168 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 2, MAY 1988

Performance Analysis and Advancement of Self-steering Arrays for Nonsinusoidal

Waves-I1

Abstract-In the first of the two companion papers, a self-steering array system for beam forming with nonsinusoidal waves was analyzed and computer simulated. It was shown that in the presence of thermal noise, the electronic beam-steering mechanism of the array system, which is based on a threshold-detection scheme, yields a considerable error in pointing a main beam toward a desired look direction from which a wavefront is received. In this paper, an alternative beam-steering mechanism is described and computer simulated for performance evalua- tion. It employs a linear-regression processor (LRP) that is less sensitive to thermal noise than the threshold-detection scheme. In the presence of thermal noise, the LRP yields a negligible error in pointing a main beam toward a desired look direction. A design of a variable delay circuit (VD) is presented for the advancement of the delay-time-adjustment mecha- nism of a self-steering monopulse array system that has been developed in theory.

Key Words-Nonsinusoidal waves, self-steering array, beam forming, angular resolution, linear regression.

Index Code--HlBd/j/c, 17d/j/c, 18d/j/c.

I. INTRODUCTION

N Part I [l] of this set of two papers, performance anal- I ysis and computer simulation of a self-steering array system that was developed in theory [2] for optimal reception of nonsinusoidal signals were presented. The main objective of the work is to investigate means for the advancement and feasibility of the electronic beam-steering mechanism of the array system. It was shown in [l] that the signal processor of the array system, which is called slope processor (SP), is sensitive to thermal noise superimposed on the processed signals. In the presence of thermal noise, the SP yields a random error in angle measurement, and as a consequence, there will be inaccuracy in steering a main beam toward the direction of the source from which the wavefront is received. In this paper, we shall present a signal processor that is less sensitive to additive noise than the SP, and can improve the accuracy of the beam-steering mechanism of the array system described in [2].

In Section 11, the principle of a linear-regression processor (LRP) for angle measurement and electronic beam steering is described, and results of its computer simulation are

Manuscript received August 24, 1987; revised November 21, 1987. This work was supported by the Research Management Unit of Kuwait University under Project EE 027.

The author is with the Electrical and Computer Engineering Department, Kuwait University, 13060 Safat, Kuwait. Tel. 4834506.

IEEE Log Number 8820090.

presented. In Section 111, the principle of a monopulse self- steering array system, first described in [2], is reviewed briefly to introduce a circuit design for the variable delay circuits (VD’s) employed in the array system for spatial filtering. Also, a new mechanism of delay-time adjustment for the monopulse array system is described. Conclusions are given in Section IV.

11. PERFORMANCE EVALUATION OF LINEAR-REGRESSION PROCESSOR

The self-steering array system that is fully described in [2] and analyzed and computer simulated in [ 13 is shown in Fig. 1 with the SP replaced by the LRP. The role of the LRP, like that of the SP, is to provide an accurate measure of the slope of the rising ramp of the trapezoidal or triangular pulses resulting at the output of summer 1 . From the ramp slope, the angle of incidence 4 of the received wavefront can be determined. This information is required by the delay-adjustment circuit (DAC) to generate the driving clock, with the proper rate, for the VD,’s. The DAC and the VD;’s may be referred to as “adaptive” or “smart” spatial filters that can steer a main beam in the direction from which a wavefront is received without prior knowledge of the source’s location.

Let us describe the principle of the LRP in the array system of Fig. 1 . Linear regression is a statistical method that determines the equation of a line that best fits a set of data points (to, ro), ( t l , r , ) , * a , (t,,, r,,), where ri is a sample value taken at time ti; the line that best fits these data points has the equation [3], [4]

R = $t -I B. (1)

This equation is called the least squares line of the given data points. The slope s’ of the least squares line is given by the equation

The value of the constant B can be determined by the

0018-9375/88/0500-0168$01.00 O 1988 IEEE

HUSSAIN: SELF-STEERING ARRAYS FOR NONSINUSOIDAL WAVES-11 169

Fig. 1 . A self-steering array system for beam forming with nonsinusoidal waves.

following equation:

B= [(i r;)( i = O i t : ) - (i r = O t i ) ( i = O i t i c ) ] / i = O

According to [ l , fig. 21, for on-axis reception (6 = 0), the input to the LRP in Fig. 1 is a rectangular pulse with peak amplitude (2m + 1)E and duration AT. For off-axis reception (+ > 0), the input to the LRP is a trapezoidal pulse if p sin + # 1 , and a triangular pulse if p sin + = 1 . The parameter p = 2md/cAT, where d is the interelement spacing of the array sensors, AT is the duration of the received rectangular pulse, c is the speed of light, and m is an integer corresponding to the number of sensors (M = 2m + 1 ) . The rising and falling ramps of the trapezoidal or triangular pulse formed by summer 1 in Fig. 1 are staircase functions with a uniform step duration that decreases if the number of array sensors is increased. For an infinite number of array sensors (m %- 1) the ramps are linear [ 13. In this case, the slope SR of the rising and falling ramps is given by [4]

S R = a , for p sin + = 0

= (2m + l ) E / ( p sin + ) A T,

for m %- 1 , and p sin + > O . (4)

In the case where the rising and falling ramps are staircase functions with uniform step duration, the ramp slope is given by 111

S R = a , for p sin + = O

=2mE/(p sin +)AT, for p sin + > O . ( 5 )

Since SR is a function of 9, a measure of SR is desirable for determining the angular position of the source from which the wavefront is received. This task can be accomplished by the LRP .

The LRP in Fig. 1 samples the pulse delivered to its input by summer 1 to generate a set of data points (to, ro), (tl, r l ) , (t2, r2), e . If the sampled pulse has a trapezoidal or a triangular time variation, which is the case for + > 0, the samples that represent the rising ramp of the processed pulse are used in (2) to calculate the ramp slope s". The equation of the least squares line that best fits the rising ramp can be determined by calculating B given in (3) and inserting s" and B into (1). If the processed pulse has a rectangular time variation, for n samples the LRP yields s" = 0 and R = B, which indicates that + = 0. In this case, the output of the LRP is an impulse with infinitesimally small duration to indicate to the DAC that no clock is necessary to drive the VDi's since the source of the received signals is located on the array axis. With no driving clock, the VDi's apply no delay to the incoming signals and the sum of their outputs yields a main beam in the direction of the source [l].

When the source of the received wavefront is located off the array axis, the angle of incidence + can be determined in the LRP by equating the calculated ramp slope to one of the relations given in (4) and (5) for p sin 4 > 0. We shall denote the angle of incidence that is obtained from the calculated ramp slope s" in the LRP by 8. For an array with an infinite number of sensors, equating s"to the relation in (4) for p sin 9 > 0 yields

&=sin-'[(2m+ l)E/ATpS], m %- 1 . (6)

The ratio (2m + l)E/AT equals the slope of the linear ramps of the triangular pulse resulting at the output of summer 1 in the array system of Fig. 1 for the special case p sin + = 1 ; it can be denoted by Sll for simplicity: '

SII = (2m + l ) E / A T, m %- 1 . (7)

Insertion of (7) into (6) results in the following relation:

qT= sin-' [ (~ /~ / s" )p- ' ] . (8)

I The subscript I1 of S,, indicates linear ramp for p sin 6 = 1 , while the subsequent subscript sl of S,, indicates staircase ramp for p sin 6 = 1 .

170 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 2. MAY 1988

For an array with a finite number of sensors, equating S to the relation in (5) for p sin 4 > 0 yields

&=sin-' [2mE/A Tps']. (9) The ratio 2mE/AT equals the slope of the staircase ramps of the triangular pulse resulting at the output of summer 1 in the array system of Fig. 1 for the special case p sin 4 = 1; it can be denoted by Ssl for simplicity:

S, = 2m E / A T. (10)

Insertion of (10) into (9) results in the following relation:

&= sin- * [(s, / S ) p - 11. (1 1)

The parameters S I ] , Ssr, and p are known in advance for a given array design and operating pulse characteristics E and AT. Thus, it is necessary to calculate Sonly in order to obtain the angle 6.

In order to apply the same principle of delay-time adjust- ment described in [ 1 , sec. IV] to the array system in Fig. 1, the LRP must generate and deliver to the DAC a rectangular pulse of duration

(12)

For 6 = 4, the true value of the angle of incidence, the duration ATo is the relative delay between the voltage signals of any two adjacent sensors. The DAC receives the rectangu- lar pulse with duration AT0 from the LRP and generates the driving clock with period AT0 for the VDi's to apply the proper delay to the incoming pulses [l]. Once this task is achieved, the array system yields a main beam in the direction 6. If the calculated angle & does not equal the true value of the angle of incidence 4, there will be an error associated with the above mechanism of pointing a main beam in the direction of the source from which signals are received. Computer- simulation results of the array system in Fig. 1 will be presented shortly.

The duration AT, given in (12) can be expressed in terms of S by substituting p = 2md/cAT into (6) and (9). Insertion of the above ratio for p into (6) results in

A TO = ( d / c ) sin 6.

A T o = ( d / c ) sin &=(2m+ 1)E/(2rn)S

=E/S, m 9 1 (13)

and the insertion into (9) yields

(14) A To = ( d / c ) sin &=E/$.

Hence, the LRP in the array system of Fig. 1 calculates s according to (2) and generates a rectangular pulse with duration A To = E/S for the DAC. In practice, the tasks of the LRP and the DAC can be accomplished with the aid of a special-purpose digital computer, which is often employed by the electronically steered array systems.

It is possible that the calculated ramp slope S according to (2) may not yield the true value of the angle of incidence 4 when used in (8) or (11). This possibility may arise when thermal noise or undesirable distortions are associated with the

processed pulses in the LRP. The staircase ramp of the processed pulses may also affect the accuracy of angle measurement based on the linear-regression algorithm given in ( 2 ) and the relation in (1 1). We have carried out computer simulation of the array system in Fig. 1 to study the performance of the LRP for comparison with that of the SP presented in [l]. As we have done in [l], a normalized error ~ ( 4 ) in angle measurement associated with the LRP is calculated in the computer simulation as follows:

x(4) = (6- 4Y4 (15)

where & is the calculated angle according to (8) or (1 1) and 4, the true value of the angle of incidence of the received wavefront. The angle 4 is an input to the computer simulation

Three-dimensional plots of the error percentage in angle measurement (% ~ ( 4 ) ) versus 4 and the number of array sensors M = 2m + 1 are shown in Figs. 2-5. The plots in Figs. 2 and 3 are derived for noise-free pulses, while the ones in Figs. 4 and 5 are for pulses with superimposed white Gaussian noise samples having a zero mean and variance u2 = 0.01. The signal-to-noise power ratio at each sensor is 20 dB. The angle 6 in (1 5) is calculated from the relation given in (8) for deriving the plots of Figs. 2 and 4, and from (11) for deriving the plots of Figs. 3 and 5. Comparison of the plots in Figs. 2 and 3 with those derived for the SP in [ l , figs. 4 and 51, shows that the performance of the LRP almost equals the performance of the SP in the absence of additive thermal noise. However, comparison of the plots in Figs. 4 and 5 with those in [ l , figs. 6 and 71, shows that the LRP yields a better performance than the SP when thermal noise is present with the processed signals. Note that the plots in Figs. 2-5 are derived for the range 0" I 4 5 5 5 " , while the ones in [ I , figs. 4-71 are derived for the range 0" I 4 I 35". The advancement of the LRP in performance over the SP is due to the fact that the principle of the LRP is not based on a threshold-detection scheme like the SP. Any threshold-detec- tion device is highly sensitive to random noise, and it is often characterized by a false-alarm rate.

Computer simulation of the beam-steering mechanism discussed above for the array system in Fig. 1 is depicted in Fig. 6. The simulation is conducted based on (2) and (1 1) for the input values of the angle of incidence 4 = O " , 4.5", 14", 26", and 43 O . The plots in Fig. 6 are the normalized energy patterns of the array system in Fig. 1 for 33 sensors and a noise-free rectangular pulse that is represented by 101 samples, each with unity amplitude. According to Fig. 6, the array system in Fig. 1 is capable of accurately steering its main beam in the direction from which the wavefront is received, only for a certain angular sector. Beyond this angular sector, there is an error associated with the mechanism of beam steering based on linear regression, as shown by the plot in Fig. 3. If the maximum scan (or steering) angle of the array system in Fig. 1 satisfies the relation p sin +,,, = 1 , the resulting error in beam steering will be negligible for the angular sector 0 I 4 5 4,,, [l]. This fact can be deduced from the plots in Figs. 2-5 and the ones in [ l , figs. 4-71 derived for the SP.

r 1 , eq. (2)l.

HUSSAIN: SELF-STEERING ARRAYS FOR NONSINUSOIDAL WAVES-11 171

Fig. 2. Percentage error (% ~ ( 4 ) ) in angle measurement associated with the LRP in the array system of Fig. 1 versus the number of array sensors M =

2m + 1 and the angle of incidence 4, The plot is derived based on (8) for noise-free signals.

Fig. 3. Percentage error (% ~ ( 4 ) ) in angle measurement associated with the LRP in the array system of Fig. 1 versus the number of array sensors M = 2m + 1 and the angle of incidence 4. The plot is derived based on (1 1) for noise-free signals.

111. PRINCIPLE OF DELAY-TIME ADJUSTMENT FOR A

SELF-STEERING MONOPULSE ARRAY

A self-steering monopulse array system for beam forming with nonsinusoidal waves is fully described in [2, fig. 41 and shown in Fig. 7 with the SP replaced by the LRP. Its basic

.65

.55

45

.35

.2 5

.1 5

. S 0

0

Fig. 4. Percentage error (% ~ ( 4 ) ) in angle measurement associated with the LRP in the array system of Fig. 1 versus the number of array sensors M = 2m + 1 and the angle of incidence 4, The plot is derived based on (8) for signals with a superimposed sample of a white Gaussian random process having a zero mean and variance u* = 0.01. The signal-to-noise power ratio at each sensor is 20 dB.

Fig. 5 . Percentage error (% ~ ( 4 ) ) in angle measurement associated with the LRP in the array system of Fig. 1 versus the number of array sensors M = 2m + 1 and the angle of incidence 4. The plot is derived based on (1 1) for signals with a superimposed sample of a white Gaussian random process having a zero mean and variance u2 = 0.01. The signal-to-noise power ratio at each sensor is 20 dB.

172 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 2, MAY 1988

Fig. 6. The main beam of the energy pattern of the array system in Fig. 1 with 33 sensors receiving noise-free rectangular pulses with normalized unity amplitude and duration. The main beam is steered via computer simulation toward the different look directions from which a wavefront is received.

SENSORS VARIABLE DELAY CIRCUITS

LINEAR REGRESSION PROCESSOR

Fig. 7. A self-steering monopulse array system for beam forming with nonsinusoidal waves. The principle of the array system is fully described in [2, fig. 41.

principle is similar to that of the conventional monopulse tracking radar [5, ch. 211. Monopulse antenna peak-amplitude and slope patterns for nonsinusoidal waves are derived in [4]. The self-steering monopulse array system in Fig. 7 is capable of pointing a main beam in the direction of a source that can be located in either one of the angular sectors, - 7r/2 I r#~ I 0 or 0 I 4 I 7r/2, without prior knowledge of the source’s

location. The roles of the LRP, DAC, VD,’s, and summer 5 in Fig. 7 are the same as in the array system of Fig. 1. The objective of summers 1-4 is to produce a monopulse signal for determining the angular sector in which the source is located, and to form the sum of the voltage signals from the sensors for processing in the LRP.

According to Fig. 7, the voltage signals from subarray 1 are

HUSSAIN: SELF-STEERING ARRAYS FOR NONSINUSOIDAL WAVES-I1 173

Buf fer B F $1 Switch OUtDUt

Fig. 8. A VD, for the self-steering monopulse array system in Fig. 7. TAD is tapped analog delay line with M = 2m taps; SW is switch; I is inverter; BF is buffer; PR is polarity reversal circuit. The tap delay is linearly variable with the rate of the driving clock, clk, which is generated by the DAC in Fig. 7.

summed in summer 1, while the voltage signals from subarray 2 are summed in summer 2. The output of summer 2 is subtracted from that of summer 1 by summer 4 to produce a monopulse signal. The function of summer 3 is the same as that of summer 1 in Fig. 1 ; it sums the rectangular voltage signals from the sensors of subarrays 1 and 2 to produce a rectangular, a trapezoidal, or a triangular pulse for processing in the LRP. If a wavefront, with a rectangular time variation, is arriving from the direction 0 < 4 s n/2, it will be received by the sensors of subarray 1 first, and after a delay of (md sin 4)/c, it will arrive at the sensors of subarray 2. In this case, the monopulse signal at the output of summer 4 is a positive pulse followed by an equal pulse amplitude reversed. The above situation is reversed if the wavefront is arriving from the direction - n / 2 I I$ < 0; the monopulse signal will be a negative pulse followed by an equal pulse amplitude reversed [2, fig. 51. For 4 = 0, the wavefront will arrive at the sensors of subarrays 1 and 2 at the same time, and the monopulse signal at the output of summer 4 will be zero. Hence, the time variation of the resulting monopulse signal is an indication of the angular sector in which the source of the received wavefront is located. We shall shortly describe how the DAC in Fig. 7 utilizes such information for providing the proper time-delay adjustment for the VD;’s.

A circuit diagram of a VD; for the self-steering monopulse array system in Fig. 7 is shown in Fig. 8. The VD; in Fig. 8 is an advanced version of the one described in [l, fig. 81. It consists of a tapped analog delay line (TAD) with M = 2m taps, a switch (SW), an inverter (I), a buffer (BF), and a polarity reversal circuit (PR). A square-wave clock (clk 1) and its complement (clk 2) are required to derive TAD. The driving clock (clk) is generated by the DAC. The delay time of a single tap is linearly variable with the period T, of the driving clock. The relative delay between the outputs of any two adjacent taps along the TAD equals T,. The output of the VD; is taken from either tap 1 (point A ) , tap i (point C) , or tap M - i + 1 (point B) via the SW, depending on the direction of arrival of the received wavefront. It is assumed that tap 1 applies no delay to the input signal.

Initially, the output terminal of each VD; is connected via the SW to point A . For on-axis reception, 4 = 0, the output of summer 4 is zero and the response of the LRP is an impulse with an infinitesimally short duration. The above indicates to

the DAC that no delay adjustment is necessary for the VD;’s; the driving clock (clk) will not be generated by the DAC. For - n / 2 I 4 < 0, the response of the LRP is a rectangular pulse with the duration ATo given in (14), and the monopulse signal from summer 4 is a negative pulse followed by an equal positive pulse. The DAC receives this information and generates a driving clock that is composed of square pulses with negative unity amplitude and a pulse repetition interval (PRI) T, = ATo. The negative square pulses are passed through the BF to the SW and the PR. The PR converts the negative amplitude of a pulse to positive, and it passes a positive input pulse with no changes. The negative clock pulses are needed to trigger the SW to connect the output terminal of VD; to point C when - n /2 I I$ < 0. As long as (the negative) clk is being run by the DAC, the output of each VD; is the voltage signal passed from tap i of the TAD. Hence, for the sector - a12 I 4 < 0, each VD; yields a delay time

T i = ( i - l )ATo=(i - l)(d/c) sin 4

7; - 7; + , < 0

(16) i = l , 2, e . . , M.

If a rectangular wavefront is received from the direction 0 < 4 5 n/2, the monopulse signal resulting from summer 4 in Fig. 7 will be a positive pulse followed by an equal negative pulse. The DAC receives the monopulse signal along with the response of the LRP and generates a driving clock that is composed of square pulses with positive unity amplitude and PRI T, = AT,. In this case, the positive clock pulses trigger the SW of each VD; to switch to point B. As long as (the positive) clk is being run by the DAC, the voltage signal at tap M - i + 1 of the TAD will be passed to the output terminal of VD;. Hence, for the sector 0 < 4 I a/2, each VD; yields a delay time

7;=(M- i )ATo=(M- i ) (d / c ) sin 4

7;-7,+,>0

i = l , 2 , * - * , M . (17)

The mechanism of time-delay adjustment described above is more advanced, and practically attractive, than the one described in [2], since it does not involve the generation of M

174 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 2, MAY 1988

separate control voltages for the VD;’s. With the above mechanism, the array system in Fig. 7 is capable of electroni- cally steering its main beam in the direction of a source that can be located at any angular position in the sector - a/2 < $ < nl2.

The self-steering array systems shown in Figs. 1 and 7, and the one described in [ 1, fig. 11, need to be advanced further to have the capability of operating in a signal environment where multiple sources are present at different angular positions. Such a development would involve a proper modeling of the signal environment, advanced signal processing, and multiple beam-forming circuitry.

IV. CONCLUSIONS

The accuracy of electronic beam steering based on the linear regression processor equals that based on the slope processor when no thermal noise is superimposed on the processed signals. In the presence of thermal noise, the performance of the LRP is better than that of the SP. For linear arrays with the design specification P sin $,,, I 1, the error in beam steering

associated with the LRP is negligible. Hence, the LRP is a better candidate than the SP for the development of self- steering array systems for the optimal reception of nonsinu- soidal waves. An advanced design of a variable delay circuit is presented for improving the mechanism of delay-time adjust- ment of the self-steering monopulse array system described in [ l , fig. 41.

111

121

131

[41

151

REFERENCES M. G. M. Hussain, “Performance analysis and advancement of self- steering arrays for nonsinusoidal waves-I,” IEEE Trans. Electro- magn. Cornpat., this issue, pp. 161-167. M. G. M. Hussain, “A self-steering array for nonsinusoidal waves,” IEEE Trans. Electromagn. Compat., vol. EMC-28, no. 2, pp. 96- 104, May 1986. C. F. Gerald, Applied Numerical Analysis, 2nd ed. New York: Addison-Wesley, 1980. M. G . M. Hussain, “Line-array beam forming and monopulse techniques based on slope patterns of nonsinusoidal waveforms,” IEEE Trans. Electromagn. Cornpat., vol. EMC-27, no. 3, pp. 143- 151, Aug. 1985. M. I. Skolnik, Ed., Radar Handbook. New York: McGraw-Hill, 1970.