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This is a problem from the book Discrete-Time Speech Signal Processing by Thomas Quatieri, Chapter 6 Problem 13:
Suppose $h[n]$ is all-pole (minimum-phase) of order $p$. Show that the all-pole (predictor) coefficients can be obtained recursively from the first $p$ complex cepstral coefficients of $h[n]$.
I was only able to solve for the opposite. i.e. complex cepstral coefficients from the pole coefficients.
If $h[n]$ is the impulse response of an all-pole minimum-phase filter of order $p$, then we have
$$\mathcal{Z}\{h[n]\}=H(z)=\frac{G}{1-\displaystyle\sum_{k=1}^pa_kz^{-k}}\tag{1}$$
where $G$ is some gain value, and $a_k$ are the predictor coefficients. Let $\hat{h}[n]$ be the complex cepstral coefficients of $h[n]$. If you know how to compute $\hat{h}[n]$ from $a_k$, then you're probably familiar with the following recursion:
$$\hat{h}[n]=a_n+\frac{1}{n}\sum_{k=1}^{n-1}ka_{n-k}\hat{h}[k],\quad 0<n\le p\tag{2}$$
From $(2)$ you can formulate a recursion for $a_k$:
$$a_n=\hat{h}[n]-\frac{1}{n}\sum_{k=1}^{n-1}ka_{n-k}\hat{h}[k],\quad 1<n\le p\tag{3}$$
with the initial value $a_1=\hat{h}[1]$.
