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Math · Advanced Math
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4. (Rank, Column Space, Null Space, MIMO Channel) (18 points) As we saw in the Homework, a MIMO Channel comprising of M transmit antennas and N receive antennas can be expressed as an N×M matrix. The input-output relation can be Where y is the data at the N receive antennas. H is the channel matrix and χ is the data at the M transmit antennas. Answer the following questions a. What is the maximum rank of the channel matrix H? [1 point] b. Using your knowledge of the rank as the number of linearly independent rows (and columns) of the matrix, what does the rank of channel matric H physically denote? [4 points] Now consider a MIMO system with three transmit and three receive antennas. Assume that the channel matrix can be represented as 3 5-1 Answer the following questions. c. Find the rank of the channel. 3 points] d. Find the range (column space) of the channel. [3 points] e. Find the kernel (null space) of the channel. Briefly state the physical interpretation of this null space. [4 points] f. Now suppose the channel matrix in Eqn 1 denotes the channel between our transmitter and an enemy eavesdropper. For obvious reasons, it is desirable that the eavesdropper is unable to obtain the transmitted data. What signal vector would you choose to transmit in such a scenario? [3 points]

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