In this problem we explore the use of singular value decomposition (SVD) as an alternative to the discrete Fourier transform for vector coding. This approach avoids the need (or a cyclic prefix, with the channel matrix being formulated as where the sequence h0, h1,. . . , h2, denotes the sampled impulse response of the channel. The SVD of the matrix H is defined by h = U [? 1 ON,v]V+ where U is an N-by-N unitary matrix and V is an (N + v)-by-(N + v) unitary matrix; that is, UU+ = I VV+ = 1 where I is the identity matrix and the superscript + denotes Hermitian transposition. The ? is an N-by-B diagonal matrix with singular values ?n, n = 1, 2, . . , N. The On,v is an N-by-v matrix of zeros.
(a) Using this decomposition, show that the N sub-channels resulting from the use of vector coding are mathematically described by Xn = ?n An + Wn
The Xn is an element of the matrix product U+x, where x is the received signal (channel output) vector, the An is the nth symbol an + jbn and Wn is a random variable due to channel noise.
(b) Show that the signal-to-noise ratio for vector coding as described herein is given by where N* is the number of channels for each of which the allocated transmit power is nonnegative, (SNR)n, is the signal-to-noise ratio of sub channel n, and F is a prescribed gap.
(c) As the block length N approaches infinity, the singular values approach the magnitudes of the channel Fourier transform. Using this result, comment on the relationship between vector coding and discrete multitone