SU Wei(蘇為)，HUANG Xiao-yan(黃曉燕)，CHENG En(程恩)，YUAN Fei(袁飛)，SUN Hai-xin(孫海信)

(Key Laboratory of Underwater Acoustic Communication and Marine Information Technology，Ministry of Education，Xiamen University，Xia’men 361005，Fujian，China)

Introduction

Some phase coherent modulation and demodulation techniques，such as PSK，can improve data rate greatly in underwater acoustic(UWA)communication，because the bandwidth of UWA channel is seriously limited.But compared with those incoherent techniques，such as FSK，they need higher SNR of the received signals for decoding the signals［1］.The use of beam-forming technique and the SNR gain obtained by spatial filters are demonstrated as an effective method to improve the communication performance if SNR is relatively low.And，switching-beam algorithms or adaptive beam-forming algorithms must be used in the receiver to track the direction of arrivals(DOA)of the incident signals.Compared with the adaptive beamforming algorithms，the switching-beam algorithms are more robust［2］.And，in order to receive undistorted PSK signals，a time domain wideband beam-forming structure with constant beam-width is proposed in Ref.［3］.It combined tapped-delay-lines with real coefficients FIR filters.And the real coefficients FIR filters are designed to realize amplitude and phase weighting at different frequency points.But，simulation results show that the real coefficients FIR filters proposed in Ref.［3］are unable to meet the requirements for complicated frequency response and the communication system with a large ratio of communication band to carrier frequency.In order to solve this problem，a time domain wideband beam-forming structure based on complex coefficient FIR digital filters is proposed in this paper.It is more suitable for PSK underwater acoustic communication than the structure proposed in Ref.［3］.

1 Design of Weights for Wideband and Constant Beam-width

An adaptive beam synthesizing algorithm is proposed in Ref.［4］，and an improved method is proposed in literature Ref.［5］.

Afterpth iteration，the designed weights of the al-gorithm proposed in Ref.［4］is

wherefp(θm)is the energy of pre-defined virtual interference source.

From Eq.(2)，it can be seen that the constraint of the mainlobe shape is not taken into account in the original algorithm.Thus，its stability is poor.An improved algorithm replacesV(θd)in Eq.(1)with(Vm+Wp－1)［5］.The formula of beam-forming weights afterpth iteration is

and，Vmcan be expressed as

whereθican be taken equally in the mainlobe，Aiis the constraint coefficient of the mainlobe.

The error between the array shape calculated after(p－1)th iteration and referenced array shapeD(θi)inθidirection is

According to Eq.(5)，the update constraint coefficientAiof the mainlobe is

個體對自身正當利益的追求是無可厚非的，畢竟“任何人如果不同時為了自己的某種需要和為了這種需要的器官而做事，他就什么也不能做”[3]286。 但在市場逐利本質的驅使下，受利益最大化原則的影響，個體極易因一己之私而產生短視、功利的思想和行為，淪為自身私欲的奴隸。 倘若各個利益主體都對一己之私錙銖必較，不惜犧牲他人或集體的利益，那么利益主體間就生成一種互抑狀態(tài)。 在這種互抑狀態(tài)下，各個利益主體看似享有爭取自身利益的自由，實則經常性的由于多方牽制而難以真正實現自身的利益訴求。 這種多元利益主體之間自發(fā)形成的互抑狀態(tài)若不加以引導和調試，必將激化本就錯綜復雜的利益沖突。

whereμis the iteration step.

2 Proposed Wideband Beam-forming Structure

2.1 Beam-forming Structure Based on Complex Coefficient FIR Digital Filter

The designed constant beam width weight of thekth sensor at the frequencyflcan be expressed as

whereaklandφklare the amplitude and phase of the weight respectively.

In literature Ref.［3］，the wideband signals received bykth sensor are weighted by real coefficient FIR digital filters with frequency responsesHk(ωl)=wkl .But，simulation results show that the real coefficient FIR digital filters can not meet the design requirements，because the frequency and phase responses of the required filters are complicated in PSK underwater acoustic communication.

Therefore，a time domain wideband beam-forming structure with constant beam width based on complex coefficient FIR digital filters is proposed in this paper，as shown in Fig.1.It compensates the time delay of integer sampling intervals with tapped-delay-lines first，calculates the complex envelopes of the received wideband signals by using spectrum shifting then，and weights the received signals with a set of complex coefficient FIR digital filters finally.

Fig.1 Time domain wideband beam-forming structure based on complex coefficient FIR digital filters

Obviously，the complex coefficient FIR digital filters can realize more complicate frequency responses than real coefficient FIR digital filters with the same order.The signals are processed as follows.

First，assuming that the received signal is the sum of a group of different frequency CW signals，thus，the compensated signal ofkth sensor can be written as

whereτkis the delay time introduced bykth tapped-de-lay-line.

Then，after spectrum shifting，the complex representation of the received signal envelop which contains high-frequency component can be expressed as

The complex envelopes of the received signals ofkth sensor can be achieved by low pass filter.

Finally，the signalsk(t)is weighted by complex coefficient FIR digital filters with responses shown in Eq.(7)，and the beam output signal is achieved by weighted sum of different sensor signals.

2.2 Design Algorithm of Complex Coefficient FIR Digital Filters

It can be proved that，if the input signal of filter is the complex envelop of received real signal，the frequency response of the complex coefficient FIR digital filter can be expressed as

wherehnis the coefficient of the designed filter，Nis the order of filter，andfsis the sampling rate of the system.

The adaptive［6］or optimization method［7］can be used to design the coefficients of the required FIR filters.For the frequencyfloutside the pass band，if using the adaptive method，the pre-defined frequency responses，including amplitude and phase，must be provided.It increases extra computational volume and has disadvantages of aberration and bad robustness.While using the optimization method，the amplitude frequency response at different frequencies will be constrained only.Therefore，it is more robust than the adaptive method.

The optimization method can be described as follows.

First，a group of frequencies in the range of［0，fs/2］is selected to make the frequency interval smaller in the communication band and larger outside the band.

Second，the optimization relational expression can be established by normL∞for the frequencyflin the pass band.

And the optimization relational expression established by normL2is

For the frequencyfloutside the pass band，it is necessary to add a constraint condition

whereais a constant used to constrain the side-lodes of FIR filters.

Finally，solving the above optimization problem can be solved.

In this paper，the software Sedumi and Yalmip are adopted to solve the above optimization problem.

3 Simulation Results

In the simulation，a volumetric array composed of 12 sensors，as shown in Fig.2(c)，is adopted.The communication band is 4 －10 kHz.The intersection angle between the direction of main beam and the base array normal is 15°.The weights are calculated by using the algorithm proposed in the Section 1 first.Then，the frequency band will be shifted from 4－10 kHz to 0－6 kHz to obtain the complex envelopes of the received signals.Finally，the real and complex coefficient FIR digital filters are used to fit the beam weights，respectively.The simulation results are shown in Fig.2.

Fig.2 Simulation results

Fig.2(a)and Fig.2(b)show the results of adopting 61-order real coefficient FIR digital filter and 31-order complex coefficient FIR digital filter to fit the weights of 0th sensor，respectively.the solid line in figure represents the expected frequency responses，i.e.the weights for wideband beam with constant width.The line labeled by‘* ’shows the fitting result of complex coefficient FIR digital filter.It essentially coincides with the expected value.And，the line labeled by‘△’shows the result of real coefficient FIR digital filter greatly.It differs from the expected value.It can be seen that the shape of the beam weighted by real coefficient filters have great distortion when the ratio of communication band to carrier frequency is larger than 0.85.

The beam shape weighted by the complex coefficient filters are shown in Fig.2(d).It can be seen clearly that the shapes of mainlobes meet the requirements well and the errors are less than 0.3 dB.

4 Conclusions

A novel wideband beam-forming structure with constant beam width based on complex coefficient FIR digital filters is proposed in this paper.The design algorithm of the complex coefficient FIR filters is presented also.The simulation results for a volumetric array composed of 12 sensors in the frequency range of 4－10 kHz show that the proposed structure is more suitable for the underwater acoustic communication situation where there is a larger ratio of communication band to carrier frequency.

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