How can I measure the harmonicity of an audio spectrum?

Question

I would like to estimate the "harmonicity" (harmonics-to-noise ratio ?) of an audio signal from its spectrum. What kind of algorithm can I use?

Answer 1

Please take a look at this previous answer. Using the semantics and mathematical definitions from that answer...

Let $x[n]$ be the input signal from which you want to measure harmonicity or periodicity. Assume $x[n]$ has no DC component (like it is the output of a DC blocking filter). And $n_0$ is the sample index of the neighborhood of signal around where you want the harmonicity to be measured. $N$ is a large integer defining how wide your rectangular window is.

Average Squared Difference Function, ASDF:

$$ Q_x[k, n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} \left(x[n+n_0-\left\lfloor \tfrac{N+k}{2}\right\rfloor] \ - \ x[n+n_0-\left\lfloor \tfrac{N+k}{2}\right\rfloor + k] \right)^2 $$

$\left\lfloor \cdot \right\rfloor$ is the floor() function and, if $k$ is even then $ \left\lfloor \frac{k}{2}\right\rfloor = \left\lfloor \frac{k+1}{2}\right\rfloor = \frac{k}{2} $.

From that define an alternative form of Autocorrelation:

$$ R_x[k,n_0] = R_x[0,n_0] - \frac12 Q_x[k, n_0] $$

where

$$ R_x[0, n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} (x[n+n_0-\left\lfloor \tfrac{N}{2}\right\rfloor])^2 $$

Since $Q_x[0, n_0] = 0$ and $Q_x[k, n_0] \ge 0$ for all lags $k$, that means that $ R_x[k, n_0] \le R_x[0, n_0] $ for all lags $k$.

You will find a period $P$ such that $P > 0$ and $R_x[P, n_0] > R_x[k, n_0]$ for all lags $k>k_0$, other than the highest peak at $k=0$.

$k_0$ is the smallest value of $k>0$ such that $ R_x[k, n_0]<0 $.

Once that peak location $P$ is found, the harmonicity is

$$ r_x(n_0) \triangleq \frac{R_x[P, n_0]}{R_x[0, n_0]} $$

If $r_x(n_0) = 1$, then $x[n]$ is perfectly periodic (or "harmonic") in the vicinity of sample index $n_0$. If $r_x(n_0) = 0$, it's noise, no periodicity to be found.

Answer 2

To answer the broad question "what algorithms you could use": There are different ways of computing an harmonics-to-noise ratio, in the time or in frequency, from basic interval cutting in the frequency domain to more involved cepstral/liftering techniques with local peak detection, using a limited number of partials, etc. Advantages and drawbacks of different methods are described for instance in Temporal and spectral estimations of harmonics-to-noise ratio in human voice signals, Qi and Hillman, 1997

Other potential sources :

• Improvements in estimating the harmonics-to-noise ratio of the voice
• An analysis of iterative algorithm for estimation of harmonics-to-noise ratio in speech

From the sources above, you have possibilities to make your questions more precise, and help contributors to provide more accurate contributions.