# How to construct marginal Hilbert spectrum

Suppose $$f$$ is a signal of duration $$t_d$$ sampled each $$0.01 sec$$. After employing empirical mode decomposition (EMD) to $$f$$, $$n$$ number of intrinsic mode functions (IMF) would be extracted. Using Hilbert transform, instantaneous frequencies ($$\omega$$) and instantaneous amplitudes ($$H$$) were calculated. As a result the $$\omega$$ is a matrix with the number of rows equal to length of signal f and number of columns equal to $$n$$ (so is $$H$$). Now I want to construct marginal Hilbert spectrum using $$\omega$$ and $$H$$. I found the following formula in literature: $$h(\omega) = \int_0^{t_d} H(\omega,t)dt$$ What confuses me is that the result of integration is a vector of length n (that is the number of IMFs). On the other hand, the values of $$\omega$$ for each IMF differs from others and they are not increasingly sorted (like what we have in wavelet transform). I hope the above clearly explains my problem.