# Lab S-3: Beamforming with Phasors. N r k. is the time shift applied to r k

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7 2.2.2 Plot Beamformer Performance Then make a plot of the beamformer response versus angle using Matlab, and also sketch the plot by hand on the verification page. Annotate the sketched plot carefully, showing the peak value(s) of the beamformer output and the angle(s) where peak value(s) are found. Also, give values of any angles where the beamformer output is zero,2 i.e., nulling directions. Instructor Verification (separate page) Sinusoidal Output of Beamformer Finally, let the source direction be θ = 0. For beamformer designed in Sect. 2.1, determine a formula for the output signal, which is a sinusoid. Give exact values for its frequency and amplitude, but write the phase as the symbol ϕ. Explain how you can use the plot in the previous part to answer this question. Instructor Verification (separate page) 2.3 Design Beamformer with Nulling Constraints For Lab-HW, consider the following design question. Create the complex weights for a 3-receiver beamformer that satisfies the following specifications: 1. For a source at angle θ 1, or at angle θ 2, the beamformer output should be zero, i.e., α 1 (t τ 1 ) + ψ s1 (θ 1 )) + α 2 (t τ 2 ) + ψ s2 (θ 1 )) + α 3 (t τ 3 ) + ψ s3 (θ 1 )) = 0 α 1 (t τ 1 ) + ψ s1 (θ 2 )) + α 2 (t τ 2 ) + ψ s2 (θ 2 )) + α 3 (t τ 3 ) + ψ s3 (θ 2 )) = 0 2. For a source at angle θ 0, the beamformer output should be equal to 7 t). 3. The three receivers are uniformly spaced at (0, 0), (0, d) and (0, 2d). 4. Assume the propagation velocity (c) and source signal frequency (ω 0 ) are known. Write a set of simultaneous phasor equations that captures all the information in the specifications 1 4. It is only necessary to set up the equations because without numbers no easy solution is possible. The equations should be written in matrix form Aw = b where A is a 3 3 matrix with complex-valued entries, w is the unknown vector of beamformer (complex) weights, and b is a known vector. Comments: It is essential to distinguish the knowns from the unknowns. Equations such as those in step 1 above define how the beamformer processes a signal from direction θ 1 or θ 2. The direction angles {θ 0, θ 1, θ 2 } are known because we want the beamformer to have a specified output for those directions. Also, the phases such as {ψ s1 (θ 1 ), ψ s2 (θ 1 ), ψ s3 (θ 1 )} are known because these are the phases at the receivers from one source signal direction. These phases depend on the distances from the source to the receivers. Even though these distances are not known, we have shown previously that when the source is far away, only the relative phases matter for the beamformer. For example, refer to the discussion leading to (12), where the phasor for the arriving signal was written in terms of phase differences. The unknowns are the parameters of the beamformer which are the weights {α 1, α 2, α 3 } and the time shifts {τ 1, τ 2, τ 3, } for the three receivers of the array. The resulting phasors that depend on these unknowns are called the complex weights of the beamformer. Note: In this nulling design, these complex weights are not obtained using the beam steering strategy. 2Since the beamformer response vs. direction plot uses a discrete grid for direction (along the horizontal axis), there may not be exact zeros. Instead, you must look for extremely small values of the beamformer response plot. 7 McClellan, Schafer and Yoder, Signal Processing First.

8 Lab: Beamforming INSTRUCTOR VERIFICATION SHEET Turn this page in to your instructor before the end of your scheduled Lab time. Name: UserID: Date: Part 2.1 Run the BFgen( UserID,1) function to obtain the system parameters: β = d = c = f = Hz List the Complex Weights for the beamformer in polar form. Discuss your method and results. Verified: Date/Time: Part 2.2 Write a Matlab for loop to calculate the beamformer output versus direction. Verified: Date/Time: Part 2.2 Sketch a plot of beamformer output versus angle. Annotate the sketch carefully, for peaks and nulls. Verified: Date/Time: Part 2.2 When the steering angle (β) is β = 0, write a formula for the output signal from the beamformer. Verified: Date/Time: 8 McClellan, Schafer and Yoder, Signal Processing First.

9 Lab: Beamforming LAB HOMEWORK QUESTION Turn this page in to your instructor at the very beginning of your next scheduled Lab time. Name: UserID: Date: Create the complex weights for a 3-receiver beamformer that satisfies the specifications given in Part 2.3 The equations should be written in matrix form Aw = b where A is a 3 3 matrix with complex-valued entries, w is the vector of beamformer (complex) weights, and b is a vector. In addition to the final answer, give some key steps in the derivation leading to the answer. 9 McClellan, Schafer and Yoder, Signal Processing First.

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