Optimum Beamforming ECE 754 Supplemental Notes Kathleen E. Wage March 31, 29 ECE 754 Supplemental Notes: Optimum Beamforming 1/39
Signal and noise models Models Beamformers For this set of notes, we assume that the snapshot vector x consists of a planewave signal component plus a noise component, i.e., x = x sig + n = b s v s + n where b s is the complex weight applied to the planewave replica v s associated with the source. n is the noise vector. The statistics of the source and noise components are: S sig = E{x sig x H sig } = E{ b s b s }v s v H s = σ 2 sv s v H s S n = E{nn H } ECE 754 Supplemental Notes: Optimum Beamforming 2/39
Models continued... Models Beamformers When the noise consists of a single planewave interferer plus spatially white noise, the noise covariance matrix is: S n = σ 2 1 v 1v H 1 + σ2 wi where the vector v 1 is the replica for the planewave interferer and σ 2 1 is the power in the interferer. We define the interferer-to-noise ratio (INR) as: σ 2 I = σ2 1 σ 2 w Number of array elements is denoted by N in these notes. ECE 754 Supplemental Notes: Optimum Beamforming 3/39
Beamformers Background Models Beamformers In equations below,v m = replica vector for the steering direction Conventional processor (uniformly weighted): w conv = v m N Minimum variance distortionless response processor: 1 w mvdr = v H ms 1 n v m } {{ } Λ mvdr S 1 n v m = Λ mvdr S 1 Minimum power distortionless response processor: 1 w mpdr = v H ms 1 x v m } {{ } Λ mpdr S 1 x n v m = Λ mpdr S 1 x v m v m ECE 754 Supplemental Notes: Optimum Beamforming 4/39
Single interferer Multiple interferers beampattern for a single interferer Consider the case when the noise consists of a single planewave interferer plus spatially white noise. As derived in class (Lecture #8), the beampattern in this case can be written: where B mvdr (u a ) = w H mvdr v a [ = Λ mvdrn σw 2 ρ ma Nσ2 I ρ ] m1 1 + NσI 2 ρ 1a ρ ma = vh mv a N ρ m1 = vh mv 1 N ρ 1a = vh 1 v a N represent the spatial correlation of the v m, v 1, and v a vectors. Note that the vector v a is the replica associated with the directional cosine u a, which determines where we re evaluating the beampattern. ECE 754 Supplemental Notes: Optimum Beamforming 5/39
Single interferer Multiple interferers beampattern for a single interferer (cont.) Note that we can interpret the beampattern as: B mvdr (u a ) = Λ mvdrn σ 2 w ρ }{{} ma CBF: v m Nσ 2 I ρ m1 1 + Nσ 2 I ρ }{{} 1a CBF: v 1 The ρ ma and ρ 1a terms are equal to a conventional beamformer steered to v m and v 1, respectively. Thus the overall beampattern is the difference between two scaled conventional patterns. Additional insights from a look at value of BP at interferer location ECE 754 Supplemental Notes: Optimum Beamforming 6/39
Single interferer Multiple interferers beampattern for a single interferer (cont.) Consider the BP evaluated at the interferer location, u a = u 1 : B mvdr (u 1 ) = Λ mvdrn ρ σw 2 m1 Nσ2 I ρ m1 1 + NσI 2 ρ }{{} 11 =1 [ ( )] = Λ mvdrn Nσ 2 σw 2 1 I 1 + NσI 2 ρ m1 Thus the beamformer creates a partial null in the beampattern at the location of the interferer. Depth of this null depends on: σ 2 I = INR. As σ 2 I, the term in parentheses goes to 1, and an exact null is created in the beampattern. ρ m1 = value of conventional BP at interferer location. If there is already a null at this location, then ρ m1 =. ECE 754 Supplemental Notes: Optimum Beamforming 7/39
Single interferer Multiple interferers Example of beampattern for 1 interferer Plots show the and conventional beampatterns for a 1-element standard linear array at different INR s. Interferer is located at u 1 =.5, which is not a null in the conventional beampattern, i.e., ρ m1. -2 db INR: since σ 2 I is small, B conv B mvdr. db INR: σ 2 I is high enough that B mvdr (u 1 ) < B conv (u 1 ). +2 db INR: deeper null created at u a = u 1. 2log 1 B(u a ) (db) 2log 1 B(u a ) (db) 2log 1 B(u a ) (db) 2 4 6 8 1.8.6.4.2.2.4.6.8 1 u a 2 4 6 INR= db Conv 8 1.8.6.4.2.2.4.6.8 1 u a 2 4 6 INR= 2 db INR=2 db Conv Conv 8 1.8.6.4.2.2.4.6.8 1 u a ECE 754 Supplemental Notes: Optimum Beamforming 8/39
Single interferer Multiple interferers Depth of nulls in beampattern The depth of the null in the beampattern is a function of ( ) Nσ 2 1 I 1 1 + NσI 2 = 1 + NσI 2 The plot at right shows this factor 2 (in db) as a function of σi 2 Null depth factor as a function of σ I in db. 2 Interferer will have almost no 4 effect on the BP for 1 log 1 (σi 2 6 ) 2 db 8 Interferer will drive BP down 1 significantly for 1 log 1 (σi 2) db 12 4 3 2 1 1 2 3 4 2 1log (σ ) 1 I 2log 1 (1+Nσ I 2 ) ECE 754 Supplemental Notes: Optimum Beamforming 9/39
Single interferer Multiple interferers beampattern when interferer at null in B conv Plots illustrate what happens when the interferer is at the location of a null in the conventional pattern. Regardless of INR level, the and conventional beampatterns are identical in this case. Plots show results for a 1-element standard linear array, but similar behavior would be seen for other arrays. 2log 1 B(u a ) (db) 2log 1 B(u a ) (db) 2log 1 B(u a ) (db) 2 4 6 INR= 2 db Conv 8 1.8.6.4.2.2.4.6.8 1 u a 2 4 6 INR= db Conv 8 1.8.6.4.2.2.4.6.8 1 u a 2 4 6 INR=2 db Conv 8 1.8.6.4.2.2.4.6.8 1 u a ECE 754 Supplemental Notes: Optimum Beamforming 1/39
Single interferer Multiple interferers BP as fxn of interferer location: db INR As discussed in class and explored in Problem Set 5, the behavior of the beampattern changes substantially as the interferer moves inside the mainlobe. Plots show results for u I =.9,.433,.2 and INR= db. Unity gain maintained at u a =. Mainlobe is distorted by presence of ML interferer. 2log 1 B(u a ) (db) 2log 1 B(u a ) (db) 2log 1 B(u a ) (db) 1 1 2 3 4 5 INR= db, u I =.9 Conv 6 1.8.6.4.2.2.4.6.8 1 u a 1 1 2 3 4 5 INR= db, u I =.433 Conv 6 1.8.6.4.2.2.4.6.8 1 u a 1 1 2 3 4 5 INR= db, u I =.2 Conv 6 1.8.6.4.2.2.4.6.8 1 u a ECE 754 Supplemental Notes: Optimum Beamforming 11/39
Single interferer Multiple interferers BP as fxn of interferer location: 2 db INR Plots show results for u I =.9,.433,.2 and INR= db. Unity gain still maintained at u a =. Mainlobe is very distorted by presence of high INR ML interferer. In general ML interferers are very hard to deal with. 2log 1 B(u a ) (db) 2log 1 B(u a ) (db) 2log 1 B(u a ) (db) 1 1 2 3 4 5 INR=2 db, u I =.9 Conv 6 1.8.6.4.2.2.4.6.8 1 u a 1 1 2 3 4 5 INR=2 db, u I =.433 Conv 6 1.8.6.4.2.2.4.6.8 1 u a 1 1 2 3 4 5 INR=2 db, u I =.2 Conv 6 1.8.6.4.2.2.4.6.8 1 u a ECE 754 Supplemental Notes: Optimum Beamforming 12/39
Single interferer Multiple interferers beampattern for multiple interferers The following plot shows the and conventional BP for a 1-element standard linear array when the noise consists of 8 strong discrete interferers (INR=2 db) outside the mainlobe plus spatially white noise. processor has 8 deep nulls 1 2 Unlike single-interferer 3 case, here BF 4 5 has wider mainlobe 6 than conventional BF 7 part of the price 8 paid for nulls 1.8.6.4.2.2.4.6.8 1 2log 1 B(u a ) (db) INR=2 db u a Conv ECE 754 Supplemental Notes: Optimum Beamforming 13/39
Single interferer Multiple interferers beampattern for 9 interferers The following plot shows that 9 equal-power interferers result in 9 deep nulls in the beampattern of the 1-element array Mainlobe is more distorted than in the 8-interferer case. 2log 1 B(u a ) (db) 1 2 3 INR=2 db, 9 interferers 4 Note: there are 5 5 interferers to the left of 6 the mainlobe and 4 7 interferers to the right 8 of the mainlobe. 1.8.6.4.2.2.4.6.8 1 u a Conv Can we null out 1 interferers with this array? ECE 754 Supplemental Notes: Optimum Beamforming 14/39
Single interferer Multiple interferers beampattern for 1 interferers When there are 1 equal-power interferers incident on a 1-element array, the beamformer can no longer null all of them. 1-element array 1 coeff s in w mvdr Do not have enough degrees of freedom to satisfy unity gain constraint and maintain 1 exact nulls with a 2log 1 B(u a ) (db) 1 2 3 4 5 6 7 INR=2 db, 1 interferers 8 1-element array 1.8.6.4.2.2.4.6.8 1 Resulting BF may still be useful, but BP behavior is different when # of interferers is N. u a Conv ECE 754 Supplemental Notes: Optimum Beamforming 15/39
Single interferer Multiple interferers In-class problems Consider a 2-element standard linear array. Can you predict how the beampattern will differ from the conventional beampattern in the following cases? 1 The noise field consists of spatially white noise plus one discrete interferer at u I =.15 with INR=4 db. 2 The noise field consists of spatially white noise plus two discrete interferers at u I = ±.1 with INR s of 2 db. 3 The noise field consists of spatially white noise plus a discrete interferer at u I =.25 with INR=2 db and a discrete interferer at u I =.5 with INR= db. ECE 754 Supplemental Notes: Optimum Beamforming 16/39
Single interferer Multiple interferers In-class problem 1 solution 2-element standard linear array, u I =.15, INR=4 db 1 In class 1 Conv 1 2log 1 B(u a ) (db) 2 3 4 5 6 1.8.6.4.2.2.4.6.8 1 u a Since interferer outside mainlobe we expect to see a significant null at the interferer location in the pattern. There is no similar null in the conventional pattern. ECE 754 Supplemental Notes: Optimum Beamforming 17/39
Single interferer Multiple interferers In-class problem 2 solution 2-element standard linear array, u I = ±.1, INR=2 db 1 In class 2 Conv 1 2log 1 B(u a ) (db) 2 3 4 5 6 1.8.6.4.2.2.4.6.8 1 u a Since interferers are at locations of nulls in conventional pattern, B mvdr = B conv for this example. ECE 754 Supplemental Notes: Optimum Beamforming 18/39
Single interferer Multiple interferers In-class problem 3 solution 2-element standard linear array, u I =.25/INR=2 db and u I =.5/INR= db. 1 In class 3 Conv 1 2log 1 B(u a ) (db) 2 3 4 5 6 1.8.6.4.2.2.4.6.8 1 u a Expect to see significant null for interferer at u I =.25 since it is outside mainlobe and has high INR. Since the interferer at u I =.5 is inside the mainlobe, we expect some mainlobe distortion, though unity gain at broadside is maintained. ECE 754 Supplemental Notes: Optimum Beamforming 19/39
Single interferer Multiple interferers Summary: effect of interferers on beampattern If INR is low, beampattern resembles conventional beampattern. For modest INR ( db), processor places partial nulls in beampattern at locations of interferers. For high INR, will place exact nulls at locations of interferers, assuming the number of discrete interferers is less than the number of elements. For interferers outside the mainlobe, the mainlobe of the processor is well-behaved (resembles conventional processor for N >>#interferers. For interferers inside the mainlobe, the mainlobe of the processor may be significantly distorted (esp. for high INR) as the processor tries to maintain the unity gain constraint and null out an interferer close to the desired signal direction. ECE 754 Supplemental Notes: Optimum Beamforming 2/39
Definition quantifies the improvement in signal-to-noise ratio due to using the array. It is defined as the ratio of the SNR at the output of the beamformer to the SNR at an individual input sensor: = A = SNR out SNR in Note that in general the noise includes white and colored noise, as well as planewave interferers. For that reason SNR is often called SINR, which stands for Signal-to-Interference-and-Noise Ratio. How to compute these quantities? ECE 754 Supplemental Notes: Optimum Beamforming 21/39
Definition How to compute SNR in The diagonal elements of S sig and S n are the signal powers and noise powers, respectively. Since the diagonal elements of v s v H s are equal to 1, the signal power at each sensor is σs. 2 We obtain the average noise power by summing the diagonal elements of S n and dividing by the number of sensors: σ 2 n = tr(s n) N Thus, the input SNR is: where tr denotes the trace operation SNR in = σ2 s σ 2 n ECE 754 Supplemental Notes: Optimum Beamforming 22/39
Definition How to compute SNR out If we process the snapshot using the weight vector w, the power in the signal component of the output is P sig = E{ w H x sig 2 } = E{w H x sig x H sig wh } = w H S sig w = σ 2 s w H v s 2 Similarly, the noise power at the output of the beamformer is P noise = w H S n w = σ 2 nw H ρ n w, where ρ n = normalized noise covariance matrix ρ n = S n /σ 2 n. Thus the output SNR is: SNR out = σ2 s w H v s 2 σ 2 n(w H ρ n w) ECE 754 Supplemental Notes: Optimum Beamforming 23/39
Definition Formula for array gain Thus the array gain for the BF with weight vector w is equal to: A = SNR out SNR in = σ2 s w H v s 2 σ 2 n(w H ρ n w) σ2 n σ 2 s = wh v s 2 w H ρ n w Note that if the weight vector is designed with a unity gain constraint for replica v s, then w H v s = 1 and the array gain reduces to A = 1 w H ρ n w Examples ECE 754 Supplemental Notes: Optimum Beamforming 24/39
Definition example 1: interferer in sidelobes The figures on the right show the and conventional beampatterns and array gains for a standard 1-element linear array with a single interferer. The top plot shows the results for 1 db INR, and the bottom plot shows the results for 5 db INR. The interferer is located at u I =.5. The conventional beamformer obviously has lower array gain because the interferer is located near one of the sidelobe peaks, whereas the beamformer has a partial null at the inteferer location. 2log 1 B(u z ) 2log 1 B(u z ) 2 4 6 8 u I =.5, INR=1 db, A mvdr =2.3 db, A conv =15.6 db 1 1.8.6.4.2.2.4.6.8 1 u z 2 4 6 8 Conventional Conventional u I =.5, INR=5 db, A mvdr =59.9 db, A conv =17. db 1 1.8.6.4.2.2.4.6.8 1 u z What happens when interferer is in the mainlobe? ECE 754 Supplemental Notes: Optimum Beamforming 25/39
Definition example 2: interferer in mainlobe and conventional beampatterns and array gains for a standard 1-element linear array with a single interferer in the mainlobe. Top plot shows results for 1 db INR, and bottom plot shows results for 5 db INR. The conventional beamformer has very low array gain because the interferer is very close to the steering direction. The beamformer is able to put a null at the interferer location, so its array gain is relatively high, even though the beampattern is quite distorted. As we see on the following slides, the white noise gain is seriously affected with an interferer in the mainlobe. 2log 1 B(u z ) 2log 1 B(u z ) 2 4 6 8 u I =.2, INR=1 db, A mvdr =6.6 db, A conv =.5 db 1 1.8.6.4.2.2.4.6.8 1 u z 2 4 6 8 Conventional Conventional u I =.2, INR=5 db, A mvdr =45.1 db, A conv =.1 db 1 1.8.6.4.2.2.4.6.8 1 u z ECE 754 Supplemental Notes: Optimum Beamforming 26/39
Definition is the array gain when S n = σ 2 wi (and ρ n = I). A w = wh v s 2 w H ρ n w = wh v s 2 w H w If w is constrained to give unit gain in the look direction, w H v s = 1: A w = 1 w H w = 1 T se is the inverse of the 2-norm-squared of the weight vector. Recall from Ch. 2 of Van Trees that white noise gain is the inverse of the sensitivity function T se. Thus, the lower the white noise gain, the higher the sensitivity of the beamformer to mismatch. ECE 754 Supplemental Notes: Optimum Beamforming 27/39
Definition continued... Note that the weight vector used in the white noise gain calculation may be computed using a non-white noise covariance, e.g., w = w mvdr = Λ mvdr S n v m In this case S n may contain white noise plus colored noise plus one or more planewave interferers. The white noise gain is still A w mvdr = 1 w H mvdr w mvdr Examples ECE 754 Supplemental Notes: Optimum Beamforming 28/39
Definition example 1: interferer in sidelobes Beampatterns for a standard 1-element array with interferer in the sidelobes at 1 db or 5 db INR. 1 db INR 5 db INR A mvdr 2.3 db 59.9 db A conv 15.6 db 17. db A w mvdr 9.9 db 9.9 db A w conv 1. db 1. db With interferer in the sidelobes the white noise gain for the and conventional processors are approximately equal. 2log 1 B(u z ) 2log 1 B(u z ) 2 4 6 8 INR=1 db, A mvdr =2.3 db, A conv =15.6 db, A w mvdr =9.9, A w conv =1. db 1 1.8.6.4.2.2.4.6.8 1 u z 2 4 6 8 Conventional Conventional INR=5 db, A mvdr =59.9 db, A conv =17. db, A w mvdr =9.9, A w conv =1. db 1 1.8.6.4.2.2.4.6.8 1 u z ECE 754 Supplemental Notes: Optimum Beamforming 29/39
Definition example 2: interferer in mainlobe Beampatterns for a standard 1-element array with interferer in the mainlobe at 1 db or 5 db INR. 1 db INR 5 db INR A mvdr 6.6 db 45.1 db A conv.5 db.1 db A w mvdr -2.7 db -4.9 db A w conv 1. db 1. db With interferer in the mainlobe the white noise gain for the processor is substantially decreased. The conventional processor is unaffected because it does not attempt to null the strong mainlobe interferer. 2log 1 B(u z ) 2log 1 B(u z ) 2 4 6 8 INR=1 db, A mvdr =6.6 db, A conv =.5 db, A w mvdr = 2.7, A w conv =1. db 1 1.8.6.4.2.2.4.6.8 1 u z 2 4 6 8 Conventional Conventional INR=5 db, A mvdr =45.1 db, A conv =.1 db, A w mvdr = 4.9, A w conv =1. db 1 1.8.6.4.2.2.4.6.8 1 u z ECE 754 Supplemental Notes: Optimum Beamforming 3/39
Definition example 3: signals The following plots illustrate the effect of white noise gain on the output signals of a beamformer. The top plot shows the output (real part only) of an beamformer with white noise gain of -4.7 db and the bottom plot shows the output of an beamformer with white noise gain 7.1 db. The lower A w is, the more white noise comes through, making it harder to distinguish the underlying signal (shown in red). u I =.2 u I =.9 2 INR=2 db, blue=, red=true signal 2 1 2 3 4 5 6 7 8 9 2 2 1 2 3 4 5 6 7 8 9 m (sample number) Note that the difference in white noise gain in these two plots is caused by the interferer moving closer to the desired signal in the mainlobe. ECE 754 Supplemental Notes: Optimum Beamforming 31/39
Definition In-class problems 4 Suppose that you have a 3-element linear array. What is the maximum possible value for the white noise gain for this array? What beamformer is guaranteed to achieve this white noise gain? 5 Suppose that you use two conventional beamformers to process data. The data contains the desired planewave signal plus spatially white noise. Beamformer #1 has a white noise gain of 1 db and Beamformer #2 has a white noise gain of 2 db. How do you expect the two outputs to differ? How do you build conventional beamformers with different white noise gains? ECE 754 Supplemental Notes: Optimum Beamforming 32/39
Definition In-class problem 4 solution Consider deriving a beamformer by maximizing white noise gain given a unity gain constraint, i.e., { } 1 max w H subject to w H v m = 1 w This problem is equivalent to minimizing w H w, i.e., { } min w H w subject to w H v m = 1 Solution via LaGrange multipliers: Q = w H w + λ(w H v m 1) w HQ = w + λv m = λ Q = w H v m 1 = ECE 754 Supplemental Notes: Optimum Beamforming 33/39
Definition In-class problem 4 solution continued... Setting the gradients equal to zero and solving yields: and w HQ = w + λv m = w = λv m (1) λ Q = w H v m 1 = Substituting 2 into 1 and solving yields: Thus w H v m = 1 (2) λ = 1 v H mv m w = 1 v H v m = v m mv m N ECE 754 Supplemental Notes: Optimum Beamforming 34/39
Definition In-class problem 4 solution continued... Thus the beamformer that maximizes white noise gain is the conventional beamformer. The maximum white noise gain is A w max = 1 v H m N v mn = N2 v H mv m = N2 N = N (The Schwartz inequality can also be used to show that this is the maximum white noise gain.) For a 3-element array, the maximum white noise gain in db is 1 log 1 3 = 14.8 db. This white noise gain is achieved by the conventional beamformer with uniform weighting. ECE 754 Supplemental Notes: Optimum Beamforming 35/39
Definition In-class problem 5 Suppose that you use two conventional beamformers to process data. The data contains the desired planewave signal plus spatially white noise. Beamformer #1 has a white noise gain of 1 db and Beamformer #2 has a white noise gain of 2 db. How do you expect the two outputs to differ? Will the true signal coming through each beamformer differ in amplitude? Answer: No. Both beamformers are designed with a unity gain constraint. How will the noise coming through each beamformer differ? Answer: The power of the noise at the output of Beamformer #1 will be higher (by 1 db) than the power of the noise at the output of Beamformer #2. ECE 754 Supplemental Notes: Optimum Beamforming 36/39
Definition In-class problem 5 continued How do you build conventional beamformers with different white noise gains? Answer: recall that the white noise gain of a conventional beamformer is equal to 1 log 1 N in db. To change the white noise gain of this beamformer, you must change the number of sensors. In this case Beamformer #1 must have 1 sensors since it has a white noise gain of 1 db. Beamformer #2 must have 2 sensors since it has a white noise gain of 2 db. ECE 754 Supplemental Notes: Optimum Beamforming 37/39
Definition Summary: array gain quantifies the improvement in signal-to-noise ratio between the input (i.e., a single sensor) and the output of an array. measures the improvment in SNR between the array s input and output when the input noise is assumed to be spatially white. The white noise gain is an indicator of the sensitivity of the beamformer to mismatch. The lower the white noise gain, the higher the sensitivity. The white noise gain of the beamformer is substantially decreased when the interferer moves inside the mainlobe. The white noise gain of the conventional beamformer is constant regardless of the interferer location since it s beampattern is fixed. ECE 754 Supplemental Notes: Optimum Beamforming 38/39
Definition Summary: array gain continued... The array gain of the beamformer is always greater than or equal to that of the conventional beamformer: A mvdr A conv The equality holds when the noise is spatially white, i.e., ρ n = I. The white noise gain of the conventional beamformer is always greater than or equal to the white noise gain of the beamformer: A w conv = N A w mvdr The maximum value of white noise gain is equal to the number of sensors in the array. ECE 754 Supplemental Notes: Optimum Beamforming 39/39