# Chapter 3 Antenna Arrays and Beamforming Array Beam Forming Techniques

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### Transcript of Chapter 3 Antenna Arrays and Beamforming Array Beam Forming Techniques

29CHAPTER 3ANTENNA ARRAYS AND BEAMFORMINGArray beam forming techniques exist that can yield multiple, simultaneouslyavailable beams. The beams can be made to have high gain and low sidelobes, orcontrolled beamwidth. Adaptive beam forming techniques dynamically adjust the arraypattern to optimize some characteristic of the received signal. In beam scanning, a singlemain beam of an array is steered and the direction can be varied either continuously or insmall discrete steps.Antenna arrays using adaptive beamforming techniques can reject interferingsignals having a direction of arrival different from that of a desired signal. Multi-polarized arrays can also reject interfering signals having different polarization statesfrom the desired signal, even if the signals have the same direction of arrival. Thesecapabilities can be exploited to improve the capacity of wireless communication systems.This chapter presents essential concepts in antenna arrays and beamforming.An array consists of two or more antenna elements that are spatially arranged andelectrically interconnected to produce a directional radiation pattern. The interconnectionbetween elements, called the feed network, can provide fixed phase to each element orcan form a phased array. In optimum and adaptive beamforming, the phases (andusually the amplitudes) of the feed network are adjusted to optimize the received signal.The geometry of an array and the patterns, orientations, and polarizations of the elementsinfluence the performance of the array. These aspects of array antennas are addressed asfollows. The pattern of an array with general geometry and elements is derived inSection 3.1; phase- and time-scanned arrays are discussed in Section 3.2. Section 3.3gives some examples of fixed beamforming techniques. The concept of optimumbeamforming is introduced in Section 3.4. Section 3.5 describes adaptive algorithms thatiteratively approximate the optimum beamforming solution. Section 3.6 describes theeffect of array geometry and element patterns on optimum beamforming performance.3.1 Pattern of a Generalized ArrayA three dimensional array with an arbitrary geometry is shown in Fig. 3-1. Inspherical coordinates, the vector from the origin to the nth element of the array is givenby ) , , ( m m m mr L and

( , , ) k 1 is the vector in the direction of the source of an30incident wave. Throughout this discussion it is assumed that the source of the wave is inthe far field of the array and the incident wave can be treated as a plane wave. To findthe array factor, it is necessary to find the relative phase of the received plane wave ateach element. The phase is referred to the phase of the plane wave at the origin. Thus,the phase of the received plane wave at the nth element is the phase constant 2multiplied by the projection of the element position mrL on to the plane wave arrivalvector k . This is given by mr k LL with the dot product taken in rectangularcoordinates.Figure 3-1. An arbitrary three dimensional arrayIn rectangular coordinates, r z y x k cos sin sin cos sin + + andz y x r m m m m m m m m m cos sin sin cos sin + + L, and the relative phase of theincident wave at the nth element is) cos sin sin cos sin () cos cos sin sin sin sin cos sin cos (sin m m mm m m m m mm mz y xr k+ + + + LL (3.1)3.1.1 Array factorFor an array of M elements, the array factor is given byzyxr mm elementthincidentwave-k31+Mmjm m me I AF1) () , ( (3.2)where mI is the magnitude and m is the phase of the weighting of the mth element.The normalized array factor is given by{ f AFAF( , ) ( , )max ( , ) (3.3)This would be the same as the array pattern if the array consisted of ideal isotropicelements.3.1.2 Array patternIf each element has a pattern ) , ( mg , which may be different for each element,the normalized array pattern is given by( )( )''++Mmjm mMmjm mm mm me g Ie g IF11) , ( max) , () , ( (3.4)In (3.4), the element patterns must be represented such that the pattern maxima are equalto the element gains relative to a common reference.3.2 Phase and Time ScanningBeam forming and beam scanning are generally accomplished by phasing the feedto each element of an array so that signals received or transmitted from all elements willbe in phase in a particular direction. This is the direction of the beam maximum. Beamforming and beam scanning techniques are typically used with linear, circular, or planararrays but some approaches are applicable to any array geometry. We will considertechniques for forming fixed beams and for scanning directional beams as well asadaptive techniques that can be used to reject interfering signals.Array beams can be formed or scanned using either phase shift or time delaysystems. Each has distinct advantages and disadvantages. While both approaches can beused for other geometries, the following discussion refers to equally spaced linear arrays32such as those shown in Fig. 3-2. In the case of phase scanning the interelement phaseshift is varied to scan the beam. For time scanning the interelement delay t is varied. (a) (b)Figure 3-2. (a) a phase scanned linear array (b) a time-scanned linear array3.2.1 Phase scanningBeam forming by phase shifting can be accomplished using ferrite phase shiftersat RF or IF. Phase shifting can also be done in digital signal processing at baseband. Foran M-element equally spaced linear array that uses variable amplitude element excitationsand phase scanning the array factor is given by [3.1]+10) cos2() ( Mmdjmme A AF (3.5)where the array lies on the x-axis with the first element at the origin. The interelementphase shift is00cos2 d (3.6)...1 2 MA1 A2ejAMej(M-1)to receiver...1 2 MA1A2AMto receivert (M-1)t33and 0 is the wavelength at the design frequency and 0 is the desired beam direction. Ata wavelength of 0 the phase shift corresponds to a time delay that will steer the beamto 0.In narrow band operation, phase scanning is equivalent to time scanning, butphase scanned arrays are not suitable for broad band operation. The electrical spacing(d/) between array elements increases with frequency. At different frequencies, thesame interelement phase shift corresponds to different time delays and therefore differentangles of wave propagation, so using the same phase shifts across the band causes thebeam direction to vary with frequency. This effect is shown in Fig 3-3. This beamsquinting becomes a problem as frequency is increased, even before grating lobes start toform.f0 1.5f02f0 0 50 100 150 200012345678phi, degrees|AF(phi)|Figure 3-3. Array factor of 8-element phase-scanned linear array computed for threefrequencies (f0, 1.5f0, and 2f0), with d=0.37 at f0, designed to steer the beam to o=45 atf0.343.2.2 Time scanningSystems using time delays are preferred for broadband operation because thedirection of the main beam does not change with frequency. The array factor of a time-scanned equally spaced linear array is given by +10) cos2() ( Mmtdjmme A AF (3.7)where the interelement time delay is given by0coscdt (3.8)Time delays are introduced by switching in transmission lines of varying lengths.The transmission lines occupy more space than phase shifters. As with phase shifting,time delays can be introduced at RF or IF and are varied in discrete increments. Timescanning works well over a broad bandwidth, but the bandwidth of a time scanning arrayis limited by the bandwidth and spacing of the elements. As the frequency of operation isincreased, the electrical spacing between the elements increases. The beams will besomewhat narrower at higher frequencies, and as the frequency is increased further,grating lobes appear. These effects are shown in Fig. 3-4.35f0 1.5f02f0 0 50 100 150 200012345678phi, degrees|AF(phi)|Figure 3-4 Array factor of 8-element time-scanned linear array computed for threefrequencies (f0, 1.5f0, and 2f0), with d=0.37 at f0, designed to steer the beam to o=45 atf0.3.3 Fixed Beam Forming TechniquesSome array applications require several fixed beams that cover an angular sector.Several beam forming techniques exist that provide these fixed beams. Three examplesare given here.3.3.1 Butler matrix The Butler matrix [3.2] is a beam forming network that uses acombination of 90 hybrids and phase shifters. An 8x8 Butler matrix is shown in Fig 3-5.The Butler matrix performs a spatial fast Fourier transform and provides 2n orthogonalbeams. These beams are linearly independent combinations of the array element patterns.36Figure 3-5. An 8x8 Butler matrix feeding an 8-element array. Circles are 90 hybridsand numbers are phase shifts in units of /8When used with a linear array the Butler matrix produces beams that overlap atabout 3.9 dB below the beam maxima. A Butler matrix-fed array can cover a sector of upto 360 depending on element patterns and spacing. Each beam can be used by adedicated transmitter and/or receiver, or a single transmitter and/or receiver can be used,and the appropriate beam can be selected using an RF switch. A Butler matrix can alsobe used to steer the beam of a circular array by exciting the Butler matrix beam ports withamplitude and phase weighted inputs followed by a variable uniform phase taper.3.3.2 Blass MatrixThe Blass matrix [3.3] uses transmission lines and directional couplers to formbeams by means of time delays and thus is suitable for broadband operation. Figure 3-6shows an example for a 3-element array, but a Blass matrix can be designed for use withany number of elements. Port 2 provides equal delays to all elements, resulting in abroadside beam. The other two ports provide progressive time delays between elementsand produce beams that are off broadside. The Blass matrix is lossy because of theresistive terminations. In one recent applicati

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