Criterion for narrowband beamforming
According to the MVDR beamformer we can easily obtain array output SNR
T. Qin, H. Zhang and X. Zhang SNR1 ¼ A new criterion sensor narrowband equivalent efficiency (SNEE) is presented for array beamforming techniques. SNEE directly shows the performance when a band signal is processed with a narrowband model, and so it can be used to judge whether a band signal should be seen as narrowband.
Introduction: When discussing adaptive array beamforming techniques, narrowband signal beamforming is different from wideband signal beamforming. Clearly, there is a problem: how narrow is narrowband and how wide is wideband? [1] provides a definition for the notion of narrowband and derives an expression which is useful in determining whether a particular scenario qualifies as narrowband. The expression [1] derived correctly predicts where the narrowband assumption fails for some super resolution algorithms. The disadvantage of the definition in [1] is that it cannot directly show the performance of the array beamforming. In this Letter, we propose a criterion named sensor narrowband equivalent efficiency (SNEE). This definition is derived from array output SNR, and it can directly show the performance of a beamforming algorithm when a band signal is seen as narrowband. Data model: Consider a uniform linear array (ULA) of M sensors with an inter-element spacing d, (the results in this Letter are easily extended to arbitrary array geometry). We assume that only one signal inspects on the array in the direction y. The carrier frequency is fc, and the base band of the signal is [b=2, b=2]. The steering vector at frequency f is að f ; yÞ ¼ ½1; e
j2pð fc þf Þd sinðyÞ=c
j2pð fc þf ÞðM 1Þd sinðyÞ=c T
;...;e
The array vector of received signals is ð b=2 x¼ sð f Það f ; yÞdf þ n
ð2Þ
where n is the additive noise and s( f ) is the power of the base signal at frequency f. Generally speaking, s( f ) is a constant function to make good use of frequency, and then we get ð b=2 að f ; yÞdf þ n ð3Þ x¼s
Z¼
Rx ¼ EfxðtÞx H ðtÞg
ð4Þ
According to a minimum variance distortionless response (MVDR) beamformer [2], the optimal weight vector is the solution to the following constrained minimisation problem: H
min½o Rx o
H
subject to: o a1 ðyÞ ¼ 1
SNR2 SNR1
ð10Þ
Now we calculate SNR2 to derive the mathematical form of SNEE. Comparing (7) with (3), we define the extended steering vector b(y) as Ð b=2 M b=2 að f ; yÞdf bðyÞ ¼ ð11Þ Ð b=2 b=2 að f ; yÞdf With extended steering vector b(y), the data correlation matrix (4) can be rewritten as Rx ¼ s21 bðyÞbH ðyÞ þ s2n I
ð12Þ
Again according to the MVDR beamformer, we get output SNR for the nonzero-band case SNR2 ¼
1 2M s21 =s2n þ M 2 s41 =s4n M s21 1=g 2M s21 =s2n þ M 2 s41 =s4n s2n
ð13Þ
where g is the square of the correlated coefficient of the constraint vector a1 and extended steering vector b H a1 g ¼ aH ð14Þ 1 bb M2 with (9) and (13), we get SNEE SNR2 1 2M s21 =s2n þ M 2 s41 =s4n ¼ SNR1 1=g 2M s21 =s2n þ M 2 s41 =s4n 1=g 1 ¼1 1=g 2M s21 =s2n þ M 2 s41 =s4n
Z¼
b=2
The data correlation matrix is
ð9Þ
However, the general signal has a certain band. In this case, the constraint vector is the same as (6), but the steering vector is a function of frequency. The constraint vector mismatches the steering vector in such condition. Clearly, the output SNR for nonzero-band signal SNR2 is smaller than SNR1, and while the bandwidth of the signal is increasing, SNR2 will become worse, i.e. for a same array, the beamforming efficiency will degrade for wideband signal. To directly show the impact of the bandwidth of the signal, we define sensor narrowband equivalent efficiency (SNEE) as
ð1Þ
b=2
M s21 s2n
ð15Þ
Analysis: At first glance, we cannot obtain any correlation between SNEE and the bandwidth of the signal. In fact g depends on FBW (base signal bandwidth=carrier frequency), and of course SNEE also depends on signal bandwidth. 1.0
M = 20
ð5Þ 0.9
where o denotes the weight vector and a1(y) is the constraint vector.
M = 30 g
0.8
SNEE: Zero-band (b ¼ 0) is the standard and ideal narrowband signal. In this case, the steering vector is very simple and the same as the constraint vector
0.7
a1 ðyÞ ¼ að f ; yÞ ¼ aðyÞ ¼ ½1; ej2pfc d sinðyÞ=c ; . . . ; ej2pfc ðM 1Þd sinðyÞ=c T
0.6
ð6Þ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 f0 /fc
The array vector (3) can be rewritten as x ¼ saðyÞ þ n
ð7Þ
Then we get the array correlation matrix Rx ¼ EfxðtÞxH ðtÞg ¼ s21 aðyÞaH ðyÞ þ s2n I
M = 40
ð8Þ
in which s21 is the power of the signal, s2n is the power of the noise, and the sensor noise is uncorrelated with each other and signal. Because the constraint vector exactly matches the steering vector, we achieve the best beamforming performance in this case.
Fig. 1 g curves (DOA ¼ 0.3, input SNR ¼ 10 db)
Fig. 1 shows several g curves. From the Figure we see that g drops while the signal bandwidth is increasing. Fig. 2 shows some SNEE curves. Again, while signal bandwidth is increasing, SNEE drops dramatically. SNEE is also correlated with some other factors, such as input SNR, DOA. From Figs. 1 and 2, we can see that g and SNEE drops when the sensor number is increasing.
ELECTRONICS LETTERS 8th July 2004 Vol. 40 No. 14
1.0
0.9 sensor number
given threshold, the signal should be processed with a wideband model, e.g. when the FBW is larger than 0.08, the SNEE is less than 0.8 in Fig. 2. The beamforming efficiency is too low for 0.8, and the signal should be processed with a wideband model.
30 50
40
SNEE
0.8
Conclusion: We propose a new criterion, SNEE, for array beamforming. SNEE can directly show the efficiency of narrowband beamforming, compared with the best performance. With SNEE we can easily decide which model (narrowband model or wideband model) is better for a signal.
0.7
0.6
0.5 0
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 FBW
# IEE 2004 Electronics Letters online no: 20040577 doi: 10.1049/el:20040577
Fig. 2 SNEE curves (DOA ¼ 0.3, input SNR ¼ 20 db)
There are relationships among the bandwidth of the signal, input SNR s21=s2n and sensor number M. From (15), while input SNR is increasing, the impact of the bandwidth of signal is decreasing. This is reasonable. Since g is a function of FBW and sensor number, the impact of the bandwidth of signal and the impact of sensor number for SNEE is dependent. With SNEE, we can easily decide whether a signal is suitable to be processed with a narrowband model. If the SNEE is smaller than a
14 March 2004
T. Qin, H. Zhang and X. Zhang (Electronic Engineering Department, Tsinghua University, Beijing, People’s Republic of China) E-mail:
[email protected] References 1 2
Zatman, M.: ‘How narrow is narrowband?’, IEE Proc. Radar, Sonar Navig., 1998, 145, (2), pp. 85–91 Capon, J.: ‘High-resolution frequency-wavenumber spectrum analysis’, Proc. IEEE, 1969, 57, pp. 1408–1418
ELECTRONICS LETTERS 8th July 2004 Vol. 40 No. 14
