Spatial undersampling of a wavefield by an array gives rise to visible
grating lobes. If you think of the wavenumber, *k*, as analogous to angular
frequency, then you must sample the signal at spatial intervals smaller than
*π/k*_{max} (or
*λ*_{min}/2) in order to remove aliasing. The
appearance of visible grating lobes is also known as spatial aliasing. The variable
*k*_{max} is the largest wavenumber value present
in the signal.

The directions of maximum spatial response of a ULA are determined by the peaks of the
array’s *array pattern* (alternatively called the *beam
pattern* or *array factor*). Peaks other than the mainlobe
peak are called grating lobes. For a ULA, the array pattern depends only on the wavenumber
component of the wavefield along the array axis (the *y*-direction for the
`phased.ULA`

System object). The wavenumber component is related to the look-direction of an arriving
wavefield by *k*_{y} = –2π sin φ/λ. The angle
*φ* is the broadside angle—the angle that the look-direction makes
with a plane perpendicular to the array. The look-direction points away from the array to
the wavefield source.

The array pattern possesses an infinite number of periodically-spaced peaks that are
equal in strength to the mainlobe peak. If you steer the array to the
*φ*_{0} direction, the array pattern for a ULA has
its mainlobe peak at the wavenumber value of *k*_{y0} = –2π sin
φ_{0}/λ. The array pattern has strong grating lobe peaks
at *k*_{ym} = k_{y0} + 2π m/d, for
any integer value *m*. Expressed in terms of direction cosines, the grating
lobes occur at *u*_{m} = u_{0} +
mλ/d, where *u*_{0} = sin
φ_{0}. The direction cosine,
*u*_{0}, is the cosine of the angle that the
look-direction makes with the *y*-axis and is equal to *sin
φ*_{0} when expressed in terms of the
look-direction.

In order to correspond to a physical look-direction,
*u*_{m} must satisfy,* –1 ≤
u*_{m} ≤ 1. You can compute a physical look-direction angle
*φ*_{m} from *sin φ*_{m}
= u_{m}
as long as* –1 ≤ u*_{m} ≤ 1. The spacing of
grating lobes depends upon *λ/d*. When *λ/d* is small
enough, multiple grating lobe peaks can correspond to physical look-directions.

The presence or absence of visible grating lobes for the ULA is summarized in this
table.

Element Spacing | Grating Lobes |
---|

*λ/d ≥ 2* | No visible grating lobes for any mainlobe direction. |

*1 ≤ λ/d < 2* | Visible grating lobes can exist for some range of mainlobe
directions. |

*λ/d < 1* | Visible grating lobes exist for every mainlobe direction. |