# In this problem we explore the use of singular value

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