# Can I "reverse" spectral leakage by solving a linear system?

When I compute a spectrogram of (say) a piece of music, there is a lot of frequency "smearing." Often we can reasonably expect that the "true" generating process is much sparser in frequency (i.e. maybe 10-20 active frequencies instead of hundreds).

We also know that plain old sine waves become smeared in spectrograms.

So, can't we model the corrupting "frequency leakage" at a specific timestep as a linear equation $$Ax = b$$ Where

• $$b$$ is the known vector representing the STFFT-computed spectral densities at the various frequencies
• $$x$$ is the unknown, sparse vector of generating frequencies
• $$A$$ is the (computable) "smearing matrix" which maps power from sparse generating frequencies to the "smeared" set of frequencies which STTFT produces (We can just compute this by taking the STFFT of various sine wave snippets)

And naturally one could extend this to further encourage sparsity, e.g. $$\min_{x} \|Ax - b\| + λ \|x\|_1$$

I have not seen this approach discussed, but I am very new to DSP. Is this a known technique, or is there some reason why it would not work in "sharpening" a spectrogram?

• FFT spectral leakage shouldn't be conceptually confused with STFT boundary effects, though they're sometimes related. Synchrosqueezing may be of interest. Commented Jun 28, 2022 at 17:47
• @OverLordGoldDragon thank you for the pointer, this is a beautiful exposition (have only read a bit of it thus far). You are right that I am conflating fft leakage and boundary effects, will have to look into this more. I had come across reassignment methods but this seems a clearer intro than others I've seen. Commented Jun 28, 2022 at 19:34
• @MichaelKayser The assumption that music is spectrally "sparse" is highly questionable. Can you clarify whether you want to work with actual music or that you know for sure that just have a sum of sine waves? Commented Jun 28, 2022 at 21:03
• @Hilmar fair point. I've been looking at simple stuff like solo instruments (e.g. a short piano piece or cello suite). Even this may have weird parts e.g. when notes are being struck, but for now I'm seeing how much mileage I can get with a sparsity assumption which I expect does explain a fair percentage of such recordings. Very open to feedback based on your experience though. Commented Jun 28, 2022 at 23:54
• I'm doubtful. IMO the sparsity assumption will discard a lot of important details. This may be ok if you are ONLY interested in pitch detection but if want to maintain the difference between, say, a clarinet and a violin this is unlikely to work. Commented Jun 30, 2022 at 17:15

This looks sort of like the standard approach to the maximum likelihood estimator for tones.

Kay (1) starts with the signal $$x[t] = A \cos(\omega_0 t + \phi) + \epsilon[t]$$ and assumes $$\epsilon[t]$$ is an independent, identically distributed Gaussian noise source with variance $$\sigma_\epsilon^2$$.

The likelihood function for this is: \begin{align} \newcommand{\bx}{\mathbf{x}} p(\bx ; \mathbf{\theta}) = \frac{1}{(2\pi \sigma_\epsilon^2)^{T/2} } \exp\Big( -\frac{1}{2\sigma_\epsilon^2} \sum_{t=0}^{T-1} \big(x[t] - \tilde{A} \cos(\tilde{\omega}_0(t-\nu) + \tilde{\phi}) \big)^2 \Big) \end{align} where our parameters of interest are $$\mathbf{\theta} = \Big [ \tilde{A}, \tilde{\omega}_0, \tilde{\phi} \Big ]^T$$.

Using the log-likelihood, and forming the vector version gives: \newcommand{\bc}{\mathbf{c}} \newcommand{\bs}{\mathbf{s}} \newcommand{\bH}{\mathbf{H}} \begin{align} L(\theta') &= \frac{T}{2}\log_e(2\pi \sigma_\epsilon^2) + (\bx -\tilde{\alpha}_c \bc - \tilde{\alpha}_s \bs )^T (\bx -\tilde{\alpha}_c \bc - \tilde{\alpha}_s \bs )\\ &= \frac{T}{2}\log_e(2\pi \sigma_\epsilon^2) + (\bx - \bH\underline{\alpha} )^T (\bx - \bH \underline{\alpha} ) \end{align} where \begin{align} \bx &= \left [ x[0], x[1], \ldots, x[T-1] \right]^T\\ \bc &= \left [ \cos\left(-\tilde{\omega}_0 \frac{T-1}{2}\right), \ldots , \cos(-\frac{\tilde{\omega}_0}{2}), \cos(\frac{\tilde{\omega}_0}{2}), \ldots \cos\left(\tilde{\omega}_0 \frac{T-1}{2}\right) \right]^T\\ \bs &= \left [ \sin\left(-\tilde{\omega}_0 \frac{T-1}{2}\right), \ldots , \sin(-\frac{\tilde{\omega}_0}{2}), \sin(\frac{\tilde{\omega}_0}{2}), \ldots \sin\left(\tilde{\omega}_0 \frac{T-1}{2}\right) \right]^T\\ \bH &= \left [ \bc \ \bs \right ]\\ \underline{\alpha} &= \left[ \tilde{\alpha}_c\ \tilde{\alpha}_s\right ]^T\\ \tilde{\alpha}_c &= \tilde{A} \cos \phi\\ \tilde{\alpha}_s &= -\tilde{A} \sin \phi \end{align} assuming $$T$$ is even.

Like I said, sort of. The $$\underline{\alpha}$$ here is analogous to your $$x$$, but $$\underline{\alpha}$$ is in no way sparse.

Interesting question. I'll think about it some more.

(1) S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall, 1997.