Digital simulation of lognormal random processes

INT. J. ELECTRONICS,

1995, VOL. 78, NO. 3, 477- 482

Digital simulation of lognormal random processes J. A. DELGADO-PENÍNt and J. SERRAT-FERNÁNDEZt Stationary lognormal random processes find application in radar and transmission systems as models of signa) generators. This paper describes a simulation model for generating either complex or real-valued lognormal, wide-sense stationary, random processes. The basic model considers a nonlinear processing of a complex-valued gaussian random process. The accuracy of the model is assessed through the simulation of realistic test cases.

l . Introduction The generation of stationary lognormal random processes plays an important role in designing electrical communication systems. In fact, radar systems design often considers the clutter modelled as a complex lognormal process (Farina et al. 1986) and mobile radiocommunication systems design characterizes the slow fading as a real lognormal process (Gilhousen et al. 1990). The generation of correlated processes with lognormal statistics is not trivial-it follows from different approaches appearing in the literature. Farina et al. ( 1986) presented a canonical model which carries out the autocorrelation shaping by means of a linear operator on batch blocks of samples. A quasi-deterministic approach to obtain real 1ognormal processes has been considered by An et al. (1990). An approach for real processes generation has been described by Gi1housen et al. (1990) and Brehm et al. (1986). This approach is based on a system constituted by a filter and an exponential nonlinearity, fed by white gaussian noise. In this case, the output process autocorrelation function is fitted to a gaussian shape by means of the minimum-squared error criterion. The purpose of this paper is to describe a new approach intended to generate either comp1ex or real-va1ued 1ognormal processes with gaussian autocorrelation shape. The approach is to be used in computer simulation, or in real-time simulators, of clutter or fading phenomena. Section 2 is devoted to describing the basic model assumed to represent the samples of a lognormal random process generator. In particular, this model is made up of a white gaussian noise generator, a linear time-invariant filter and a nonlinear memoryless device. Section 3 is focused on the problems arising from the adopted procedure to shape the output autocorrelation function. This result of § 3 is the specification of the linear filter with parameters strictly dependent on the variables assumed in the basic model. Finally, the last section of the paper describes a summary of results from statistical trials carried out on the lognormal process generator. · 2.

Basic model A schematic diagram of the proposed lognormal random process generator model is depicted in Fig. 1, where u(t) is a wide-sense stationary complex-valued Received 24 June 1994; accepted 26 July 1994. tDepartment of Signa! Theory and Communications, Universitat Politecnica de Catalunya, Barcelona, Spain. 0020--7217/95 $ 10.00 © 1995 Taylor & Francis Ltd.

478

J. A. Delgado-Penín and J. Serrat-Fernández

u(t) Figure l.

~'---H-(f_)___,¡ w(t)1

~g~(t]

ex p( ) 1 o

Functional block diagram of the proposed lognormal process generator.

white-gaussian random process with its real and imaginary components mutually uncorrelated. The random process u(t) feeds a linear time-invariant system whose transfer function is H(f), such that w(t) is a correlated gaussian process (Proakis 1989)

w(t)=x(t)+jy(t)

(1)

Rww(r) = E{ w*(t+ r)w(t)} = Rxx(r) + Ryy(r) -j(Rxy(r) - Ryx(r))

(2)

with autocorrelation function

The complex correlated lognormal process g(t) is finally obtained by means of an exponential memoryless nonlinearity so that (3)

g(t) = exp [x(t) + j y(t)]

Note that from this equation, when making y(t) equal to zero, g(t) becomes real. Therefore, the model of Fig. 1 is general and the discussion and conclusions hereafter can be applied either to complex or real-valued lognormal processes. As the statistics of g(t) only involve first and second-order moments, it will be completely characterized by the mean and the autocorrelation functions given by Farina et al. (1986) and Harger (1970) E{g(t)} = exp (

Rxx(O) - R (O)) 2

YY

.. (cosRxy(O) + JsmRYxCO))

R 99 (r) = exp [Rxir) + Ryy(r) + Rxx(O)- Ryy(O) - j(Rxy(r)- Ryx(r))]

(4) (5)

The major problem concerning the proposed model lies in the fact that, from a specified autocorrelation function, R 9 g(r) it is necessary to find the corresponding function Rww(r) for the input process. In other words, it is necessary to solve the inverse of (5), which leads to determining the magnitude of H(f). However, the existence and uniqueness of the solution cannot be guaranteed for an arbitrary R 9 g(r) . 3.

Simulation model

The present approach considers the fact that the autocorrelation function of the output process g(t) can be approximated to a gaussian shape as (6) whose Fourier transform, namely the power spectral density (PSD) of g(t), is given by (7)

- -- - - - - - - -

-----

--------- - -

- - - - - - - - - - - -- -- - - - - - - - - -

Digital simulation of lognormal random processes

479

y

Figure 2. Functional block diagram of the proposed simulation model.

where Pis the output average power (P = R 99(0) = exp[2Rxx(O)]) and 0" 8 is related to the bandwidth of the above-mentioned PSD. On the other hand, and to simplify the outstanding model it will be assumed that (8)

and that (9) with y being a real-valued factor greater or equal to O. Therefore, (5) becomes (10) 1

and the generic model shown in Fig. 1 becomes the mode1 of Fig. 2 where up(t) and uq(t) are the real and imaginary components of the process u(t) . The autocorrelation Rxx(r) can be found now equating (7) and (10). Then (11)

Unfortunately, this autocorrelation function is unrealizable beca use Rxx( oo ) oo (Proakis 1989). Therefore, sorne kind of approximation is mandatory and , of course, it will be necessary to determine how it restricts the ranges of the involved parameters. From this point of view, (11) is approximated by means of the exponentia1 function of the K 1 exp ( - K 2 r 2 ) type. This is reasonable beca use the right-hand si de of (11) can be associated with the first two terms of the series expansion of K 1 exp ( - K 2 r 2 ). Therefore, identifying the coefficients K 1 and K 2 and noting Rxx(r) as the approximation of Rxx(r), = -

(12)

is obtained.

J. A . Delgado-Penín and J. Serrat-Fernández

480

The effects on the output autocorrelation function, when Rxir) (see equation (12)) is used instead of Rxir) (see equation (11)), are pointed out in Figs 3 and 4. In these figures the function evaluated with (10) is denoted 'ideal', and the function evaluated with the same equation, but using Rxir) instead of Rxx(r), is called 'approximated'. Because a smaller K 2 factor was used in Fig. 3, the resulting fit is much better than that of Fig. 4. It is interesting to see that in (12), K 2 can be made small enough by means of y, and therefore, independent of the ratio a~/ RxxCO). This is how the value of the factor y is fixed except that g(t) is real. In this case y would be equal to zero. Once the autocorrelation function of x(t) (and hence of y(t)) has been determined, the squared magnitude of the transfer function H(f) is readily determined by computing its Fourier transform (Proakis 1989). Therefore (13) where (14) That is, the filters in the Fig. 2 model are gaussian filters with an appropriated gain and with the standard deviation a A given in (14). 4.

Results

In order to validate the proposed lognormal process generator approach, the simulation of the model of Fig. 2 was undertaken. To generate g(t), white gaussian noise sources were used followed by two equal gaussian filters. Each of these filters was an infinite impulse response filter (IIR) whose recurrency equation was obtained by resorting to the series expansion of

dB 0,-~~------------------------------,

-5

.

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.....

~o

-15

..

-20

. .. ... .... .

. ... ....

.. .. .. .. .

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

·25L--L--L--L--L-~--~~--~~--~~--~

o

2

3

4

5

6

7

8

9

10

11

12

TIME UNITS

- Approxlmated - - Ideal

Figure 3.

Computed output autocorrelation function, solid curve, when RxxCr:) is used with K 2 = 10- 3 . The broken curve is the specified autocorrelation .

481

Digital simulation of lognormal random processes dB 0 .--===----------------------------~

-5

-10

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

.. . .... . ... .... ,..

',

-15

.. . ...... .. .. ... ... . . . .. ..... .. . . .. ........ ·' ..

-20

-25L__L__L __ L_ _ o 2 3 4

L_-L--~~--J__J_ _J_~~~

6

6

7

8

9

10

11

12

TIME UNITS

- Approxlmated - - Ideal

Figure 4. Computed output autocorrelation function , solid curve, when R xx('r) is used with K 2 = 1O- 2 • The broken curve is the specified autocorrelation.

H(f) magnitude in Laguerre polynomials (Jones 1970). Specifically, a ten-pole filter was used (five cascaded cells of order two) which leads to a gaussian transfer function in a dynamic range of about 80 dB. On the other hand, all the specific assumptions or choices related throughout this paper were considered. Figure 5 shows the output PSD of a complex process g(t) (solid curve) with o-B = 0·042 frequency units ( - 3 dB bandwidth of 0·1 frequency units) and Rxx(O) = 0·346. In this case, the approximation was done with K 2 = 0·00l. In the same figure the ideal gaussian PSD is shown as a broken curve. As can be observed, the fit is good in a dynamic range up to 30 dB. Figure 6 was achieved with the same parameters but with K 2 = 0·0 l. As expected, the differences are now significan t.

-60 L_--~

0.8

_____ L_ _ _ _L __ _

0.85

0.95

0.9

~_ _ _ __ L_ _ _ _J __ _ _ _L __ _~

1

1.05

1.1

1.15

1.2

FREQ. UNITS

-

Slmulated

----- Ideal

Figure 5. The solid curve is the output power spectral density (PSD) estimated by means of the simulation model with K 2 = 10 - 3 . The broken curve is the ideal gaussian PSD.

Digital simulation of lognormal random processes

482

-50L---~----~-----L----~----~----L---~----~

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

FREO. UNITS -

Slmulated

----- Ideal

Figure 6. The solid curve is the output power spectral density (PSD) estimated by means of the simulation model with K 2 = 1O- 2 • The broken curve is the ideal gaussian PSD.

5.

Conclusions

This paper describes a new approach intended to allow the sequential generation of complex-valued or real-valued stationary lognormal processes whose frequency dispersion can be accurately controlled by means of gaussian filters and an unbalancing factor of the two inputs of an exponential memoryless operator. The main constraint is that the values adopted by the frequency dispersion of the output process must be of the order of magnitude of the filter's bandwidth, so as to allow the synthesis of such filters, with the same sampling frequency selected, m order to carry out the simulation. REFERENCES

AN, J. F. , TURKMANI, A. M. D. , and PARSONS, J. D., 1990, Implementation of a DSP-based frequency non-selective fading simulator. 5th International lEE Conference on Radio Receivers and Associated Systems, Record No . 325, pp. 20-24. BREHM, H. , STAMMLER, W., and WERNER, M., 1986, Design of a highly flexible digital simulator for narrowband fading channels. Signa! Processing III: Theory and Applications, edited by I. T. Young et al. , (North-Holland: Elsevier). FARINA, A., Russo, A. , and STUDER, F. A., 1986, Coherent radar detection in log-normal clutter. Proceedings of the lnstitution of Electrical Engineers, 133, Part F, 39-54. GILHOUSEN, K. S., JACOBS, I. M., PADOVANI, R., and WEAVER, L. A., JR . 1990, Increased capacity using CDMA for mobile satellite communication. IEEE Journal on Selected . Areas in Communictions, 8; 503-514. HARGER, O. R., 1970, On the characterization and likelihood functional of log-normal random processes. IEEE Transactions on Information Theory, 16, 630-632. JONES, N. B., 1970, A Laguerre series approximation to the ideal Gaussian filter. The Radio and Electronic Engineer, 40, 151-155. PROAKIS, J. G. , 1989, Digital Communications, second edition (New York: McGraw-Hill).