In this discussion I make various ad hoc assumptions about a priori statistics, measurement errors, 

 etc. to produce tractable illustrative examples, but these assumptions should not be construed as being 

 my opinions of the behavior of actual signeds and noise. 



One implicit assumption is that the distribution of noise intensity in azimuth is stationary, and 

 especially, that its expected value is independent of when the measurement is made. While this may be 

 nearly true over a short time interval, it is certainly not true for longer periods. Inasmuch as directional- 

 ity measurements with linear arrays normally must be sequential, attempts to improve ultimate results by 

 increasing the number of measurements inevitably extend the observation period and increasingly violate 

 the implicit stationarity assumption. The consequences of this dilemma will not be explored. 



AMBIGUITY IN A LINEAR ARRAY 



Observations with a Single Array Heading 



Suppose we have a linear horizontal array of omnidirection sensors and two plane waves with the 

 same wave shape arriving from horizontal directions of different azimuths. If the two azimuths axe 

 equally spaced in opposite senses from the axis of the array, the signals received at the receiving ele- 

 ments will be identical. Given only the sensor outputs, there is no signal processing whatsoever that 

 would distinguish one of these plane waves from the other. 



Suppose that the signeils from the sensors are combined in a beam former which produces a narrow 

 main lobe beEim response and low side lobes. Inevitably, the total array response pattern is symmetric 

 vnth respect to the axis of the array. Therefore, for any steering angle other than end fire, each main 

 beam will have a twdn symmetrically displaced to the opposite side of the axis. 



Suppose we ctre given a stationary two-dimensional sound intensity field where the sound intensity 

 received in any direction characterized by the angle is 1(d) (Fig. 1). Suppose, furthermore, that this 

 field is examined with a linear array having a beam former which produces (symmetrically paired) beams 

 of negligible width with negligible side lobes. What is actually measured with such ein instrument? 



The easiest way to answer that question is to divide the function 1(6), representing the field, into a 

 symmetric and an antisymmetric component. Let us define the symmetric part (Fig. 2). 



Fig. 1. Example of a two-dimensional angular 

 intensity distribution 1(8 ) 



Fig. 2. The symmetric part /s(0) of /(9) 



891 



