Modified MAXIMIN Adaptive Array Algorithm for Frequency-Hopping System Raja D Balakrishnan, Bagawan S. Nugroho and Hyuck M. Kwon Wichita State University, Department of Electrical and Computer Engineering Wichita KS 67260-0044 Tel: (316) 978-6308, Fax: (316) 978-5408 Email: [email protected] Abstract - The MAXIMIN algorithm in [1] is a blind adaptive-array algorithm that provides simultaneous interference suppression and beam forming in a frequency-hopping communication system. This algorithm is so named because one set of correlators serves to maximize the desired signal while the other serves to minimize the interference plus noise. Though MAXIMIN algorithm is robust and provides improvement in signal-to-interference-plus-noise ratio (SINR), the convergence is reached only after a considerable time. In this paper we propose a modification to the MAXIMIN algorithm, which can provide improvements in the rate of convergence. Computer simulation results under various combinations of arrival of interferences reveal the robustness of the proposed modification.1 I. INTRODUCTION The MAXIMIN algorithm of Torrieri and Bakhru [1] is a blind adaptive-array algorithm that suppresses interference before it enters the demodulator of a frequency-hopping communication system and thereby providing a spatial processing gain. The algorithm discriminates between the desired signal and the interference on the basis of the distinct spectral characteristic of the frequency-hopping signals. The algorithm maximizes the signal-to-noiseplus-interference-ratio (SINR) by maximizing the desired signal power while simultaneously minimizing the interference-plus-noise power. Torrieri and Bakru [1] suggested the adjustment of the loop gain as a function of the estimated SINR at each iteration, for rapid convergence of the algorithm. They also suggested the suitable estimates for various parameters for the implementation of the algorithm. However these estimates may not be suitable when 1

This material is based upon work supported by the U. S. Army Research Laboratory and the U. S. Army Research Office under grant number DAAD19-01-10537. This paper will be presented in parts at the IEEE ISSSTA, Prague, Czech Republic, September 25, 2002.

the SINR is too low. Hence to alleviate the problems faced under low SINR and also to improve the rate of convergence we propose the estimation of correlation matrices of the signal and noise by means of a recursive algorithm. We also propose the computation of variable adaptation parameter in the weight vector update equation of [1] for stability of the algorithm and expediency of convergence. Simulation results indicate robustness of the algorithm. The choice of the recursive estimates is justified by observing the variance of SINR under steady-state conditions, which is considerably smaller than that of the algorithm of [1]. This paper is organized as follows. In section II we will describe the basic MAXIMIN algorithm of [1]. In section III we will describe the proposed modifications of MAXIMIN algorithm. In sections IV and V, we will describe the implementation of the algorithm and the simulations and will demonstrate the improvements, respectively. Finally in Section VI we will draw some conclusions and will discuss the room for further improvements. II. MAXIMIN ALGORITHM The MAXIMIN algorithm of [1] is described as follows. The desired signal and interference are assumed to be stationary stochastic processes. Let X (i ) denote the vector of complex envelopes of the N antenna outputs at a discrete time instant i. The vector X (i ) can be decomposed as X (i ) = s (i ) + n(i ) , (1) where, s (i ) is the vector of complex envelops of the desired signal and n(i ) is the vector of complex envelops of the interference-plus-noise. Let W denote the complex weight vector, R ss and Rnn are the desired-signal and interference-plus-noise correlation matrices. Then the SINR is given by P WHR W ρ = s = H ss . (2) Pn W Rnn W

To use the method of steepest descent in deriving the adaptive algorithm, [1] employs maximizing SINR as the optimization criterion. The method of steepest descent for discrete-time systems gives the following recursive equation for the weight vector: ˆ ρ (k ) , (3) W (k + 1) = W (k ) + µ 0 (k )∇ W where the gradient of the SINR with respect to the complex weight vector W = W R − jW I is given as, R W R W  ∇ W ρ = ρ  ss − nn  . Pn   Ps

 R W (k ) Rnn W (k )  − W (k + 1) = W (k ) + µ 0 (k ) ρ (k )  ss  Pn  Ps  (5) Torrieri and Bakru in [1] proposed suitable estimates for Rss W (k ) , Rnn W (k ) , Ps and Pn under the assumption that the interference occupies only a small fraction of the hopping band. Hence we have from [1],  s * y sR (k ) n * y nR (k )  − W (k + 1) = W (k ) + µ 0 (k ) ⋅ ρˆ (k )   Pˆn (k )   Pˆs (k ) (6) and under the assumption that noise power is much less than the desired signal power the estimates are given by [1], km 1 s * y sR (k ) = X * (i ) y R (i ) , (7) ∑ m i =( k −1) m +1

where y R (i ) is the real part of the adaptive filter output y (i ) and m is the number of samples per iteration. Similarly, km 1 nˆ * (i ) yˆ nR (i ) , m i =( k −1) m +1

∑

(8)

where nˆ (i ) denote the vector of N discrete-time outputs of the monitor filters and define yˆ n (i ) = W T (i ) ⋅ nˆ (i ) , (9) and yˆ nR = Re[ yˆ n ] . Therefore,

and

km 1 Pˆs (k ) = y R2 (i ) , m i =( k −1) m +1

(10)

km 1 2 Pˆn (k ) = yˆ nR (i ) , m i =( k −1) m +1

∑

(11)

Pˆs (k ) . Pˆ (k )

(12)

∑

ρˆ (k ) =

In the following section we propose a modification to the MAXIMIN algorithm to perform even under low SNR but still being robustness. III. PROPOSED MODIFICATIONS

(4)

Hence

n * y nR (k ) =

approximates the desired signal correlation vector by the use of the received signal vector. This may hold under high SNR but will fail when SNR is low leading to longer convergence or none at all!

Regarding the noise-plus-interference to have almost the same statistics in both the main channel and monitor channel, we propose a new approximation to the time domain estimates of [1]. Since noise, interference and desired signal are statistically independent, and assuming the identical statistics in the main and monitor channels, we have, R xx = R ss + Rnn , (13) where R xx denotes the autocorrelation matrix of the received signal with noise and interference in the main channel. Using the main channel and monitor channels we propose the following estimates: n −1

R xx (k ) = ∑ γ n −i X (i )⋅ X H (i ) + X (k ) ⋅ X H (k ) (14) i =1

where γ is the forgetting factor and k is the iteration index. Equation (23) can be rewritten as, R xx (k ) = γ ⋅ R xx (k − 1) + X (k )⋅ X H (k ) . Similarly,

(15)

Rnn (k ) = γ ⋅ Rnn (k − 1) + n(k )⋅ n H (k ) . (16) Using (24) and (25) it readily follows that R ss (k ) = R xx (k ) − Rnn (k ) . (17) The choice of γ has its effect on the convergence and response of the system to input variations. Convergence and stability of the new update scheme in (15) – (17) are guaranteed by evaluating the optimal step-length µ 0 (k ) for every iteration k, based on the SINR given in (8). The SINR at any iteration (k + 1) is given by

ρˆ (k + 1) =

A1 (k ) µ 2 (k ) + B1 (k ) µ (k ) + C1 (k ) A2 (k ) µ 2 (k ) + B 2 (k ) µ (k ) + C 2 (k )

, (18)

where A1 (k ) = ρˆ 2 (k ) ⋅ Z H (k ) ⋅ R ss (k ) ⋅ Z (k ) A2 (k ) = ρˆ 2 (k ) ⋅ Z H (k ) ⋅ R nn (k ) ⋅ Z (k ) ,

n

With these estimates and (6) the weight vector is updated every iteration. As seen equation (7)

{ B (k ) = ρˆ (k ) ⋅ {Z

} ⋅ Z (k )}

B1 (k ) = ρˆ (k ) ⋅ Z H (k ) ⋅ R ss (k ) ⋅ W (k ) + W H (k ) ⋅ R ss ⋅ Z (k ) 2

H

(k ) ⋅ R nn (k ) ⋅ W (k ) + W

C1 (k ) = W H (k ) ⋅ R ss ⋅ W (k ) ,

H

(k ) ⋅ R nn

C 2 (k ) = W H (k ) ⋅ R nn ⋅ W (k )

and R ss (k ) ⋅ W (k ) R nn (k ) ⋅ W (k ) . − Pˆs (k ) Pˆn (k ) Differentiating (27) with respect to µ (k ) and equating the derivative to zero will yield us, A(k ) ⋅ µ 2 (k ) + 2 B (k ) ⋅ µ (k ) + C (k ) = 0 , (19) where A(k ) = A1 (k ) B 2 (k ) − A2 (k ) B1 (k ) , B (k ) = A1 (k )C 2 (k ) − A2 (k )C1 (k ) C (k ) = B1 (k )C 2 (k ) − B 2 (k )C1 (k ) . and Clearly the solution of (19) gives us the optimum µ (k ) for a given W (k ) and ρˆ (k + 1) . We can choose one of the two possible solutions of (19) as the optimum µ (k ) . Simulation results clearly indicate the faster convergence of this approach. Z (k ) =

IV. IMPLEMENTATION Following [1], the implementation of the MAXIMIN algorithm and its proposed improvement are shown in Figures 1 and 2. The front-end devices include a band pass filter and a low noise amplifier. The received signal is de-hopped and down converted to a fixed intermediate frequency (IF). The filtered IF signal is then sampled by an analog-to-digital converter (ADC). It is then applied to a base band converter. The discrete-time output of the converter is then passed through a base band filter with a bandwidth equal to that of the frequency channel. The complex vector of base band filter outputs denoted as X (i ) , is then applied to the adaptive filter. The monitor filter extracts the interference and noise information from the frequency-translated signal. The monitor filter has a pass band offset by f 0 from the base band filter and coinciding with a down converted frequency channel. The monitor filter lets the adaptive filter to monitor the interference and noise that will be present in the base band filter when the carrier frequency of the desired signal eventually coincides with that of the interference. The monitor filter is set at a center frequency of f 0 = 2 B , where B is the bandwidth of both the base band filter and the frequency channel. V. SIMULATION AND RESULTS To enable comparisons, the system parameters used are the same as [1] and are given below in table I. The desired signal arrives from a direction perpendicular

to one of the edges, and this direction is defined to be 0o Table I: System parameters Parameter

Value

Array antennas

4, omni, at vertices of a square

Array edge length

1 wavelength

Center frequency

3 GHz

Hop Dwell Time

1 ms

Data Rate

100 Kbps

Frequency Modulation Signal-to-noise ratio

MSK

Hopping bandwidth

30 MHz

Number of frequency Channels Monitor filter offset

M=300

Sampling rate

800 ksamples/s

Weight iterations per Hop Total interference-tosignal ratio Interference type

8

Number of hops per Experiment

50

20 dB per antenna and channel

200 kHz

10M Tones in channels

Perfect synchronization between the frequency hopping signals at all the antenna outputs and the frequency synthesizer in the receiver is assumed. In practice the algorithm will have a cold start and hence we set the initial weight vector W (0) = [1 0 0 0] (20) The parameter α is set at 0.2 for [1]. The proposed scheme is found to work for any choice of 0 < µ (0) ≤ 1 . So we set it at 0.1, for convenience. We also scale µ(k) by a factor of 10 for faster convergence. The forgetting factor γ is set at 0.999. Also for fair comparison the weight vectors in [1] and in the proposed scheme are normalized at every iteration. Each experiment consists of 20 trials with 50 hops per trial. As in [1], the final SINR is calculated as the average of the last 40 iterations of all the trials and is used to calculate the corresponding standard deviations. Crossing number, defined in [1], as the number of weight iterations required for the average SINR over all the trials to exceed a threshold value equal to 3 dB less than the final SINR, provides a rough measure of the relative time required for convergence to steady state. Tables II and III tabulates the simulation results under various conditions and compares the proposed scheme with

[1]. Odd numbered figures show the improvement in convergence time and also in the final SINR when compared with [1], while the even numbered figure shows the gain in the array response when compared with [1]. Table II. Simulation results with edge length = λ , Bandwidth = 30 MHz for scheme in [1]. Input SINR (dB) 20

Interference Angle (degrees) 40, 70

Final SINR (dB) 25.4

Standard Deviation (dB) 1.5

Crossing Number

10

40, 70

15.68

0.5

239

10

20, -20

14.76

0.5

160

2

20, -20

8.20

0.35

240

928, September 1984. [3] D. Torrieri, Principles of Secure Communication Systems, 2nd ed. Boston: Artech House, 1992. [4] R. T. Compton, Adaptive Antennas: Concepts and Performance. New York: Prentice-Hall, 1988. [5] J. G. Proakis, Digital Communications, 3rd ed. New York: McGraw-Hill, 1995. [6] S. Haykin, Adaptive Filter Theory, 3rd ed. New York: Prentice-Hall, 1996. [7] S. S. Rao, Optimization Theory and applications, 2nd ed. New Delhi: Wiley Eastern, 1991. Dehopped signal Front-end devices

199

Baseband converter

Baseband filter

X1R

} X1 X1I

FH replica

Table III. Simulation results with edge length = Bandwidth = 30 MHz for the proposed scheme.

IF samples

A/D converter

IF filter

Timing and control

Baseband filter

λ,

Input SINR (dB) 20

Interference Angle (degrees) 40, 70

Final SINR (dB) 24.3

Standard Deviation (dB) 1.2

Crossing Number

10

40, 70

16.39

0.6

24

10

20, -20

16.13

0.5

61

2

20, -20

8.81

0.25

22

n^1R

} n^1

n^1I

Figure1. Dehopping and initial processing.

24

Auto correlation estimator

^ Rxx

W1

X

{

X1

Σ

yR yI

XN W1

VI. CONCLUSION In this paper we have proposed a modification that improves the performance of the MAXIMIN algorithm of [1]. It is shown that the proposed modification can perform well under low SINR scenarios, expedites the convergence and improves the robustness of the MAXIMIN algorithm under different combinations of arrival of the interference, thereby making it more reliable in all situations. Though the computational load of the proposed modification might be higher than [1], but the reliability of the communication link guaranteed which is crucial especially military communications. REFERENCES [1] D. Torrieri and K. Bakhru, “The Maximin Algorithm for Adaptive Arrays and FrequencyHopping Systems”, ARL – TR – 2026, December 1999. [2] _____, “The maximin algorithm for adaptive arrays and frequency-hopping communications”, IEEE Trans. Antenna Propag., vol. 32, pp. 919-

WN

Weight processor

W1

^ n

{

WN

n^1

Σ ^ nN WN

^ ynR ^ ynI

Auto correlation estimator

^ Rnn

Figure 2. Adaptive algorithm with proposed modification.

20

30

T orrieri P roposed

T orrieri P roposed

15

25 10

5 Array Gain (dB)

SINR in dB

20

15

0

-5

10 -10

5

0

-15

-20

0

50

100

150

200 Iterations

250

300

350

-80

400

Figure 3. Average SINR for 20 trials with two interference signals arriving at 40o and 70o with input SINR=20 dB.

-60

-40

-20 0 20 Incident Angle (degrees)

40

60

80

Figure 6. Array gain pattern at the end of trial with two interference signals arriving at -20o and 20o with input SINR=10 dB. 30

20

T orrieri P roposed

T orrieri P roposed

15

25

10

SINR in dB

Array Gain (dB)

20

5 0 -5

15

10

-10

5

-15 0

-20

-80

-60

-40

-20

0

20

40

60

0

50

100

80

Incident Angle (degrees)

Figure 4. Array gain pattern at the end of trial with two interference signals arriving at 40o and 70o with input SINR=20 dB.

150

200 It erat ions

250

300

350

400

Figure 7. Average SINR for 20 trials with two interference signals arriving at -20o and 20o with input SINR=2 dB.

30

20

T orrieri P roposed

T orrieri P roposed 15

25

10

5 Array Gain (dB)

SINR in dB

20

15

10

0

-5

-10 5

-15 0

0

50

100

150

200 It erations

250

300

350

400

Figure 5. Average SINR for 20 trials with two interference signals arriving at -20o and 20o with input SINR=10 dB.

-20

-80

-60

-40

-20 0 20 Incident Angle (degrees)

40

60

80

Figure 8. Array gain pattern at the end of trial with two interference signals arriving at -20o and 20o with input SINR=2 dB.