306 IEEE TRANSACTIONS ON EDUCATION, VOL. 37, NO. 3, AUGUST 1994

The Look-Up Table for Deriving the Fourier Transforms of Cosine-Pulses

Chang E. Kang, Senior Member, IEEE, and Yong S . Oh, Member, IEEE

Abstract-This paper presents a new and easy method of obtaining the Fourier transforms of the nth order cosine-pulses which have uniform amplitudes. This new method focuses on de- riving formulas which are recursively related to their orders, and thus making it also applicable to numerical solutions. Concerning the procedures needed to obtain the analytical solutions, this new method proves to be simpler than conventional methods, because the results consist of a sum of two functions which can be easily calculated recursively. It must be noted that the formula can be represented as a complete recursion by separating the coefficients in the manner originated by the authors. The resulting equation is the sum of the original "sinc" functions shifted by some symmetrical factors and then multiplied by several constants. The constants are easily determined by the binomial coefficients and the shifting factors from the corresponding exponential differences in the expansion of (a + b)". Furthermore, a look- up table is obtained, making it possible to get all the coefficients and factors needed for the Fourier transform of the cosine-pulses of any order.

develop a more compact form. Fortunately, all the parameters needed for the formula can be obtained from the expansion of (a + b)" and a coefficient directly determined from its order. A look-up table containing the parameters is included in section IV. These formulas, which are expressed as propositions, are proven by means of induction and are located in the Appendix.

11. PROBLEM DESCRIP~ON AND BRIEF REVIEW OF CONVENTIONAL METHODS

The functions of interest are the cosine-pulses which have the same amplitude for any order n = 1,2,3, . . .. We define the nth order cosine-pulse as follows:

(Definition 1 : The nth order cosine-pulse)

p n ( t ) = Acos (g),n(;), for n = 1,2,3;.. . where

I. INTRODUCTION HE nth order cosine-pulses are often used to represent T the transfer functions or the time responses of several

communication systems [ 11-[3]. Nevertheless, if the order of the function is augmented, the process to derive the Fourier transform of it using conventional methods becomes complex and difficult. Although the method of consecutive differentia- tions or by the convolution theorem can be used to derive the transforms, the procedure becomes increasingly tedious as the order of the function increases [2]. In addition, these methods fail to offer the benefits of a recursive relationship.

On the other hand, the class-I PRS (partial response sig- naling) system was introduced in 1963 under the name of duobinary [4]. And in 1975, P. Kabal and S . Pasupathy generalized this concept and proposed a model separated into two parts, namely, the transversal filter and the bandlimiting filter [5 ] .

We present a recursive procedure to derive the Fourier transforms of the nth order cosine-pulses by means of the class-I PRS system which has been modified for our purposes. Throughout this paper, we will define the problems while briefly describing the procedures of the conventional methods in section II. Then a recursive formula will be derived for the transform of the function using the modified model in section 111, and separate the coefficients from the formula in order to

Manuscript received April 1992; revised June 1992. C.-E. Kang is with the Electronic Engineering Department, Yonsei Univer-

sitv. Seoul 120-749 Korea.

Deriving the frequency-domain function P, (f) from (1) will be the focus of this paper.

In addition, we describe the 'sanc' function and the newly defined 'cosinc' function which can be used as a foundation for later developments.

(Definition 2 : The sine function and the cosinc function) sin RX

sinc(x) - RX

(3) cos RX

RX cosinc(x) -= -

We can obtain the Fourier transform of function (1) through the method of consecutive differentiations or by the convolu- tion theorem when the order of the function is relatively low P I .

As for the method of consecutive differentiations, a deriva- tive which contains the original function or some previous derivatives as a part of it must be found in the process of obtaining the derivatives of the function. For the 1st order, the second derivative can be written with a constant multiple of the original function p l ( t ) and constant multiples of shifted impulses. We have

cosinc(2 f T ) 8 A f r 2

P l ( f ) = 1 - (4f.)2 (4)

Y.-S. Oh is with the Department of Information and Communication

IEEE Log Number 9403 115. Engineering, Mokwon University, Taejon, Korea. where the function cosine(.) has been defined in (3). For the

2nd order, we need the first three derivatives of pZ( t ) . It can be

0018-9359/94$04.00 0 1994 IEEE

KANG AND OH LOOK-UP TABLE FOR DERIVING FOURIER TRANSFORMS OF COSINE-PULSES 307

As the order of the function increases, the situation com- pletely changes. In other words, the above processes and the

seen that the third derivative contains only a constant multiple of the first derivative and some shifted impulses. Therefore, after some painstaking calculations, we can find the Fourier transform of p z ( t ) in the compact form of

c - A

Kn -- Gn * c > Jh

more and more complicated to manipulate. The convolution integral, introduced in many textbooks, is

yet another method to derive the Fourier transforms of the nth order cosine-pulses [l], [2]. When the order of the function is relatively low (n = 1,2,3, or 4), the Fourier transform can be obtained if we consider (1) as a product of two functions so that they are convolved in the frequency-domain.

for n = 1 , 2 , 3 , - . . where

zn(t) = A COS - ( ;:)n for n = 1 , 2 , 3 , - - . and

(7)

for all n. With this manipulation, it can be seen that only the form of

zn(t ) is varied whereas y(t) is fixed. The Fourier transform of y(t) is easily obtained as

Y (f) = 2rsinc(2rf) (9)

For n = 1 ,2 and 3, it is relatively simple to derive the transforms of zn(t). Convolving X,(f) with Y ( f ) , we obtain the Fourier transform of p , (t ). PI ( f ) and PZ ( f ) are the same as the (4) and (5), respectively, and the 3rd order function P3(f) is given by

48Ar2 f cosinc(2rf) (10) P3(f) = (1 - (4Tf)2}{9 - ( 4 T f ) 2 }

Generally speaking, as the order of function (1) increases, the form of Xn(f) tends to be more complicated, and the processes of convolving and calculating will be tedious and difficult. Moreover, all recursive properties are absent from the trends of X,(f) for n = 1 , 2 , 3 , . - . . However, if we ignore the coefficients of X,(f), P,(f) only includes shifted sinc functions for any n. This property will guide us in the separation of coefficients presented later in section IV.

Fig. 1. Modified model of the class-I PRS for the nth order function

111. DERIVING THE RECURSIVE FORMULA

The transfer function of the class-I PRS system has the form of a truncated cosine function [4], [5]. Presented, is a modified model of the class-I PRS in order to obtain a recursive formula for deriving the Fourier transforms of the nth order cosine-pulses as shown in Fig. 1.

The model consists of a tapped delay line that corresponds with the polynomial K,(D) = (1 + D) in cascade with a filter whose frequency response H,-l(f) is equivalent to the overall transfer function of the system for the (n - 1)st order. We designate the first-half marked K, as the transversal filter and the second-half marked G, as the bandlimiting filter. We systematically alter the response of the bandlimiting filter along the order n, while the transversal filter remains fixed for every n. It is assumed that the frequency response of the bandlimiting filter is the same as that of the minimum- bandwidth PRS system for the 1st order function [5]. Simply stated. we set

G(f) = Ho(f) = Tz r p 1 . f ) (1 1)

which shapes a Fourier transform pair with

g1( t ) = ho(t) = -smc Tz Tl ’ (a) With the above assumptions and the model in Fig. 1, we

(Proposition 1) have the following proposition.

P(n) : &(f) = 2n~z(cos~fT1)n r p l f ) (13)

e h,(t) ( t + - ;) +h,-1 ( t - - s> (14)

for all n = 1 , 2 , 3 , - . - with Ho(f), ho(t) in ( l l ) , (12), respectively .

The symbol ‘e‘ denotes that the two functions on either side form a Fourier transform pair. The proof of Proposition 1 is discussed in Appendix A.

Because functions Ho(f) and ho(t) are even, it can be easily shown that &(f) and h,(t) are also even for all n = 1 , 2 , 3 , . - . using mathematical induction [6]. We are now ready to derive a recursive relationship for the Fourier transform of funtion (1).

308 IEEE TRANSACTIONS ON EDUCATION, VOL. 37, NO. 3, AUGUST 1994

From the duality of the Fourier transform pair[l-31 and because (13) and (14) are even functions, another transform pair is shown as follows:

with

If we set the parameters as TI = & and T2 = $ in (1% (16), and (17) for n = 1 , 2 , 3 , + . ., an essential recursive formula appears

= hn-1 (f + ;) + hn-1 (f - ;) (19)

for n = 1 , 2 , 3 , - . . with

Applying the equation in (19) with (20) to derive the Fourier transforms of functions in (18), it can be seen that the analytical processes to derive the transforms are much simpler than that of the conventional methods discussed in Section 11. Also, (19) with (20) can be easily solved with computer-aided numerical methods due to its recursive characteristics [7].

Iv. SEPARATION OF COEFnCIENTS AND THE LOOK-UP TABLE

After considering the clue which has been hinted from the method by the convolution theorem in section I1 and performing several inductive steps with relations in (19) and (20), we observe that the construction of the transforms is a product of a constant and a sum of shifted sinc functions for all n = 1,2 ,3 , . . .. Therefore, the relationship given in (19) can be written in the following manner:

where

for n = 1 ,2 ,3 , . - e with

Ro (f) = sine( 27f) (24)

Due to the fact that the relations (23) and (24) contain no coefficients, we are not hindered from solving the difference

equation as a complete recursion. After developing the re- lationship in (23) for various values of n with (24), then comparing it with the expansion of ( ~ + b ) ~ , we discover some interesting facts.

The binomial theorem, which can be easily proved by mathematical induction, gives the general expression for the expansion of (U + b)" [8].

(U + b)" = 2 ($rib' = (;),nbo +

i=O

+. . .

+ (n)uob" n

where the binomial coefficients are given by

n! (1) = q i q ! for i = 0 ,1 ,2 , . . . , n.

tween "U" and "b" for the ith term of expansion (25) as Furthermore, if we define the exponential difference be-

di E (exponent of a) - (exponent of b) = n - 2 i (27)

for i = 0 , 1 , 2 , . . . , n . a simple and useful representation of Rn(f) is obtained. All the above mentioned details are arranged in the following proposition:

(Proposition 2)

Q(n> : &(f) = 9 (:)Ro(f+ 2) i = O

= 2 (1) sinc (2rf + $) (28) a=O

for n = 1 ,2 ,3 , . . with d; in (27). This proposition is proved in Appendix B. We now arrive at the final result. The Fourier transform of

the nth order cosine-pulse given in (1) can be rewritten as

for n = 1 ,2 ,3 , . . . with d; in (27). Result (29) can be divided into three parts: the constant,

the binomial coefficients, and the shifted sinc functions. The constant is obtained directly from (22) for any n. Although the binomial coefficients differ with each power of (U + b), the coefficients of the successive powers of (U + b) can be arranged with Pascal's triangle [8]. Lastly, each part containing the shifted sinc functions can be completely determined by the values of d;'s since the remainings of the part are all fixed. Also we have already defined the values of di's as the exponential differences in (27) in the expansion of (U + b)". From these analyses, Table I is submitted for clarification. This table can be easily expanded to any order through simple arithmetic.

KANG AND OH: LOOK-UP TABLE FOR DERIVING FOURIER TRANSFORMS OF COSINE-PULSES 309

Order Constant n

0 2 AT 1 AT 2 AT / 2 3 AT/^^ 4 ~ ~ 1 2 3 5 AT /24 6 AT / 2 5

Pascal's Triangle Exponential difference dz

1 0 1 1 1 -1

1 2 1 2 0 -2 1 3 3 1 3 1-1 -3

1 4 6 4 1 1 5 10 105 1

1 6 15 20 15 6 1

4 2 0 -2 -4 5 3 1 -1 -3 -5

6 4 2 0 -2 -4 -6

Ultimately, painstaking calculations are no longer needed to obtain the Fourier transform of function (1). All that is needed is the order of the function. Otherwise meaning, we can find all the parameters that are needed to fabricate the Fourier transform of the problem function (1) of any order on the corresponding row of Table I, using the relationship represented in (29). For example, given the fifth order nature of (l), all the necessary parameters can be located on the fifth row of Table I resulting in

+ 5sinc (27f - !) + sinc (2rf - i) } (30)

V. CONCLUSION

In this paper, we have proposed a new and easier recursive method to derive the Fourier transforms of the nth order cosine-pulses using a modified class-I PRS system model. The process of this method is considerably simpler and more compact than that of the conventional methods which utilize consecutive differentiations or the convolution theorem. Also, the difference equation derived for each order consists of the sum of two functions which can be easily obtained from the step of the preceding order. Furthermore, because the formula consists of sinc functions shifted by symmetrical factors, a transform containing a single term can be easily produced.

In addition, the final result has been divided into three parts by separation of coefficients described in section IV. The parameters of each part can be easily obtained from a table that has been constructed from Pascal's triangle along with the coefficients in (22) and the exponential differences in the expansion of (a + b)". Thus, it is no longer necessary to perform complex calculations as in the case of conventional methods because the parameters of each order can be obtained from the corresponding row of the look-up table.

APPENDIX A PROOF OF PROPOSITION 1

From the structure of the model for the nth order function given in Fig. 1, we have a Fourier transform pair for the

transversal filter

k, ( t ) = 6 ( t + +) + 6(t- +) * K"(f) = 2cos7rfT1

for all n = 1 , 2 , 3 , - . . (A.1) and from the convolution theorem for the system theory [7] ,

the system functions for the model constitute another pair

hn(t) = k ( t ) * g n ( t ) = k ( t ) * h,-l(t)

= hn-l (t + +) + hn-l ( t - +) * Hn(f) = Kn(f)Gn(f ) = Kn(f)Hn- l ( f )

= 2 cos TfT1 H,- 1 ( f ) for all n = 1 , 2 , 3 , . . . (A.2)

where Ho( f ) and ho (t) are given in (1 1) and (12), respec- tively, and the symbol '*' denotes the convolution integral of the two functions on either side.

Next, the proposition is proven by the principle of mathe- matical induction in the following manner [6] .

Basis step When n = 1, using assumptions (1 l), (12), and the relation (A.2), we have

Hl(f) = 2cos7rfT1Ho(f) = 2T2cosafT1 rI(T1f)

which indicates that statement P( 1) is true.

for some integer m > 1, we have the pair: Induction step If it is assumed that statement P(m) is true

Hm(f) = 2nT2(cos.rrfTl)" fl(T1.f)

*h,(t)=h,-1 ( t + - ?) +h,-1 ( t - - :) After one more step with (A.l) and (A.2), the relation for

(m + 1) is obtained as follows:

H,+l(f) = 2COS~fTlHm(f) = 2"+1T2(cosafT1)m+l r I (T1f )

e h,+1(t)=hm t + - +h , t - - ( ?) ( s> which proves the validity of P(m + 1).

(Q.E.D.). By induction, statement P ( n ) is true for all n = 1,2,3, . . .

APPENDIX B PROOF OF PROPOSITION 2

This proposition can also be proved by the principle of

Basis step When n = 1, the following relationships are mathematical induction in the following manner [6] .

obtained using (23) and (24)

with Ro( f ) = sinc(2rf)

310 IEEE TRANSACTIONS ON EDUCATION, VOL. 37, NO. 3, AUGUST 1994

and they are rewritten as and an identity

where di = 1 - 22(2 = 0 , l ) from (27). Thus, statement Q ( l ) is true.

Suppose that statement Q(m) is true for some m(m > l ) , then we have

Induction step

where d; = m - 22(2 = 0 , 1 , 2 , . . . , m) from (27).

for ( m + 1) results Applying (23) and (24) once again, the following expression

where di = m - 2i, for 2 = 0,1 ,2 , . . . , m. After expanding the summations in the above expression

and binding the terms which have the same delay factors, we obtain

Rm+l(f) = { ( y ) + (;)}+ + y) + { ( Y ) + (Y) }RO( f + F) +. . .

+ { ('c) + (.: l)}RO ( I + -:; + ) +. . ,

Using the theorem

('r) + (A) = (T') for s = 1 , 2 , 3 , . . . ,m.

we obtain a simple result

where d; = ( m + 1) - 22, for 2 = 0 , 1 , 2 , . .. , ( m + 1). With the help of (24), we arrive at the final result as follows:

These prove that statement Q ( m + 1) is also true.

(Q.E.D.). Therefore, statement Q ( n ) is true for all n = 1 , 2 , 3 , . . .

REFERENCES R. E. Ziemer and W. H. Tranter, Principles of Communications-Sqstems, Modulations and Noise, 2nd ed., New York: Houghton Mifflin, pp. 37-1 11, 1985. A. B. Carlson, Communication systems- An Introduction to Signals and Noise in Electrical Communication, 3rd ed., Englewood Cliffs, NJ: Prentice-Hall, pp. 31-67, 1986. J. W. Cooky, P. A. Lewis, and P. D. Welch, "The fast Fourier transform and its applications," IEEE Trans. Education vol. E-12, pp. 27-34, Mar. 1969. A. Lender, "The duobinary technique for high speed data transmission," IEEE Trans.Commun.Electron., vol. 82, pp. 214-218, May 1963. P. Kabal and S. Pasupathy, "Partial-response signaling," IEEE Trans. Commun., vol. COM-23, pp. 921-934, Sep. 1975. B. Kolman and R. C. Busby, Discrete Mathematical Structures for Computer Science, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 47-58, 1987. E. 0. Brigham, The Fast Fourier Transform. Englewood Cliffs, NJ: Prentice-Hall, pp. 148-223, 1974. S. Lipschutz, Theory and Problems of Finite Mathematics. Schaum Publishing Co., pp. 141-145, 1966.

Chang-Eon Kang (S'68-M'74-SM'88) was bom in Cheju, Korea. in 1938. He received his B.S. degree in electrical engineering from Yonsei University, Seoul, Korea, in 1960, and his M.S. and Ph.D. degree in electrical engineering from Michigan State University, MI, in 1969 and 1973, respectively.

From 1973 to 1982 he was an associate profes- sor in the Department of Electrical Engineering at Northem Illinois University, Dekalb, IL. Since 1982 he has been a professor of the Electronic Engi- neering Department and Dean of the Information

Systems Center at Yonsei University, Seoul, Korea. During 1988-1989 he served as vice president of the IEEE Korea Section,

and from 1989-1991, served as president of Korea Institute of Communication Sciences (KICS). Afterward, he was nominated the Honorary President of KICS by the Board of Directors.

Dr. Kang has published over 120 papers in the area of communications. He is the author of seven books: Communication Engineering, Communication Systems, Digital Communication Systems, Data Communications, Information Theory, Communication Experiments and Telematics & New Media. He was the recipient of the Academic Excellence Award by KICS on two occasions and also the recipient of the National Honorary Medal by the President of the Republic of Korea Tae-Woo Roh. His research interests include digital and data communications, information theory and their applications with an emphasis on mobile Communications.

He is a member of Eta Kappa Nu and Tau Beta Pi.

KANG AND OH: LOOK-UP TABLE FOR DERIVING FOURIER TRANSFORMS O F COSINE-PULSES

Yong-Sun Oh (S’88-M’93) was born in Taejon, Korea, in 1957. He received the B.S., M.S., and Ph.D. degree in electronic engineering from Yonsei University, Seoul, Korea, in 1983, 1985, and 1992, respectively.

He worked as a research engineer at the Research and Development Department of Samsung Electron Co. Ltd., Kiheung, Kyungki-Do, Korea, from 1984 to 1986. He joined the faculty of the Department of Information and Communication Engineering, Mokwon University, Taejon, Korea, in 1988, where

he is currently an assistant professor. His research interests include digital and data communication systems, information theory and their applications with an emphasis on short-pulse transmission and bandwidth-efficient coded modulation.

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