Channel-Adaptive Waveforms, Part I: Singular Value Decomposition of a Matrix-Valued Impulse Response

Orthogonal frequency division multiplexing (OFDM) is becoming the modulation method of choice for dealing with wideband, delay-spread (frequency dependent) wireless channels. In effect, the technique divides the channel into a multiplicity of parallel narrowband flat-fading subchannels. Perfect orthogonality among the subchannels is assured by using a cyclic prefix whose duration exceeds the delay-spread, i.e., the memory, of the channel. However for certain applications, the cyclic prefix occupies an inordinate fraction of the transmission time. This inefficiency can occur when short OFDM symbol intervals are used, in ad-hoc networks, or for MIMO multiple-messaging that uses multiple basestations. If the transmitters have timely knowledge of the channel response then it may be possible to obviate the need for a cyclic prefix by using channel- adaptive waveforms. This paper lays the mathematical foundation for channel- adaptive waveforms by developing a space-time extension of the singular value decomposition (svd). The svd, a standard tool for theoretical and computational matrix analysis, represents a matrix as a finite sum of outer products, each of which comprises an input eigenvector, a nonnegative, real singular value, and an output eigenvector. Our representation for a matrix-valued impulse response, either time-invariant or time-varying, involves a countably infinite sum of outer products comprising an input vector eigenfunction, a nonnegative- real scalar singular value, and an output vector eigenfunction, and it provides the most concise and elegant description of the action of the channel when driven by finite-duration input signals.