Linear fractional shift invariant (LFSI)

LINEAR FRACTIONAL SHIFT INVARIANT (LFSI) SYSTEMS Okay Akay Dokuz Eyliil University, Department of Electrical and Electronics Engineering Kaynaklar Campus, 35160 Buca I IZMIR TURKEY [email protected]. tr ABSTRACT

is obtained as another special case'

In this paper, we formulate continuous time linear fractional shift invariant (LFSI) systems that generalize the well-known linear time invariant (LTI) systems by means of an angle parameter 4. LTI systems are obtained as a special case of LFSl systems for 4 = 0. LFSl systems belong to the large class of time-varying systems. Whereas LTI systems commute with time shifts, LFSl systems commute with fractional shifts defined on the time-frequency plane. Just as the conventional Fourier transform (FT) diagonalizes LTI systems, an LFSl system associated with angle 4 is diagonalized by the fractional Fourier transform (FrFT) defined at the perpendicular angle 1 + (r/Z).We show that the eigenfunctions of an LFSI system at angle 4 are linear FM (chirp) signals with a sweep rate of tan 4. Finally, we demonstrate via a simulation example that, in certain cases, LFSI systems can outperform LTI systems.

Additionally, in the limit as 4 approaches TT or odd multiples of K , the FrFT reduces to a simple axis-reversal operation,

(lF"s)(t)= s ( - t ) .

The FrFT can alternatively be interpreted as a unitary signal representation with respect to a fractional domain. r, as illustrated in Fig. I .

'I

Figure I: Fractional domain T, at angle 1,between time t, (4 = 0) and frequency f, (4 = 5) domains.

1. INTRODUCTION

The fractional Fourier transform (FrFT) is a generalization of the identity transform and of the conventional Fourier transform (FT) into fractional domains ofthe time-frequency plane (see Fig. I). Formal definition ofthe F ~ F To f a time domain signal s ( t ) is given [ I , 21 as (lF49)(r)=

(2)

Motivated by the interest shown for the FrFT, we intraduced [3] the unitary fractional-shift operator, which is denoted by R$ and defined as

(R$s)(t)= s ( t

sqr)=

-

cos 4

p w 2 cos 4 sin de@rtpsin 4 , (3)

R$ shifts the support of the signal s ( t ) radially in the timefrequency plane by an amount equal to p along the radial orientation 4, measured counterclockwise from the time axis as seen in Fig. I . Moreover, R$ simplifies to the unitary time-shift, ( T , s ) ( t )= s(t - T ) , and frequency-shift, (F,s)(t) = eJ2""'s(t), operators as special cases for 4 = 0 and 4 = I, respectively;

4 = 2na, $4 = (zn+1)7r,

where n is an integer, FQ denotes the FrFT operator and Sm(r)represents the FrFT of s ( t ) . A , = J1 - j c o t 4 is a constant term which guarantees that the FrFT preserves the signal energy. As seen in the second line of the definition, the identity transform is obtained in the limit as 4 approaches zero or even multiples of K. By substituting 4 = ~ / in2 the definition of the FrFT, the conventional FT

0-7803-7946-2/03/$37.00 02003 IEEE

(R:s)(t) = (T,s)(t) and (R%s)(l) = (F,s)(t). (4) 'We note that. in denoting conventional FT signals. we will use as a superscript to explicitly lndicate the angle parameter ofthe FT within the general framework of Le FrFT.

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LTI systems are fundamentally related to the time (Q = 0) domain, LFSI systems are associated with the fractional domain, T , at angle $. It is shown in [6] that fractional convolution in (5) can equivalently he written in terms of the FrFT signals S ~ ( T ) and H"r) as

2. LFSI SYSTEMS Since considerably many physical systems possess the key system attributes of linearity and time-shift invariance, they can be modeled appropriately as linear time invariant (LTI) systems. In signal processing, the study of LTI systems is of paramount importance because of the fact that LTI systems are fundamentally related with the conventional FT which forms the core of signal and system analysis [4]. In this paper, we formulate linear fractional-shift invariant (LFSI) systems which generalize conventional LTI systems. LFSI systems are invariant to fractional shifts along radial directions of the time-frequency plane (see Fig. 1) as defined by R$ in (4). Since time shifts correspond to a special case of fractional shifts for Q = 0 as shown in (4), LFSI systems simplify to LTI systems for Q = 0. Furthermore, LFSI systems can be considered as time-vaving with a special type of time variance. We use unitary fractional-shift operator R$ in (3) within an inner product in order to formulate LFSI systems. Accordingly, representing the LFSI system associated with the angle Q by the operator Km, the input/output relation of an LFSI system is derived as

-p

{ K m ( s m ( r ) )( 7} ) =

Sm(B)[ff% - B)l*dB

(7)

where So(,) and IT$(,) are the FrFT at angle Q of s ( t ) and h(t),respectively. Thus, according to (7), the output of an LFSI system can alternatively be found by computing the time domain convolution of the FrFT signals So(?) and

Hm(r). 2.1. Fractional Impulse Response

At this point of our exposition, the following question can be posed. What is the function that characterizes an LFSI system just as an LTI function is characterized by the impulse response function h(t)? To answer this question, let us assign S"(r) = & ( T ) in the input/output relation in (7). 6(r) is a unit impulse function in the fractional, T , domain and thus, could suitably he called a "fractional unit impulse function"in analogy with the time domain unit impulse function 6(t). Then, the corresponding system output becomes

{ K m ( 4 t ) )(}r ) = (s,R!Fwh) (5) s(b)h*(Tcos,++ - ~ ) ~ - 3 2 7 @sinQdp r

r 2 cosQsin

/

J

{Km(a(r))l (T) =

where s ( t ) is the system input and h(t) is a function representative of the LFSI system. More will be said about h(t) shortly. In (S), IF' is the axis reversal operator given in (2) and B is used as the dummy integration variable. Previously, (5) was derived within a different context [6], where it was called fractional convolution of two time domain signals s(t) and h(t). Note that, for the special case of Q = 0 (time domain), ( 5 ) reduces to the inpudoutput relation of an LTI system2 as

[ffWI*.

(8)

For Q = 0, (8) simplifies to its LTI counterpart as

{K0(W))}( t ) = h'(+ (9) Thus, since the complex conjugate of h(t) is obtained as the system output in response to an input of unit impulse function, h(t) is called the system impulse response which fully identifies the LTI system. Since the output of the LFSI system in response to the fractional unit impulse function is found as [H"(r)]*in (8), analogous to the impulse response of an LTI system, HO(r) could be called the "fractional impulse response" of an LFSI system. Similar to the role played by the impulse response of an LTI system, the fractional impulse response H"T) completely characterizes the LFSI system. It is of interest to determine the time domain function that correspondsto the fractional unit impulse function, S(r). To accomplish that task, we compute the inverse FrFT of S(r). Thus, using the FrFT tables [ l , 21 and denoting the time domain counterpart of fractional unit impulse function by it is found as

J

{Ko(s(t))}( t ) = ( s , T t F w h )= s(b)h*(t- B)dP (6) where s ( t ) is the system input and h ( t ) is called the system impulse response which fully characterizes the LTI system [4, 51. The inpdoutput relation in (6) is well known as the LTI (time domain) convolution operation. In this paper, we look at fractional convolution defined in (5) from a systems perspective as the defining input/output relation of an LFSI system. It should be noticed from (5) and (6) that whereas the output of an LTI system is a time, t, function, the output of an LFSI system is a function of the fractional variable, T. This is a result of the fact that while

am(,),

a q t ) = (F-ma(T)) ( t )=

Jm e-J"tZ

Cot

Q. (10)

2AAssumingthe most general case including complex functions, we obtain the complex conjugation ofthe Unit impulse response h ( t ) in (6).For physical systems with real impulse respollse functions, (6) simplifies to the more familiar form [SI.

As a result, the time domain counterpart of the fractional unit impulse function 6 ( ~is) a linear FM with a sweep rate of - cot Q.

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Applying the change of variable P' = r - p, we obtain

2.2. Perpendicular Fractional Response

In this section, we derive a simpler expression to compute the output of an LFSl system. We proceed by computing the conventional FT of both sides of (7) to obtain

/ -

- elllaw

{F?(Km(S#(c,))) ( r ) }( U ) = S*+f(,)[Hm+5(-,,)]*.

[~m(p~)1*~-~2at$'d~~

-e~2ru~p~+f(-U)1*.

(11) In the special case f o r d = 0. ( I I ) reduces to

Thus, we can write the eigenequation as

{IFS (Ku(s(t)))( t ) }(f)= S f ( f ) [ H $ ( - f ) ] '

(12)

{K4 ( S m ( r ) ) (] r ) = [H"$(-u)]'S*(r).

where Sf(/) and H ; ( f ) are the conventional FTs of the input signal s ( t ) and of the system impulse response, h(t), respectively. The relationship in (12) expressing the FT of the output of an LTI system as a multiplication of the FT ) [ H 5 ( - f ) ] *is well known. In the functions S % ( fand signal processing literature, Hi ( f ) is called the frequency response of the LTI system [4]. Due to the uniqueness of the FT, any LTI system is also identified by its frequency response, H % ( f ) . Comparing ( 1 1 ) with (12). H * + ~ ( u ) can be termed the "perpendicular fractional response" of an LFSI system i n analogy with the frequency response of an LTI system. Just as the fractional impulse response H 4 ( r ) , the perpendicular fractional response H*+$(U) also completely characterizes the LFSl system associated with angle

d. 3. EIGENANALYSIS OF LFSI SYSTEMS

S,(t;,')

= J

m

p

(

t

s,(t;u) = {IF-# (S@(,))}( t ; u )

=

d

m p r ( t 2 + U?

(16)

tan C e J 2 T & .

Finally, from ( I 5), the eigenvalues of the LFSI system are given by the FrFT signals [H"f((-u)]*, where Hm+$(u) is the perpendicular fractional response of the LFSI system K4. The eigenequation for the LTI system Ku can be obtained as a special case of (I 5 ) by substituting d = 0, (17)

where s ( t ) = e J z x f t .Thus, as is well known, complex exponentials s,(t; f ) = eJZaftare the eigenfunctions of LTI systems, with [H%( - f ) ] * denoting the corresponding eigenvalues. Note that H %(f)is the frequency response of the LTI system KO. Various quantities defined in this paper in relation to LFSI systems and their corresponding LTI counterparts are summarized in Table 1. 4. SIMULATION EXAMPLE

We demonstrate that LFSl systems can be advantageously used in separating multicomponent signals. Our time domain signal is given as

2 +712)tan4eJ2e7.

(13) The FrFT signal [ H @ t + 4 ( - ~ )is] the * corresponding eigenvalue. Proof: We use the equivalent alternative formulation of an LFSl system given in (7) to derive the eigenfunctionsand the eigenvalues. First, let us assign Sm(r) = eJZnur in (7). Thus, we have

{ K m ( S e ( r ) )(}r ) = / e ~ Z n 7 r f l [ H-P)]'dO. o(~

Consequently, S b ( r ) = eJZnur are the eigenfunctions of the LFSI system Kmas expressed in the fractionaldomainr. To obtain the time domain expression for the eigenfunctions, we have to compute the inverse FrFT of S @ ( r )Thus, . using the tables of the FrFT [ 1,2] we find that

{KU(5(t))J ( t ) = [Hf(-f)l"

Complex exponentials are fundamentally important in the study of LTI systems. This stems from the fact that the response of an LTI system to a complex exponential input is the same complex exponential with only a change in amplitude. Accordingly, complex exponentials are called the eigenfunctions of LTI systems and the amplitude factor is referred to as the corresponding system eigenvalue 141. In this section, we derive the eigenfunctions and eigenvalues of LFSl systems and show that they generalize LTI system eigenfunctions and eigenvalues. Proposition I: Time domain eigenfunctions of an LFSI system Km defined in ( 5 ) are complex linear FM chirp functions given as

(15)

z(t) = e -7&

+ ,J("mt2

+ Znfot) +

(18)

where m = -cot(%),fo = 2.5. Thus, z(t) is composed of a Gaussian function and a linear FM which is multiplied by a sinusoidal tone. The signal is cormpted by complex white Gaussian noise of 20 dB SNR represented as w(t). We sample ~ ( tby) substituting t = nT where the sampling interval T = l/mand n = -127, -126,. .., 126,127,128. Hence, the sampled signal 4 7 4 has N = 256 samples.

(14)

587


LFSI, ($) @ ( t ) = ,/-

System Eigenfunctions System Eigenvalues

= j 1- 3 t a n ~ e 3 n ( t 2 + u 2 ) t a n $ e 3 2 ? r ~s . ( t ; f ) = eJ2nft

[H”f”(-v)]’

-

I)

I

,

. e

-

4

4

-

2

6. REFERENCES

[ l ] H. M. Ozaktas, Z. Zalevsky and M. A. Kutay, The Fractional Fourier Transfoim with Applications in Optics and Signal Processing. Wiley & Sons, 2001.

.

.

0

2

1

[3] 0. Akay and G. F. Boudreaux-Battels, “Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform,” IEEE Sig. Pmc. Letters, pp. 3 12-314, Dec. 1998.

B

@I

14 --*lndxnh 8 21

(-f 11’

[2] L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. on Sig. Pmc., vol. 42, pp. 3084-3091,Nov. 1994.

........ . .................. . 4

IHf

We introduced LFSI systems which generalize LTI systems with an angle parameter $. LFSI systems are invariant to fractional shifts defined on the fractional domains of the time-frequency plane. The FrFT diagonalizes LFSI systems just as LTI systems are diagonalized by the conventional FT. We also defined various quantities such as fractional impulse response and perpendicular fractional response which characterize LFSI systems. All LFSI quantities simplify appropriately to corresponding LTI quantities for 4 = 0. Eigenfunctions and eigenvalues of LFSI systems are also derived. Finally, a simulation example is performed to demonstrate a scenario in which an LFSI system could be preferable over LTI systems.

”””” ’ ’

0

h(t)

Hi(f)

5. CONCLUSIONS

+.

05

e-Jntz

H W H”*(u)

m-

*,:r*l -3

4

Fractional Impulse Function Fractional Impulse Response Perpendicular Fractional Response

In Fig. 2 (a), sampled time domain signal z ( t )is plotted. Fig. 2 (b) shows the fractional domain representation of z(t) obtained by computing the FrFT of z ( t )at angle + = E3.. S’imilarly, conventional FT ($ = 5) of z(t) is shown in Fig. 2 (c). We emphasize that the angle parameter q5= E of the FrFT matches the sweep rate, n,of the linear FM signal. As a result, the FrFT is able to represent the linear FM as nearly concentrated as an impulse and separated from the Gaussian component as seen in Fig. 2 @). Thus, the linear FM and Gaussian components can be separated via a simple masking in the fractional domain. It can be proven mathematically that the FrFT followed by a masking operation forms an LFSI system. Fig. 2 (d) shows the inverse FrFT at $ = -f of the masked FrFT signal which demonstrates that the Gaussian function is recovered to a great extent and the linear FM signal is suppressed. On the other hand, looking at Fig. 2 (a) and (c), we can see that the linear FM and Gaussian signals overlap and cannot be separated by a simple masking operation.

I I’

LTI, ($ = 0)

. . . . . . . . . .. . .. . . . .

...................... . . .. . : . ...~... .: . .

1 r : 2

[4] L. B. Jackson, Signais, Systems, and Transforms. Reading, MA: Addison-Wesley, 1991.

A 0

4.5

+-*

0

2

4

[5] R. G. Baraniuk and D. L. Jones, “Unitary equivalence: A new twist on signal processing,” IEEE Trans. on Sig. Pmc., vol. 43, pp. 2269-2282, Oct. 1995.

8

ldl

Figure 2: a) Signal in time domain (4 = 0) b) Signal in frequency domain ($ = ./2) c) Signal in fractional domain (4 = r / 3 ) d) Recovered Gaussian signal in time domain.

[6] 0. Akay and G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. on Sig. Proc., vol. 49, pp. 979-993, May 2001.

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