3.2 Data Model

Most of the modern approaches to signal processing are model-based, in the sense that they rely on certain assumptions on the data observed in the real world. The prevailing data model used in the remainder of this book is discussed in this section.

The following scenario is assumed and maintained throughout our description of DOA estimation algorithms [2]:

•Isotropic and linear transmission medium: There are d sources that emit d signals; the d signals travel through a medium and then impinge onto and an M-element antenna or sensor array. The transmission medium between the sources and the array is assumed to be isotropic and linear; in other words, the medium has its physical properties the same in all different directions and the signals or waves at any particular point can be superposed linearly. The isotropic and linear property of the medium ensures:

(1)that the propagation property of the waves do not change with the DOAs of signals, and (2) that the signals traveling through the medium and then impinging on or received by any element of the M-element array can be computed as a linear superposition of d signal wavefronts generated by the d sources. In addition, the gain of each antenna or sensor element is assumed to be one.

•Far-field assumption: The d signal sources are located far from the array such that the wavefront generated by each source arrives at

all the elements at an equal direction of propagation and the wavefront is planar (far-field approximation). Thus, the propagating fields of the d signals arrived at the array are considered as parallel to each other. This assumption can, in general, be realized by making the distance between the signal sources and the array much larger than the dimension of the antenna array. A rule of thumb is that the distance is larger than 2D2/λ, with D being the dimension of the array and λ being the wavelength of the signals [3].

•Narrowband assumption: The d signals from the d sources have the same carrier frequency such that their frequency contents are concentrated in the vicinity of carrier frequency fc. Then,

τ

Figure 3.1 The scenario under consideration in this book.

between the source and the first (rightmost) element is dd. The configuration is shown in Figure 3.2.

Suppose that a plane wave signal generated by source i impinges on the array at an angle θi and the signal generated by source i is a

ith source

s r(t)

Figure 3.2 Data model for DOA estimation of d sources with a linear array of the M element.

narrowband signal sir (t ); it then travels over a distance of dd at a speed of c

[3] and reaches the first (rightmost) element. In other words, the signal received by the rightmost element is the delayed version of the sir (t ) with a

Because the ULA positions the array elements on a line, the signal traveling to the mth element will take an extra distance in comparison with the signal arriving at the rightmost element (see Figure 3.2). This extra distance can be found as:

Therefore, the signal arriving at the mth element will take the additional delay, which is equal to:

The signal received by the mth element is then the delayed version of the signal si1(t) (which is received by the first element) with the additional delay of τmi:

that is associated with the ith source that generates the signal of incident

angle θi. λ = c/fc denotes the wavelength corresponding to the carrier frequency fc . In deriving (3.7), approximation (3.2) is applied.

In the complex phasor form, the above received signal corresponds

to:

s im (t ) ≈ αi (t − τ d )e j [2πf c (t − τ d )+ βi (t )]e j (m −1) μi = s i (t )e j (m −1) μi (3.8)

Equation (3.8) shows that the signal received by the mth element from the ith source is the same as that received by the first (rightmost) element but with an additional phase shift factor of e j (m −1) μi ; this factor is dependent only on spatial frequency μi and the position of the element relative to the first element. For each incident angle θi that determines a source, there is a corresponding spatial frequency μi. Therefore, the whole objective of estimating a DOA (i.e., θi) is to extract this spatial frequency μi from the signals received by the array.

In order to be able to determine θi uniquely from μi, one-to-one correspondence is desired between them. As a result, the spatial frequencies μi are limited to −π ≤ μi ≤ π and the range of possible DOAs is restricted to the interval of −90° ≤ θi ≤ 90°. This, in turn, requires that the element spacing satisfies ≤ λ/2. If the element spacing does not satisfy this relation, there is an ambiguity in DOA determination, that is, there will be two solutions for the angles from a specific value of μi; this will result in an array having the grating lobes: lobes other than the main lobe permits the signals from undesired directions. Technically this is called spatial aliasing. It is analogous to the Nyquist sampling rate for a fre- quency-domain analysis of a signal.

Now consider that the all the signals generated by all the d sources, si(t), 1 ≤ i ≤ d, the overall signal and noises received by the mth element at time t can be expressed as:

i =1