An (N-1)^{th} order differential beamforming can be utilized to for an endfire beamformer with arbitrary beamwidth by cascading \frac{N (N-1)}{2} first order differential beamformers together to synthesize a single output.Consider  N microphone array as shown in Figure 1:

N microphone array

Figure 1: N microphone array

Instead of performing each first order differential beamformer, it can be shown that each output can be multiplied by a coefficient which follows a binomial expansion pattern. A first order beamformer is obtained by y(w) = -x_1(w) + x_2(w) e^{-jw\frac{d}{c}}, where y(w) is the output and x_i(w) is the input from the microphone i all in frequency domain.  It is easy to show that for an N^{th} order beamformer, the output will be:

y(w) = \sum\limits_{i=0}^{N-1>0} (-1)^{N+i-1} \binom{N}{i} \psi(w)^{i} x_i (w)

\psi(w) = e^{-jw\frac{d}{c}}.

Figure 2 illustrates the differing output beam patterns using different orders but identical settings such as sampling rate and  consecutive microphone distance.  The effective beam-patterns for a frequency of 2kHz is shown in Figure 2 to illustrate spatial noise suppression.

Theoretical spatial filtering beam patterns. 

Figure 2: Theoretical spatial filtering beam patterns.

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