Generating Swerling Random Sequences

Transcription

1 Generating Swerling Random Sequences Mark A. Richards January 1, 1999 Revised December 15, 8 1 BACKGROUND The Swerling models are models of the robability function (df) and time correlation roerties of the radar backscatter from a comle target [1]. Develoed in the early days of radar, the Swerling models aly to a finite grou of ulses. They were develoed with the model of a rotating surveillance radar in mind. As the radar beam swees ast a target (a single scan), it collects echoes from that target in the aroriate range bin for several ulses. Once the beam moves ast the target, no more echoes are received until the net scan, when the beam has swung back around to the target osition again; another grou of several ulses is then received. Detection is assumed to be attemted using all of the ulses from a single scan. Thus, the joint statistics of a grou of target echo samles from contiguous ulses of a single scan are of interest. The Swerling models are formed from the four combinations of two robability functions (df s) for the individual echo owers and two assumtions regarding the decorrelation time, or indeendence, of ulses within a single scan according to the following table: SWERLING TARGET MODELS robability decorrelation function of ower scan-to-scan ulse-to-ulse eonential 1 chi-square, degree The individual echo owers (roortional to radar cross section) are assumed to ehibit either an eonential df (Swerling 1 and ) or a 4 th -degree chi-square df (Swerling 3 and 4). The corresonding voltage distributions (square root of ower) are the Rayleigh and the 4 th - degree chi distributions. The Rayleigh voltage/eonential ower df, which is obtained from a law of large numbers argument, is aroriate for a target comosed of a large number of aroimately equal-strength scatterers, with no one scatterer dominant. It is often alied to large (with resect to wavelength), comle targets, esecially when viewed over changing asect angles. Page 1 of 6

2 The 4 th -degree chi voltage/4 th -degree chi-square ower df is an aroimation to the df obtained in the case of a large number of equal strength scatterers lus a single, steady dominant scatterer, with the ower of the dominant scatterer equal to 1+ times the total ower of all the smaller scatterers. 1 The eact distribution for this case is the Rice or Rician distribution, which can model any ratio of the dominant to lesser scatterers. However, the Swerling aroimation is well-entrenched, artly because it is more analytically tractable. Note that the voltage models roduce only non-negative values. It is assumed that what is being modeled is the outut of a linear detector, i.e. the magnitude of the real or comle video receiver voltage. The ower models corresond to a square-law detector. Concerning decorrelation, the term scan-to-scan decorrelation imlies that all the echoes within a given scan have the same value, drawn from the aroriate df. All of the ulses on the net scan are again equal to one another, but not to those in the revious scan. Instead, they have a new, indeendent value drawn from the aroriate df. In ulse-to-ulse decorrelation each individual echo ower samle is a new, indeendent random variable drawn from the aroriate df. Physically, decorrelation is tyically assumed to be caused by changing radartarget asect angle from one measurement to the net. It can also be forced through the use of frequency agility [1]. Reference is sometimes made to a Swerling or Swerling 5 model. This is the case of a single, nonfluctuating scatterer and thus does not require the generation of random variables. ALGORITHMS The only issue in generating Swerling random sequences is the generation of indeendent random variables having the various robability functions required for voltage or ower samles. Decorrelation models are imlemented by either using one random variable from the desired df for all N samles in a scan (scan-to-scan decorrelation, i.e. the Swerling 1 or 3 models), or generating searate random variables of the desired df for each of the N samles (ulse-to-ulse decorrelation, i.e. the Swerling or 4 models)..1 EXPONENTIAL DISTRIBUTION The eonential robability function of mean µ is given by [] ( ) 1 µ e, = µ, < (1) 1 The articular dominant/small scatterer ratio of 1 + is the ratio that causes the first two moments (mean and variance) of the Rice distribution to match the chi-square distribution. Page of 6

3 The standard deviation of this eonential random variable is µ. A very simle way to generate eonential-distributed random variables from uniform [,1) rv s u is by the following transformation [3]: = µ ln u () Figure 1 is a histogram obtained by alying this transformation with µ = 1 to a vector of 1, uniform rv s generated in MATLAB. Also shown is the theoretical robability function. The theoretical standard deviation for this distribution is also 1. The samle mean and standard deviation were 1.7 and 1.19; the differences are about.% in the standard deviation and.3% in the mean Figure 1. 1-bin histogram of a unit-mean eonential samle sequence generated in MATLAB using the native MATLAB uniform random number generator and Eqn. (). The corresonding theoretical df of Eqn. (1) is also shown.. RAYLEIGH DISTRIBUTION The Rayleigh robability function of mean µ is given by [] ( ) π 4, e π µ = µ, < The standard deviation of this Rayleigh random variable is ( 4 ) µ π π =.57µ. (3) A simle way to generate Rayleigh-distributed random variables from uniform [,1) rv s u is by the following transformation [3]: Page 3 of 6

4 4µ = ln u (4) π The logarithm converts the uniform distribution to an eonential distribution; the scale factor adjusts the mean; and the square root converts the eonential distribution to a Rayleigh. Figure is a histogram obtained by alying this transformation with µ = 1 to a vector of 1, uniform rv s generated in MATLAB. Also shown is the theoretical robability function. The theoretical standard deviation for this distribution is.573. The samle mean and standard deviation were 1. and.511; the differences are about.1% in the standard deviation and.% in the mean Figure. 1-bin histogram of a unit-mean Rayleigh samle sequence generated in MATLAB using the native MATLAB uniform random number generator and Eqn. (4). The corresonding theoretical df of Eqn. (3) is also shown..3 4 TH -DEGREE CHI-SQUARE DISTRIBUTION The 4 th -degree chi-square robability function of mean µ is given by [] ( ) 4, e µ = µ, < (5) The variance σ of this random variable is µ. A simle way to generate 4 th -degree chi-square random variables is by the following transformation [3]: Page 4 of 6

5 µ = ln ( uu 1 ) (6) where u 1 and u are indeendent uniform (,1] rvs. Notice the similarity to Eqn. (). Figure 3 is a histogram of 1, 4 th -degree chi samles obtained by alying Eqn. (6) with µ = 1 to, uniform rv s generated in MATLAB. Also shown is the theoretical robability function. The theoretical standard deviation for this distribution is.771. The samle mean and standard deviation were 1.3 and.7114; the differences are about.3% in the mean and about.6% in the standard deviation Figure 3. 1-bin histogram of a unit-mean 4 th -degree chi-square samle sequence generated in MATLAB using the native MATLAB uniform random number generator and the method of Eqn. (6). The corresonding theoretical df of Eqn. (5) is also shown..4 4 TH -DEGREE CHI DISTRIBUTION The 4 th -degree chi distribution having mean µ is given by [] ( ) 4 α 3 α µ e = 4 µ <,, where α Γ( 5 ) Γ( ) The variance σ of this random variable is ( µα) ( 4 α ). (7) Page 5 of 6

6 Just as a Rayleigh rv could be obtained as the square root of an eonential rv with aroriate scaling to get the desired mean, a simle way to generate 4 th -degree chi-distributed random variables from uniform [,1) rv s u is by the following transformation: µ = ln ( uu 1 ) (8) α where u 1 and u are indeendent uniform (,1] random variables and α is as described in the revious aragrah. Notice the similarity to Eqn. (4). Figure 4 is a histogram of 1, 4th-degree chi samles obtained by alying Eqn. (6) with µ = 1 to, uniform rv s generated in MATLAB. Also shown is the theoretical robability function. The theoretical standard deviation for this distribution is.363. The samle mean and standard deviation were.999 and.3616; the differences are about.1% in the mean and about.4% in the standard deviation Figure 4. 1-bin histogram of a unit-mean 4 th -degree chi samle sequence generated in MATLAB using the native MATLAB uniform random number generator and the method of Eqn. (8). The corresonding theoretical df of Eqn. (7) is also shown. 3 REFERENCES [1] M. A. Richards, Fundamentals of Radar Signal Processing. (McGraw-Hill, New York, 5.) [] A. Paoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, fourth edition. (McGraw-Hill, New York,.) [3] J. E. Gentle, Random Number Generation and Monte Carlo Methods, nd ed. (Sringer, New York, 3.) Page 6 of 6