Frequency truncated discrete-time system norm

F requency truncated discrete-time system norm ∗ Han uman t Singh Shekha w at Indian Institute of T ec hnology Guw ahati, Guw ahati, India h.s.shekha w at@iitg.ac.in Abstract Multirate digital signa l process ing and model re- duction applications require co mputation of the fre- quency trunca ted nor m of a discrete- time system. This pap er explains how to compute the frequency truncated norm of a discrete-time sy stem. T o this end, a m uch-generalized problem of integrating a transfer function of a discrete-time s ystem giv en in the descriptor form over an int erv al o f limited fre- quencies is also discussed along with its co mputa- tion. 1 In tro duction The frequency truncated discrete-time system no rm of a linear discrete-time-inv ariant G (with the transfer function G ( z ) in z -domain) defined as kG k 2 [ θ 1 ,θ 2 ] := 1 2 π tr Z θ 2 θ 1 G ∼ (e j θ ) G (e j θ ) dθ, (1) where the conjugate system G ∼ ( z ) := G ∗ ( ¯ z − 1 ) ( ∗ is the adjoint op eration and ¯ z is complex conjugate of z ). The need for the freq uency truncated discrete- time system nor m arises naturally in the multi-rate discrete signal pro cess ing . F or example, consider a simple setup of multi-rate dis crete s ignal pro cessing as shown in Figure 1. Here, the input discrete s ignal G e w y y h u e  χ ↓ M ↑ M G - Figure 1: A s etup for mult i-ra te discr ete signal pro- cessing y is rea l and repres ent ed as the output of a sy stem G driven by the discrete white Gaussian noise with ∗ The material in this paper was partially presented at 22th In ternational Symposium on Mathematical Theory of Net- wo rks and Systems (MTNS 2016) , July 11-15, 2016, Minneapo- lis, USA. zero mean. The transfer function (in z -domain) of G is represented b y G ( z ). Hence, the power s pec tr al density o f y is given by | G (e j θ ) | 2 for a ll fr equencies θ ∈ [ − π , π ]. χ is the analy sis filter who se o utput is down-sampled by a factor M . The down-sampled output is aga in upsampled by a factor M follow ed by a synthesis filter  . The reconstructed output u is compared with the input signal y . The aim is to de- sign b oth the analysis and sy nt hesis filter given G ( z ) in suc h a wa y that time av erag ed mean square erro r J = lim N →∞ 1 2 N + 1 N X n = − N E ( e 2 [ n ]) is minimised [9]. Here, E is the exp ectation op erator . Assume that the G is stable and G (e j θ ) is do mina nt in the frequency band [ − π M , π M ] that means | G (e j θ 1 ) | > | G (e j θ 2 ) | if θ 1 ∈ [ − π M , π M ] and θ 2 ∈ [ − π , π ] \ [ − π M , π M ]. In this case, optimal synthesis and a nalysis filter give the error (see [9] for details) J = kG k 2 L 2 − 1 2 π tr Z π M − π M G ∼ (e j θ ) G (e j θ ) dθ . (2) Here, kG k L 2 := kG k [ − π ,π ] represents the L 2 norm of the system. Thus, we can see that truncated system norms natur a lly aris e s in mu lti-rate discrete sig nal pro cessing. F or a g eneral G (e j θ ), computation of (1) is needed [9]. It is further assumed that the discrete-time sys- tem G in (1) is linear discrete- time-inv ariant (LDTI) and its tra ns fer function G ( z ) is a pro pe r ratio - nal transfer function with real co efficients. These t yp e o f systems can b e repr e sented in state space a s G ( z ) = D + C ( z I − A ) − 1 B where A, B , C and D are real matric es. F or simplicity of exp osition, it is assumed that D = 0 thro ughout this pape r. It will be shown later in the pap er that ev aluation of the frequency tr uncated discr ete-time system norm is a sp ecial case of the generalized problem of integration of a transfer function given in the descriptor form i.e Z θ 2 θ 1 C (e j θ E − A ) − 1 B dθ (3) 1 where A, B , C , and E are real matrices. The ab ove int egra l is also h elpful in the ev aluation of the frequency-domain controllabilit y and the observ abil- it y Gra mmian o f a system with transfer function C ( z I − A ) − 1 B [10, 5, 8, 2]. An expressio n for the frequency truncated discre te- time s y stem norm for a stable discrete-time system is given in [8, Theorem 3 .8] which dep e nds upon in- vertibilit y of the A matr ix. This pap er provides a mo dification which res olves this problem for a stable discrete-time system. This paper further genera lizes the result of [8, Theo rem 3.8 ] and provides an expres- sion for (1) with minimal restriction o n the sy stem po les. Integral of a tra nsfer function (of a discrete- time system) in the descriptor form and a n umerically viable expres sion fo r its computation is also given in the pap er. As a by-pro duct, similar r esults for a con- tin uous time system given in the descriptor form are briefly men tioned. Section 2 con tains an expres sion for the freq uency truncated discre te- time system nor m for a stable sys- tem a nd Section 3 co nt ains a n expr e ssion for the fre- quency tr unca ted discr ete-time sys tem norm for a generic case. In tegra tion of a transfer function given in the descriptor for m is discus s ed in Section 3 and a metho d for its co mputation is g iven in Section 4. Section 5 cont ains results rela ted to freq uency trun- cated norm of a co nt inuous time system given in the descriptor form. Notation: R and C denote the s et of r eal and co m- plex num b ers re sp e ctively . ¯ R − denotes the closed negative real axis (i.e. the nega tive real a xis includ- ing zer o). Arg( z ) denotes the pr incipal arg umen t i.e. argument of the c o mplex n umber z in ( − π , π ] a nd wrap( θ ) := Ar g (e j θ ) for all θ ∈ R . F or square complex matrices A and E , the matrix p encil ( A, E ) is called regular if αE + β A is invertible for at-lea st o ne set o f complex n um b ers α and β . An eigenv alue λ ∈ C of a matrix p encil ( A, E ) satisfies det( A − λE ) = 0. W e define j := √ − 1. σ ( A, E ) deno tes spectr um (the s et of eig e n v alue s ) o f the matrix p encil ( A, E ). A func- tion f is define d on σ ( A ) if it follows definition 1 . 1 of [3]. If A ∈ C n × n do es not hav e an y e ig env alues on the ¯ R − then there is a unique log a rithm Q = log ( A ) whose all eigenv alues lie in the op en horizontal strip { z ∈ C | π < imag( z ) < π } o f the complex plane [3, Theorem 1.31]. The Q is known a s the princip al lo g- arithm . The F r´ echet deriv a tive of a matrix function f : C n × n → C n × n at A in the direction X is denoted by L f ( Q, X ) [3, § 3.1]. 2 Stable Case A discrete-time system G is defined s table if A is Sch ur (i.e., all eigen v alues of A a re strictly inside the unit circle in the co mplex plane). It is w ell known that for a stable discrete-time system G , the squared L 2 norm is given by kG k 2 L 2 := 1 2 π tr Z π − π G ∼ (e j θ ) G (e j θ ) dθ = tr B T P B (4) where P is the unique solution of the discrete Lya- punov equation A T P A − P + C T C = 0 . (5) The frequency trunca tion discr ete-time system norm can be calculated by ev aluating an anti-deriv ative (or primitive) R G ∼ (e j θ ) G (e j θ ) dθ first. Theorem 2.1 L et discr ete-time system G b e stable and st rictly pr op er and G (e j θ ) = C (e j θ I − A ) − 1 B with A, B, C r e al matric es. Then, an anti-derivative R G ∼ (e j θ ) G (e j θ ) dθ e quals B T P B θ + 2 imag( B T P log( I − e − j θ A ) B ) wher e P is the unique solution of (5) and log denotes the princip al lo garithm and θ ∈ [ − π , π ] . Pr o of. Consider function f ( z , θ ) = ( − j z − 1 log(1 − e − j θ z ) if z 6 = 0 je − j θ if z = 0 F o r a given θ ∈ [ − π , π ], f ( z , θ ) is analytic in the op en unit disk ar ound zer o (in the complex plane) a s 1 − e − j θ z never lies on the clos e d nega tive real a xis ¯ R − . Clearly , ∂ f ( z , θ ) ∂ θ = (e j θ − z ) − 1 is also a nalytic (for a given θ ∈ [ − π , π ]) in the op en unit disk aro und zero. Hence, it follows from [4, Theorem 6.2.27] that f ( A, θ ) is an a nt i-deriv ative of (e j θ I − A ) − 1 . Also, note that G ∼ (e j θ ) G (e j θ ) = B T ( I − e j θ A T ) − 1 P B + B T P A (e j θ I − A ) − 1 B = B T P B + e j θ B T ( I − e j θ A T ) − 1 A T P B + B T P A (e j θ I − A ) − 1 B . Now, using the anti-deriv ative of (e j θ I − A ) − 1 and the fact that integration (w.r.t. θ ) o f the complex conjugate is the conjugate of integration, we hav e the result. 2 The pro of of the Theo rem 2.1 is essentially similar to the pro of o f [8, Theorem 3.8] without the need for inv er sion o f the A matrix. Using [3, Theore m 1.31], we hav e kG k 2 [ − π ,π ] = tr B T P B = kG k 2 L 2 . 3 General case In the previo us sectio n, the pole s of a discrete-time system m ust b e in the unit circle. W e know that in te- gration of a meromorphic function is po ssible as long as we are not integrating ov er a p ole. Hence, systems with p oles o n the unit cir cle as well as inside and outside o f the unit circle (apart from p oles within the limits of in tegra tio n) would b e a more genera l case. This sectio n is ab out the integration o f a trans- fer function given in the descriptor form (see (3) ) for the general case. This will further help in obtaining an expression for the freq uency truncated discre te- time system norm in the gener al case. It is as sumed that eigenv alue o f the matrix p encil ( A, E ) ca n lie o n the unit circle as well a s inside a nd outside of the unit circle. The re s ult needs logarithm of matrices as exp ected. How ever, there ar e few mathematical techn icalities which w e ha ve to take ca re. The first issue is R (e j θ E − A ) − 1 dθ is a function of t wo matrices E a nd A . Hence, the definition of a ma tr ix function given in [3 ] do es not help here as it is. If A or E matrix is in vertible then R (e j θ E − A ) − 1 dθ can b e written as a function o f E A − 1 or E − 1 A r esp ectively . How ever, the s itua tion is a little complicated if b o th E and A ar e singular. Things can b e simplified if w e assume that the matrix p encil ( A, E ) is reg ula r. T o illustrate this further, assume that the ( A, E ) is regular and α 6 = 0 then ( z E − A ) − 1 = α ( αE + β A ) − 1 ( z ( I − β Q ) − αQ ) − 1 where Q := A ( αE + β A ) − 1 . Hence, we need R α ( z ( I − β Q ) − αQ ) − 1 dθ which is a function of just one matrix Q . If α = 0 then A is inv ertible. This case ha s b een discussed already . The second issue is r elated to the principal lo g- arithm of a matrix as it do es not exist if eigenv al- ues of the matrix lie on the closed negative rea l axis ¯ R − . The critica l task here is to choos e the r ight anti- deriv ative of α ( z ( I − β Q ) − αQ ) − 1 such that the prin- cipal lo garithm is defined. F urthermore, Q can hav e eigenv alues on the unit cir cle as well as inside and outside of the unit cir cle. Hence, obtaining the rig ht anti-deriv ative is quite challenging. F or example, as- sume β = 0 , α = 1 a nd A has a t least one eig en- v alue outside the unit circle. In this case, Q = A and log( I − e − j θ A ) (which w e obtained in the stable case) is not a v a lid anti-deriv ative of ( z I − A ) − 1 as there exists a v alue of θ ∈ [ − π , π ] where eig e n v al- ues of I − e − j θ A lie o n ¯ R − . In this work, the right anti-deriv ative is obta ined by the well-kno wn tangent half-angle substitution i.e. e j θ = 1 + j t 1 − j t = − t − j t + j (6) Here t = tan( θ 2 ). Selec tio n of the right an ti-deriv ative is also an issue in [7] which was solved by taking j out of the integrand whenever necessa ry . This technique is also used her e alo ng with the half- a ngle substitu- tion. Theorem 3.1 Assume that a discr ete-time s yst em K c an b e r epr esente d in the descriptor form as K ( z ) = ( z E − A ) − 1 with r e al matric es A and E . Also , assume that θ 1 , θ 2 ∈ ( − π , π ) and the matrix p encil ( A, E ) is r e gular i.e. W := αE + β A is invert- ible for at-le ast one set of c omplex numb ers α and β . If e j φ not an eigenvalue of ( A, E ) for any φ ∈ R and wrap( φ ) ∈ [ θ 1 , θ 2 ] then Z θ 2 θ 1 K (e j θ ) dθ = lim ǫ → 0 1 j A − 1 ǫ  − η I + log  Γ( θ 2 , ǫ )Γ( θ 1 , ǫ ) − 1  (7a) = lim ǫ → 0 1 j  − η I + log  Γ( θ 1 , ǫ ) − 1 Γ( θ 2 , ǫ )  A − 1 ǫ (7b) wher e A ǫ := A + αǫI , E ǫ := E − β ǫI , ǫ ∈ C , η := log  tan(0 . 5 θ 2 ) − j tan(0 . 5 θ 1 ) − j  and Γ( θ , ǫ ) := ( E ǫ + A ǫ ) tan(0 . 5 θ ) − j( E ǫ − A ǫ ) . Pr o of. Assume that α 6 = 0. Define f d ( z , θ ) := ( 1 j z  − η + lo g Ω( z, t ) Ω( z, t 1 )  , z 6 = 0 α j (e − j θ 1 − e − j θ ) z = 0 where t := tan( θ 2 ) and t 1 := ta n( θ 1 2 ) and Ω( z , t ) := t (1 − β z + αz ) − j(1 − β z − αz ) for a real t . If z is an eigenv alue o f Q := A ( αE + β A ) − 1 then z α 1 − β z is also an eig env alue of ( A, E ). Hence, for all θ ∈ [ θ 1 , θ 2 ], f d ( z , θ ) is defined on σ ( Q ) as long as e j φ not an eigenv alue of ( A, E ) for a ny wrap( φ ) ∈ [ θ 1 , θ 2 ] (see Theorem A.1). Also, ∂ f d ( z , θ ) ∂ θ = α 1 e j θ (1 − β z ) − αz . Now, it follows from Theorem A.2 and [4, Theorem 6.2.27] that R θ 2 θ 1 K (e j θ ) dθ = W − 1 R θ 2 θ 1 α ( z ( I − β Q ) − αQ ) − 1 dθ = W − 1 f d ( Q, θ 2 ). F rom [3, Theore m 3.8] 3 and Theo rem A.1 , the F r ´ echet deriv ative L f d ( Q, X ) of f d at Q in the dire ction X exists. Hence, f d ( Q + ǫαW − 1 ) = f d ( Q ) + ǫL f d ( Q, αW − 1 ) + o ( | ǫα |k W − 1 k ). Hence, we hav e (7a). Equation (7b) follows from [3, Theorem 1.13.c]. On the o ther hand, if α = 0 then A is inv ert- ible and β 6 = 0 b y the regula rity of ( A, E ). Hence, R θ 2 θ 1 K (e j θ ) dθ = A − 1 R θ 2 θ 1 (e j θ E A − 1 − I ) − 1 dθ . Define for a complex z f i ( z , θ ) := 1 j  − η + log t ( z + 1) − j( z − 1) t 1 ( z + 1) − j( z − 1)  where t := tan( θ 2 ) and t 1 := tan( θ 1 2 ). If z is an eigenv alue o f E A − 1 then 1 z is an eig env a lue of ( A, E ). Hence, for all θ ∈ [ θ 1 , θ 2 ], f i ( z , θ ) is defined on all eigenv alues o f E A − 1 as long as e j φ not an e igenv alue of ( A, E ) for any wrap( φ ) ∈ [ θ 1 , θ 2 ] (see Theore m A.2). Now, it follo ws from Theorem A.2 and [4, Theo - rem 6 .2.27] that R θ 2 θ 1 K (e j θ ) dθ = A − 1 R θ 2 θ 1 (e j θ E A − 1 − I ) dθ = A − 1 f i ( E A − 1 , θ 2 ). Eq uiv alence to the limits can be pr ov ed in a manner similar the α 6 = 0 case. Equation (7) can be extended for θ = π as shown in the following result. Corollary 3. 2 L et k ( z ) b e as in The or em 3.1. As- sume t hat θ 1 ∈ ( − π , π ] and t he matrix p encil ( A, E ) is re gular i.e. W := αE + β A is invertible for at- le ast one set of c omplex numb ers α and β . If e j φ not an eigenvalue of ( A, E ) for any φ ∈ R and wrap( φ ) ∈ [ θ 1 , π ] then R π θ 1 K (e j θ ) dθ e quals lim ǫ → 0 1 j A − 1 ǫ  η f I − log  t 1 I − jΦ − ( ǫ )Φ + ( ǫ ) − 1  = lim ǫ → 0 1 j  η f I − log  t 1 I − jΦ + ( ǫ ) − 1 Φ − ( ǫ )  A − 1 ǫ (8) wher e t 1 := tan( θ 1 2 ) , A ǫ := A + αǫI , E ǫ := E − β ǫ I , ǫ ∈ C , η f := log( t 1 − j) , Φ + ( ǫ ) = E ǫ + A ǫ and Φ − ( ǫ ) = E ǫ − A ǫ . Pr o of. Since − 1 is not an e ig env alue of the matrix penc il ( A, E ), E + A is inv ertible. Define t := tan( θ 2 ), Q ǫ := A ǫ W − 1 and ˜ Q := ( I − β Q ǫ − αQ ǫ )( I − β Q ǫ + αQ ǫ ) − 1 . Now, log  tI − j( E ǫ − A ǫ )( E ǫ + A ǫ ) − 1  = log  tI − j ˜ Q  F o r sufficiently small | ǫ | , the above logar ithm ex its if e j φ not an eigenv alue of ( A, E ) for any φ ∈ R a nd wrap( φ ) ∈ [ θ 1 , π ] (the pr o of is similar to the pro o f Theorem A.1). Hence, [3, Theor e m 11 .3 ,Theorem 11.2] implies that log  t − j t 1 − j  = log( t − j) − log( t 1 − j) log( Q t ) = log ( tI − j ˜ Q ) − log( t 1 I − j ˜ Q ) . where Q t := ( tI − j ˜ Q )( t 1 I − j ˜ Q ) − 1 . Now, us ing R π θ 1 K (e j θ ) dθ = lim θ → π R θ 2 θ 1 K (e j θ ) dθ the res ult fo l- lows from [7, Lemma 5(2)]. Similar res ults can b e o btained if integral limits a re [ − π , θ 1 ] or [ − π , π ]. Note that if A is inv ertible then (7) and (8) ca n be simplified by taking ǫ = 0 . Oth- erwise, (7) and (8) needs a pro p e r limit. Section 4 contains other forms of (7) a nd (8) which ar e inde- pendent of a ny limit. Ho wev er, the current form is useful in obtaining a simple express ion for the fre- quency truncated discrete-time system norm as ex - plained in the following result. Theorem 3.3 Supp ose a discr ete-time system G c an b e r epr esente d in st ate- sp ac e as G ( z ) = C ( z I − A ) − 1 B with r e al matric es A, B, and C. Define A h :=  A 0 C T C I  , E h :=  I 0 0 A T  , C h :=  0 − B T  , and B h :=  B 0  . Assume that θ 1 , θ 2 ∈ ( − π , π ) . If e j φ not an eigen- value of A for any φ ∈ R su ch that w r ap ( φ ) ∈ [ θ 1 , θ 2 ] ∪ [ − θ 1 , − θ 2 ] then Z θ 2 θ 1 G ∼ (e j θ ) G (e j θ ) dθ = 1 j C h log  Γ d ( θ 1 ) − 1 Γ d ( θ 2 )  B h wher e Γ d ( θ ) := ( E h + A h ) tan(0 . 5 θ ) − j( E h − A h ) . Pr o of. The sy s tem G ∼ ( z ) G ( z ) ca n b e express ed as G ∼ ( z ) G ( z ) = z C h ( z E h − A h ) − 1 B h . Clearly , Z θ 2 θ 1 G ∼ (e j θ ) G (e j θ ) dθ = Z θ 2 θ 1 C h  E h − e − j θ A h  − 1 B h dθ = Z − θ 2 − θ 1 C h  e j θ A h − E h  − 1 B h dθ Assume that λ max and λ min represents the ma x im um and minim um absolute v alues of eigenv alues of A . Now, det( E h − µA n ) = det( I − µA ) det( A T − µI ) 6 = 0 if µ and 1 /µ is not an eigen v a lue of A . Hence, the matrix p encil ( E h , A h ) is regular. It also shows that the e ig env alues of ( E h , A h ) a re e ig env alues of A and A − 1 . This means, for an y φ ∈ [ θ 1 , θ 2 ], if e j φ is not an eigenv alue of ( A h , E h ) then e − j φ not an eigen v alue of 4 ( A h , E h ). This implies e j φ not an eigenv alue of A for any φ ∈ [ θ 1 , θ 2 ] ∪ [ − θ 1 , − θ 2 ]. Note that lim ǫ → 0 ( E h + ǫαI ) − 1 B h = B h and C h B h = 0. Finally , equiv alence to the limit needed in (7b) can be prov ed in a ma nner given in the pro o f o f Theorem 3.1. Note that the ab ov e r esult does not need a ny limits. 4 Computation Equation (7) can b e conv erted into ano ther for m (in- depe ndent of ǫ ) which uses exp o nent ial of matrices and ψ 1 ( A ) := ∞ X j =0 1 ( j + 1 )! A j (9) The adv antage is that both of these functions hav e nu merically accurate and r eliable implementation [3, § 10.5, § 10.7.4]. Theorem 4.1 L et ψ 1 b e as in (9) . Using n ota- tions and c onditions of The or em 3.1, we have that R θ 2 θ 1 K (e j θ ) dθ e quals 1 j W − 1  − αL (e − j θ 2 I − e − j θ 1 e Y ) + β Y  (10) wher e Y ǫ := − η I + log  Γ( θ 1 , ǫ ) − 1 Γ( θ 2 , ǫ )  , Y := lim ǫ → 0 Y ǫ = − η I + log  Γ( θ 2 , 0)Γ( θ 1 , 0) − 1  and L := lim ǫ → 0 (e Y ǫ − I ) − 1 Y ǫ = ψ 1 ( Y ) − 1 . Pr o of. Y = lim ǫ → 0 Y ǫ due to Theorem A.1. Note that ( E ǫ + A ǫ ) t − j( E ǫ − A ǫ ) = ( t − j)( E ǫ − e − j θ A ǫ ) (11) where t := tan( θ 2 ). Now, exp  − η I + log  Γ( θ 2 , ǫ )Γ( θ 1 , ǫ ) − 1 )  = e Y ǫ . Using [3, Theorem 10.2,Theore m 1.17 ], we hav e t 1 − j t 2 − j  Γ( θ 2 , ǫ )Γ( θ 1 , ǫ ) − 1 )  = e Y ǫ where t i := tan( θ i 2 ). Assume α 6 = 0. Us ing (11), we hav e α (1 − e Y ǫ ) E ǫ W − 1 = α (e − j θ 2 I − e − j θ 1 e Y ǫ ) A ǫ W − 1 (1 − e Y ǫ ) α ( E − β ǫI ) W − 1 = αM A ǫ W − 1 where W := αE + β A and M ǫ := e − j θ 2 I − e − j θ 1 e Y ǫ . F ur ther , simplifying using αE W − 1 = I − β AW − 1 , Q ǫ := A ǫ W − 1 = AW − 1 + αǫW − 1 and αE ǫ W − 1 := I − β Q ǫ , we have I − e Y ǫ = ( αM ǫ + ( I − e Y ǫ ) β ) Q ǫ . Using [3, Theorem 1.13.a] and Z θ 2 θ 1 K (e j θ ) dθ = lim ǫ → 0 1 j W − 1 Q − 1 ǫ Y ǫ , we hav e the result. E quiv alence of L and ψ is stan- dard [3, § 10.7.4]. Note that L is inv ertible becaus e it has no zero e ig env alues. If α = 0 then A is invertible a nd β 6 = 0 by the regular ity o f ( A, E ). Hence, R θ 2 θ 1 K (e j θ ) dθ = 1 j A − 1 Y . Equation (8) can b e also mo dified in a manner sim- ilar to Theorem 4.1. Corollary 4.2 L et ψ 1 b e as in (9) . Using notations and c onditions of Cor ol lary 3.2, we have that Z π θ 1 K (e j θ ) dθ = 1 j W − 1 L f  e Y ( α − β ) + (e − j θ 1 α + β ) I  wher e Y ǫ := − η f I + log  tI − jΦ − ( ǫ )Φ + ( ǫ ) − 1  , Y := lim ǫ → 0 Y ǫ = − η f I + log  tI − jΦ − (0)Φ + (0) − 1  and L f := lim ǫ → 0 (e Y ǫ − I ) − 1 Y ǫ = ψ 1 ( Y ) . 5 A brief note on the c on tin u- ous time system Similar to Theorem 3.1 and Theorem 4.1, the results of [7] can b e extended to the contin uo us time descrip- tor systems. The s e results a r e useful in the mo del reduction applica tions [6]. The pro of is similar to Theorem 3.1 and T heo rem 4.1. Theorem 5.1 L et ψ 1 b e as in (9) . Assume that a c ontinu ou s time system K c an b e r epr esente d in the descriptor form as K ( s ) = ( sE − A ) − 1 with r e al ma- tric es A and E . Also, assume that ω 1 , ω 2 ∈ R and the matrix p encil ( A, E ) is r e gular i.e. W := αE + β A is invertible for at-le ast one set of c omplex numb ers α and β . If the matrix p encil ( A, E ) has no imaginary eigenvalue j λ with λ ∈ [ ω 1 , ω 2 ] then Z ω 2 ω 1 K (j ω ) dω = lim ǫ → 0 1 j E − 1 ǫ log  ˜ Ω( ω 2 , ǫ ) ˜ Ω( ω 1 , ǫ ) − 1  = lim ǫ → 0 1 j log  ˜ Ω( ω 1 , ǫ ) − 1 ˜ Ω( ω 2 , ǫ )  E − 1 ǫ = − W − 1  β ˜ L ( ω 2 I − e ˜ Y ω 1 ) + j α ˜ Y  wher e A ǫ := A + αǫI , E ǫ := E − β ǫI , ǫ ∈ C , ˜ Ω( ω , ǫ ) := ω E ǫ + j A ǫ , ˜ Y ǫ := log  ˜ Ω( ω 2 , ǫ ) ˜ Ω( ω 1 , ǫ ) − 1  ˜ Y := lim ǫ → 0 ˜ Y ǫ = lo g  ˜ Ω( ω 2 , 0) ˜ Ω( ω 1 , 0) − 1  and ˜ L := lim ǫ → 0 (e ˜ Y ǫ − 1 ) − 1 ˜ Y ǫ = ψ 1 ( ˜ Y ) − 1 . 5 6 Conclusions Computation of the frequency truncated discrete- time s y stem no rm a r ises in different signal pr o cess- ing a nd model reduction applications. This paper contains expressio ns for in tegral of the transfer func- tion of a discr ete-time system given in the descripto r form. The re sult for the descriptor system is used in obtaining the frequency tr uncated nor m of a discrete- time system in the general c ase. Simplified results in case of stable systems ar e a lso given in the pa p er . Similar results for the contin uous time systems given in the descriptor form, are also mentioned briefly . Ac kno wledgemen ts The author w ould like to thank Pr of. Nicholas J . Higham (The Universit y of Manchester, UK) and Prof. R. Alam (Indian Institute of T echnology Gu wa- hati, India) for many useful sugg estions. A App endix The results in this section explain when the func- tions required in the pro of of Theo rem 3.1 a r e de- fined. Note that an analytic function in domain D is alwa ys defined a t all z ∈ D . Theorem A.1 L et α and β b e any two c omplex numb ers and θ 1 , θ ∈ ( − π , π ) . Defin e a c omplex func- tion f d ( z , θ ) := ( 1 j z  − η + log Ω( z, t ) Ω( z, t 1 )  , z 6 = 0 α j (e − j θ 1 − e − j θ ) elsewher e wher e η := log  tan(0 . 5 θ ) − j tan(0 . 5 θ 1 ) − j  , t := tan( θ 2 ) and t 1 := tan( θ 1 2 ) and Ω( z , t ) := t (1 − β z + αz ) − j(1 − β z − αz ) for a r e al t . Ass u me that α 6 = 0 . Then, f d ( z , θ ) is analytic in C \ ˆ D wher e ˆ D := { z ∈ C | αz 1 − β z = e j θ , wrap( φ ) ∈ [ θ 1 , θ ] } . Her e, C \ ˆ D is an op en set. Pr o of. It is straig htf orward to verify that η is well defined. It is now sho wn that C \ ˆ D is a n op en set. Assume β = 0, then the result is trivial. Assume β 6 = 0 then g ( z ) := αz 1 − β z is co nt inuous apar t from the p oint z = 1 β . Since the set Y := { e j φ | φ ∈ [ θ 1 , θ ] } is closed and g − 1 ( Y ) do es not contain z = 1 β , contin uit y of g in C \{ 0 } implies that C \ ˆ D is a n open s e t. T o chec k whether log Ω( z, t ) Ω( z, t 1 ) is well defined or not, first assume that 1 − β z 6 = 0 . Then, Ω( z , t 1 ) = ( t 1 + j)( − e j θ 1 (1 − β z ) + αz ). Hence, Ω( z , t 1 ) is inv ertible as z / ∈ ˆ D . Assume that the pr incipal log do e s not e xist for a z / ∈ ˆ D . This means log Ω( z , t ) Ω( z , t 1 ) = t (1 + a ) − j(1 − a ) t 1 (1 + a ) − j(1 − a ) = − ρ for all ρ ≥ 0. Here, a := αz 1 − β z . The a bove implies that a must b e on unit circle i.e. a = e j ψ . If a = − 1 (i.e. ψ = π ) then it is trivia l to s ee that the pr incipal log exist. On the other hand if a = e j ψ and a 6 = − 1 then (1 + a ) t − j(1 − a ) (1 + a ) t 1 − j(1 − a ) = t − tan ψ 2 t 1 − ta n ψ 2 = − ρ iff tan ψ 2 = ρt 1 + t ρ +1 . This means ψ ∈ [ θ 1 , θ ]. Hence, z ∈ ˆ D . Contradiction. There fo re, the principal log exists for all z / ∈ ˆ D . Now, as sume 1 − β z = 0. Then, β 6 = 0, z 6 = 0 and Ω( z , t 1 ) = ( t 1 + j) αz . Hence, Ω( z , t 1 ) is inv ertible as α 6 = 0. Also, lo g Ω( z, t ) Ω( z, t 1 ) = lo g( t +j t 1 +j ). It is straig htfor- ward to see that this logarithm exists. The ab ov e analysis implies that 1 j z  − η + log Ω( z, t ) Ω( z, t 1 )  is analy tic on an op en set C \ ˆ D apa rt from z = 0 where it has a remov able sin- gularity . Clear ly , lim z → 0 f d ( z , θ ) = f d (0 , θ ). Hence, f d ( z , θ ) is analytic in C \ ˆ D (see e.g. [1, § 16.20 ]). 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