#### 2.1. Background

The normalized array factor of an

N element, TMLA when receiving a signal from direction

θ can be written as [

16]

where

k is the wavenumber (rad/m),

n is the element number,

d is the spacing between elements in metres, and

${U}_{n}\left(t\right)$ is a time-switching function that turns each element either off or on in time defined by,

where

${\tau}_{n,on}$ and

${\tau}_{n,off}$ are the switch on and off times respectively for the

nth element in the array.

If each element is switched on then off uniformly in sequence,

${U}_{n}\left(t\right)$ becomes a periodic function and can be formulated as an infinite sum of

m Fourier components each with coefficient

${c}_{m,n}$ [

17]

where the coefficients can be calculated as [

11]

The signal power is distributed among the harmonics of the switching pattern frequency

${f}_{s}=1/{T}_{s}$, centred on a main frequency

${f}_{0}$. Using the above equations, the array factor for a particular harmonic

m can be calculated as,

and if

R is defined as the ratio of element spacing to signal wavelength

λ,

then the normalized array factor for each harmonic can be written as

Figure 2 shows the normalized response of the dominant harmonics (i.e.,

${f}_{c}\pm 2{f}_{m}$,

${f}_{c}\pm {f}_{m}$,

${f}_{c}$) in a five-element array with each harmonic having a maximum amplitude at a different angle (i.e.,

$\pm 53.{1}^{\circ}$,

$\pm 23.{6}^{\circ}$,

${0}^{\circ}$). From Equation (

7), it can be shown that the direction in which the maximum response occurs for each harmonic can be calculated as:

It can also be observed that in the case of a uniformly switched array (i.e., (${\tau}_{n,off}-{\tau}_{n,on})\times N={T}_{s}$, and each element is switched on in sequence), the point at which each harmonic’s maximum amplitude occurs also marks the point where all other harmonics are at a minimum. The signal power associated with each frequency can be easily identified using a Discrete Fourier Transform (DFT). In practical terms this means that by analytical study of the amplitudes of each harmonic’s DFT bin, a value for a given received signal’s DoA can be derived.

Array factors associated with each harmonic depend on the value of

R as defined in Equation (

6). As

R becomes smaller, the main beam angles of each harmonic as calculated in Equation (

8) will be more widely spaced. If

R is greater than 0.5 (half-wavelength spacing), harmonics will obtain grating lobes due to the effect of spatial aliasing; in these cases, there are multiple angles in which harmonics become responsive, and the problem of direction-finding becomes more numerically complex. Throughout this paper,

R is chosen to be 0.5 to ensure that each of the harmonics has a single main beam within the range of

$\pm {90}^{\circ}$.

#### 2.2. Proposed Method of Determining the DoA

The approach taken in this paper is realised by finding the associated harmonic frequencies relating to both the signal frequency and the array switching frequency. For example, an array with five elements (

$N=5$) will distribute the power of an incoming signal into frequencies at

${f}_{c}-2{f}_{m}$,

${f}_{c}-{f}_{m}$,

${f}_{c}$,

${f}_{c}+{f}_{m}$ and

${f}_{c}+2{f}_{m}$. As shown in Equation (

8), each of these harmonics has a known main beam angle

${\theta}_{m}$ (i.e.,

$-53.{1}^{\circ}$,

$-23.{6}^{\circ}$,

${0}^{\circ}$,

$23.{6}^{\circ}$ and

$53.{1}^{\circ}$).

Consider a single frequency sinusoidal signal impinging on the array at an angle

θ measured relative to the broadside direction. This signal will cause one or more frequency bins to become filled (as demonstrated by

Figure 2) according to which harmonics are detected at the receiver. In the case where

θ matches exactly a value of

${\theta}_{m}$, only one frequency is expected to be present in the output. It can then be assumed that if the value of the largest DFT bin is dominant, while the adjacent bins are small, then the signal angle is close to the main beam angle of that harmonic. Conversely, if the amplitudes of two adjacent bins are similar and also the largest measured, then the signal is approximately half way between the two main beams. The DoA can be estimated by looking at the average angle created by two beams, as weighted by the harmonic power level the array receives; estimation of the DoA can be expressed as:

where

${X}_{\alpha}$ and

${X}_{\alpha +1}$ are the two largest DFT bins that are adjacent to each other, and

${\theta}_{\alpha}$ and

${\theta}_{\alpha +1}$ are the main beam angles associated with the corresponding harmonics.

The steps required to perform DoA estimation using the proposed method can be summarised as follows:

Collect a fixed number of samples from each microphone in turn (this can be repeated multiple times, however it is often adequate to do this just once). The harmonic frequency can be calculated as the reciprocal of the time taken to sample the set of all sensors.

Perform a DFT on the fundamental and expected harmonics of the switching frequency.

Select the DFT bins with the two largest amplitudes and identify their main beam angles.

Calculate a DoA using Equation (

9)

For example, if an unknown signal source produces the frequency spectrum shown in

Figure 3 when sampled with a five-element array, then the power levels of harmonics 1 and 2 are taken as

${X}_{\alpha}$ and

${X}_{\alpha +1}$ respectively and angles

${\theta}_{\alpha}$ and

${\theta}_{\alpha +1}$ are found from Equation (

8) as

$23.{6}^{\circ}$ and

$53.{1}^{\circ}$. Using Equation (

9), the estimated signal DoA is

$44.{9}^{\circ}$.

#### 2.2.1. Error Reduction Technique

Figure 4 shows the amplitudes of each harmonic plotted against the associated main beam angles. The source in this example is located at

$25.{0}^{\circ}$ and the solid lines indicate the result for (

$N=5$). In this situation, the choices of

${X}_{\alpha}$ and

${X}_{\alpha +1}$ are not clear. Harmonic 1 (

$m=1$) is the dominant frequency, but the two adjacent harmonics (

$m=0$,

$m=2$) are very similar in amplitude; both make possible candidates for the selection of

${X}_{\alpha}$ or

${X}_{\alpha +1}$. Using the method described, harmonics 0 and 1 will produce an estimation of

$22.{0}^{\circ}$ whereas using harmonics 1 and 2 will produce an estimation of

$25.{0}^{\circ}$. Clearly, harmonic 2 is the correct value to use, however determining which of the two small harmonics can be challenging especially in the presence of noise. It can be concluded following this argument that the worst error case occurs when the incident angle of the signal is close to one of the harmonics’ main beam angles.

This issue can be mitigated by processing the sampled data in a different way. If a number of samples has been taken from each element in turn but not repeated, it is possible to remove the samples recorded by the outer elements so that data from an

$N-2$ element array is obtained. This produces a different frequency spectrum due of the effect of removing elements, the modulation frequency becomes higher and the resultant harmonic beam directions change according to Equation (

8). By selecting the number of elements used, errors caused by estimating the source direction using harmonic amplitudes close to the null regions can be avoided. Using the data provided, if harmonics 0 and 1 are taken as

${X}_{\alpha}$ and

${X}_{\alpha +1}$ in the three-element case then the method produces an estimation of

$24.{6}^{\circ}$.