key based on the channel state information which is reciprocal. The following quantization techniques allow for error correction of the quantization process.
CQG
First of all Alice and Bob agree on how many diﬀerent symbols shall be extracted from a single value of the signal. This is done by deﬁning M diﬀerent equally probable quantization sectors (QS), each representing one symbol. Methods for determining such sectors are described by J. Wallace in another publication [3].
Similar to quadrature amplitude modulation (QAM), these sectors are deﬁned by M' = √M/ 2 one-sided (positive axis) and symmetric quantization intervals (QIs) for both the real (QIRe) and the imaginary (QIIm) part of the signal. The gap between the diﬀerent QIs is called the guardband. If a value lies within the guardband g, it is dropped by both parties. It is apparent that largening the guardband increases robustness but decreases the rate extractable information. The QIs and the guardband are displayed in figure 1. There are two diﬀerent methods of communicating on whether or not a measured value is valid.
- One-way handshake: Alice sets a guardband indicator bit (GIBAlice) to either 0 if the value is valid or 1 if the value lies within the guardband, The GIB is then transmitted to Bob, who discards the current sample if GIBAlice = 1. One issue arises if Alice accepts a value while Bob does not since only Alice can communicate possible errors.
- Two-way handshake: If a two way handshake is used, both Alice and Bob send a GIB and drop the value if either GIBAlice = 1 or GIBBob = 1. Else, the value is accepted.
CQA
A major issue with CQG is the loss of information through dropping all bits within the guardband. Wallace et al. provide an alternate solution for this issue. The idea is to have two diﬀerent quantization maps, based on Alice’s measurement. Similar to CQG, M equally probable QS are deﬁned. These QS again consist of a range of complex numbers. Alice then analyzes the values of the signal step-by-step. For each real and imaginary part of the value, Alice sends a QM bit to Bob which is deﬁned by the exact position of the real/imaginary part within the QS. If the value lies within the lower half of the QS, QM is set to 0. Otherwise it is set to 1. He then uses one of two overlapping mapping tables, that have been defined before the start of the quantization, to determine the according quantization, according to ﬁgure 2. The overlap of these two tables serves as a more efficient alternative for the guardband of CQG. This process ensures, that every single value is clearly identiﬁable by Bob, as one can see for the following examples, compare figure 2:
- Value 1: Alice measures QS = 1 and sets QM = 0. Bob uses mapping table QSQM=0 and measures the QS to be 0. In this case, both mapping tables would work.
- Value 2: Alice measures QS = 1 and sets QM = 1. Bob uses mapping table QSQM=1 and measures the QS to be 1. If mapping table QSQM=0 would have been used, the resulting value could have been either 0 or 1.
- Value 3: Alice measures QS = 2 and sets QM = 0. Bob uses mapping table QSQM=0 and measures the QS to be 2. If mapping table QSQM=1 would have been used, the resulting value could have been either 0 or 1.
- Value 4: Alice measures QS = 3 and sets QM = 0. Bob uses mapping table QSQM=0 and measures the QS to be 0. In this case, only mapping table QSQM=0 yields the correct result as QSQM=1 would have measured a 3.
Once the real and the imaginary part of the complex value are analyzed by both Alice and Bob, both parties receive the same QS and thereby symbol for the value. The major advantage of CQA over CQG is that no values need to be dropped as the guardband has been replaced by a ﬂexible mapping scheme. Intuitively, this increases the eﬃciency of the quantization.