Array design is critical to successful phased array measurements. Array mea-surements start with and are ultimately constrained by the placement of array sensors. When aero acoustic phased array measurement capabilities were de-veloped in the mid 1990s at the Boeing Company and NASA Ames it became evident that conventional planar array designs (Johnson and Dudgeon 1993) were inadequate for aero acoustic testing applications. The broad frequency range requirements of aero acoustic applications inspired application of novel array designs based on sampling of logarithmic spirals (Underbrink 1995, Underbrink and Dougherty 1996). These designs have proven successful in a number of aerodynamic test facilities.
Background
An array is a sampled aperture made up of discrete receivers; in the case of an aero acoustic array, these receivers are microphones or dynamic pressure sen-sors. In an aero acoustic phased array, phase delays are applied to the signals received by the individual acoustic sensing elements to steer or focus the ar-ray. Steering the array means selecting phase delays to "look" in a particular direction and implies plane wave (far-field) propagation. Focusing the array means selecting phase delays to look at a particular point in space and implies spherical wave (near-field) propagation. Most aero acoustic applications re-quire the focused array approach. The array is typically in close proximity to the source due to either the constraints of the test facility (wind tunnel walls relatively close to the model), or the size of the array relative to its distance from the source.
Certain array performance characteristics are of primary concern when de-signing arrays. These characteristics include array resolution, spatial aliasing, and array sidelobe suppression. Array resolution describes the ability of the array to resolve spatially where a source of noise is located. Spatial aliasing is a side effect of poorly designed arrays manifested by apparent (i.e. false) noise sources at locations other than that from which the noise is actually being gen-erated. Array sidelobe suppression addresses the ability of the array to sup-press energy from sources coming from directions other than where the array is being focused (or steered).
In a simplified view of the array design process, there are two primary steps:
1. Come up with candidate array designs; and
2. Evaluate the candidate designs to determine the best one.
The process is both iterative and ad hoc. Step 1 can be an exercise in creativity and ingenuity. However, some guidance can be helpful in the process. Step 2 is procedural and requires application of simple array processing algorithms.
The following sections present tools for array evaluation and then explore the design process in detail.
Beamforming
Array processing algorithms that focus or steer an array are commonly re-ferred to as beamforming algorithms. The directional response of an array is called its directivity pattern. A directivity pattern is determined by both the array design and the beamforming algorithm used. The term, beamforming, has its origins in arrays that transmit energy rather than receive it. Neverthe-less, whether the directivity pattern of an array is derived from the transmis-sion or reception of energy (or both), the pattern has characteristics that look like beams. There is always a main beam, also called the mainlobe of the array pattern. There are also secondary lobes called sidelobes. In contrast to the mainlobe of an array directivity pattern, the sidelobes are smaller and do not correspond to the direction in which the array is being steered. In other words, sidelobes indicate an array's ability to filter out signals propagating from other directions.
Characteristics of the mainlobe and sidelobes of an array directivity pat-tern are the focus (no pun intended) of investigation when evaluating the goodness of an array design. The width of the mainbeam, called the beam-width, determines the array resolution; its ability to localize sources and to re-solve closely spaced sources. The strength (or height) of the array directivity pattern sidelobes relative to the mainlobe establishes the array's effective dy-namic range. This means that sources that are different in amplitude by more than the difference between the strength of the mainbeam and the sidelobes will not be discernable (without using advanced beamforming techniques, that is). To investigate the characteristics of an array directivity pattern it must first be produced. This can be done using the simple beamforming algorithm developed below (Johnson and Dudgeon 1993).
Begin with an array of M sensors whose locations are given by -->
xm ' m
=
1,2, ... , M , (3.1)where
x
= (x, y, z) is a spatial position vector with arbitrary origin. Now as-sume a monopole point source with signals (t) located atxs'
The signal at some other arbitrary pointx
is given byf(--> )
=
s(t-Ix - xsl/c)
x, t 1--> --> I X-Xs ' (3.2)
where c is the speed of sound. This is a form of the solution to the wave equa-tion in spherical coordinates. The received signal at the m th sensor is
() s(t-Tmlc)
Ym t = ,
Tm (3.3)
where Tm
=
IXm - xsi is the distance between the source and the m th sensor. To cophase the signal received at all sensors, the signal is delayed by an amount equal to the difference between the time it takes the source signal to propagate to the origin and the time it takes to propagate to the sensor. For the m th sen-sor, the delayed signal is( Ts - T m ) 1 ( T m Ts - T m) 1 ( Ts )
Ym t- =-s t--- =-s t-- ,
C Tm C C Tm C (3.4)
where Ts is the distance between the source and the origin. The sum of all the delayed signals yields the beamformer response
z(t)=s(t-
2
C)[i
m= 1~].
Tm (3.5)Multiplying by TslTs gives
1 ( Ts)
[~
Ts]z(t)=-s t-- £.. - •
Ts C m= 1 Tm (3.6)
The beamformer output for the source is the signal received at the origin weighted by the part of the above equation in square brackets. If the origin is at the phase center of the array defined by
--> M ~-->
Xo = £.. X m , (3.7)
m=l
and if the source is a long distance away from the array such that T m '=' Ts, then the beamformer output can be seen to be the signal received at the origin weighted by the number of sensors.
When focusing at some arbitrary point in space, different from where the source is, the signals that go into the beamformer are not cophased and the output is less than that from focusing at the point where the source is located. Let Tp represent the distance between the origin and the arbitrary focus point xp, and let T:n represent the distance between the m th sensor and the arbitrary focus point. Then the delayed signal at the m th sensor is
( Tp - T:n) _ 1 ( Tp - T:n T m)
Ym t- --s t- - - .
C Tm C C (3.8)
The sum of all the delayed signals now yields
(3.9)
This is the beamformer output for any focal point xp and actual source loca-tion
xS'
The signal may be written in terms of its Fourier Transform5 (t- rp - (::n - rm)) = _[ S(m) exp {jm[t- rp - (::n - rm)]} dm.
(3.10) Adding and subtracting rsf c in the exponent yields
(3.11)
=J { ( rs)} { [ (rs - rp) - (r m - r:n) ]}
=_=
S(m) exp jm t-~ exp jm c dm.Now the beamformer can be written in terms of frequency as z(t) =
~
~ _= ] S(m) exp {jm (t -!i)}
c(3.12)
~
rs {. [(rs-rp)-(rm-r:n)]}dx£..- exp }m m.
m-l rm c
The Fourier Transform of the propagating signal at the origin is ( ) _ ( ) exp(-jmrsfc)
So m -S m . (3.13)
rs Now define W(m, xp,xs), such that
z(t) = =
J
So(m) W(m, xp,xs) exp (jmt) dm, (3.14)which has the form of an inverse Fourier Transform. The spectrum of the ar-ray output may then be written as
(3.15)
al !'..
N
-
D-u Q)
0
-5 l::! ttl -10
§ o z - 15
20
Main Lobe
Sidelobes < -15 dB
20
y (in.) -20 - 20 x (in.)
Fig. 3.1. Example near-field array pattern shows main lobe and sidelobes for a single fre-quency on a plane parallel to the array at a selected distance from the array. Sidelobes below a certain level (-15 dB in this case) are cut off to aid in the display and interpreta-tion of the array pattern. The plot is normalized to the peak level of the main lobe of the ar-ray pattern
where the effect of the beamformer on the signal spectrum is W( W,Xp,X-> -» -s -
~
£..,2
exp{ . [(
JW rs - rp) - (r m - r:n) }] •m= 1 rm C (3.16)
Equation 3.16 will be used to produce many of the proceeding array patterns in this chapter. It may be used in practice for evaluating candidate array designs for near field beamforming applications. Taking the magnitude squared of the array pattern, normalizing the result, and expressing in decibels (3.17) will further aid in the evaluation of the array patterns. The array pattern peak (of the main lobe) will now be displayed as 0 dB and the sidelobes will be displayed as some number of dB down from the main lobe (for example, 10 dB down will be expressed as -10 dB). An example array pattern is shown in Figure 3.1.
For simplicity in evaluating and comparing general array design strategies, it is convenient to use plane wave beamforming (Underbrink and Dougherty 1996). Plane wave beamforming assumes the sources are in the far field and is thus only concerned with direction of propagation and angular resolution 2. A
2 When designing arrays for particular near-field applications the appropriate parameters to consider are point-in-space location of the source relative to the array and dimensional resolution of the array at that location. Hence, the near-field array pattern (Equation 3.16) should be used to evaluate candidate array designs for a specific application.
co
:!::.'" a.
"0 Q)
0 -5
~ «I - 10 E (5
z -15
k x Ik
Fig. 3.2. Example far-field array pattern shows main lobe and sidelobes for a single fre-quency. k)k and kylk specify look direction from the array. Sidelobes below a certain level (-15 dB in this case) are cut off to aid in the display and interpretation of the array pattern.
The plot is normalized to the peak level of the main lobe of the array pattern
method for characterizing a planar design is to postulate a normal-incidence, unit-pressure wave. The beamforming amplitude then reduces to
1
~
ejlc .xm I.
m=! M (3.18)
~ ~
where k is the wave number of the incident plane wave. The direction of k is the direction of propagation; the length is the angular frequency divided by the speed of sound. For each frequency, Equation 3.18 is evaluated o...rer a range of directions given by k)k and kylk, the direction components of k cor-responding to the in-plane microphone coordinates, Xm and Y m' ofxm • As with the near-field array patterns, taking the magnitude squared of the far-field array patterns, normalizing the result, and expressing in decibels will aid in array pattern evaluation. An example far-field array pattern is shown in Figure 3.2.
Evaluating Array Performance
The above beamforming equations (Equations 3.16 and 3.18) provide the ba-sis for evaluating array performance. Equation 3.16 is used for a point source at
Xs
where the array is specified by the sensor positionsx m.
The array evalua-tion region is specified by the set of pointsx
P' and OJ encompasses the fre-quency range of interest. EquatiQ!13.18 is used for a normal-incidence far-field source with frequency given by k and array sensor positions given byx
m • Thedirections of propagation for array evaluation are given by a range of kxfk and kylk. The evaluation process for a proposed array design consists of:
1. Generate the array patterns using the beamforming equation for the desired evaluation region(s) and across the desired frequency range.
2. Interrogate the array patterns to determine array resolution and array dynamic range.
In the first step of the evaluation process, the evaluation region may need to tend beyond the immediate region surrounding the device under test. The ex-tent of the required evaluation region depends on the test facility, the device un-der test, and any significant interactions that the device unun-der test might have due to its presence in the test environment. The primary point of concern here is whether there are potential extraneous noise sources outside the immediate region of the device under test that the array must be capable of suppressing. In a closed-circuit wind tunnel, the array performance must be evaluated over 360 degrees because of the reverberant environment and multiple facility-opera-tion-induced noise sources such as fans, motors, and flow-induced noise.
Array Resolution
Array resolution specifies how well an array is able to resolve direction of propagation. It is typically specified in terms of a 3 dB down point, that is, the point at which the spatial filter formed by an array processing algorithm reaches 3 dB down from the peak. The width of the spatial filter at the 3 dB down point, called the beamwidth, determines the array resolution. In general terms, the half-width of the mainlobe of an array pattern is the angle between the direction of maximum response of the mainlobe and the direction in which the response is reduced by
1/{2
(corresponding to 3 dB down). For near-field source mapping applications, it is convenient to express array reso-lution in terms of linear units (e. g., centimeters) at a reference distance from the array corresponding to that of the device under test. However, since for a fixed size aperture array resolution varies linearly with wavelength, the reso-lution may be expressed in terms of wavelength. The specification for array resolution may thus be stated asBWd=kA, (3.19)
where BWd is the beamwidth at a distance d from the array, k is a constant, and A is the wavelength. The value of k will depend on the size of the array aper-ture, the placement of sensors within the array aperaper-ture, and d.
Also of consequence when determining array resolution is the incident angle of the propagating waves. For maximum resolution, an array should be positioned such that the incident waves propagate along the array axis (i.e., perpendicular to the array). There are however some situations where it is
ArrayC i Array B Array A
Fig. 3.3. Multiple arrays used to assess source directivity of a flap-edge noise source.
Dashed lines show emission angles (Oe) for each array. Solid lines show direction of energy propagation (Or) for each array
either not feasible or not desirable to position an array such that the device un-der test is broadside to the array. Some examples are:
1. The test facility design constrains where an array panel may be installed.
2. The desire to obtain source directivity requires multiple arrays mounted along the wall of a closed-circuit wind tunnel test section corresponding to various source emission angles (and hence different angles of incidence for each array) as shown in Figure 3.3.
3. A device under test is large relative to the array such that different sources emanating from the device under test have different incident angles with re-spect to the array. Figure 3.4 illustrates one such example where the flap edge and slat gap are at a relatively large separation given the proximity of the array.
As Figure 3.5 illustrates, the far-field effective aperture of the array is related to the cosine of the direction from which the incident waves are propagating.
The far-field effective aperture dimension Dfis given by
(3.20) where D is the actual aperture dimension and cfJ is the angle relative to the axis of the array from which the incident waves are propagating. Hence, the maxi-mum resolution is obtained at broadside to the array. The near-field effective aperture dimension is slightly smaller or larger than the far-field effective aperture dimension depending upon the focus point. Referring again to
i
\'Array B
-<E-Flow
Array A
Fig. 3.4. Example of substantially different emission angles for two sources on the same de-vice under test. The slat source has an emission angle of
eel.
The flap-edge source has an emission angle ofee2
z /
I Source I / (xs,Zs)
I I
( I
I ~ /
//~ ''II
~~;/
Df~1/ I
/ / /
! I
I /
x
Fig. 3.5. Effective aperture dimension Df , for a far-field source is shown by the dashed line representing the wave front of a plane wave cutting through center of the array aperture (shaded rectangle). Effective aperture dimension Dn , for a near-field source at (x" z,) is shown by a solid curve representing the spherical wave front of a point source cutting through the center of the array aperture
70
Fig. 3.6. Three dB down isosurface for a point source in the near field of an array. The source is on axis (x, y) = (0,0) and 60 in. (152.4 em) away from a 58-in. (147.3 em) aperture array. Note that the resolution in depth (perpendicular to the array) is about five times less than the lateral resolution
Figure 3.5, if the source is located at the point (xs, zs) in a coordinate system with the origin at the center of the array, then the near-field effective aperture dimension Dn is given by3
[
A final array resolution consideration for near-field beamforming is that of depth resolution. Since a near-field beamformer focuses on a point in space, it has the ability to resolve source locations in three dimensions. However, the ability to resolve a source position in depth (i. e., perpendicular to the array), is far less than the lateral resolution capability. Figure 3.6 shows a three dB down isosurface for a point source located on the axis of a 58-in. (147.3 cm) diameter array at a distance of 60 in. (152.4 cm). The major axis of the ellip-soid is about five times that of the minor axis.
Spatal Aliasing
Spatial aliasing is an undesirable side effect from spatially undersampling the aperture of an array. Spatial aliasing has its analog in the time domain. Just as temporally undersampling a time domain signal results in the inability to
dis-3 The reader may wish to verify lim Dn = Df .
r->=
til ~
Fig. 3.7. Far-field beam pattern at 3 kHz for a six-by-six square lattice array. Sensor spac-ing is 8 in. (20.3 cm). Since the aperture is undersampled for the beamformed frequency, spurious lobes appear in the array pattern making it impossible to determine the location of the true source.
tinguish between multiple frequency components, spatially undersampling an array aperture results in an inability to distinguish between multiple direc-tions of propagation 4. In the time domain, frequency components that would alias into the adequately sampled region may be filtered using either analog or digital filtering techniques. However, in the spatial domain, there is no analo-gous process that would filter directional components that alias into the ade-quately sampled directional region.
To see the effect of spatial aliasing consider the square lattice array of Fig-ure 3.7 and the associated array directivity pattern. Clearly, there are several dominant lobes. In time domain signal processing, sampling at a rate at least twice the highest frequency prevents aliasing. This sampling rate, called the Nyquist rate (Oppenheim and Schafer 1975), may be applied in spatial domain signal processing by sampling at an interval not to exceed one-half wave-length. Figure 3.8 demonstrates the effect of an adequately sampled array for the beamformed frequency. Clearly, there are no ambiguous lobes when the Nyquist sampling frequency criterion is met. Note however, the potential for substantial sensor count requirements to meet the Nyquist criterion. Even a relatively small 20-by-20 cm square planar array would require more than 8700 sensors on a 0.214 cm grid spacing to prevent aliasing at 80 kHz. It is not very practical to deploy an array of such magnitude.
4 The discussion here is focused on spatial aliasing. It is assumed that the temporal signal has been appropriately filtered and sampled to avoid temporal aliasing.
co
Fig. 3.8. Far-field array pattern at 3 kHz for a ten-by-ten square lattice array. Sensor spac-ing is 5 in. (12.7 em). Since the aperture is adequately sampled for the beamformed fre-quency, no spurious lobes exist and the source location is apparent
Array Design Strategy
A phased array installation at the NASA Ames 7- by 10-Foot Wind Tunnel where arrays were mounted in both the wall and ground plane is shown in
A phased array installation at the NASA Ames 7- by 10-Foot Wind Tunnel where arrays were mounted in both the wall and ground plane is shown in