How do subspace methods work for frequency and DOA estimation?

Subspace methods exploit eigenstructure of signal covariance matrix. Principle: Covariance matrix R = signal subspace + noise subspace (orthogonal). MUSIC: Find noise subspace eigenvectors, search for frequencies/angles where steering vector is orthogonal to noise subspace (pseudospectrum peaks). ESPRIT: Uses shift-invariance structure, estimates frequencies from signal subspace rotation. Advantages: Super-resolution (resolves closely spaced components), statistical consistency. Limitations: Requires number of sources known (or estimated via MDL/AIC), assumes uncorrelated sources, and noise must be white. Applications: Radar DOA, spectral analysis, and communications channel estimation.