Once identified, MUSIC uses the columns of the noise subspace \(\textbf{V}_r\) to identify incident signal directions orthogonal to that subspace. This involves sweeping the steering vector (as a function of the DOA) against these noise subspace basis vectors to compute the strength of the so-called MUSIC pseudo-spectrum given in the following equation. The index \(P\) represents the index of the first noise singular value (assuming they have been sorted in descending order). Also \(\textbf{V}_r(j)\) denotes the \(j\)-th column of \(\textbf{V}_r\). Once again, \(P\) may be assumed known or estimated using various techniques. The \(\times\) operator below represents an inner product. The \(\textbf{s}(k)\) represents the steering vector which is a function of the array manifold and the presumed direction of arrival.
MUSIC may solve the above equation either directly by looking for the peaks of the pseudo-spectrum that occur when the steering vector becomes orthogonal to the noise subspace. This occurs in the preceding equation when its denominator goes to zero. Computing these peaks requires a costly division operator. Instead, the denominator may be inspected instead for its nulls. This generally gives a very similar result but requires no division operator. This tutorial uses the latter approach to reduce the compute workload.