DESIGN FOR DIGITAL COMMUNICATION SYSTEMS VIA SAMPLED-DATA H CONTROL

Transcription

1 DESIG FOR DIGITAL COMMUICATIO SYSTEMS VIA SAMPLED-DATA H COTROL M agahara 1 Y Yamamoto 2 Department of Applied Analysis and Complex Dynamial Systems Graduate Shool of Informatis Kyoto University Kyoto JAPA Abstrat: The design proedure for the equalization of digital ommuniation hannels is developed based on the sampled-data H ontrol theory The proedure provides transmitting/reeiving filters so as to minimize the error between the original signal and the reeived signal with a time delay and to redue the noise added to the hannel While the system has an ideal sampler a zero-order hold and a time delay the design problem an be redued to a finite-dimensional disrete-time problem using the FSFH (fast-sample and fast-hold) approximation umerial examples are presented to illustrate the effetiveness of the proposed method Copyright 21 IFAC Keywords: Sampled-data systems; Digital ommuniations; Digital filters; H-infinity optimization; Time delay 1 ITRODUCTIO owadays the importane of digital ommuniations is inreasing owing to the rapid growth of the Internet the ellular phones and so on(proakis 1989) Iigital ommuniation espeially in pulse amplitude modulation (PAM) or in pulse ode modulation (PCM) the analog signal whih is to be transmitted is sampled and beomes a disrete-time signal In the onventional way the harateristi of the analog signal is not onsidered and hene the whole system is regarded as a disrete-time system For example one usually assumes that the original analog signal is bandlimited up to the yquist frequeny But in reality no signals are entirely band-limited This paper proposes a new design methodology based on the sampled-data ontrol theory(chen and Franis 1995b) that takes aount of intersample behaviors or frequeny omponents beyond the yquist frequeny iisrete-time Re- ently the sampled-data ontrol theory is applied to some digital signal proessing systems(chen and Franis 1995a; Khargonekar and Yamamoto 1996; Yamamoto et al 1997; agahara and Yamamoto 2) We propose a design for the digital ommuniation system via the sampled-data ontrol In (Erdogan et al 2) a disrete-time H design for reeiving filters or equalizers is introdued but no design for transmitting filter is mentioned However it is diffiult to attenuate both the signal reonstrution error and the additive noise by only an equalizing after the signal is reeived Therefore we show the transmitting filter design whih is exeuted in the same way as the reeiving filter design Moreover we introdue the H method whih takes aount of a tradeoff between the signal reonstrution error and the energy of transmitted signal with an appropriate weighting funtion Design examples are presented to illustrate the effetiveness of the proposed method 1 nagahara@asikyoto-uajp 2 yy@ikyoto-uajp; author to whom all orrespondene should be addressed

2 w F y S y h d u K T v d C d K R H h P ++ Fig 1 Digital ommuniation system 2 DESIG PROBLEM FORMULATIO The blok diagram Figure 1 shows a digital ommuniation system whih is known as PAM or PCM The inoming signal w L 2 [ ) goes through an analog low-pass filter F and beomes y whih is nearly (but not entirely) band limited The filter F governs the frequeny-domain harateristi of the analog signal y 3 The signal y is then sampled by the sampler S h to beome a disrete-time signal (or PAM signal) y d with sampling period h Then the signal is shaped or enhaned by the transmitting digital filter K T to the signal v d to be transmitted to a ommuniation hannel The transmitted signal v d is orrupted by the ommuniation hannel C d and the additive noise In PCM ommuniation is also onsidered as the noise generated by quantizing and oding error The reeived signal goes through the reeiving digital filter K R whih tries to attenuate the orruption and the noise then beomes an analog signal u by the hold devie H h with sampling period h and smoothed by an analog low-pass filter P and finally we have the output signal z Our objetive is to reonstrut the original analog signal y by the transmitting filter K T and the reeiving filter K R against the orruption aused by the hannel C d and the additive noise Therefore onsider the blok diagram Figure 2 whih is the signal reonstrution error system for the design In the diagram the following points are taken into aount: The time delay e Ls is introdued beause we allow a ertain amount of time delay for signal reonstrution The transmitted signal v d is estimated with a weighting funtion W z beause the energy or the amplitude of the transmitted signal v d is usually limited The noise has a frequeny harateristi W n Then our design problem is as follows: Problem 1 Given stable analog filters F (s) and P (s) digital filters (weighting funtions) W n (z) and W z (z) and a hannel model C d (z) find digital filters K T (z) and K R (z) whih minimizes J 2 := z e 2 + z d 2 L2 l2 w L 2 l 2 w 2 L + n 2 d 2 (1) l 2 3 In the onventional design F is onsidered as an ideal filter whih has a ut-off frequeny up to the yquist frequeny w e Ls y + F e + y S d v h K d u T C d K d u R H h P z + W z W n z d Fig 2 Signal reonstrution error system w e Ls y + F e + y S d v h K d T u C d K d u R H h P z + W n Fig 3 Error System T R for reeiving filter design 3 DESIG ALGORITHM 31 Deomposing Design Problems Problem 1 is a simultaneous design problem of a transmitting filter and a reeiving filter and it is diffiult to solve the problem diretly Therefore we introdue a deomposition of the design problem into two steps that is the design for the reeiving filter and that for the transmitting filter Obviously the transmitting filter K T annot attenuate the additive noise hene the reeiving filter K R has to play that role Moreover K R has to reonstrut the original signal from the orrupted signal (if K R did not have to reonstrut the optimal filter will be learly K R = ) Therefore we first design the reeiving filter K R in order to reonstrut the original signal and to attenuate the noise by the blok diagram Figure 2 with W z = and with K T = 1 Theesign the transmitting filter by the blok diagram Figure 2 with W n = and with K R whih is obtained the previous design that is we onsider the hannel as K R C d The design proedure is as follows: Step 1(Design for reeiving filter) Find a reeiving filter K R whih minimizes T R 2 := e 2 L2 w L 2 l 2 w 2 L + n 2 d 2 (2) l 2 in Figure 3 with fixed K T (the initial filter is K T = 1) Step 2(Design for transmitting filter) Find a transmitting filter K T whih minimizes T T 2 := w L 2 e 2 L2 + z d 2 l2 w 2 L 2 (3) in Figure 4 with K R whih is obtained in the previous step

3 w e Ls + y e F y d v d u d u S h K T C d K R H h P z W z z d Fig 4 Error System T T design for transmitting filter e d e w w d S h/ T H h/ Fig 5 fast sample/hold disretization 32 Fast Sample/Hold Approximation The design problems (2) and (3) involve a ontinuous time delay omponent e Ls and hene they are infinite-dimensional sampled-data problems To avoid this diffiulty we employ the fast sample/hold approximation method (Keller and Anderson 1992; Yamamoto et al 1999) By the method our design problems (2) and (3) are approximated to finite-dimensional disrete-time problems assuming that the delay time L to be mh where m is a positive integer: Theorem 2 Assume that L = mh m Then (1) for the error system T R in Step 1 there exist finite-dimensional disrete-time systems {T R : = 1 2 } suh that lim T R = T R (2) for the error system T T in Step 2 there exist finite-dimensional disrete-time systems {T T : = 1 2 } suh that lim T T = T T PROOF By the fast sample/hold method we approximate ontinuous-time inputs and outputs to disrete-time ones via the ideal sampler and the zero-order hold that operate in the period h/ (Figure 5) Then apply the disrete-time lifting(yamamoto et al 1997) L to the disretized input/output signal e d and w d we an get the lifted signals ẽ d := L (e d ) w d := L (w d ) Then we an approximate the ontinuous signal as h h e L 2 ẽ d l 2 w L 2 w d l 2 Moreover define T R 2 := T T := w d l 2 w d l 2 ẽ d 2 l 2 w d 2 l + 2 h 2 l 2 ẽ d 2 l + 2 h z d 2 l 2 w d l 2 where the systems T R and T T are approximated to finite-dimensional disrete-time systems then we an show T R T R T T T T as by using the method as shown in (Yamamoto et al 1999) under the assumption L = mh One the problems have been redued to disretetime problems they an be solved by a ontrol design toolbox suh as those given by MATLAB The resulting disrete-time approximant is given by the following: Theorem 3 The approximated disrete-time systems T R and T T are given as follows: T R := F l (G R K R ) T T := F l (G T K T ) [ [ z m F G R := d ] ] [ ] P d Cd K T JF d W n [ z m ] [ ] F d Pd K G T := R C d W z F d J := [I ] F d (z) := A F d A 1 F d B F d A 2 F d B F d B F d C F C F A F d C F B F d C F A 1 F d C F A 2 F d B F dc F d A 3 F d B F d A P d A k P d B P d k=1 C P D P P d (z) := C P A P d C P B P d + D P C P A 1 P d C P A k P d B P d + D P k=2 A F d := e AF h BF d := h h e AF t B F dt A P d := e AP h BP d := e AP t B P dt [ ] [ AF B F (s) =: F AP B P (s) =: P C F C P D P ] where F l (G K) denotes the linear frational transformation of plant G and filter K

4 ẽ d w d 3 G R 2 1 K R Fig 6 Disrete-time H design problem for reeiving filter K R 1 2 ẽ d z d G T w d K T Fig 7 Disrete-time H transmitting filter K T design problem for Then our design problems (2) and (3) are redued to finite-dimensional disrete-time H problems whih are shown in Figure 6 and Figure 7 Fig 8 Gain response of filters:sampled-data H design (transmitting filter: solid reeiving filter: dots) and disrete-time H design (dash) Design for W z = 4 DESIG EXAMPLES We present a design example for 1 F (s) := 1s + 1 P (s) := 1 W n(z) := 1 C d (z) := z 1 52z z 3 with sampling period h = 1 and time delay L = mh = 2 An approximate design is exeuted here for = 8 Here we design without onsidering the transmitting signal that is W z (z) = For omparison the disrete-time H design(erdogan et al 2) is also done Figure 8 shows the gain responses of the filters and Figure 9 shows the frequeny response of T ew whih is the system from the input w to the error e and Figure 1 shows that of T zn from the additive noise to the output z Compared with the disrete-time design the sampled-data one shows better frequeny response both in T ew and in T zn Moreover we an say that only an equalizer is not able to attenuate the orruption aused by the hannel and the additive noise that is we need an appropriate transmitter for transmission To explain this fat we show a simulation of these ommuniation systems The input signal y is the retangular wave whose amplitude is 1 and the noise is the disrete-time sinusoid [k] = sin(2k) Figure 11 shows the output z Fig 9 Frequeny response of T ew : sampled-data H design (solid) and disrete-time H design (dash) with the reeiving filter and the transmitting filter designed via sampled-data method and Figure 12 shows that with the reeiving filter designed iisrete-time (and without any transmitting filter) We see that the former shows muh better reonstrution against the noise than the latter 42 Design for W z (z) Then we onsider the design with the estimation of the transmitting signal v d that is W z (z) We observe from Figure 8 that the transmitting filter shows high gain around the yquist frequeny (ie ω = π) and hene we take W z (z) = r z 1 z + 5 as the weighting funtion of the transmitting signal whose gain harateristi is shown in Figure

5 '& %$ " #! Fig 1 Frequeny response of T zn : sampleddata H design (solid) and disrete-time H design (dash) Fig 13 Gain harateristi of the weighting filter W z (z) 15 Time Response H norm [db] Fig 11 Time response with sampled-data design r Fig 14 Relation between r and T ew (solid) T vw (dash) 15 Time Response 3 Fequeny Response Fig 12 Time response with disrete-time design 13 where the parameter r = 21 The other design parameters are the same as the example above Figure 14 shows the H norm of T ew and T vw whih is the system from w to v d in Figure 2 whih varies with r [ 5] We an take aount Fig 15 Gain response of transmitting filters designed for r = 21 (solid) and r = (dash) of a trade-off between the error attenuation level and the amount of the transmitting signal with Figure 14 For example we hoose r = 21 in order to attenuate the error less than 26dB

6 funtion By the fast sample/hold method the design is redued to a finite dimensional disrete time design whih an be easily implemented to CAD (eg MATLAB) Fig 16 Frequeny response of T ew designed for r = 21 (solid) and r = (dash) Fig 17 Frequeny response of T vw designed for r = 21 (solid) and r = (dash) Figure 15 shows the gain response of transmitting filters designed for r = and r = 21 We an see that the new filter shows better attenuation than the filter designed for r = at high frequeny Figure 16 shows the frequeny response of the error system T ew We see that the attenuation level of T ew designed for r = 21 is less than 26dB Figure 17 shows the frequeny response of T vw We an see that the amount of the transmitting signal is attenuated at high frequeny 6 REFERECES Chen T and B A Franis (1995a) Design of multirate filter banks by H optimization IEEE Trans on Signal Proessing SP Chen T and B A Franis (1995b) Optimal Sampled-Data Control Systems Springer Erdogan A T B Hassibi and T Kailath (2) On linear H equalization of ommuniation ahnnels IEEE Trans on Signal Proessing SP-48(11) Keller J P and B D O Anderson (1992) A new approah to the disretization of ontinuoustime ontrollers IEEE Trans on Automati Control AC-37(2) Khargonekar P P and Y Yamamoto (1996) Delayed signal reonstrution using sampleddata ontrol Pro of 35th Conf on Deision and Control pp agahara M and Y Yamamoto (2) A new design for sample-rate onverters Pro of 39th Conf on Deision and Control pp Proakis J G (1989) Digital Communiations MGraw Hill Yamamoto Y A G Madievski and B D O Anderson (1999) Approximation of frequeny response for sampled-data ontrol systems Automatia Yamamoto Y H Fujioka and P P Khargonekar (1997) Signal reonstrution via sampleddata ontrol with multirate filter banks Pro of 36th Conf on Deision and Control pp COCLUDIG REMARKS We have presented a new method of designing transmitting/reeiving filter iigital ommuniation An advantage here is that an analog otimal performane an be obtained and this an be advantageous in audio/speeh signal transmission Another advantage is that the trade-off between the attenuation of the reonstrution error and the energy of the transmitting signal is onsidered by the H design with an appropriate weighting