ON THE CAPACITY OF LINEAR TIME-VARYING CHANNELS Sergio Barbarossa, Anna Scaglione Infocom Dept., Univ. of Rome “La Sapienza”, Roma, ITALY {sergio , annas}Qinfocom. ing . uniromai . it ABSTRACT Linear time-varying (LTV) channels are often encountered in mobile communications but, as opposed to the linear time-invariant (LTI) channels case, there is no a well e s tablished theory for computing the channel capacity, or providing simple bounds to the maximum information rate based only on the channel impulse response, or predicting the structure of the channel eigenfunctions. In this paper, we provide: i) a method for computing the mutual informa- tion between blocks of transmitted and received sequences, for any finite block length; ii) the optimal precoding (de- coding) strategy to achieve the maximum information rate; iii) an upper bound for the channel capacity based only on the channel time-varying transfer function; iv) a time- frequency representation of the channel eigenfunctions, re- vealing a rather intriguing, but nonetheless intuitively jus- tifiable, bubble structure. 1. INTRODUCTION The knowledge of the channel capacity has a clear impor- tance in digital communications because it establishes the value of the maximum information rate that can be trans- mitted through the channel with an arbitrarily low bit error rate (BER), provided that sufficient redundancy is added to the transmitted sequence via precoding. For linear time- invariant (LTI) channels with additive Gaussian noise, it is well known how to compute the channel capacity, how to obtain an upper bound based only on the channel transfer function and which is the form of the channel eigenfunc- tions (complex exponentials with linear phase). Unfortu- nately, due to inherent mathematical tractability problems, the theoretical framework for LTV channels, in spite of their increasing importance in mobile communications, is not as well established as in the case of LTI channels. Indeed, in [4] the general case of linear channels is considered in the continuous-time domain but, due to evident difficulties to solve the integrals, no closed form solutions are provided neither for the capacity, nor for an easy form for its bound, nor for the channel eigenfunctions. There are more recent works addressing the computation of the capacity of LTV channels, where the channel variability is modeled in a sta- tistical sense and the capacity is evaluated using ensemble averages. However, in some cases, e.g. [5] and [6], the LTV channels are memoryless, whereas in [7], to simplify This work was supported by ARL contract no. DAAL01-98-Q-1053. the mathematical tractability, only the two-ray propaga- tion model is considered. In this work, we assume that the channel is modeled, as usually in mobile communications, as the superposition of a discrete number of paths, each one characterized by a complex amplitude, a time delay and a Doppler frequency shift and we provide: i) an explicit for- mula for the mutual information, for any block length P and its asymptotic behavior for P going to infinity; ii) the optimal precoding (decoding) strategy maximizing the in- formation rate; iii) an upper bound for the channel capacity, based only on the knowledge of the channel time-varying transfer function; iv) the properties of the channel eigen- functions. To evaluate the impact of the channel variability on the capacity, we compare the performance of equivalent LTI and LTV channels (where the LTI channel is derived from the LTV channel simply setting the Doppler frequen- cies to zero). 2. CHANNEL MODEL AND OPTIMAL CODING STRATEGY The most general form for a continuoustime (CT) LTV channel is given by the following input/output relationship m y(t) = / h(t; T)2(t - T)dT + v(t) (1) -m where v(t) is additive noise and the channel is completely characterized by the time-varying impulse response h(t; 7) or its counterpart, the time-varying transfer function H(t; f), defined as the Fourier ’lkansform (FT) of h(t;~) with re- spect to T. In mobile communications, where the channel is often characterized by multiple propagations, with the generic kth path described by a complex amplitude hk, a delay Tk and a Doppler frequency shift fk, the timevarying impulse response and transfer function are: K-1 and k=O (3) k=O respectively. The equivalent discrete-time model of a causal time-varying channel is y(n) = h(n, k)z(n-k)+w(n). In this paper, we assume that the channel memory is finite, that is h(n,k) = 0 for k > L. The most convenient form for expressing the 1/0 relationship in the DT case is the matrix form. Specifically, considering a block of M received symbols y, the output is related to the input as y = Txtv, where T is the M x P channel matrix, x is the P x 1 input vector, v is the M x 1 noise vector and P = M + L. 0-7803-5041-3/99$10.00 0 1999 IEEE 2627