1 On the Channel Capacity of Wireless Fading Channels C. D. Charalambous and S. Z. Denic School of Information Technology and Engineering, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada. {chadcha, CDC 2002

2 Introduction The channel capacity is one of the most important parameters of any communication channel because it gives the maximal theoretical rate at which one can transmit data. Present an encoding and decoding strategy with feedback, which achieves the channel capacity of additive Gaussian wireless fading channels Although the feedback does not increase the channel capacity, it can be used to reach the channel capacity. CDC 2002

3 Introduction cont. The approach considered, which involves computation of channel capacity, optimal encoding and and decoding gives an interesting insight into simultaneous source and channel coding for wireless channels. The results are applicable to problems in control Engineering which employ wireless communication links CDC 2002

4 Introduction cont. Assumption: The channel is modeled as a random process, where the attenuation, phases, and delays are random processes. The derivation, employs the mutual information in a general setting, defined through the Radon-Nikodym derivative. Results: Optimal encoding/decoding strategies when the information sources are Non-stationary Gaussian source; Random variable source. CDC 2002

5 Introduction cont. It turns out that the encoder, which reaches the channel capacity uses linear feedback and that the optimal decoder minimizes mean square error. CDC 2002

6 Channel Model and Mutual Information in Presence of Feedback is a complete probability space with filtration and finite time on which all random processes are defined - source signal - is state channel process CDC Wiener process independent of X, representing thermal noise

7 The received signal can be modeled as is a delay,is a Doppler shift, r is an amplitude, is a phase, M is a number of resolvable paths, and - - is the non-anticipatory functional representing encoding is a carrier frequency CDC 2002 (1) Channel Model and Mutual Information in Presence of Feedback

8 Also, we defined the following measurable spaces associated with stochastic processes, their filtrations, and truncations of filtrations CDC 2002 Channel Model and Mutual Information in Presence of Feedback

9 The following assumption is make (1) has the unique strong solution. Definition 1. The set of admissible encoders is defined as follows CDC 2002 Channel Model and Mutual Information in Presence of Feedback

10 Theorem 1. Consider the model (1). The mutual information between the source signal X and received signal Y over the interval [0,T], conditional on the channel state , I T (X,Y |F  , is given by the following equivalent expressions CDC 2002 i) ii) Channel Model and Mutual Information in Presence of Feedback

11 CDC 2002 iii) Channel Model and Mutual Information in Presence of Feedback Definition 2. Consider the model (1). The Shannon capacity of (1) is defined by subject to the power constraint on the transmitted signal (2)

12 Upper Bound on Mutual Information Theorem 2. Consider the model (1). Suppose the channel is flat fading. The conditional mutual information between the source signal X, and the received signal Y is bounded above by It can be proved that this upper bound is indeed the channel capacity, by observing that there exists a source signal with Gaussian distribution (3) CDC 2002 such that the mutual information between that signal X and received signal Y is equal to the upper bound (3).

13 Upper Bound on Mutual Information The capacity is CDC 2002

14 Optimal Encoding/Decoding Strategies for Non-Stationary Gaussian Sources We assume that the channel is flat fading (M=1), it is known to both transmitter and receiver, and the source is Gaussian, nonstationary given by F t and G t are Borel measurable and bounded functions, G t G t tr >0,  t  [0,T], W is a Wiener process independent of Gaussian random variable X 0 ~ CDC 2002 (4)

15 Optimal Encoding/Decoding Strategies for Non-Stationary Gaussian Sources CDC 2002 Definition 3. The set of linear admissible encoders L ad, where L ad  A ad, is the set of linear non-anticipative functionals A with respect to source signal X, which have the form The received signal is then The processes W, and N are independent, and the power constraint (2) becomes (6) (5)

16 Optimal Encoding/Decoding Strategies for Non-Stationary Gaussian Sources Decoding. The optimal decoder in the case of mean square error criteria is the conditional expectation, while the error covariance is Encoding. The optimal encoder is derived by using the equation for the optimal decoder, and the equation for power constraint (6). CDC 2002

17 Theorem 3 (Coding theorem for stochastic source). If the received signal is defined by the equation (5), the source by (4), then the encoding reaching the upper bound, optimal decoder, and corresponding error covariance are respectively given by Optimal Encoding/Decoding Strategies for Non-Stationary Gaussian Sources CDC 2002

18 Optimal Encoding/Decoding Strategies for Random Variable Sources Theorem 4 (Coding theorem for random variable source). If a source signal X, which is Gaussian random variable is transmitted over a flat fading wireless channel, then the optimal encoding and decoding with feedback reaching the channel capacity are CDC 2002

19 Conclusions In the case of non-stationary Gaussian source - encoder, and decoder depend of channel process Z and received signal Y - error covariance on the statistics of the source through functions F t and G t - the encoding consists of two terms, the first plays the role of power control, and the other is the decoding error - this result is consistent with water pouring argument - the result tells us that the optimal way of communications is by only sending the error and not the whole signal - the fact that error covariance depends on the source suggests that the power of transmitted signals for different types of sources should be adjust in order to have the same performance for different sources

20 In the case of random variable source - If we set F t = G t =0, in the result for error covariance of the non-stationary source, the result will be the same as for the random variable source, and that was expected - For the flat fading channel, it can be proved that the error covariance is meaning that the error decreases exponentially with time and the transmitted power but also increases if the attenuation is large. Conclusions

21 Future Work Future work can include the following topics -Joint source and channel coding for wireless channels -Computation of the channel capacity for multipath channels -Application of the results to some problems in control and estimation