## In this problem we explore the use of singular value decomposition (SVD) as an alternative to the…

In this problem we explore the use of singular value decomposition (SVD) as an alternative to the DFT for vector coding. This approach avoids the need for a cyclic prefix, with the channel matrix being formulated as

where the sequence h0, h1, ¼, hv denotes the sampled impulse response of the channel. The SVD of the matrix H is defined by

where U is an N-by-N unitary matrix and V is an (N + v)-by-(N + v) unitary matrix; that is,

where I is the identity matrix and the superscript denotes Hermitian transposition. The is an N-by-N diagonal matrix with singular values is an N-by-v matrix of zeros. a. Using this decomposition, show that the N subchannels resulting from the use of vector coding are mathematically described by

The Xn is an element of the matrix product , where x is the received signal (channel output) vector. An is the nth symbol an + jbn and Wn is a random variable due to channel noise.

b. Show that the SNR for vector coding as described herein is given by

where is the number of channels for each of which the allocated transmit power is nonnegative, (SNR)n is the SNR of subchannel n, and Γ 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 multitoned