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plotGratingLobeDiagram(H,FREQ)
plotGratingLobeDiagram(H,FREQ,ANGLE)
plotGratingLobeDiagram(H,FREQ,ANGLE,C)
plotGratingLobeDiagram(H,FREQ,ANGLE,C,F0)
hPlot = plotGratingLobeDiagram(___)
plotGratingLobeDiagram(H,FREQ) plots the grating lobe diagram of an array in the u-v coordinate system. The System object™ H specifies the array. The argument FREQ specifies the signal frequency and phase-shifter frequency. The array, by default, is steered to 0° azimuth and 0° elevation.
A grating lobe diagram displays the positions of the peaks of the narrowband array pattern. The array pattern depends only upon the geometry of the array and not upon the types of elements which make up the array. Visible and nonvisible grating lobes are displayed as open circles. Only grating lobe peaks near the location of the mainlobe are shown. The mainlobe itself is displayed as a filled circle.
plotGratingLobeDiagram(H,FREQ,ANGLE), in addition, specifies the array steering angle, ANGLE.
plotGratingLobeDiagram(H,FREQ,ANGLE,C), in addition, specifies the propagation speed by C.
plotGratingLobeDiagram(H,FREQ,ANGLE,C,F0), in addition, specifies an array phase-shifter frequency, F0, that differs from the signal frequency, FREQ. This argument is useful when the signal no longer satisfies the narrowband assumption and, allows you to estimate the size of beam squint.
hPlot = plotGratingLobeDiagram(___) returns the handle to the plot for any of the input syntax forms.
H |
Antenna or microphone array, specified as a System object. |
FREQ |
Signal frequency, specified as a scalar. Frequency units are hertz. Values must lie within a range specified by the frequency property of the array elements contained in H.Element. The frequency property is named FrequencyRange or FrequencyVector, depending on the element type. |
ANGLE |
Array steering angle, specified as either a 2-by-1 vector or a scalar. If ANGLE is a vector, it takes the form [azimuth;elevation]. The azimuth angle must lie in the range [-180°,180°]. The elevation angle must lie in the range [-90°,90°]. All angle values are specified in degrees. If the argument ANGLE is a scalar, it specifies only the azimuth angle where the corresponding elevation angle is 0°. Default: [0;0] |
C |
Signal propagation speed, specified as a scalar. Units are meters per second. Default: Speed of light in vacuum |
F0 |
Phase-shifter frequency of the array, specified as a scalar. Frequency units are hertz When this argument is omitted, the phase-shifter frequency is assumed to be the signal frequency, FREQ. Default: FREQ |
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. |