In almost all applications of beamforming, there is a requirement to form a null in a articular direction corresponding to noise sources like fans, air conditioners or heating vents. An $m^{th}$ order differential beamforming is possible if the null direction is known apriori, $m < N-1$. It should be noted that even though differential beamforming produces narrow beamwidth pickups, the parameters can be chosen to ensure that all the desired beam directions are picked up without undue attenuation.

Consider a far field source impinging N UCA microphones as shown in Figure 1:

Figure 1: N UCA microphones

For example, for a $16$ microphone, circular array, a $9^th$ order differential array can be used to produce the beam pattern shown in Figure 2 below.

Figure 2: Beam pattern at $\sim 2kHz$

As shown in Figure 2, the beamwidth is about $240^{\circ}$ whilst a deep null is formed at the desired direction.

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