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OBTAINING TIME DOMAIN EXPRESSIONS OF BAND LIMITED SIGNAL PULSES FOR ZERO ISI.

Byline: Ejaz A. Ansari

ABSTRACT: In this paper, we derive time domain expressions of band limited signal pulses used to obtain zero inter symbol interference (ISI) for digital communication. Although time domain expressions of raised cosine (RC) and square root raised cosine (SRRC) pulses have been reported in many Electrical Engineering books on Digital Communications but however, their closed forms have not been computed. We thus, make use of Fourier transform properties along with its tables in a sensible way to derive these expressions in time domain. We consider only four band limited pulses in frequency domain which are rectangular, triangular, raised cosine and square root raised cosine. We start with the Fourier transform pair of unit step function and compute the time domain expression of rectangular band limited pulse using time shift and linearity properties of Fourier transform. Then, we make use of auto correlation / convolution property of Fourier transform to obtain time domain expression of triangular band limited pulse. Finally, we compute the time domain expressions of (RC) and (SRRC) band limited pulses using linearity, differentiation and even properties of Fourier transform.

Key Words: Closed forms, Raised Cosine, Square Root Raised Cosine, Time domain expressions, Zero ISI

1 INTRODUCTION

Due to infinite absolute bandwidth of multilevel pulses and their improper filtration through a communication system, spreading takes place in time domain which causes the pulses corresponding to each symbol to smear into adjacent time slots and results into inter symbol interference (ISI) [1]-[2]. Restriction on the bandwidth of the pulses is needed so that pulses could have rounded tops instead of flat ones in order to avoid ISI. Due to two significant problems present in the overall amplitude transfer characteristic of the signal pulse that causes ISI; i) flat spectrum over the range of frequencies, |f| less than B and zero elsewhere which results into physically unrealizable impulse response of the system, ii) the synchronization of the clock in the decoding sampling circuit needs to be almost perfect which results into ISI due to inaccurate synchronization [3] [5].

Because of these difficulties, we are forced to include other signal pulse shapes that have a slightly different bandwidth. The basic idea is to find pulse shapes that pass through zero at adjacent sampling points and yet to have an envelope that decays much faster than 1/t so that clock jitter in the sampling times does not cause appreciable ISI. Plots of the frequency responses and their corresponding impulse responses of raised cosine (RC) and square root raised cosine (SRRC) pulses for various roll off factors, AY are sketched in section 6 of this paper. It is seen from these graphs that as the absolute bandwidth of the pulse is increased, the filtering and clock timing requirements are relaxed a little bit since the envelope of the impulse response decays as 1/t3 which is faster than that of the order 1/t for large values of t. The baud rate of these pulses is dependent on the rolloff factor, AY and absolute bandwidth of the system which was not the case with either rectangular or triangular shape signal pulse [1 5].

In this paper, we thus, compute time domain expressions /closed forms of band limited signal pulses using Fourier transform properties sensibly. We consider four types of signal pulses in the paper which are of rectangular, triangular, raised cosine and square root raised cosine shapes. Time domain expressions of raised cosine and square root raised cosine shape pulses have been reported in many books on digital communications for elimination of inter symbol interference (ISI) without their derivation.This paper is organized as follows: Section 2 describes the computation of time domain expression of rectangular shape pulse using Fourier transform pair of unit step function, duality, time shift and linearity properties of Fourier transform. Computation of time domain expression of triangular shape pulse is done in section 3. Section 4 and 5 describes the detailed procedure of computing time domain expressions of raised cosine and square root raised cosine shape pulses. We provide the sketch of these pulses and their time domain expressions in section 5. Finally, we present our conclusions in section 6.

2 COMPUTATION OF TIME DOMAIN EXPRESSION OF RECTANGULAR SHAPE SIGNAL PULSE

Rectangular shaped signal pulse, r(f) as per [6] [8] is defined as; it is equal to T for |f| = 0.5t and zero otherwise. Shifted form of unit step functions permit us to express this pulse as, r(f) = T x{u(f + 0.5t) u(f 0.5t). We know from Fourier transform tables given in [9] [12] that u(t) (j2pf)-1 + 0.5d(f) and time reversal property of Fourier transform immediately allows us to write u(-t) (-j2pf)-1 +0.5d(f), where we have used the even property of Dirac delta function. Now, duality property of Fourier transform is utilized to express the inverse Fourier transform of u(f) as (- j2pt)-1 + 0.5d(t) u(f). If X(f) denote the Fourier transform of x(t), then x(t)exp(j2pf0t) denote the Fourier transform ofX f f0 using frequency translation property. We applythis property along with linearity of Fourier transform and compute time domain expression of r(f) as,Equation (1) has been derived in various books of Signalsand Systems using either the definition of Fourier Transform or its differentiation property. But, in this case, it has been derived with a different approach.

Two cases are of interest, one is for AY = 0 and the other is forAY = 1. When AY = 0, then t = AY/T = 0 and the pulse in timedomain from eq.(4) simply reduces to xrc(t) = sinc(t/T) which corresponds to the spectrum Xrc(f) = 2k, for 0 = |f| = 1/2T infrequency domain. And when AY = 1, then t = 1/T and the product of sinc and cos functions simply reduces toWe'll apply here the definition of inverse Fourier transform to find out the corresponding pulse in time domain. First, we see that root raised cosine pulse spectrum is a real and even function of frequency, f. It means that its corresponding pulse in time domain will also be a real and even function of time, t when inverse transformed in time domain. Using inverse definition of Fourier transform, we have

6 PLOTTING OF TIME DOMAIN EXPRESSIONS OF SIGNAL PULSESIn this section, we sketch the time domain expressions ofband limited signal pulses described in eqs. (1) (5) for different values of t and excess bandwidth parameter, AY. Figure 1 (a) and (b) shows the variation of rectangular and triangular signal pulses for different values of t and the variation of their corresponding time domain expressions for these values of t. The phenomenon of compression / expansion in frequency domain is clearly visible in the sketches of these time domain expressions.

corresponding time domain expressions for these values of AY. The variation of raised cosine and its square root pulses in frequency domain is clearly visible in the sketches of these time domain expressions.

Moreover, the rate of decay of closed forms is t and t2 and both functions in time domain are non-causal which require a time delay of significant length to be able to become physically realizable.Figure 2 (a) and (b) shows the variation of raised cosine and its square root signal pulses for different values of excess bandwidth parameter, AY and the variation of their both functions realizable. Moreover, the bandwidth of the functions in figure 2 depends upon the parameter, AY which was not the case in figure 1.

7 CONCULUSIONS

In this paper, we derived closed forms or time domainexpressions of four band limited signal pulses such asrectangular, triangular, raised cosine and square root raisedcosine. Although closed forms for rectangular and triangular pulses have been computed in many books on Signals andSystems, however, we used a different approach here in obtaining the closed form expressions of these two pulses. The closed forms of raised cosine and square root raised cosine pulses which are extensively used in digital communications for zero ISI have been reported in many books on Digital Communications without their derivation. We used Fourier transform properties along with its tables in a sensible manner to compute the closed form expressions of these important pulses. Finally, we showed the variation of these pulses and their corresponding time domain expressions with respect to the parameters T and AY in the form of sketches.

ACKNOWLEDGMENT

We are extremely grateful to the Department of Electrical Engineering of COMSATS Institute of Information Technology (CIIT) for carrying out this work. Moreover, we are also thankful to the anonymous reviewers for their valuable suggestions towards the improvement with respect to the quality of the paper. Finally, we appreciate the efforts of Mr. Saad Aslam, faculty member in the Department ofElectrical Engineering, CIIT, Lahore for assisting in providing the desired sketches.

REFERENCES

1.Simon Haykin and Michael Moher, CommunicationSystems, 5th edition, John-Wiley and Sons edition, India-2010.2.Leon W. Couch, II, Digital and Analog CommunicationSystems, 7 edition, Pearson Education, India 2009.3.B.P. Lathi and Zhi Ding, Modern Digital and AnalogCommunication Systems, 4Press, India 2010. edition, Oxford University4.John G. Proakis, Digital Communications, 3rd edition,McGraw Hill International Editions, 1995.

5.A. B. Carlson, P. B. Crilly and J. C. Rutledge, Communication Systems. 4th Edition, McGraw Hill,2002, New York, NY 10020.

6.A. V. Oppenheim, A. S. Willsky with S. H. Nawab, Signals and Systems, 2nd edition, Prentice Hall, New Delhi, India, 2004.

7.Hwei P. Hsu, Schaum's Outlines on Signals andSystems, 3rd Edition, Tata McGraw Hill, New Delhi,2004.

8.Simon Haykin and Barry Van Veen, Signals andSystems, 2nd Edition, John Wiley and sons, 2007.

9.M. H. Hayes, Digital Signal Processing, Tata McGraw- Hill, 2005.

10. A. V. Oppenheim, R. W. Schafer and J. R. Buck, Discrete-time Signal Processing, Pearson edition, 2nd edition, 1994.

11. J. G. Proakis and M. Dimitris, Digital Signal Processing, (Principles, Algorithms and Applications), Pearson son edition, 4th edition, 2011.

12. S. K. Mitra, Digital Signal Processing, Hill, 3 edition, 2006.
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Title Annotation:intersymbol interference
Author:Ansari, Ejaz A.
Publication:Science International
Article Type:Report
Date:Mar 31, 2014
Words:1677
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