Associated to an Hadamard matrix $H\in M_N\left(ℂ\right)$ is the spectral measure μ ∈ [0,N] of the corresponding Hopf image algebra, A = C(G) with $G\subset S{⁺}_N$. We study a certain family of discrete measures ${\mu }^r\in [0,N]$, coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type ${\int }_0^N{(x/N)}^{p}d{\mu }^r\left(x\right)={\int }_0^N{(x/N)}^{r}d{\nu }^p\left(x\right)$, where ${\mu }^r,{\nu }^r$ are the truncations of the spectral measures μ,ν associated to $H,H^t$. We also prove, using these truncations ${\mu }^r,{\nu }^r$, that for any deformed Fourier matrix $H=F_M{\otimes }_Q{F}_N$ we have μ = ν.