Dynamic range and other fundamentals of the complex ... - IEEE Xplore

397

IEEE TRANSACTIONS ON COMMUNICATIONS. VOL. 37. NO 4 . APRIL 1989

111. THEORIGIN OF THE APPROXIMATING EXPRESSION A N D THE LINKBETWEEN PARAMETER CY AND THE DYNAMIC RANGE

Dynamic Range and Other Fundamentals of the Complex Bessel Function Series Approximation Model for Memoryless Nonlinear Devices M. S . O’DROMA

Abstract-This communication clears some of the confusion in the application of the complex Bessel function series approximation model of the memoryless amplitude and phase characteristics of devices such as TWT’s, MESFETS, Lasers, mixers, etc. The origin and a new derivation of this approximation model i s given and the link between the operating dynamic range of the nonlinear device characteristic under study and a parameter in the approximation is shown.

I. INTRODUCTION Some confusion has arisen in the application of the complex Bessel function series approximation of memoryless nonlinear devices such as TWTA’s, [ l ] and [2]. A source of this confusion has been in the choice of a value of a parameter in the approximating expression. The practice has been to choose this simply as though it was just an “arbitrary constant which scales the input,” [ 3 ] , or as an optimizing parameter in the effort to get a best fit of the approximation to the measured data. By examining the derivation of the particular approximation a determining and important link between this parameter and the desired operating dynamic range of the nonlinear characteristic being approximated is shown. How values for this parameter are calculated in terms of input overdrive relative to device saturation is also shown. The link to the dynamic range is also made in [4] where the Chebyshev transformation is employed. In examining the origin of the approximating expression an alternative and simpler derivation to that given in [5] is shown. It is a development on work described in [6]. 11. APPLICATION OF THE COMPLEX BESSELFUNCTION SERIES APPROXIMATION MODEL

The useful approximating expression is given [ 5 ] K

g ( A ) e L f f ( A ) = bk

. Jl(akA)

(1)

k= I

where g ( A ) and f ( A ) are the actual measured single unmodulated carrier envelope amplitude and phase transfer characteristics. In the approximation on the right-hand side of the equation b k are complex coefficients, J1(.) is a Bessel function of the first kind and parameter a , which has been a source of the confusion, is treated below. A particular difficulty has been the effort to relate the dynamic range of the approximation to the dynamic range of the single or multicarrier input signal. This difficulty is cleared up when it is realized that the parameter “a” in (1) is much more than simply an optimizing parameter, useful for obtaining a good fit to the measured data when this is done using a Fletcher-Powell or similar type of numerical technique, o r just an arbitrary constant for scaling the input level.

Paper approved by the Editor for Satellite Communications and Coding of the IEEE Communications Society. Manuscript received October 30, 1987; revised December 28. 1987. M. S. O’Droma is with the Electronic and Computer Engineering Department, University of Limerick, Limerick, Ireland. IEEE Log Number 8926516.

The parameter a is in fact directly related to the dynamic range of the instantaneous input signal to the nonlinear device, (c.f., [4] also). This can be seen clearly when it is realized that (1) is readily derived by approximating the periodic extension of the instantaneous voltage transfer characteristic by a complex Fourier series expansion

c m

e z ( t )=

ckeJakel(‘)

(2)

k= -m

where e l ( t )is the input signal (general form), and ez(t) is the output signal. The parameter a is determined by the maximum dynamic range D of the input to be approximated by this model

D = 2a/a.

(3)

This in turn defines the period of the periodic extension. Following this approach it is readily shown, (c.f., the Appendix). that the zonal output q ( t ) of a general multicarrier input (including noise, m signals in total), el(t)

(4)

. ex;!

I J n / ( u p + d/(t))

(5)

with the condition Cy:, n l = 1. The output in any other harmonic zone, e . g . , for mixer outputs and detector outputs, is similarly derived from (2) by setting the relevant condition on E n l , I = 1, m . Thus, knowing the coefficients bk [from ( l ) ] the output signal for any input signal for this type of nonlinear device (i.e., high frequency, bandpass, and memoryless devices) is completely described.

Iv. CALCULATING o( I N TERMSOF DEVICE OVERDRIVE When an approximation models the nonlinearity over a dynamic range up to x dB input overdrive with respect to saturation, the value of a is readily given in terms of x as follows. Assuming, as is usual in these situations, that the nonlinear characteristics are normalized with respect to an input saturation power P, of a single unmodulated carrier. This could arbitrarily be chosen in those cases where the saturation point is not clearly defined. Let A , be the peak instantaneous voltage amplitude corresponding to P, (i.e., P, = l / T j l ( A ssin t ) * dt = A f / 2 ; when measurements are assumed to be scaled to a 2 input impedance). This is taken as the value of the 1 f normalizing voltage. In the device measurements x dB input overdrive (upper limit of the dynamic range) corresponds to an input power of P,, watts, which in turn corresponds to a peak instantaneous sinusoidal voltage amplitude of A , . Thus A , i s the actual dynamic range (scaled again to a 1 Q input impedance and also assuming a class C type device). Hence, A , / A , is the normalized dynamic range, i.e., D in (3). Thus, a is given

C Y = ~ ~ / D = ~ T A , / A , ~ = ~ ~ . ~ O ~ ~ ’ ~ ~ . This would also explain why a = 0.6 corresponds to an overdrive of x = 20 dB, [2] and [4], and provides quite

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 37, NO. 4, APRIL 1989

satisfactory approximation, while very poor results were obtained in [ 11 with CY = 200, as this corresponds to a dynamic range reaching only up to a backoff from saturation of 30 dB, i.e., x = -30 dB! APPENDIX Derivation of the complex Bessel function series approximation of the single carrier envelope transfer characteristics from a complex Fourier series expansion of the instantaneous voltage transfer characteristic. Assuming the instantaneous voltage transfer characteristic can be represented by a complex Fourier series expansion of its periodic extension where the period D, ( = ~ T / c Y )is, defined by the maximum dynamic range of the input signal el([), then the output is m

e z ( t )=

ckeJahel(‘). k=

(‘41)

-m

This of course will produce components at all the harmonics of the frequency (or frequencies) contained in el(t). However, for microwave and mm-wave amplifiers which are followed by zonal filters only fundamental components in the frequency zone of e , ( [ )will appear in the output. Consider a general expression for a multicarrier input m

el ( t )=

cos

M))

(U,+

(A21

/=I

where A I ( t )and/or c$[(t)would contain the information which would be intelligible or unintelligible, such as narrow-band noise and distorted multipath signals. The angular frequencies wI are real or arbitrarily defined carrier frequencies, the latter applying in the case of noise or multipath. Now the output can be written

Equation (A7) describes the zonal output of the nonlinear amplifier in “analytic” form. Taking the real part would give the signal as it appears at the output of the real system. The desired Bessel function series approximation to the unmodulated single carrier envelope transfer characteristics is now readily obtained from the single carrier zonal response, derived from (A7) by setting m = 1, (letting A , ( t ) = A , wI = w . and 4 l ( t ) = 0)

REFERENCES X. T . Vuong and H. J. Moody, “Comments on a general theory for intelligible crosstalk between frequency-division multiplexed anglemodulated carriers,” IEEE Trans. Commun., vol. COM-28, 1 I , pp. 1939-1943, NOV. 1980. B. Pontano and 0. Shimbo. Reply to [l], ibid, pp. 1943-1944. 0. Shimbo and B. Pontano, “A general theory for intelligible crosstalk between frequency-division multiplexed angle-modulated carriers, IEEE Trans. Commun., vol. COM-24, pp. 999-1007, Sept. 1976. 0. Shimbo and L. N. Nguyen, “Further clarification on the use of the Bessel function expansion to approximate TWTA nonlinear characteristics,” IEEE Trans. Commun., vol. COM-30, pp. 418-419, Feb. 1982. J. C. Fuenzalida, 0. Shimbo, and W. L. Cook, “Time-domain analysis of intermodulation effects caused by nonlinear amplifiers,” COMSAT TR, vol. 3, no. I , Spring 1973. M. J. Eric. D. A. George, and A. R. Kaye, “Analysis and compensation of bandpass non-linearities for communications,” IEEE Trans. Commun., vol. COM-20, pp. 965-972. Oct. 1972.

m

ez(t)=

CkeJak E;“=

(w+$/(o)

A/(f)cos

(‘43)

A General Analysis of Bit Error Probability for Reference-Based BPSK Mobile Data Transmission

k= - m

A. BATEMAN

Applying the Bessel function series expansion m

e J ~cos e =

jnJn(x)eJne

(A5)

n= - m

where J,,(x) are Bessel functions of the first kind. Then m

ez(t)= k= - m

JCk

i

[(-j)

fi

[Jn(akA/(t))

Abstract-This paper seeks to present a unified analysis of the performance of binary phase shift keying under static and mobile operating conditions, when a separate reference tone is used for channel sounding and subsequent “coherent” data detection. The paper shows that under both Rician and Rayleigh fading conditions, the use of a reference can eliminate the irreducible error rate phenomenon, with minimal sacrifice in BER performance over an ideal BPSK system.

I= I

nI,nz’.’nm= - m

I. INTRODUCT~ON The current interest in reference-based coherent data systems has resulted in a number of papers being published on nl = 1 will yield a signal e,(t) the subject in recent months, [1]-[5]. By using a reference Applying the condition consisting only of all the components which fall into the tone to facilitate coherent detection, both power and band“zonal” frequency band (i.e., output band corresponding to width efficient data modulation formats can be operated in the hostile mobile radio environment, an environment in which the input band) conventional “data-derived’’ coherent detection techniques such as Costas or N t h power tracking loops fail catastrophi-

-

where

bk =

( j ) n / j ] ejxE

j(ck - c - k ) , for k

=

1,

I n/(a/f+d/(r)).

. - . , 00.

(A6)

Paper approved by the Editor for Modulation Theory and Nonlinear Channels of the IEEE Communications Society. Manuscript received May 30, 1987; April 10, 1988. The author is with the Centre for Telecommunications Research, University of Bristol, Bristol BS2 8TR, England. IEEE Log Number 8926528.

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1989 IEEE