Theory of Classical Beamforming (Optimal Beamforming) Prepared by Natalia Schmid (ver. 1.0 from June 15, 2015) 1. Background information Environment = some medium through which waves propagate, = sources of radiation energy = boundaries, reflectors. We have a passive system (telescope). Associated with the medium is a field f (t , x ) , t = time and x = space (3D) f is a measurable physical quantity, scalar‐ or vector‐ values. Sensor = device which measures the field. The ideal sensor is a “field sampler.” Located at x 0 , it produces output signal z (t )  c f (t , x0 ) . For more information, one can use multiple sensors or sensor array. Collection of outputs forms a vector‐valued function of time:  f (t , x1 ) 
 z1 (t ) 
 f (t , x ) 
 z (t ) 
2 
z (t )   2   c 
  
  




 f (t , x M )
 z M (t )
Traveling waves: Solutions to wave equation (in general).  kT x 
 Plane wave: w(t , x)  s  t 
c


k = unit vector in the direction of propagation c = propagation velocity s (t ) = field value at the origin ( x  0 ). Can be any function. ( k / c) = slowness vector. Monochromatic plane wave: s (t )  A cos( 2ft   ) w(t , x)  A cos(2 f (t 
kT x
)  ) c
1  A cos(2 ft  2 f
kT x
  ) c
Complex solution: s (t )  A exp( j ( 2ft   )) kT x
kT x
w(t , x)  A exp( j (2f (t 
  ))  s(t ) exp( j 2f
) c
c
Note, the field at all points x is the same ( s (t ) ) except for a position‐dependent multiplicative kT x
) . phase factor exp( j 2f
c
Simple Model: 


Field consists of a superposition of complex exponential plane waves. In the neighborhood of the sensors, there is no attenuation, dispersion, or refraction. Sensors are linear and time‐invariant, but may have responses which are frequency and direction dependent. The array response: Consider the response of an array of ideal isotropic sensors (i.e. field samplers) to a single complex monochromatic plane wave: θ
Let z i (t ) be the output of the i‐th sensor: k T x1
z1 (t )  s (t 
)
c

z M (t )  s(t 
k T xM
)
c
2 k T xi
Define  i 
(delay relative to origin) c
z1 (t )  s (t   1 )  A exp( j ( 0 (t   1 )   ))

z M (t )  s(t   M )  A exp( j (0 (t   M )   ))
In general,  z1 (t )   exp( j 1 ) 
 A exp( j (t   )) z (t )      


 z M (t ) exp( j M )
Note: Vector of observations is a constant complex vector (depends on k , 0 , but not on t ) times a time‐varying complex scalar. Now let s (t ) be a bandpass signal with center frequency 0 s(t )  sb (t ) exp( j 0 t ) sb (t ) = baseband envelope z1 (t )  sb (t   1 ) exp( j 0 (t   1 ))

z N (t )  sb (t   N ) exp( j 0 (t   N ))
If the bandwidth of s (t ) is sufficiently small (signal varies but slow), we can involve the narrowband assumption: sb (t   )  sb (t ) for   worst‐case transit time across the array. Under this assumption:  exp( j0 1 ) 
 s (t ) exp( j t )  a ( ) s (t ) exp( j t ) 
z (t )  
0
b
0
 b
exp( j0 N )
a ( ) = direction vector Note: a ( ) depends on 0 , k (i.e.  ) , possibly non‐ideal sensor characteristics, but not time dependent. 3 Multiple signals: K
z (t )   a( k ) s k (t ) k 1
 s1 (t ) 
 a(1 ) a ( 2 )  a( K )     A() s(t )  s K (t )
The array manifold A propagation vector k is determined by two angles:  (elevation) and  (azimuth). A narrowband plane wave from direction ( ,  ) at frequency 0 induces a direction vector a( ,  ,  0 ) . Array manifold is a set of all direction vectors. For a narrow band array processing {a( ,  ), all  and  , fixed f 0 } . Array processing objectives: 


Detection of signals Estimation of signal parameters (including direction of arrival) Estimation of signal s (t ) 

Tracking of signal sources Characterization of signal sources and, before any of the above can happen, calibration of the instrumentation. Statistical Models for Sensor Array Data We will attempt to use the following notation consistently M = number of sensors N = number of snapshots (data points) K = number of signals (when appropriate) Model equation: z (t )  a ( ) s(t )  n(t )
( K  1) a ( ) = direction vector (array response in the direction of  ) , M‐by‐1 vector s(t ) = source signal, scalar, time varying. Now we need to add a statistical description to this model. 4 s(t ) is circular Gaussian stochastic process n(t ) is a zero‐mean circular, Gaussian stochastic process z (t ) is a circular Gaussian stochastic process. Furthermore, if s(t ) , n(t ) , and z (t ) are finite power, we will assume that they are wide sense stationary (WSS). 2. Beamforming (BF) Model equation: z (t )  As (t )  n (t ) One signal (K=1): z (t )  a ( ) s (t )  n(t ) Beamforming: estimation of s (t ) by sum of weighted and/or delayed sensor outputs. Σ
(t)
M
M
i 1
i 1
sˆ(t )   wi* z i (t ) or sˆ(t )   wi* z i (t   i ) Time delays are typically used for wideband signals. We will concentrate on narrowband processing and thus M
sˆ(t )   wi* z i (t )  w H z (t ) i 1
5 The wi are called beamformer weights, and the vector w  [ w1 ,, wM ]T is called the beamformer weight vector. Choosing a w to estimate s (t ) from a particular  is known as “steering a beam.” The first and most obvious choice of w is wi e jc ( ,i ) . We have zi (t )  s(t ) exp( jo i ( ))  n(t ) M
M
M
i 1
i 1
i 1
 zi (t )wi*   s(t ) exp( jo i ( )) exp( jo i ( ))   ni (t ) exp( jo i ( )) M
M
M
i 1
i 1
i 1
  s (t )   ni (t ) exp( j o i ( ))  Ms(t )   ni (t ) exp( j o i ( )) If n(t ) ~ CN (0,  n2 I ) , then so is n(t ) w and the second term is w H n(t ) ~ CN (0, M n2 ) . The SNR at the BF output is SNRout 
 s2
signal power M 2 s2


M
noise power
M n2
 n2
 s2
is the SNR at each individual sensor, SNRin . Thus the array gain, defined as  n2
SNRout
G
is G  M (the number of sensors). SNRin
But Suppose there are other signals, at other directions, which we can view as interference. What is the response of the array to a signal at  1 , given that the beam is steered at  0 ? z (t )  a ( 1 ) s (t ) (model for observed data) Instead of sˆ(t ) we will use y (t ) y (t )  w H z (t )  a H ( 0 ) a (1 ) s (t ) For a fixed w , the function w H a ( )  W ( ) is called the beampattern. It is like the frequency response of an LTI, where  plays the role of frequency. An ideal BF is one for which W ( 0 )  1 at some “look direction”  0 , and is small for all other  . Example: (Uniform Linear Array, ULA) Beamformer design 6 The problem of choosing a beamformer weight vector w , to meet certain criteria placed on W ( ), is much like FIR filter design. For ULAs, the two problems are equivalent. Engineering trade‐off: main lobe width (BW) vs. side lobe level. Often one can use w  ha ( ) , where h is a “window.” The beampattern, viewed as a function of  , is the Fourier transform of h , shifted to  0 w( ) 
M 1
h e  e
j i
i 0
 j 0 i
i
 H (   0 ) . 3. Optimal Beamforming Conventional beamforming – signal add constructively, noise adds destructively, SNR improvement (array gain) = M when Rnn   n2 I . When Rnn   n2 I , we will need to do a little bit more. Model: z  a ( ) s  n z = M  1 observation vector a ( ) = direction vector (known) n = noise vector ~ CN (0, Rnn ) , known and independent of s . Four main approaches: 1) s ~ CN (0,  s2 ) , find Minimum Mean Square Error (MMSE) of s . 

2) Minimize E | w H z |2 such that w H a ( )  1 . 3) Maximize SNR. 4) Find maximum likelihood (ML) estimate of s . Problem 1 z  a ( ) s  n s and n are independent and their parameters  s2 and Rnn are known n ~ CN (0, Rnn ) s ~ CN (0,  s2 ) Thus, z ~ CN (0 , s2 a( )a H ( )  Rnn ) 7 Note that any estimate is a function of observed data: sˆ  f ( z ) . The MMSE estimate of a parameter is obtained by minimizing E{| s  sˆ | 2 } The solution is Wiener filter: sˆ  Rsz R zz1 z Rsz  E{sz H }  E{s ( s * a H ( )  w H )}   s2 a H ( ) Therefore, sˆMMSE  w H z , where w   s2 R zz1 a ( ) . Comment: this estimate requires knowledge of  s2 . Also, solution is biased (conditional on s ): E{| sˆ | 2 }  E{| s | 2 } , in general. Problem 2: Minimum Variance Distortionless Response (MVDR) beamformer Choose weight vector to minimize BF output power such that w H a ( )  1 . Note that the output of beamformer is ŝ . We always wish for an unbiased estimate with minimum variance. Since s and n are independent, w H z  w H a ( ) s  w H n  s  w H n and E{| w H z | 2 }  E{| s | 2 }  E{| w H n | 2 }  a zero cross term This is a constrained optimization problem with an equality constrained. The Method of Lagrange Multipliers is used to find the solution to the problem. min E{| w H z | 2 } such that w H a ( )  1 . The Langrangian is given as w
J (w)  w H Rzz w  2 Re{ ( w H a ( )  1)}  w H Rzz w   ( w H a ( )  1)  * (a H ( ) w  1) Differentiating with respect to w H yields J
 R zz w  a ( )  0 w H
The optimal beam former weights: 8 w  Rzz1 a ( ) . To find the Lagrange multiplier,  , use the equality constraint:  * a H ( ) Rzz1a ( )  1 resulting in: *    
1
. a ( ) R zz1 a ( )
H
The optimal weights are wMVDR 
R zz1 a ( )
a H ( ) R zz1 a ( )
and the MVDR signal estimate: sˆ MVDR 
a H ( ) R zz1 z
a H ( ) R zz1 a ( )
Comments: (1) The estimate depends on R zz and not on  s2 or Rnn (2) R zz can be estimated from data (adaptive beamforming) (3) ŝMVDR can be written as sˆMVDR 
(a H ( ) R zz1 / 2 )( R zz1 / 2 z )
, (a H ( ) R zz1 / 2 )( R zz1 / 2 a ( ))
where Rzz1/ 2 z ‐ whitened observation and Rzz1 / 2 a ( ) ‐ direction vector in whitened observation space. Problem 3: Max SNR E{| w H a ( ) s | 2 } E{w H a ( ) ss * a H ( ) w}  s2 | w H a ( ) | 2
SNR 


E{| w H n | 2 }
E{w H nn H w}
w H Rnn w
1/ 2 1/ 2
1/ 2
Let Rnn  Rnn
Rnn . Define u  Rnn
w or w  Rnn1 / 2 u , then u H Rnn1 / 2 a ( ) a H ( ) Rnn1 / 2 u
uHu
This is maximized when u  kRnn1 / 2 a ( ) (arbitrary k ) SNR   s2
w  Rnn1 / 2 u  kRnn1a ( ) 9 sˆ MSNR  ka H ( ) Rnn1 z Problem 4: Maximum Likelihood (ML) Suppose s is unknown but deterministic. The probability density function (also known as likelihood function) of the observe data is L( z : s)    M (det Rnn ) 1 exp(( z  a ( ) s) H Rnn1 ( z  a ( ) s)) and the log‐likelihood: l ( z : s)  ( z  a ( ) s) H Rnn1 ( z  a ( ) s )  const. Keeping the terms that involve s l ( z : s)  s * a H ( ) Rnn1 a ( ) s  2 Re{z H Rnn1 a ( ) s} l
 2a H ( ) Rnn1 z  2a H ( ) Rnn1 a ( ) s  0 s
a H ( ) Rnn1 z
Rnn1 a ( )
sˆML  H
w
and 
ML
a ( ) Rnn1 a ( )
a H ( ) Rnn1 a ( )
Same as ŝ MSNR when k 
1
a ( ) Rnn1 a ( )
H
Note: MMSE, MVDR depend on R zz MSNR, ML depend on Rnn R zz  Rnn   s2 a ( )a H ( ) . Using Matrix Inversion Lemma, [ Rnn   s2 aa H ] 1  Rnn1 
 s2 Rnn1 aa H Rnn1
1   s2 a H Rnn1 a
Rnn1 a
R zz1 a
we can arrive to the following result H 1  H 1 and thus a R zz a a Rnn a
10 sˆMVDR  sˆML All four solutions for w are the same to within a constant. References: 1. Van Trees, H., [Detection, Estimation, and Modulation Theory, Part IV, Optimum Array Processing],
John Wiley and Sons (2002). 2. Muirhead, R. J., [Aspects of Multivariate Statistical Theory], Wiley Interscience, New York (2005).
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