Bandwidth expansion in joint source- channel coding and twisted modulation

Transcription

1 Banwith expansion in joint source- channel coing an twiste moulation Pål Aners Floor Tor A Ramsta

2 Outline. Introuction to nonlinear mappings. Banwith expansion an Twiste moulation. Some history. Optimum noise immunity for low intensity noise. Noise immunity in presence of strong noise. Calculation an simulation of the performance for a : mapping when using an Archimeean spiral. Summary an conclusions. References.

3 Source-channel mapping in general. ρ ψ x = ρ y

4 Performance measure. Compare the mappings to OPTA to etermine their performance: W log b + c n = B log X OPTA = SNR CSNR = = D b D X + c n k m

5 Banwith expansion an Twiste moulation. The following figure shows a general :n banwith expaning system.

6 Banwith expansion an Twiste moulation. If the source has an infinite alphabet, the mapping operation will be a piecewise continuous curve. For a linear system this curve will be a straight line trough the origin BPAM. These systems are quite far from the optimum boun except for very low CSNR. To make the system get closer to the optimum boun, the curve nees to be nonlinear. If we have a nonlinear signal curve an moulates it s given values onto n orthonormal waveform functions, we get what is referre to as twiste moulation.

7 Some History. The Russian V. A.Kotel nikov consiere optimum noise immunity in the mse sense in his octoral thesis [Kotel nikov 59]. This thesis gives some pretty goo ieas on how to construct banwith expaning systems that get quite close to the optimum bouns, although information theoretical consierations are absent. Kotel nikov s thesis ates back to 947, but was not known in the west until 959, when a translation of his thesis was publishe. Shannon treate this type of schemes in 949 [Shannon49]! Kotel nikov consiere optimum noise immunity for: i Amplitue an time iscrete sources. ii Amplitue continuous an time iscrete sources. iii Amplitue an time continuous sources. In this presentation we will be concerne with ii only.

8 Influence of noise when transmitting separate parameter continuous alphabet values. In its shown that the conitional probability for receiving the parameter value, when the signal xt is receive, is given by Gaussian noise case: p x = K x e T n x t s, t Where s is the channel signal some signal curve, an is some constant epenent on xt, but inepenent of an t. We recognise this as the a posterior probability. This will be have a maximum for some. xm K x

9 Optimum noise immunity in the presence of weak noise. In the low intensity noise case signal curve s can be approximate by the tangent space at the transmitte point: Minimising the mse: gives the minimum low noise error:,,, xm xm xm t s t s t s + = = m m x m x p ε min ' xm n s ε ρ =

10 Optimum noise immunity in the presence of weak noise. This result tells us, that given a small noise variance, the mse can be mae smaller by increasing the signal curves stretch factor length of tangent vector. So for a imension n an a given channel power constraint, we nee to make the signal curve as long as possible, i.e. it has to be twiste aroun in the given constraine region. We have a nonlinear system!

11 Noise immunity in presence of strong noise. There is a certain limit CSNR where the low noise approximation is no longer vali not the case for linear systems. We will encounter what is calle the threshol effect. The threshol effect makes the system break own an is funamentally unavoiable when using nonlinear signal curves. One nee to keep two parts of the curve large enough, so that this effect occur with a very small probability. This gives a trae-off between weakan strong noise consierations.

12 : Banwith expansion. Archimeean spiral. As we have seen previously the spiral works very well as a banwith reucing system spiral in the source space. Its therefore tempting to see if this is the case for banwith expansion as well spiral in the channel space. This was motivate by a comment in [Cover & Thomas9]: A goo rate- istortion coe is in general a goo channel coe, an vice versa uality Its also a convenient structure for a power constraine Gaussian channel because of its circular symmetry. From the previous low noise analysis it shoul also be convenient.

13 Banwith expansion. Archimeean spiral. Structure in the receivers channel space. Approx. source signal Pos signal + noise Neg signal + noise Threshol effect ρ ρ ρ n r = ϕ cosϕ i + sinϕ j π ρ ρ ρ n r = ϕ cosϕ + π i + sinϕ + π j π

14 Banwith expansion. Archimeean spiral. I will use the approximation to the inverse curve length function to map the source onto the spiral. ϕ = a a 0.6 is an amplification factor. This will make the velocity vector s norm equal for all. This again yiels inepenence between signal an noise after reception. = q ρ s q x' q a q ϕ - Inverse curve length function

15 System Reference system. HSQLC [Cowar00] HSQLC structure in channel space.

16 Archimeean spiral. Performance. This system outperforms the HSQLC system for CSNR above 3B. Simulation n=0.88 Comparison several n OPTA Archemeian spiral, n=0.88 Optimal point OPTA : HSQLC Archemeian spiral Linear system BPAM SNR B 30 SNR CSNR B CSNR

17 Calculation of the performance of the Archimeean spiral. The channel CSNR is given by: The receive SNR is given by: CSNR = C ms n x SNR = We nee to calculate: ε l + i The mean square carrier power. ii The low noise error. iii The threshol error. the calculations are base on [Thomas et. al. 75]. ε a

18 Calculation of the performance of the Archimeean spiral Will assume a Gaussian input signal. The signal will be truncate at ±. The truncation error can be neglecte if we choose the signals stanar eviation small enough. We assume an AWGN channel. i The mean square carrier power: ii Low noise error: = = 5 π e a p s C ms ρ [ ], in the interval the curve length for Where L is ' = = = = ε ε ε ε L s p p n l n wn wn l ρ

19 Calculation of the performance of the Archimeean spiral. iii The threshol error is quite ifficult to calculate. The following figure shows how it s one approximately :

20 Calculation of the performance of the Archimeean spiral. Since the spiral is uniform, the probability for the threshol effect to occur, is the same for all given. We get the following expression: an where = + = + = + + a s a s p s s erf n a π π ε

21 Calculate vs. simulate results for the Archimeean spiral OPTA Calculate performance SNR CSNR

22 Comparison to other techniques. [McRae7] This comparisation is for a recieve SNR of 40B.

23 Summary/conclusions. In a banwith expaning system we must consier both weak an strong noise immunity, an there is a trae-off between them. To get close to the optimum boun, one nee to use nonlinear systems. When using nonlinear expaning systems the threshol effect is funamentally unavoiable. A systems has been foun, an verifie both mathematically partly an by simulation for : expansion.

24 References [Cowar00] Cowar Helge. Joint Source- Channel Coing: Utilization in Image Communications. PhD thesis. NTNU 00. [Kotel nikov59] Kotel nikov V. A. translate by Silverman R. A The Theory of Optimum Noise Immunity first eition, McGraw-Hill, New York-Toronto-Lonon 959. [McRae7] McRae, Daniel D. Performance Evaluation of a New Moulation Technique, IEEE Trans. Comm. Tech. Vol: Com-9 No. 4, Aug 97 [Shannon49] Shannon Claue E. Communications in the presence of noise, Proc. IRE, 37 No. 0- Jan 949. [Thomas et. al.75] Thomas, C. Melvil; May, Curtis L an Welti, George R. Hybri Amplitue-an- Phase Moulation for Analog Data Transmission, IEEE Trans. Comm. Tech. Vol: Com-3 No. 6, June 975 [Cover&Thomas] Cover, T. M. an Thomas, J. A. 99, Elements of Information Theory, John Wiley & sons, Inc., New York, NY, USA. [Wozen&Jacobs65] Wozencraft John M an Jacobs Mark J. Principles of Communication Engeneering secon eition, Wiley & Sons, New York-Lonon-Syney 965.