- Email: [email protected]
Adaptive wideband beamforming with sensor delay-lines
ARTICLE IN PRESS Signal Processing 89 (2009) 876–882
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Adaptive wideband beamforming with sensor delay-lines Wei Liu Communications Research Group, Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, UK
a r t i c l e i n f o
abstract
Article history: Received 8 October 2007 Received in revised form 8 August 2008 Accepted 5 November 2008 Available online 18 November 2008
A novel adaptive wideband beamforming method is proposed, where the tap delays in the traditional method are replaced and only one adaptive coefficient is required for each sensor. In this way, the complex wideband beamforming system with temporal filtering can be avoided by implementing the proposed beamformer with simple analogue circuits for very high frequency and bandwidth applications. Another advantage of this new method is its potential to avoid the beam widening effect at higher off-boresight angels. The effectiveness of the proposed method is verified by simulations. & 2008 Elsevier B.V. All rights reserved.
Keywords: Adaptive beamforming Wideband Sensor delay-lines Tapped delay-lines
1. Introduction Beamforming is an array signal processing technique to form beams in order to receive signals illuminating from specific directions, whilst attenuating signals from other directions. The sensors in an array system can be positioned in space according to different patterns, e.g. along a straight line, around a circle, or on a plane. Such arrangements lead to linear arrays, circular arrays and planar arrays, respectively [1]. Fig. 1 shows a linear array system with M sensors, where a signal sl ðtÞ arrives from the direction fl . The propagation delay between the sensor 0 and the sensor m is denoted by tm. The received array signals are given by x0 ðtÞ, x1 ðtÞ; . . . ; xM1 ðtÞ. For narrowband signals [2], to form a beam pointing to the direction of arrival (DOA) of the signal of interest, we can apply one coefficient to each of the received signals and then sum them up directly. For wideband signals, however, such a configuration will not work well and normally we need to employ a tapped delay-line (TDL) system [3,4], as shown in Fig. 2, where each of the received array signals xm ðtÞ, m ¼ 0; 1; . . . ; M 1 is processed by the following TDL, with an adjacent tap
Tel.: +44 114 2225813; fax: +44 114 2225834.
E-mail address: w.liu@sheffield.ac.uk 0165-1684/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.11.005
delay of D and the corresponding tap coefficients given by wm;j , m ¼ 0; 1; . . . ; M 1, j ¼ 0; 1; . . . ; J 1. There are in total J taps for each TDL. The length J of the TDLs is dependent on the bandwidth of the impinging signals [5]. In general, the larger the bandwidth, the longer the TDLs [6]. The TDL coefficients can be designed for maintaining a fixed specified response for all signal/ interference scenarios, which leads to the concept of a data independent beamformer [4]. Alternatively, they can be chosen based on the statistics of the array data for optimizing the array’s response, which forms a statistically optimum beamformer [4]. Since the statistics of the array data are often not known or may change over time, adaptive algorithms may be used to determine these coefficients. In its discrete form, the TDL system is replaced by a finite impulse response (FIR) filter and adaptive algorithms can be realized by digital circuits. In both of the cases, the delays between samples/taps get shorter and shorter with increasing signal bandwidth and very high speed circuits need to be employed. As an example, consider a general ultrawideband system with frequency range from 0 Hz to 6 GHz. If we perform beamforming in digital form, then the sampling rate should be at least 12 GHz. With 16 bits per sample, it is almost impossible to build such a high-speed circuit with current technology. To avoid these difficulties associated with the traditional TDL system and also provide an alternative to it, in
ARTICLE IN PRESS W. Liu / Signal Processing 89 (2009) 876–882
this paper we propose a novel method for adaptive wideband beamforming, i.e. we replace the tapped delays by spatial propagation delays by positioning more sensors behind the original array system. We consider this new system as adaptive wideband beamforming with sensor delay-lines (SDLs). This method can be applied to any kind of array system. For the original linear array, we will have a rectangular array, as shown in Fig. 3, to replace the structure in Fig. 2; for an original rectangular array, we will have a cubic array, as shown in Fig. 4; for a semicircular array, we will have a new system shown in Fig. 5, or a cylindrical form for a circular array, etc. Moreover, these SDL sensors could be positioned not necessarily in a regular spacing, but arbitrary. The advantage of this arrangement is twofold: Since only one coefficient is required for each received array signal, the function of the original temporal filtering process is now achieved by spatial filtering using multiple sensors so that a complicated wideband beamforming system is able to be implemented by simple analogue circuits. This is useful especially when the signal frequency (and bandwidth) are very high and a digital implementation becomes extremely difficult or even not viable at all, as mentioned in the case before.
x0 (t )
sl ( t ) φl
xm(t ) τm
x M−1(t ) Fig. 1. A linear array system with M sensors, where a signal sl ðtÞ arrives from the direction fl .
877
The second advantage is its potential to avoid the beam widening effect at high off-boresight angels. For example, for the traditional TDL structure in Fig. 2, the beamwidth will increase significantly when a broadside main beam is steered to an angle f closer to 90 . However, for the corresponding SDL structure in Fig. 3, we can rotate the set of coefficients by 90 to form a main beam pointing to the direction f ¼ 90 and the new beam will have the same beamwidth, because its effective aperture will not go to zero as in the linear array case and it has a full coverage over the 3601 azimuth range. Note if we want to implement this beamformer in digital domain, we will still require the same sampling rate as the traditional one. However, we may need the proposed structure for those applications where we want to avoid the beam widening effect at high off-boresight angles and using more sensors may well be justified by meeting the requirements of the corresponding applications. This idea is derived from a recently proposed frequency invariant beamforming design method for rectangular arrays [7]. A special characteristic of this fixed beamformer is that there are no TDLs involved and only one single weight is attached to each sensor. By examining its beam pattern, we can find that this rectangular array is approximating the response of a frequency invariant linear array with TDLs [8–10] by employing spatial propagation delays instead of wired delays. We therefore generalize the idea to the adaptive case and also to arbitrary beam patterns and sensor position patterns, as discussed before. Here we will only focus on the simple but widely used traditional linear array case, which corresponds to the rectangular array in the new approach, and provide some initial investigation results to show its effectiveness. An extensive study of this new approach for
x 0(t) w0.0
w0.1
w0.2
w0,J −1
w1.0
w1.1
w1.2
w1,J −1
sl (t ) φl
x1(t)
y (t)
xM−1(t) wM−1.0
wM−1.1
wM−1.2
wM−1.J −1
Fig. 2. A general structure for wideband beamforming.
ARTICLE IN PRESS 878
W. Liu / Signal Processing 89 (2009) 876–882
x 0.0(t)
x 0.1(t)
w0.0
x 0.2(t)
w 0.1
x 0.N−1(t)
w0.N−1
w0.2
sl (t ) φl
x 1.0(t)
x 1.1(t)
w1.0
x 1.2(t)
w1.1
x 1.N−1(t)
w1.N−1
w1.2
y (t)
x M−1.0(t)
wM−1.0
x M−1.1(t)
x M−1.2(t)
wM−1.1
x M−1.N−1(t)
wM−1.2
wM−1.N−1
Fig. 3. The proposed rectangular array to replace the traditional linear array with TDLs.
z
2. Adaptive wideband beamforming with SDLs
signal
θ y φ x Fig. 4. The proposed cubic array to replace the traditional rectangular array with TDLs.
sl(t) φl
As discussed in the previous section, we propose to use sensor delays to replace the tapped delays in the traditional adaptive wideband beamforming structure. For the linear array shown in Fig. 2, the corresponding new structure will be a planar array and it has been shown in Fig. 3, where each of the TDLs with length J is replaced by an SDL with N sensors, and the delay between adjacent taps is thus replaced by the spatial propagation delay between adjacent sensors. In this new structure, there are in total M N sensors, with the received sensor signals denoted by xm;n ðtÞ, m ¼ 0; 1; . . . ; M 1, n ¼ 0; 1; . . . ; N 1. As there is only one coefficient for each received sensor signal, there are no TDLs involved and adaptive beamforming can be performed by very simple analogue circuits. Note that there is no temporal information used in the beamforming process. Therefore it can be considered as a wideband beamforming structure with spatialonly information. 2.1. Wideband response of the proposed structure In Fig. 3, the output yðtÞ can be expressed as
Fig. 5. The proposed planar array to replace the traditional semi-circular array with TDLs.
all kinds of beamforming scenarios and array sensor position patterns will be left for future study. This paper is organized as follows. In Section 2, a theoretical explanation about why the proposed structure can form a wideband beam response is first given and then two adaptive beamforming implementations are discussed; simulation results are provided in Section 3, and conclusions drawn in Section 4.
yðtÞ ¼ wT xðtÞ,
(1)
where w ¼ ½w0;0 . . . wM1;0 w0;1 . . . wM1;1 . . . wM1;N1 T , xðtÞ ¼ ½x0;0 ðtÞ . . . xM1;0 ðtÞ x0;1 . . . xM1;1 ðtÞ . . . xM1;N1 T . (2) Suppose all of the signals are on the same plane as the rectangular array, i.e. the elevation angle y ¼ 90 . Then the response of the array with respect to temporal frequency
ARTICLE IN PRESS W. Liu / Signal Processing 89 (2009) 876–882
879
o rad=s and angle of arrival ðy ¼ 90 ; fÞ of the impinging
ω1
signal is given by P ðo; fÞ ¼
M1;N1 X
jmo sin fdy =c jno cos fdx =c
wm;n e
e
,
ω 2d 2 c2
(3)
m;n¼0;0
ω1 = ω 2 tanφ
φ ω2
where c is the wave propagation speed, dx is the spacing between adjacent SDL sensors, and dy is the spacing between sensors ðm; nÞ and ðm þ 1; nÞ with m ¼ 0; 1; . . . ; M 1 and n ¼ 0; 1; . . . ; N 1. With the following substitutions:
o1 ¼
o sin fdy
Fig. 6. The location of the response of the array to the signal with frequency o and direction of arrival angle f.
, c o cos fdx , o2 ¼ c
(4)
r(t)
we have P ðo1 ; o2 Þ ¼
M1;N1 X
+
wm;n ejmo1 ejno2 .
(5)
m;n¼0;0
x(t) Then, given a set of coefficients wm;n , the array’s response can be obtained by first applying the two-dimensional Fourier transform to them as in (5) and then the two substitutions as in (4). From (4), we have
o1 d x ¼ tan f. o2 d y
(6)
Thus, f is given by
o d f ¼ arctan 1 x . o2 d y
(7)
From (4), we also have
o¼
co1 . sin fdy
(8)
Then, given any desired wideband response P ðo; fÞ, we can use the substitutions (7) and (8) to express it in the form of P ðo1 ; o2 Þ, then the desired coefficients wm;n can be obtained by applying the inverse Fourier transform to P ðo1 ; o2 Þ. This provides a theoretical explanation about why this structure can form a wideband beam response. Note a desired frequency invariant response P ðfÞ can be considered as a special case and the corresponding coefficients wm;n can be obtained in the same way. For simplicity, we consider a special case with dx ¼ dy ¼ d. From (4), we have
o21 þ o22 ¼
o2 d 2 c2
(9)
On the ðo1 ; o2 Þ plane, the response of the array to the signal with frequency o and DOA angle f is located on the point shown in Fig. 6. Corresponding to the traditional TDL systems, the coefficients for this new structure can be determined in different ways, depending on the specific situation. Here we give two examples in the following.
y(t)
2.2. Reference signal-based beamformer The first one is the case for which a reference signal rðtÞ is available and the weights are adjusted to minimize the mean square error between the beamformer output yðtÞ and the reference signal rðtÞ. This is the simplest case and its structure is shown in Fig. 7. It is a classical adaptive filtering problem and can be solved by any existing adaptive algorithms such as the least mean square (LMS) or recursive least squares (RLS) algorithms, or their subband implementations [11]. 2.3. Linearly constrained minimum variance beamformer If a desired reference signal is not available, but we know the DOA angle of the signal of interest and their bandwidth range, then we can impose some constraints on the array coefficients and then adaptively minimize the variance EfyðtÞ yðtÞg of the beamformer output subject to the imposed constraints. This leads to the well-known linearly constrained minimum variance (LCMV) beamformer [12]. The LCMV problem can be formulated as w
(10)
w
Fig. 7. A general structure for the first implementation of the proposed wideband beamformer.
min wH Rxx w
and
o1 ¼ o2 tan f.
e(t)
−
subject to CH w ¼ f,
(11)
where Rxx is the covariance matrix of observed array data in x, C is the constraint matrix and f is the response vector. The constraint will ensure that no matter how to adjust the array coefficients, the resultant beamformer will have the desired response set out by the constraint equation CH w ¼ f. Fig. 8 shows a general structure for such an LCMV beamformer. The constrained adaptive optimization in (11) can be conveniently solved using a generalized sidelobe canceller (GSC) [13], which performs a projection
ARTICLE IN PRESS 880
W. Liu / Signal Processing 89 (2009) 876–882
z
x (t)
y(t)
w
signal θ
C Hw = f Fig. 8. A general structure for the LCMV beamformer, where CH w ¼ f is the constraint imposed on the adaptive process.
y φ x
d (t)
wq
y (t)
+
Fig. 10. A planar array in the three-dimensional space.
−
x (t )
u (t ) B
wa
^y (t) 10 Traditional structure with TDLs Proposed structure with SDLs
Fig. 9. A general structure for the GSC.
of the data onto an unconstrained subspace by means of a blocking matrix B and a quiescent vector wq as shown in Fig. 9. Thereafter, standard unconstrained adaptive algorithms such as the LMS and RLS algorithms can be invoked to minimize the variance of yðtÞ, which is given by yðtÞ ¼ dðtÞ
wH a uðtÞ,
(12)
ensemble mean square residual error/[dB]
5 0 –5 –10 –15 –20 0
where
0.5
1
1.5 iterations
dðtÞ ¼ wH q xðtÞ
with wq ¼ CðCH CÞ1 f
(13)
and H
uðtÞ ¼ B xðtÞ.
(14)
The blocking matrix B must satisfy CH B ¼ 0
(15)
to block the signal of interest in the lower branch. 2.4. Discussion In appearance, the proposed rectangular array system for wideband beamforming is the same as a traditional rectangular array working for narrowband beamforming, because neither of them has TDLs involved and they have the same structure as shown in Fig. 3. Then, how to tell the difference between the two systems? To explain it in simple terms, we put the rectangular array in the coordinate system shown in Fig. 10. In a traditional narrowband rectangular array system, we normally assume that the signal of interest is from the broadside, i.e. y ¼ 0 in Fig. 10, and the interfering signals are from other directions with different y and f. If the signal of interest are not from the broadside, we can impose appropriate time delays, or phase shifts immediately after
2
2.5 x 104
Fig. 11. The two learning curves for the reference signal-based beamformer with a stepsize of 0:45.
each sensor output, such that the signals incident on the array from directions of interest other than broadside appear as identical replicas of one another at the outputs of the steering delay elements. In our proposed approach, we shall assume the signal of interest is from the direction f ¼ 0 , y40, and the interfering signals are from other azimuth angles fa0. As a result, the constraint matrix C is different from that of the traditional approach. However, in the reference signal-based case, we will not really see much difference in the implementation except that one is for wideband signals and one is for narrowband signals.
3. Simulations In our simulations, the proposed SDL beamforming structure with M ¼ N ¼ 10 is compared with the traditional TDL structure with M ¼ J ¼ 10. To run the simulations on a computer, we discretized the continuous signals and the sampling frequency is twice the highest frequency of the signal of interest. The spacing between adjacent sensors is half wavelength of the signal component with the highest possible frequency (corresponding to a normalized frequency p). The signal of interest comes from the broadside and four interfering signals from the
ARTICLE IN PRESS W. Liu / Signal Processing 89 (2009) 876–882
881
0 –10 –20
gain/[dB]
–30 –40 –50 –60 –70 –80 0.4 0.6 ]
[π Ω/
0.8 1 –80
–60
–40
–20
20
0
60
40
80
DOAφ Fig. 12. The resultant beam pattern for the bandwidth of interest ½0:4p; p for the proposed structure with a reference signal available.
4. Conclusions A novel adaptive wideband beamforming method has been proposed, where the TDLs in the traditional wideband beamforming structure are replaced by SDLs and
20 Traditional GSC with TDLs Proposed GSC with SDLs
ensemble mean square residual error/[dB]
directions y ¼ 0 and f ¼ 20 ; 40 ; 30 ; 60 , respectively. All of the signals have a bandwidth of ½0:4p; p. The signal to interference ratio (SIR) is about 20 dB and the signal to noise ratio (SNR) is about 20 dB. We use a normalized LMS algorithm with a stepsize of 0:45 for the reference signal-based case and 0:20 for the LCMV case. In the first set of simulations, we assume a reference signal is available and the implementation in Fig. 7 is used. The learning curves for the ensemble mean square output error are shown in Fig. 11 and we can see that although the convergence rate of the proposed structure is lower than the traditional structure, it has reached a lower steady state error for this specific scenario. The resultant beam pattern for the frequency range of ½0:4p; p for the proposed structure is shown in Fig. 12, where the four nulls at the interference directions of f ¼ 20 , 40 , 30 , 60 are clearly visible, indicating a successful beamforming process. For the second implementation, i.e. no reference signal is available, which is more practical, as shown in Fig. 9, both structures have a similar convergence speed, which is shown in Fig. 13 with a stepsize of 0:20. However, the SDL system again has achieved a lower steady state error for this special case. The resultant beam pattern is shown in Fig. 14. Similar to Fig. 12, we can also see the nulls at the interference directions.
15 10 5 0 –5 0
0.2
0.4
0.6
0.8 1 1.2 iterations
1.4
1.6
1.8
2
x 104
Fig. 13. The learning curves based on the GSC with a stepsize of 0.20.
only one coefficient is required for each of the received sensor signals. Due to its simple structure, this new class of beamformers can be implemented by simple analogue circuits and therefore we can avoid the high-speed analogue TDLs or digital sampling circuits in the traditional beamforming structure, which could be very expensive with increasing signal bandwidth and for some extreme cases the tapped delays would be so short that they simply could not be realized by the technique of the state of the art. Another very important advantage of the SDL-based structure is its potential to avoid the beam widening effect at higher off-boresight angels. The concept behind this new method is general and can be applied to any adaptive wideband beamforming scenarios with arbitrary beam patterns and sensor position patterns. Some simulation results for a rectangular array in
ARTICLE IN PRESS 882
W. Liu / Signal Processing 89 (2009) 876–882
0
gain/[dB]
–20 –40 –60 –80 0.4 0.5 0.6 Ω/[π ]
0.7 0.8 0.9 1
–80
–60
–40
–20
0
20
40
60
80
DOA φ
Fig. 14. The resultant beam pattern for the proposed structure based on the GSC.
place of the traditional linear array with TDLs have been provided to show that the new structure is capable of performing the beamforming task effectively. References [1] H.L. Van Trees, Optimum Array Processing, Part IV of Detection, Estimation, and Modulation Theory, Wiley, New York, USA, 2002. [2] M. Zatman, How narrow is narrowband?, IEE Proceedings—Radar, Sonar and Navigation 145 (2) (April 1998) 85–91. [3] J.T. Mayhan, A.J. Simmons, W.C. Cummings, Wide-band adaptive antenna nulling using tapped delay lines, IEEE Transactions on Antennas and Propagation AP-29 (November 1981) 923–936. [4] B.D. Van Veen, K.M. Buckley, Beamforming: a versatile approach to spatial filtering, IEEE Acoustics, Speech, and Signal Processing Magazine 5 (2) (April 1988) 4–24. [5] E.W. Vook, R.T. Compton Jr., Bandwidth performance of linear adaptive arrays with tapped delay-line processing, IEEE Transactions on Aerospace and Electronic Systems 28 (3) (July 1992) 901–908. [6] L. Yu, N. Lin, W. Liu, R. Langley, Bandwidth performance of linearly constrained minimum variance beamformers, in: Proceedings of the
[7]
[8]
[9]
[10]
[11] [12]
[13]
IEEE International Workshop on Antenna Technology, Cambridge, UK, March 2007, pp. 327–330. M. Ghavami, Wideband smart antenna theory using rectangular array structures, IEEE Transactions on Signal Processing 50 (9) (September 2002) 2143–2151. T. Sekiguchi, Y. Karasawa, Wideband beamspace adaptive array utilizing FIR fan filters for multibeam forming, IEEE Transactions on Signal Processing 48 (1) (January 2000) 277–284. W. Liu, S. Weiss, A new class of broadband arrays with frequency invariant beam patterns, in: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Montreal, Canada, vol. 2, May 2004, pp. 185–188. W. Liu, S. Weiss, Design of frequency invariant beamformers for broadband arrays, IEEE Transactions on Signal Processing 56 (2) (February 2008) 860. S. Haykin, Adaptive Filter Theory, third ed., Prentice-Hall, Englewood Cliffs, NJ, 1996. O.L. Frost III, An algorithm for linearly constrained adaptive array processing, Proceedings of the IEEE 60 (8) (August 1972) 926–935. L.J. Griffiths, C.W. Jim, An alternative approach to linearly constrained adaptive beamforming, IEEE Transactions on Antennas and Propagation 30 (1) (January 1982) 27–34.
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Adaptive wideband beamforming with sensor delay-lines Wei Liu Communications Research Group, Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, UK
a r t i c l e i n f o
abstract
Article history: Received 8 October 2007 Received in revised form 8 August 2008 Accepted 5 November 2008 Available online 18 November 2008
A novel adaptive wideband beamforming method is proposed, where the tap delays in the traditional method are replaced and only one adaptive coefficient is required for each sensor. In this way, the complex wideband beamforming system with temporal filtering can be avoided by implementing the proposed beamformer with simple analogue circuits for very high frequency and bandwidth applications. Another advantage of this new method is its potential to avoid the beam widening effect at higher off-boresight angels. The effectiveness of the proposed method is verified by simulations. & 2008 Elsevier B.V. All rights reserved.
Keywords: Adaptive beamforming Wideband Sensor delay-lines Tapped delay-lines
1. Introduction Beamforming is an array signal processing technique to form beams in order to receive signals illuminating from specific directions, whilst attenuating signals from other directions. The sensors in an array system can be positioned in space according to different patterns, e.g. along a straight line, around a circle, or on a plane. Such arrangements lead to linear arrays, circular arrays and planar arrays, respectively [1]. Fig. 1 shows a linear array system with M sensors, where a signal sl ðtÞ arrives from the direction fl . The propagation delay between the sensor 0 and the sensor m is denoted by tm. The received array signals are given by x0 ðtÞ, x1 ðtÞ; . . . ; xM1 ðtÞ. For narrowband signals [2], to form a beam pointing to the direction of arrival (DOA) of the signal of interest, we can apply one coefficient to each of the received signals and then sum them up directly. For wideband signals, however, such a configuration will not work well and normally we need to employ a tapped delay-line (TDL) system [3,4], as shown in Fig. 2, where each of the received array signals xm ðtÞ, m ¼ 0; 1; . . . ; M 1 is processed by the following TDL, with an adjacent tap
Tel.: +44 114 2225813; fax: +44 114 2225834.
E-mail address: w.liu@sheffield.ac.uk 0165-1684/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.11.005
delay of D and the corresponding tap coefficients given by wm;j , m ¼ 0; 1; . . . ; M 1, j ¼ 0; 1; . . . ; J 1. There are in total J taps for each TDL. The length J of the TDLs is dependent on the bandwidth of the impinging signals [5]. In general, the larger the bandwidth, the longer the TDLs [6]. The TDL coefficients can be designed for maintaining a fixed specified response for all signal/ interference scenarios, which leads to the concept of a data independent beamformer [4]. Alternatively, they can be chosen based on the statistics of the array data for optimizing the array’s response, which forms a statistically optimum beamformer [4]. Since the statistics of the array data are often not known or may change over time, adaptive algorithms may be used to determine these coefficients. In its discrete form, the TDL system is replaced by a finite impulse response (FIR) filter and adaptive algorithms can be realized by digital circuits. In both of the cases, the delays between samples/taps get shorter and shorter with increasing signal bandwidth and very high speed circuits need to be employed. As an example, consider a general ultrawideband system with frequency range from 0 Hz to 6 GHz. If we perform beamforming in digital form, then the sampling rate should be at least 12 GHz. With 16 bits per sample, it is almost impossible to build such a high-speed circuit with current technology. To avoid these difficulties associated with the traditional TDL system and also provide an alternative to it, in
ARTICLE IN PRESS W. Liu / Signal Processing 89 (2009) 876–882
this paper we propose a novel method for adaptive wideband beamforming, i.e. we replace the tapped delays by spatial propagation delays by positioning more sensors behind the original array system. We consider this new system as adaptive wideband beamforming with sensor delay-lines (SDLs). This method can be applied to any kind of array system. For the original linear array, we will have a rectangular array, as shown in Fig. 3, to replace the structure in Fig. 2; for an original rectangular array, we will have a cubic array, as shown in Fig. 4; for a semicircular array, we will have a new system shown in Fig. 5, or a cylindrical form for a circular array, etc. Moreover, these SDL sensors could be positioned not necessarily in a regular spacing, but arbitrary. The advantage of this arrangement is twofold: Since only one coefficient is required for each received array signal, the function of the original temporal filtering process is now achieved by spatial filtering using multiple sensors so that a complicated wideband beamforming system is able to be implemented by simple analogue circuits. This is useful especially when the signal frequency (and bandwidth) are very high and a digital implementation becomes extremely difficult or even not viable at all, as mentioned in the case before.
x0 (t )
sl ( t ) φl
xm(t ) τm
x M−1(t ) Fig. 1. A linear array system with M sensors, where a signal sl ðtÞ arrives from the direction fl .
877
The second advantage is its potential to avoid the beam widening effect at high off-boresight angels. For example, for the traditional TDL structure in Fig. 2, the beamwidth will increase significantly when a broadside main beam is steered to an angle f closer to 90 . However, for the corresponding SDL structure in Fig. 3, we can rotate the set of coefficients by 90 to form a main beam pointing to the direction f ¼ 90 and the new beam will have the same beamwidth, because its effective aperture will not go to zero as in the linear array case and it has a full coverage over the 3601 azimuth range. Note if we want to implement this beamformer in digital domain, we will still require the same sampling rate as the traditional one. However, we may need the proposed structure for those applications where we want to avoid the beam widening effect at high off-boresight angles and using more sensors may well be justified by meeting the requirements of the corresponding applications. This idea is derived from a recently proposed frequency invariant beamforming design method for rectangular arrays [7]. A special characteristic of this fixed beamformer is that there are no TDLs involved and only one single weight is attached to each sensor. By examining its beam pattern, we can find that this rectangular array is approximating the response of a frequency invariant linear array with TDLs [8–10] by employing spatial propagation delays instead of wired delays. We therefore generalize the idea to the adaptive case and also to arbitrary beam patterns and sensor position patterns, as discussed before. Here we will only focus on the simple but widely used traditional linear array case, which corresponds to the rectangular array in the new approach, and provide some initial investigation results to show its effectiveness. An extensive study of this new approach for
x 0(t) w0.0
w0.1
w0.2
w0,J −1
w1.0
w1.1
w1.2
w1,J −1
sl (t ) φl
x1(t)
y (t)
xM−1(t) wM−1.0
wM−1.1
wM−1.2
wM−1.J −1
Fig. 2. A general structure for wideband beamforming.
ARTICLE IN PRESS 878
W. Liu / Signal Processing 89 (2009) 876–882
x 0.0(t)
x 0.1(t)
w0.0
x 0.2(t)
w 0.1
x 0.N−1(t)
w0.N−1
w0.2
sl (t ) φl
x 1.0(t)
x 1.1(t)
w1.0
x 1.2(t)
w1.1
x 1.N−1(t)
w1.N−1
w1.2
y (t)
x M−1.0(t)
wM−1.0
x M−1.1(t)
x M−1.2(t)
wM−1.1
x M−1.N−1(t)
wM−1.2
wM−1.N−1
Fig. 3. The proposed rectangular array to replace the traditional linear array with TDLs.
z
2. Adaptive wideband beamforming with SDLs
signal
θ y φ x Fig. 4. The proposed cubic array to replace the traditional rectangular array with TDLs.
sl(t) φl
As discussed in the previous section, we propose to use sensor delays to replace the tapped delays in the traditional adaptive wideband beamforming structure. For the linear array shown in Fig. 2, the corresponding new structure will be a planar array and it has been shown in Fig. 3, where each of the TDLs with length J is replaced by an SDL with N sensors, and the delay between adjacent taps is thus replaced by the spatial propagation delay between adjacent sensors. In this new structure, there are in total M N sensors, with the received sensor signals denoted by xm;n ðtÞ, m ¼ 0; 1; . . . ; M 1, n ¼ 0; 1; . . . ; N 1. As there is only one coefficient for each received sensor signal, there are no TDLs involved and adaptive beamforming can be performed by very simple analogue circuits. Note that there is no temporal information used in the beamforming process. Therefore it can be considered as a wideband beamforming structure with spatialonly information. 2.1. Wideband response of the proposed structure In Fig. 3, the output yðtÞ can be expressed as
Fig. 5. The proposed planar array to replace the traditional semi-circular array with TDLs.
all kinds of beamforming scenarios and array sensor position patterns will be left for future study. This paper is organized as follows. In Section 2, a theoretical explanation about why the proposed structure can form a wideband beam response is first given and then two adaptive beamforming implementations are discussed; simulation results are provided in Section 3, and conclusions drawn in Section 4.
yðtÞ ¼ wT xðtÞ,
(1)
where w ¼ ½w0;0 . . . wM1;0 w0;1 . . . wM1;1 . . . wM1;N1 T , xðtÞ ¼ ½x0;0 ðtÞ . . . xM1;0 ðtÞ x0;1 . . . xM1;1 ðtÞ . . . xM1;N1 T . (2) Suppose all of the signals are on the same plane as the rectangular array, i.e. the elevation angle y ¼ 90 . Then the response of the array with respect to temporal frequency
ARTICLE IN PRESS W. Liu / Signal Processing 89 (2009) 876–882
879
o rad=s and angle of arrival ðy ¼ 90 ; fÞ of the impinging
ω1
signal is given by P ðo; fÞ ¼
M1;N1 X
jmo sin fdy =c jno cos fdx =c
wm;n e
e
,
ω 2d 2 c2
(3)
m;n¼0;0
ω1 = ω 2 tanφ
φ ω2
where c is the wave propagation speed, dx is the spacing between adjacent SDL sensors, and dy is the spacing between sensors ðm; nÞ and ðm þ 1; nÞ with m ¼ 0; 1; . . . ; M 1 and n ¼ 0; 1; . . . ; N 1. With the following substitutions:
o1 ¼
o sin fdy
Fig. 6. The location of the response of the array to the signal with frequency o and direction of arrival angle f.
, c o cos fdx , o2 ¼ c
(4)
r(t)
we have P ðo1 ; o2 Þ ¼
M1;N1 X
+
wm;n ejmo1 ejno2 .
(5)
m;n¼0;0
x(t) Then, given a set of coefficients wm;n , the array’s response can be obtained by first applying the two-dimensional Fourier transform to them as in (5) and then the two substitutions as in (4). From (4), we have
o1 d x ¼ tan f. o2 d y
(6)
Thus, f is given by
o d f ¼ arctan 1 x . o2 d y
(7)
From (4), we also have
o¼
co1 . sin fdy
(8)
Then, given any desired wideband response P ðo; fÞ, we can use the substitutions (7) and (8) to express it in the form of P ðo1 ; o2 Þ, then the desired coefficients wm;n can be obtained by applying the inverse Fourier transform to P ðo1 ; o2 Þ. This provides a theoretical explanation about why this structure can form a wideband beam response. Note a desired frequency invariant response P ðfÞ can be considered as a special case and the corresponding coefficients wm;n can be obtained in the same way. For simplicity, we consider a special case with dx ¼ dy ¼ d. From (4), we have
o21 þ o22 ¼
o2 d 2 c2
(9)
On the ðo1 ; o2 Þ plane, the response of the array to the signal with frequency o and DOA angle f is located on the point shown in Fig. 6. Corresponding to the traditional TDL systems, the coefficients for this new structure can be determined in different ways, depending on the specific situation. Here we give two examples in the following.
y(t)
2.2. Reference signal-based beamformer The first one is the case for which a reference signal rðtÞ is available and the weights are adjusted to minimize the mean square error between the beamformer output yðtÞ and the reference signal rðtÞ. This is the simplest case and its structure is shown in Fig. 7. It is a classical adaptive filtering problem and can be solved by any existing adaptive algorithms such as the least mean square (LMS) or recursive least squares (RLS) algorithms, or their subband implementations [11]. 2.3. Linearly constrained minimum variance beamformer If a desired reference signal is not available, but we know the DOA angle of the signal of interest and their bandwidth range, then we can impose some constraints on the array coefficients and then adaptively minimize the variance EfyðtÞ yðtÞg of the beamformer output subject to the imposed constraints. This leads to the well-known linearly constrained minimum variance (LCMV) beamformer [12]. The LCMV problem can be formulated as w
(10)
w
Fig. 7. A general structure for the first implementation of the proposed wideband beamformer.
min wH Rxx w
and
o1 ¼ o2 tan f.
e(t)
−
subject to CH w ¼ f,
(11)
where Rxx is the covariance matrix of observed array data in x, C is the constraint matrix and f is the response vector. The constraint will ensure that no matter how to adjust the array coefficients, the resultant beamformer will have the desired response set out by the constraint equation CH w ¼ f. Fig. 8 shows a general structure for such an LCMV beamformer. The constrained adaptive optimization in (11) can be conveniently solved using a generalized sidelobe canceller (GSC) [13], which performs a projection
ARTICLE IN PRESS 880
W. Liu / Signal Processing 89 (2009) 876–882
z
x (t)
y(t)
w
signal θ
C Hw = f Fig. 8. A general structure for the LCMV beamformer, where CH w ¼ f is the constraint imposed on the adaptive process.
y φ x
d (t)
wq
y (t)
+
Fig. 10. A planar array in the three-dimensional space.
−
x (t )
u (t ) B
wa
^y (t) 10 Traditional structure with TDLs Proposed structure with SDLs
Fig. 9. A general structure for the GSC.
of the data onto an unconstrained subspace by means of a blocking matrix B and a quiescent vector wq as shown in Fig. 9. Thereafter, standard unconstrained adaptive algorithms such as the LMS and RLS algorithms can be invoked to minimize the variance of yðtÞ, which is given by yðtÞ ¼ dðtÞ
wH a uðtÞ,
(12)
ensemble mean square residual error/[dB]
5 0 –5 –10 –15 –20 0
where
0.5
1
1.5 iterations
dðtÞ ¼ wH q xðtÞ
with wq ¼ CðCH CÞ1 f
(13)
and H
uðtÞ ¼ B xðtÞ.
(14)
The blocking matrix B must satisfy CH B ¼ 0
(15)
to block the signal of interest in the lower branch. 2.4. Discussion In appearance, the proposed rectangular array system for wideband beamforming is the same as a traditional rectangular array working for narrowband beamforming, because neither of them has TDLs involved and they have the same structure as shown in Fig. 3. Then, how to tell the difference between the two systems? To explain it in simple terms, we put the rectangular array in the coordinate system shown in Fig. 10. In a traditional narrowband rectangular array system, we normally assume that the signal of interest is from the broadside, i.e. y ¼ 0 in Fig. 10, and the interfering signals are from other directions with different y and f. If the signal of interest are not from the broadside, we can impose appropriate time delays, or phase shifts immediately after
2
2.5 x 104
Fig. 11. The two learning curves for the reference signal-based beamformer with a stepsize of 0:45.
each sensor output, such that the signals incident on the array from directions of interest other than broadside appear as identical replicas of one another at the outputs of the steering delay elements. In our proposed approach, we shall assume the signal of interest is from the direction f ¼ 0 , y40, and the interfering signals are from other azimuth angles fa0. As a result, the constraint matrix C is different from that of the traditional approach. However, in the reference signal-based case, we will not really see much difference in the implementation except that one is for wideband signals and one is for narrowband signals.
3. Simulations In our simulations, the proposed SDL beamforming structure with M ¼ N ¼ 10 is compared with the traditional TDL structure with M ¼ J ¼ 10. To run the simulations on a computer, we discretized the continuous signals and the sampling frequency is twice the highest frequency of the signal of interest. The spacing between adjacent sensors is half wavelength of the signal component with the highest possible frequency (corresponding to a normalized frequency p). The signal of interest comes from the broadside and four interfering signals from the
ARTICLE IN PRESS W. Liu / Signal Processing 89 (2009) 876–882
881
0 –10 –20
gain/[dB]
–30 –40 –50 –60 –70 –80 0.4 0.6 ]
[π Ω/
0.8 1 –80
–60
–40
–20
20
0
60
40
80
DOAφ Fig. 12. The resultant beam pattern for the bandwidth of interest ½0:4p; p for the proposed structure with a reference signal available.
4. Conclusions A novel adaptive wideband beamforming method has been proposed, where the TDLs in the traditional wideband beamforming structure are replaced by SDLs and
20 Traditional GSC with TDLs Proposed GSC with SDLs
ensemble mean square residual error/[dB]
directions y ¼ 0 and f ¼ 20 ; 40 ; 30 ; 60 , respectively. All of the signals have a bandwidth of ½0:4p; p. The signal to interference ratio (SIR) is about 20 dB and the signal to noise ratio (SNR) is about 20 dB. We use a normalized LMS algorithm with a stepsize of 0:45 for the reference signal-based case and 0:20 for the LCMV case. In the first set of simulations, we assume a reference signal is available and the implementation in Fig. 7 is used. The learning curves for the ensemble mean square output error are shown in Fig. 11 and we can see that although the convergence rate of the proposed structure is lower than the traditional structure, it has reached a lower steady state error for this specific scenario. The resultant beam pattern for the frequency range of ½0:4p; p for the proposed structure is shown in Fig. 12, where the four nulls at the interference directions of f ¼ 20 , 40 , 30 , 60 are clearly visible, indicating a successful beamforming process. For the second implementation, i.e. no reference signal is available, which is more practical, as shown in Fig. 9, both structures have a similar convergence speed, which is shown in Fig. 13 with a stepsize of 0:20. However, the SDL system again has achieved a lower steady state error for this special case. The resultant beam pattern is shown in Fig. 14. Similar to Fig. 12, we can also see the nulls at the interference directions.
15 10 5 0 –5 0
0.2
0.4
0.6
0.8 1 1.2 iterations
1.4
1.6
1.8
2
x 104
Fig. 13. The learning curves based on the GSC with a stepsize of 0.20.
only one coefficient is required for each of the received sensor signals. Due to its simple structure, this new class of beamformers can be implemented by simple analogue circuits and therefore we can avoid the high-speed analogue TDLs or digital sampling circuits in the traditional beamforming structure, which could be very expensive with increasing signal bandwidth and for some extreme cases the tapped delays would be so short that they simply could not be realized by the technique of the state of the art. Another very important advantage of the SDL-based structure is its potential to avoid the beam widening effect at higher off-boresight angels. The concept behind this new method is general and can be applied to any adaptive wideband beamforming scenarios with arbitrary beam patterns and sensor position patterns. Some simulation results for a rectangular array in
ARTICLE IN PRESS 882
W. Liu / Signal Processing 89 (2009) 876–882
0
gain/[dB]
–20 –40 –60 –80 0.4 0.5 0.6 Ω/[π ]
0.7 0.8 0.9 1
–80
–60
–40
–20
0
20
40
60
80
DOA φ
Fig. 14. The resultant beam pattern for the proposed structure based on the GSC.
place of the traditional linear array with TDLs have been provided to show that the new structure is capable of performing the beamforming task effectively. References [1] H.L. Van Trees, Optimum Array Processing, Part IV of Detection, Estimation, and Modulation Theory, Wiley, New York, USA, 2002. [2] M. Zatman, How narrow is narrowband?, IEE Proceedings—Radar, Sonar and Navigation 145 (2) (April 1998) 85–91. [3] J.T. Mayhan, A.J. Simmons, W.C. Cummings, Wide-band adaptive antenna nulling using tapped delay lines, IEEE Transactions on Antennas and Propagation AP-29 (November 1981) 923–936. [4] B.D. Van Veen, K.M. Buckley, Beamforming: a versatile approach to spatial filtering, IEEE Acoustics, Speech, and Signal Processing Magazine 5 (2) (April 1988) 4–24. [5] E.W. Vook, R.T. Compton Jr., Bandwidth performance of linear adaptive arrays with tapped delay-line processing, IEEE Transactions on Aerospace and Electronic Systems 28 (3) (July 1992) 901–908. [6] L. Yu, N. Lin, W. Liu, R. Langley, Bandwidth performance of linearly constrained minimum variance beamformers, in: Proceedings of the
[7]
[8]
[9]
[10]
[11] [12]
[13]
IEEE International Workshop on Antenna Technology, Cambridge, UK, March 2007, pp. 327–330. M. Ghavami, Wideband smart antenna theory using rectangular array structures, IEEE Transactions on Signal Processing 50 (9) (September 2002) 2143–2151. T. Sekiguchi, Y. Karasawa, Wideband beamspace adaptive array utilizing FIR fan filters for multibeam forming, IEEE Transactions on Signal Processing 48 (1) (January 2000) 277–284. W. Liu, S. Weiss, A new class of broadband arrays with frequency invariant beam patterns, in: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Montreal, Canada, vol. 2, May 2004, pp. 185–188. W. Liu, S. Weiss, Design of frequency invariant beamformers for broadband arrays, IEEE Transactions on Signal Processing 56 (2) (February 2008) 860. S. Haykin, Adaptive Filter Theory, third ed., Prentice-Hall, Englewood Cliffs, NJ, 1996. O.L. Frost III, An algorithm for linearly constrained adaptive array processing, Proceedings of the IEEE 60 (8) (August 1972) 926–935. L.J. Griffiths, C.W. Jim, An alternative approach to linearly constrained adaptive beamforming, IEEE Transactions on Antennas and Propagation 30 (1) (January 1982) 27–34.