LTV System Modelling

Transcription

1 Helinki Univerit of Technolog S Potgraduate Coure in Radiocommunication Fall 2000 LTV Stem Modelling Heikki Lorentz Sonera Entrum O heikki.lorentz@onera.fi Januar 23 rd 200

2 Content. Introduction 2. Time-Domain Decription for Linear Time-Varing Stem 2.. The Impule Repone 2.2. The Superpoition Integral 3. Frequenc-Domain Repreentation of Time-Varing Stem 3.. Two-Dimenional Frequenc repone 3.2. Bandwidth Relation in Time-Varing Stem 3.3. Sampling Rate 4. Propertie of Linear Time-Varing Stem 4.. Propertie of Convolution 4.2. Interconnection of Linear Time-Varing Stem 5. Model for LTV Stem 5.. Linear Differential Equation with Time-Varing Coefficient 5.2. Separable Model 5.3. Tapped Dela-Line Channel Model 6. Concluion. Introduction Linear element in a communication tem can be time-invariant (LTI) or time varing (LTV) in nature. The aumption of time invariance implie that the propertie of tem being modelled do not change over (long period of) time. Whether to ue a time-invariant or time-varing model i uuall determined b the rate at which the characteritic of the communication tem being modelled are changing in comparion to other parameter of the communication tem uch a the mbol rate. If the tem parameter are changing at a rate approaching the mbol rate, a time-varing model i appropriate. The fixed radio link channel characteritic are due to change in the atmopheric condition, which tpicall have a time contant of everal minute to hour. If the communication link i operating at a mbol rate of 00 Mbit/, then the time contant aociated with the channel variation i ver long compared to the mbol time. If the objective of imulation i BER etimation, then, during the etimation interval, the channel can be aumed to be in a tatic tate and a time-invariant model can be ued.

3 The long-term behaviour of the channel and it impact on long-term tem performance can be evaluated b analing tem performance over a erie of naphot of tatic channel condition, uing a different time-invariant model for each naphot. In the other hand, when the radio link channel i affected b multipath phenomena, then ver fat change in the channel characteritic ma occur and time-varing model mut be ued. While a time-varing model ma not be needed for BER etimation, uch a model will be necear to tud the behaviour of receiver ubtem uch a nchronier and equalier. Alo, in the mobile radio channel the movement of the mobile terminal caue fat change in the characteritic of the channel compared to the tranferred mbol rate. 2. Time-Domain Decription for Linear Time-Varing Stem 2.. The Impule Repone The commonl ued form of the time-varing impule repone i modelled on the definition of the impule repone for linear time-invariant tem a ( t τ ) = Γ[ δ ( t τ )] h, () where τ i the time (from the origin) when the impule i applied, and Γ i the linear time-varing tem operator. The mot important application of time varing model in communication tem i in the modelling of time-varing (multipath) channel. To decribe the behaviour of uch channel, Kailath introduced everal alternative formulation for the impule repone. The mot convenient for purpoe of modelling communication channel i to define c ( τ ˆ,t) a the repone meaured at time t to a unit impule applied at time t τˆ, c ( τˆ, t) = Γ[ δ ( t ( t τˆ ))] = Γ[ δ ( τˆ )] The above impule repone i referred to a the channel impule repone. The relationhip of the definition of the impule repone to the channel impule c τ, t = h t, t τ h t, τ = c t τˆ, t. repone i ( ˆ ) ( ) or ( ) ( ) (2)

4 Since the tem i time-varing, the impule repone will change a a function of both the time at which the impule i applied and the time at which the output i meaured. Hence both c ( τ ˆ,t) and h ( t,τ ) are urface in three-dimenional pace. Cro ection of thee urface are illutrated in Figure. Figure : Two-dimenional repreentation of a time-varing impule repone Figure a how two hpothetical impule repone ( t,τ ) Oberve that caualit require ( t,τ ) c ( τ ˆ,t) = h ( t, t τ ) and τ = t τ the two function. h for value of τ = 0, 2. h = 0 for t < τ. Becaue of the relationhip ˆ, it i imple to etablih a correpondence between While the impule repone of an LTI tem maintain the ame functional form irrepective of when the impule wa applied at the input, the impule repone of an LTV tem depend on when the input wa applied The Superpoition Integral The repone of the tem to an arbitrar input i determined b a uperpoition integral. Uing the impule repone, the uperpoition integral i or ( t) h( t τ ) x( τ ) =, dτ ( t) h( t τ ) x( τ ) In term of the channel repone we have (3) =, (4)

5 or ( t) c( τˆ, t) x( t τˆ ) = dτˆ ( t) = c( τˆ, t) x( t τˆ ) (5) (6) When the tem i time-invariant, h ( t, τ ) = h( t τ ) and c ( τˆ, t) = c( τˆ ), and (3),(5) i the convolution integral. Although the convolution integral i alo ometime referred to a the uperpoition integral, we will reerve the term uperpoition for the timevaring cae in (3) and (5). For caual tem thi integral ma be written a and ( ) = t h( t, τ ) x( τ ) dτ becaue h( t, τ ) = 0 for τ < 0 0 ( t) = c( τˆ, t) x( t τˆ ) dτˆ where c( τˆ, t ) = 0 for τˆ t > 0 (7) (8) 3. Frequenc-Domain Repreentation of Time-Varing Stem For an LTV tem one can develop the notion of a tranfer function, though a time-varing tranfer function C ˆτ ( f, t), b impl taking the Fourier tranform of c τ ˆ,t with repect to τˆ a ( ) C τˆ ( f, t ) c( τˆ, t ) j 2 π fτˆ = e d τ ˆ (9) If the tem i lowl time-varing, then the concept of frequenc repone and bandwidth can be applied to C ˆτ ( f, t). Wherea the LTI tem i characteried b a ingle impule repone function and a ingle tranfer function, the LTV tem i characteried b a famil of impule repone function and tranfer function, one function for each value of t, a hown in Figure. The invere tranform i given b

6 ( τ, t) C ( f t) c j2πfτˆ ˆ = ˆ, e df (0) τ The output (t) of a tem c ( ˆ,t) τ with the input x(t) can be determined b ( t) C ( f t) X ( f ) where X(f) i the Fourier tranform of x(t). j2πft = ˆ, e df () τ 3.. Two-Dimenional Frequenc repone Kailath introduced a frequenc-domain function C τˆ, t τˆ, j2πνt j2πfτˆ j2πνt ( f, ν ) = C ( f, t) e dt = c( τˆ t) e e dτ dt (2) The frequenc ν i aociated with the rate of change of the tem and f i aociated with the frequenc of the tem excitation. From (0) and () we get the tem output frequenc repone ( ν ) = Cτ ˆ, ( f, f ) X ( f ) df (3) Y t ν Equation (3) i the frequenc-domain convolution of the input and tem frequenc-domain characteritic. In the double integral of Equation (2), the frequenc variable f i aociated with the time variable τˆ and it ma be viewed a analogou to the frequenc variable f in the tranfer function H(f) of linear time-invariant tem Bandwidth Relation in Time-Varing Stem The time-varing character of tem i uuall manifeted b a frequenc pread, or a frequenc hift, or both. Thu the repone of a LTV tem to a ingle frequenc f 0 can be a frequenc pectrum or a frequenc pectrum centered about a different frequenc than f 0. The width of the pectrum can be regarded a a meaure of the variation of the tem.

7 3.3. Sampling Rate An input with bandwidth B i to a linear time-varing tem characteried b B reult in an output bandwidth not greater than B i + B. Hence, the ampling rate mut be f = 2 i + T ( B B ) (4) 4. Propertie of Linear Time-Varing Stem 4.. Propertie of Convolution The propertie of convolution for LTI tem are: aociativit, ditributivit and commutativit. For LTV tem the propertie are aociativit and ditributivit, the commutativit doe not hold: h ( t τ ) h ( t, τ ) h ( t, τ ) h (, τ ), 2 2 t (5) 4.2. Interconnection of Linear Time-Varing Stem A in the cae of LTI tem, the interconnection of LTV tem can be implified uing block diagram equivalent although the implification i much more complicated ince tranform method are not applicable. A cacade interconnection of tem i equivalent to the uperpoition of their impule repone. For LTV tem the cacade operation i not commutative. 5. Model for LTV Stem 5.. Linear Differential Equation with Time-Varing Coefficient Some LTV tem are decribed b linear differential equation with time-varing coefficient of the form ( t) n d ( ) ( t) t + L + a ( t) ( t) x( t) (6) n d an ( t) + a n n = n 0 dt dt If the tem i lowl varing it can be regarded a quaitatic. It can then be modelled a a recurive IIR filter.

8 For rapidl varing tem the impule repone i ( t τ ) = ( t τ ) u( t τ ) h,, (7) If (6) ha an analtical olution, it impule repone ha a eparable form h n ( t ) = p ( t) q ( τ ), τ i i, t τ (8) i= The above repreent a eparation model. Unfortunatel, onl firt order differential equation have a general olution Separable Model If the tem ha a eparable form of impule repone, then the tem output i t ( t) = h( t ) x( τ ) dτ = p ( t) x( τ ) q ( τ ) 0 N t τ i i d (9) i = 0, τ The realiation of uch a tem i hown in Figure 2. Figure 2: Structure of eparable model for LTV tem.

9 5.3. Tapped Dela-Line Channel Model A variet of tapped dela-line model have been developed b Kailath baed on the ampling theorem. Thee model differ depending upon whether one aume the input to the channel, the output or the channel itelf to be bandlimited or not. If the input ignal i bandlimited, a tapped dela-line can be derived either for lowpa or bandpa channel. If the channel input x(t) i bandlimited to B i, the output (t) i ( ) t = 2B m= i g 2 m B i, t x t 2 m B i (20) The repreentation (20) can be ntheied b a tapped dela-line with tap having time-varing gain g ( ) n t = g, t and tap dela 2Bi n T 2 =, a hown in the Figure 3. B i Figure 3: Sampling model for LTV tem in the form of tapped dela-line. In imulation the quaitatic approach i feaible if the channel Doppler pread i much maller than the ignal bandwidth, i.e. B << B i. The tap gain can then be regarded a contant and the imulation of fading tem can be approximated b a erie of imulation.

10 6. Concluion Linear element in a communication tem can be time-invariant (LTI) or time varing (LTV) in nature. Radio channel are uuall time varing. Slow variation are caued b change in the atmopheric condition and rapid variation occur in multipath propagation event or when the radio terminal are moving in the mobile radio network. LTV tem are more difficult to imulate than LTI tem. In LTV tem the bandwidth expand, which i called the Doppler broadening, and therefore the imulation ampling rate mut be increaed accordingl. If the input ignal i bandlimited the LTV tem can be repreented a a tapped dela-line with timevaring coefficient. Reference Michel C. Jeruchim, Philip Balaban & K. Sam Shanmugan: Simulation of Communication Stem. Second Edition. Kluwer Academic / Plenum Publiher. New York

11 Problem Show that the relationhip in Y ( ν ) = Cτ ˆ, t ( f, ν f ) X ( f ) df (tem output frequenc repone) implie the one in f = 2( B + B ) (ampling rate). T i