Appendix 6: The Effect of Surface Size and Illumination on the Polar Pattern

Chapter Contents

• A6.1 An Array of Point Sources
  • A6.1.1 The Visible Region
  • A6.1.2 The Effect of Sampling
  • A6.1.3 The Effect of a Progressive Phase Shift
• A6.2 Application to Diffuser Design
• A6.3 Application to Array Loudspeakers
  • A6.3.1 Acoustic Spatial Filtering
• A6.4 Application to Constant-Directivity Horns
• A6.5 The Effect of Mouth, or Array, Size, on Beamwidth
  • A6.5.1 The Minimum Beamwidth Frequency as a Function of Size
• References

In order to understand how the properties of sequences or the size of a loudspeaker affect polar performance, we must first look at some theory behind array polar patterns.

A6.1 An Array of Point Sources

Consider an evenly spaced, linear array of perfect point source radiators, as shown in Figure A6.1, with complex amplitudes A₀ … A_{N− 1}. This corresponds to the radiated sound from an array of speakers, and the amplitudes represent the illumination of the surface. If we are an infinite, or at least very large, distance away, we can make the following approximations:

1. 1. The wavefronts are planar, and therefore all the radiators will have the same angle of incidence (θ) to the far-off point.
2. 2. The differences in path lengths are so small that only the initial phase difference, due to θ, affects the received amplitude.

These approximations are known as the far-field assumptions and, in theory, will be satisfied, providing one is a reasonable distance from the array.

Assuming, for the moment, that the far-field assumptions are satisfied, we can say the following about our linear array of ideal point sources.

1. 3. The far-field response will be given by the sum of the individual point sources, with an additional phase delay/advance due to θ, which is the angle from the normal, as shown in Figure A6.1.
2. 4. The phase delay due to θ will be given by:

where n is proportional to the point source number, as shown in Figure A6.1.

For the example shown in Figure A6.1, this results in an equation for the far-field polar response at a frequency whose wavenumber is k, which is:

where the wave number k is given by:

This can be rewritten as:

If we make Ω = kdsinθ sin, Equation A6.4 can be rewritten as:

Equation A6.5 is in fact a discrete Fourier transform (DFT) in which Ω = kdsinθ. This means that the far-field polar pattern of an array of point sources is related to the illumination of the surface by a Fourier transform relationship. Therefore, all the theorems that apply to the Fourier transform apply to an array of point sources. In particular, these are:

1. 5. Linearity and Superposition: Weighted addition in the spatial domain is equivalent to addition in the transformed polar pattern domain.
2. 6. The Convolution Theorem: This theorem states that convolution in the spatial domain is equivalent to multiplication in the transformed polar pattern domain. The converse is also true.
3. 7. The Wiener-Khinchin Theorem: The Wiener-Khinchin theorem states that the squared Fourier transform magnitude of a sequence, in the spatial domain, that is its polar pattern, is equal to the Fourier transform of its autocovariance (or autocorrelation function).
4. 8. The Shift Theorem: A shift in the spatial domain leads to a linear (progressive) phase change in the transformed polar pattern domain and vice versa.

As we shall see later, these have some important consequences.

A6.1.1 The Visible Region

Although, in theory, the variable in equation A6.5 can range from $-\infty$ to $+\infty$, in reality it cannot. In fact, because sinθ cannot exceed ±1, there is only a limited range of θ that makes any physical sense. This region is known as the “visible region,” and, because Ω = kd sinθ, the visible region corresponds to −kd ≤ Ω ≤ +kd. The visible region corresponds to the angles between ±90° of the normal direction.

This is shown in Figure A6.2 for a 10-element array of points, with the elements spaced 4.3 cm apart, at 1khz (kd = 0.79). If we double the frequency to 2kHz, kd doubles (kd = 1.58), and the visible region also doubles, as shown in Figure A6.3.

As the visible region corresponds to the angles between ±90° of the normal direction, the effect of doubling the visible region also implies a narrowing of the main lobe if its shape does not change as the visible region increases, as in our examples.

A6.1.2 The Effect of Sampling

When the frequency gets high enough that the spacing between the point sources becomes greater than half a wavelength, the array becomes undersampled. Under these conditions, one gets spatial aliasing, which results in multiple main lobes. Figures A6.4, A6.5 and A6.6 illustrate this. Figure A6.4 shows the 1kHz example with the scale expanded. The first thing to note is that the visible region still covers the same region as that of Figure A6.2. The second thing to note is that the expanded scale reveals the multiple peaks that indicate the possibility of spatial aliasing.

Figure A6.5 shows the visible region when the frequency equals 7kHz (kd = 5.5). Here we can see that, although the aliased main lobe is not visible, there is an increase in side-lobe levels due to the spatial aliasing. Figure A6.6 shows the visible region when the frequency equals 10kHz (kd = 7.85). Here we can see that the aliased main lobe is now visible, and there is a large increase in the side-lobe levels due to the spatial aliasing.

A6.1.3 The Effect of a Progressive Phase Shift

From the shift theorem, we know that a shift in the spatial domain leads to a linear (progressive) phase change in the Fourier domain and vice versa. Thus, a progressive phase shift in the spatial domain would result in a linear shift of the function in Ω space. This would result in the main lobe moving off the central axis, thus steering the main beam to an off-centre angle. However, the visible region would remain in the same place.

A6.2 Application to Diffuser Design

For a diffuser we wish to have a P(θ_k) that is uniform with respect to angle. This corresponds to a pattern of coefficients. This corresponds to a diffuser structure that has a constant Fourier transform magnitude over the visible region. The effect of an obliquely incident wavefront is to add an additional progressive phase shift across the diffuser’s reradiated sound. This causes the visible region to be shifted in angle. Therefore, in addition, we would like the Fourier transform magnitude to be uniform outside the visible region as well to cover oblique incidence. Therefore, we need to find diffusion structures that have uniform-magnitude Fourier transforms, such as Schroeder (Schroeder, 1975) diffusers. We can use the Fourier transform relationship between the far-field polar pattern and the pattern of amplitudes at the diffuser’s surface to help us choose appropriate sequences for diffusion structures.

The Wiener-Khinchin theorem states that the squared Fourier transform magnitude of a sequence is equal to the Fourier transform of its autocovariance (or autocorrelation function). Therefore, sequences whose autocovariance either is a delta function or close to a delta function will form good diffusers, because the Fourier transform magnitude of a delta function is uniform.

The convolution theorem states that convolution in the spatial domain is equivalent to multiplication in the Fourier domain and that the converse is true. This means that multiplication, or modulation, in the spatial domain corresponds to convolution in the polar pattern domain. This allows us to use a variety of modulation techniques on short diffusers to achieve good diffuser (Angus, 1998, 2000; Cox, 2004), without the lobe narrowing that results from a repeated set of short diffusers. In fact, the Fourier relationship allows one to develop new diffusion structure, such as “binary amplitude” (Angus, 2000; Cox, 2004) and “ternary” (Cox, 2005) diffusers.

A6.3 Application to Array Loudspeakers

An early example of an array loudspeaker was the column loudspeaker. In this arrangement, a number of small loudspeakers were arranged in a closely spaced line. Because of the extended length of the source in one plane, directivity control was achieved in that plane. However, the beam pattern would get progressively more directive with frequency, as predicted by the Fourier transform. Techniques were developed to reduce this behavior, usually by applying the necessary frequency-dependent weighting, tapering or windowing using simple electrical circuits, a direct application of the convolution theorem. Methods of steering these line speakers were also developed either by using simple analogue delay techniques or by using the inherent phase shifts in the filters used to taper the array. Again, this is a direct application of the shift theorems of the Fourier transform.

They could also exhibit unwanted side lobes at higher frequencies due to aliasing, which reduced their utility. That is, above some frequency, the spacing between the drivers is greater than half the wavelength of the sound being produced. This results in spatial aliasing and results in a loss of control of the beam pattern.

Avoiding spatial aliasing requires a huge number of small loudspeakers, resulting in a prohibitive cost for the array. For example, ideally we want pattern control over the entire audio frequency range. However, even if we make the speaker spacing 4.3 cm, which is unfeasibly small because we would need a large number to achieve low-frequency pattern control, we still have significant aliasing at 10kHz.

A6.3.1 Acoustic Spatial Filtering

One way of reducing the effect of spatial aliasing is to use directive loudspeakers instead of point sources as the array elements. If one uses directive sources, their polar patterns will act as a form of spatial filter. That is, the off-axis side lobes will be reduced by the off-axis reduction in sound level that a directive source affords. Figure A6.7 shows an array response at 10kHz (kd = 7.85) with the response of a continuous line source, of length equal to the element spacing, superimposed upon it. Of particular note is that the zeros of the continuous line source fall on the aliased main lobes from the point source array. Because the far-field polar pattern of an array of point sources is related to the applied signals by a Fourier transform relationship, all the theorems that apply to the discrete Fourier transform apply to the array loudspeaker. This means that the theorem that convolution in one domain is equal to multiplication in the other domain applies to this situation. Replacing each of the point sources with a continuous line source is equivalent to convolving it with the point array. Therefore, the effect of replacing the point source with the continuous sources is to multiply their far-field patterns together.

This pattern multiplication is well known, and the effect is for our example is shown in Figure A6.8. One can see that the aliased main lobes have been eliminated. In fact, the response has become equivalent to a continuous line source of the same extent as the array. Clearly, using directional sources, such as constant-directivity horns, can also be used to achieve similar effects. It is this that results in the success of large arrays based on them, providing the horns have directivity control before spatial aliasing occurs. Once the directivity of the individual elements is considered, the need for curved arrays also becomes apparent, as the spatial filtering effect of the sources must also be factored in.

A6.4 Application to Constant-Directivity Horns

Keele (Keele, 2000, 2002, 2003a,b) extended the work of Van Buren et al. (Rogers and Van Buren, 1978; Van Buren et al., 1983) to develop the constant beamwidth theory (CBT) arrays. In his papers, the transducer is a spherical cap of arbitrary half-angle with Legendre function shading. It provides a constant beam pattern and directivity with extremely low side lobes for all frequencies above a certain cutoff frequency.

To maintain constant beamwidth behavior, CBT circular-arc loudspeaker line arrays require that the individual transducer drive levels be set according to a continuous Legendre shading function. This shading gradually tapers the drive levels from maximum at the center of the array to zero at the outside edges of the array. Keele developed approximations to the Legendre shading that both discretize the levels and truncate the extent of the shading so that practical CBT arrays can be implemented. He determined by simulation that a 3-dB stepped approximation to the shading maintained out to −12 dB did not significantly alter the excellent pattern control of a CBT line array.

Conventional CBT arrays require a driver configuration that conforms to either a spherical-cap curved surface or a circular arc. Keele also showed how CBT arrays can also be implemented in flat-panel or straight-line array configurations using signal delays and Legendre function shading of the driver amplitudes. CBT arrays do not require any signal processing except for simple frequency-independent shifts in loudspeaker level. This is in contrast with conventional constant-beamwidth flat-panel and straight-line designs, which require strongly frequency-dependent signal processing.

These results are important because they also provide a link between array loudspeakers and constant-directivity horns. Figure A6.9, reproduced from Keele, 2002, shows how the delays for a planar CBT array effectively move the driver from its position on a flat surface to a point on a circular arc. That is, it provides a delay that makes the wavefront at the planar array that is spherical. According to the Fourier theory we have described earlier, the Fourier transform of the combination of phase shifts, due to a spherical wavefront and Legendre weighting, results in a frequency independent constant beamwidth above a certain cutoff frequency. Furthermore, the cutoff frequency is a function of both the required directivity angle and the length of the array.

If we compare this to a constant-directivity horn, we observe many similarities.

1. 1. First, the conical flare of such horns results in a spherical wavefront at the horn’s mouth.
2. 2. Second, the projection of the spherical wavefront’s intensity onto the planar front of the horn results in an intensity that has a cosine rolloff from the center of the horn. This approximates a Legendre weighting at the center of the planar front of the horn, as shown in Figure A6.10. Unfortunately, the weighting is not enough at the edges of the horn aperture. However, practical constant-directivity horns have an additional, more extreme flare at the mouth. This would have the effect of more rapidly reducing the amplitude at the edge of the horn’s mouth, and so it would more closely approach Legendre weighting.

Thus, constant-directivity horns can be seen as a simple approximation to a CBT array!

A6.5 The Effect of Mouth, or Array, Size, on Beamwidth

How does the size of the horn mouth or the array size affect the lowest frequency for a given beamwidth?

The normalized polar pattern for a linear array of N equally driven point sources is given by:

and its −6 dB angle occurs when:

This can be rewritten as:

but

so

Which, if we combine all the constants, becomes:

Equation A6.10 shows that the −6 dB angle depends on the ratio of the wavelength and the number of sources times the distance between them. The number of sources times the spacing between them is simply the length of the array L. So equation A6.10 becomes:

where: L = The size of the source in meters

or

as a function of frequency.

Because of the convolution theorem discussed earlier, this equation applies to continuous sources, for example, horn mouths, as well as to speaker arrays. This means that the directivity of a uniformly driven speaker array or a uniformly illuminated horn mouth is a function of the number of wavelengths that fit into the size L.

For example, if exactly one wavelength fits across the speaker length, the −6 dB angle will be ±40° a beamwidth of 80°, and for two wavelengths the beamwidth will be 37° and so on, the beamwidth approximately halving for every doubling of frequency. On the other hand if the wavelength to length ratio is $\frac{\lambda }{L}\ge \frac{1}{0.64}\ge 1.57$ then the speaker has no directivity at all, it becomes omnidirectional!

A6.5.1 The Minimum Beamwidth Frequency as a Function of Size

It is sometimes useful to be able to calculate the minimum frequency a speaker of a given size can achieve at a particular coverage angle. We can do this by rearranging equation A6.7:

Which, if we combine all the constants, becomes

Equation A6.13 gives a simple relationship between the size of a speaker array or a constant-directivity horn mouth and the minimum frequency that sustains the desired beamwidth.

This appendix has presented the basic Fourier relationship between the far-field polar response and the near-field illumination of the aperture. It has demonstrated its utility in a variety of electroacoustic applications. In particular, it is possible to derive some useful relationships between the size of the speaker, with respect to wavelength, and its directivity performance versus frequency—in particular that for directivity, size really does matter!