Frequency estimation

Frequency estimation is the process of estimating the complex frequency components of a signal in the presence of noise[1], given assumptions about the number of the components. (This contrasts with spectral density estimation which does not make prior assumptions about the components.)

The most common methods involve identifying the noise subspace to extract these components. The most popular methods of noise subspace based frequency estimation are Pisarenko's Method, MUSIC, the eigenvector solution, and the minimum norm solution.

For example, consider a signal, ${\displaystyle x(n)}$, consisting of a sum of ${\displaystyle p}$ complex exponentials in the presence of white noise, ${\displaystyle w(n)}$. This may be represented as

${\displaystyle x(n)=\sum _{i=1}^{p}A_{i}e^{jn\omega _{i}}+w(n)}$.

Thus, the power spectrum of ${\displaystyle x(n)}$ consists of ${\displaystyle p}$ impulses in addition to the power due to noise.

The noise subspace methods of frequency estimation are based on eigen decomposition of the autocorrelation matrix into a signal subspace and a noise subspace. After these subspaces are identified, a frequency estimation function is used to find the component frequencies from the noise subspace.

Methods of frequency estimation

${\displaystyle {\hat {P}}_{PHD}(e^{j\omega })={\frac {1}{|\mathbf {e} ^{H}\mathbf {v} _{min}|^{2}}}}$
${\displaystyle {\hat {P}}_{MU}(e^{j\omega })={\frac {1}{\sum _{i=p+1}^{M}|{\mathbf {e} }^{H}{\mathbf {v} }_{i}|^{2}}}}$,

Eigenvector Method

${\displaystyle {\hat {P}}_{EV}(e^{j\omega })={\frac {1}{\sum _{i=p+1}^{M}{\frac {1}{\lambda _{i}}}|\mathbf {e} ^{H}\mathbf {v} _{i}|^{2}}}}$

Minimum Norm

${\displaystyle {\hat {P}}_{MN}(e^{j\omega })={\frac {1}{|\mathbf {e} ^{H}\mathbf {a} |^{2}}};\mathbf {a} =\lambda \mathbf {P} _{n}\mathbf {u} _{1}}$

Related techniques

If one only wants to estimate the single loudest frequency, one can use a pitch detection algorithm.

If one wants to know all the (possibly complex) frequency components of a received signal (including transmitted signal and noise), one uses a discrete Fourier transform or some other Fourier-related transform.