One-Dimensional Uniform Array. Linear Array Principle of Operation

Transcription

1 One-Dimensional Uniform Array Array Output Linear Array Principle of Operation d sin d d sin c

2 Array Output Setting the delays to [(N 1) m]d sin θ τ m =, m =0,...,N 1 c causes the received signals to add up in-sync. Signals adding in-sync result in a stronger version of the original signal. d sin d d sin c

3 By the time shift theorem of Fourier transforms, we have F F s(t) S(f) s(t τ) S(f) e i2πfτ. So for a sinusoidal or narrowband signal at frequency f 0, we can replace the delay τ m by phase shift φ m =2πf 0 τ m. Assuming narrowband waves and phase shifters with φ 0 = φ 1 = φ 2 = = φ =0 and N identical elements with effective area A e (θ) (gain G(θ)) for a wave from direction θ, it can be shown the effective area of the array is A(θ) =A e (θ) 1 N or equivalently G(θ) =G e (θ) 1 N 2 e i2πn(d/λ) sin θ sin [Nπ(d/λ) sin θ] sin [π(d/λ) sin θ] Array Length = (N 1)d Larger d implies higer resolution, but there is a price to pay. 2

4 If d>λ/2, we get grating lobes due to constructive interference at Bragg angles: 6π θ 4π 2π θ In order to reduce grating lobes, you must have d<λ/2. You can also 1. Use nonuniform spacing of elements; 2. Use an A e (θ) that reduces the most problematic grating lobes. (elements may be large) Radio astronomy arrays often have severe grating lobes. In radar they can be more problematic. Usually take d λ/ In radar they can be more problematic. Usually take d λ/2.

5 Synthetic Arrays A real array: A synthetic array: Another approach is to use a single element and move it between observations Signal processing is used to synthesize an equivalent array. v The received signal is recorded with phase information. We collect data at each position. We apply proper phase shifts to the received data. We sum to synthesize an array antenna. This arrray being sequentially generated is a little different than a real array. It does have high angular resolution like a real array. This is the approach used in Synthetic Aperture Radar (SAR).

6 Comparison of Real and Synthetic Arrays Real Array T/R XMIT RCVR Real Array The complex field at the n-th element is v n = i 2π } λ (d m + d n ) m=0 = i 2π } m=0 i 2π } λ d m, The output of the entire array is the sum V out,r = v n = i 2π } = [ i 2π } ] 2 m =0, 1,...,N 1. m=0 i 2π } λ d m

7 Synthetic Array Synthetic Array T/R XMIT RCVR The synthetic array is sequentially built, one element at a time. Synthetic Array Response The response of the n-th array element is v n = i 2π } λ (d n + d n ) Synthetic Array = i 4π } T/R XMIT RCVR The synthetic array output response is V out,s = v n = i 4π } λ d n = [ i 2π }] 2

8 Real Aperture: V out,r = [ i 2π } ] 2 (Square of Sum) cross terms! Synthetic Aperture: V out,s = [ i 2π }] 2 (Sum of Squares) no cross terms! Let s Compare with Identical Array Geometries Both arrays have N + 1 identical isotropic elements. Elements distributed along line with separation d. Reference scatterer in far-field.

9 V out,r (θ) = N/2 n= N/2 +i 2πλ } nd sin θ 2 = [ [ sin πd λ (N + 1) sin θ] sin [ πd λ sin θ] ] 2 V out,s (θ) = N/2 n= N/2 i 4πλ } nd sin θ 2πd sin θ = sin [ (N +1) λ sin [ ] 2πd sin θ λ ] A Plot of the Responses Text The synthetic array has higher resolution, but also higher sidelobes.