Ergodic Capacity of Continuous-Time, Frequency-Selective Rayleigh Fading Channels with Correlated Scattering

Transcription

1 Ergodic Capacity of Continuous-Time, Frequency-Selective Rayleigh Fading Channels with Correlated Scattering IEEE Information Theory Winter School 2009, Loen, Norway Christian Scheunert, Martin Mittelbach, and Konrad Schubert Ý Department of Electrical Engineering and Information Technology Ý Department of Mathematics Dresden University of Technology, Dresden, Germany March 31, 2009

2 Motivation, Objective Motivation how to calculate the ergodic capacity of ultra-wideband (UWB) channels physically motivated channel models are standard for UWB analytical investigations are scarcely possible with these models Objective propose a mathematically motivated UWB channel model provide a procedure to calculate the ergodic capacity present appropriate tools of second-order calculus for this task calaculate the ergodic capacity of the proposed channel Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 1 of 16

3 Outline 1 Channel Model 2 Tools 3 Ergodic Capacity Calculation 4 Conclusion Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 2 of 16

4 Channel Model

5 Channel Model Assumptions, Objective Assumptions single user, single antenna average power constraint on the channel input full channel state information (CSI) at receiver no CSI at transmitter but knowledge of fading statistics Objective stochastic multipath model for equivalent lowpass channels frequency-selective Rayleigh fading [1],[2], block fading limited bandwidth, continuous time 1 U. Schuster, and H. Bölcskei, Ultrawideband Channel Modeling on the Basis of Information-Theoretic Criteria, IEEE Trans. Wireless Commun M. Mittelbach, C. Scheunert, and F. Bruder, Impact of UWB Channel Modeling on Outage and Ergodic Capacity, Proc. of IEEE ICUWB Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 3 of 16

6 Channel Model Overview Overview when block fading is assumed, the channel impulse response (CIR) of frequency-selective fading channels may be written as tµ n k 1 «k ³ t t k µ with appropriate gain coefficients «k, and behaviour process ³ physical model: gain «k, and behaviour ³ of k-th cluster 3, 4 mathematical model: gain «k, and sampling ³ at position t k 3 J. Kunisch and J. Pamp, Measurement Results and Modeling Aspects for the UWB Radio Channel, Proc. of IEEE UWBST J. R. Foerster, M. Pendergrass, A. F. Molisch, A channel model for ultra wideband indoor communication, Mitsubishi Eletric Research Laboratory, Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 4 of 16

7 Channel Model Example, Approach Example frequency-selective Rayleigh fading channel impulse response it appears in UWB and other systems with the TX input filter taken into account 3e-4 3e-4 2e-4 2e-4 CIR 1e-4 0e+0-1e-4 modified CIR 1e-4 0e+0-1e-4-2e-4-2e-4-3e Time -3e Time Approach assume the channel to be quadratic mean (q.m.) continuous stationary process with mean zero that is attenuated by an integrable function Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 5 of 16

8 Channel Model Formulae, Discussion Channel Model Formulae let be a q.m. continuous stationary Gaußian process onê let be the Hilbert transform of let be the pre-envelope of, i.e., i let g be a Riemann integrable (attenuation) function onê the frequency-selective Rayleigh fading channel model is given by g Discussion On Infinite Continuous Time no finite bandwidth finite time dilemma integration overêmight be easier than calculating finite sums theory of processes with finite second-order moments is well known Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 6 of 16

9 Tools

10 Tools Stochastic Integrals in Q.M. Theorem: Let be a zero mean processes with finite second-order moments, and f be a Riemann integrable function onê. The q.m. integral ½ f tµ tµdt ½ exists if and only if ½ ½ I f tµf sµr t sµds dt ½ ½ exists as a Riemann integral. Further, if exists we have E µ 0 E 2 I 5 H. Cramér and M. R. Leadbetter, Stationary and Related Stochastic Processes, Wiley, E. Wong and B. Hajek, Stochastic Processes in Engineering Systems, Springer, Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 7 of 16

11 Tools Stochastic Integrals in Q.M. Corollary: If the covariance function R is bounded and continuous on Ê Ê, then the q.m. integral ½ f tµ tµdt ½ exists. Corollary: Let a b ¾Êwith a b. If the covariance function R is continuous on a b a b, then the q.m. integral exists and we have b a tµdt E µ b a E tµ dt Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 8 of 16

12 Tools Spectral Representation Theorem: (Bochner-Chintschin) Let R be the covariance function of a zero mean q.m. continuous stationary processes onê. Then, there exists a unique finite measure F on the Borel -field ofê, such that ½ R tµ e iwt F dw µ ½ Theorem: (Cramér) Let be a zero mean q.m. continuous stationary process. Then, there exists a unique random measure on the Borel -field ofê, such that ½ tµ e iwt dw µ ½ Moreover, the relation E dw µ 2 F dw µ holds. 7 M. B. Priestley, Spectral Analysis and Time Series, Academic Press, Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 9 of 16

13 Ergodic Capacity Calculation

14 Ergodic Capacity Calculation Overview Ergodic Capacity Formulae let W be the intervall of channel frequencies being considered let x be the signal to noise ratio (SNR), i. e., x 0 let be the Fourier transform of the ergodic capacity formulae is given by C x µ E ln 1 x w µ 2 dw Steps of Calculation 1 determine the properties of the process, and hence 2 make sure the Fourier transform of exists 3 find out the distribution of the process 2 4 verify if integral and expectation operator may be interchanged 5 calculate the ergodic capacity W Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 10 of 16

15 Ergodic Capacity Calculation Step 1 Theorem:, are q.m. continuous jointly stationary Gaußian processes with mean zero. Further, the random variables tµ, tµ having the same index t are independent. Proof: With the spectral representation of show that µ ¼ is a zero mean Gaußian vector process with covariance matrix function Ò ¼ Ó E tµ tµ sµ sµ R t sµ E tµ sµ E tµ sµ R t sµ where E tµ sµ 2 ½ sin v t sµ F dv µ 0 Corollary: is a q.m. continuous stationary Gaußian process with mean zero. Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 11 of 16

16 Ergodic Capacity Calculation Step 2 Theorem: (Existence of ) There exists a zero mean process with finite second-order moments being the Fourier transform of the process, i.e., the relation ½ w µ e iwt tµdt ½ holds. Proof: Since g, show that e iw g is Riemann integrable. R is bounded and continuous, as is a q.m. continuous stationary process. Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 12 of 16

17 Ergodic Capacity Calculation Step 3 Theorem: (Properties of 2 ) For all w the random variables Re w µ, Im w µ are independent and identically Gaussian distributed with mean zero. Thus, w µ 2 Re w µ 2 Im w µ 2 is exponentially distributed with the parameter 2R w w µ, where ½ ½ R w w µ cos w t sµ R t sµg tµg sµdsdt ½ ½ ½ ½ sin w t sµ E tµ sµ g tµg sµdsdt ½ ½ Moreover, 2 is a q.m. continuous process. Proof: Show that Re Im µ ¼ is a zero mean Gaußian vector process with covariance matrix function 2R diag 1 1µ. Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 13 of 16

18 Ergodic Capacity Calculation Step 4 Theorem: Let x be a nonnegative variable. Then the relation holds. E W ln 1 x w µ 2 dw W E ln 1 x w µ 2 dw Proof: Since the interval of integration W is finite, it suffices to show that the process ln 1 x 2 is q.m. continuous. As d 1 ln 1 yµ dy 1 y 1 holds for y 0 the function ln 1 µ is Lipschitz continuous. Hence E ln 1 x w µ 2 ln 1 x v µ 2 2 E w µ 2 for w v as 2 is a q.m. continuous process. v µ Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 14 of 16

19 Ergodic Capacity Calculation Step 5 Theorem: For any positive variable x the ergodic capacity might be calculated as 1 1 C x µ exp 2xR w w µ Ei 1 2xR w w µ dw W where Ei is the integral exponential function. Proof: Since 2 is exponentially distributed with the parameter 2R w w µ, the assertion immediately follows form ½ y E ln 1 x 2 1 2R with integration by part. 0 ln 1 xyµ exp 2R dy Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 15 of 16

20 Conclusion

21 Conclusion, Outlook Conclusion a mathematically motivated channel model has been proposed tools of second-order calculus have been presented a procedure to calaculate the ergodic capacity has been provided the ergodic capacity of the proposed channel has been calaculated Outlook fit the stationary part of the channel to a stationary ARMA process estimate the ARMA model order (useful for general theory) estimate the ARMA model parameters (useful for exact capacity values) investigate random fields instead of processes, i. e., avoid the block fading approach Christian Scheunert, et al., Ergodic Capacity of Continuous-Time..., IEEE IT WS, March 31, Page 16 of 16

22 Thank you for your attention! Any questions? This research was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant FI 470/7-1.