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I was confused about the double hat notation $\hat{\hat .}$ and my teacher and I thought it probably means an estimate of an estimate most of the time, but we could not be 100% sure of its meaning.

One hat $\hat .$ is an estimate but what does this double hat usually mean? Does it have many different meanings in DSP? Can it mean an estimate of unit vector sometimes? Does it have other signs whatever it means?

Example

You can see such double hat on page such as 213 in Prentice Hall - Modeling Of Dynamic Systems book. In the chapter on Periodograms: Blackman-Tukey's Method, they use this kind of spectral estimate for N length function:

$$\hat\phi_N(\omega)=\int_{-\pi}^{\pi}W_\gamma (\omega-\xi)\hat{\hat{\phi}}_N(\xi)d\xi$$

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It seems this notation is used for empirical transfer function estimates (see pg. 207, 8.4 Fourier Analysis). To wit, on pg. 79 of Modeling: identification and simulation of dynamical systems we have

The double hat denotes that the ETFE is a very rough estimate of the transfer function.

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  • $\begingroup$ I guess an estimate of an estimate would constitute a very rough estimate which seems consistent with the proposed definition in the original question. $\endgroup$
    – Bruce Zenone
    Commented Jan 9, 2013 at 22:37

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