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Lecture 8-10 - SPATIAL CHANNELS 1 Propagation Delay Across...

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1 SPATIAL CHANNELS
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2 Propagation Delay Across the Array T = d sin θ c The key element in smart antenna testing is the Delay . As the signal arrival at the antenna array travels across the different sensors, it encounters a propagation Delay d θ d sin θ wavefront Signal arrival
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3 Steering Vector s ( t ) = e - j ϖ t [1 e - j ϖ T ....... e - j ϖ ( M - 1) T ] t The signal vector: Ω = ϖ T = 2 π d λ sin θ Spatial Frequency: θ d d sin θ Reference sensor e j ϖ t e j ϖ ( t - T ) e j ϖ ( t - ( M - 1) T ) 1 M
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4 Array Pattern s ( t ) = e - j ϖ t a ( θ ), a ( θ ) = [1 e - j π sin θ ....... e - j ( M - 1) π sin θ ] t Steering vector for d / λ = 0.5 θ Array coefficient vector Array output W = [ w 0 w 1 ........ w M - 1 ] t w M - 1 w 0 y ( t ) = W H s ( t ) = e j ϖ t W H a ( θ ) e j ϖ t SUM
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5 Continue G ( θ ) = W H a ( θ ) Array Pattern: Spatial Response: W H a ( θ ) Parameters controlling the Array pattern are Array Geometry Array Coefficients Array Manifold
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6 Array Manifold Each antenna responds by a gain and phase to the incident waveform depending on its angle of arrival and frequency a ( θ ) = [ ζ 1 ( θ , f ) ζ 2 ( θ , f ) e - j π sin θ ....... ζ M ( θ , f ) e - j ( M - 1) π sin θ ] t Array Manifold Vector The complex gain can be obtained by array calibration. M i f i ,..., 2 , 1 ), , ( = θ ζ The steering vector loses its known structure.
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7 Spatial Structures The spatial structure is given by the array manifold (AM), which is a set of array response vectors indexed by the angles- of-arrival. The array manifold a( θ ) represents the response of the antenna elements relative to the first element for a wavefront arriving at the carrier frequency from a direction θ . The AM includes the effect of a number of factors, such as array geometry, elements patterns, inter-element coupling, scattering from antenna support structures.
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8 Narrowband Signals Maximum distance traveled (M-1)d ( M - 1) d sin θ Time to travel across the array is smaller than the inverse of the signal bandwidth ( M - 1) d c pp 1/ BW
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9 Narrowband Model Assumption Large signal bandwidth High data rate Large array aperture Large inter-element spacing Large number of antennas The narrowband assumption may not hold for
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