Sinusoidal Amplitude and Phase Estimation

The form of the optimal estimator (4.12) immediately suggests the following generalization for the case of unknown amplitude and phase:

That is, is given by the complex coefficient of projection [248] of onto the complex sinusoid at the known frequency . This can be shown by generalizing the previous derivation, but here we will derive it using the more enlightened orthogonality principle [109].

The orthogonality principle for linear least squares estimation states that the projection error must be orthogonal to the model. That is, if is our optimal signal model (viewed now as an -vector in ), then we must have [248]

Thus, the complex coefficient of projection of onto is given by

The optimality of in the least squares sense follows from the least-squares optimality of orthogonal projection [109,115,234]. From a geometrical point of view, referring to Fig.4.22, we say that the minimum distance from a vector to some lower-dimensional subspace , where (here for one complex sinusoid) may be found by ``dropping a perpendicular'' from to the subspace. The point at the foot of the perpendicular is the point within the subspace closest to in Euclidean distance.

figure[htbp]

---

Previous: Sinusoidal Amplitude Estimation
Next: Sinusoidal Frequency Estimation

Order a Hardcopy of Spectral Audio Signal Processing

---

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.