I'm reading the original paper on PSOLA (Pitch-Synchronous Overlap-Add), and am struggling with equation (2), in which the authors analysed the effect of a simple pitch-and-time-scale modification on a stationary noise process $n(m)$, which is super-positioned on a periodic signal $d(m)$ with period $P$.
Here, the analysis window size can be assumed to be integer multiples of $P$, with hop size $P$, with the constant overlap assumption satisfied.
Denoting the pitch modification factor by $\beta$, the time-scale modification factor by $\gamma := \beta$, the original noise process by $n(m)$, the analysis window by $h(m)$ and the synthesis noise signal by $\tilde{n}(m)$, equation (2) states: \begin{align} \tilde{n}(m) &= \sum_s h(s\beta P - m)n(m - s \beta P + s P) \tag{2a} \\ &= \tilde{R}(\tau) = \frac{1}{\beta P} \sum_{k=0}^{\beta P - 1} E(\tilde{n}(k)\tilde{n}(k-\tau)) \tag{2b} \\ \tilde{R}(\tau) &= \frac{1}{\beta P} \sum_{m} \rho(\tau - m\beta P) R(\tau + m(1- \beta P)). \tag{2c} \end{align} where $\tilde{R}(\tau)$ and $R(\tau)$ denote the original and synthesized noises' auto-correlation, and $\rho(\tau):=h(t)*h(-t)$ is the analysis window auto-correlation.
Indeed all three of the equalities stated here baffled me. Specifically,
- In (2a), why is the window $h(.)$ flipped such that it's not $h(m - s\beta P)$?
- In (2a), why is the noise term $n(m - s \beta P + sP)$ and not just $n(m - s\beta P)$?
- Given that we are modifying both the pitch and time by the same factor, why don't we simply have $\tilde{n}(m)=n(m/\beta)$ and indeed $\tilde{d}(m)=d(m/\beta)$?
- In (2b), Why is the synthesized noise $\tilde{n}(R)$ equal to the autocorrelation $\tilde{R}(\tau)$?
- How does (2c) come about?
Note that the paper provides the following footnote to explain this equation:
$\tilde{n}(m)$ is not a WSS (Wide sense stationary) process, but a WS (Wide-sense) cyclostationary process. This means that the autocorrelation sequence $R(t_1, t_2) = E(\tilde{n}(t_1)\tilde{n}(t_2))$ is not invariant on the diagonal of $t_1-t_2$ plane, but is nevertheless periodic=: $R(t_1 + m\beta P, t_2 + m\beta P) = R(t_1, t_2)$. However, there is a close relationship between WS stationary and WS cyclostationary processes: it can be demonstrated that if the time origin is subject to a random shift $\theta(t)$ that varies sufficiently slowly, then the resulting process $\tilde{n}(t)=\tilde{n}(t - \theta{t}))$ is WS stationary (Papoulis, 1984). The autocorrelation sequence of this process is equal to the "time-average autocorrelation" of process $\tilde{n}(m)$.
which is still cryptic to me.
Reference
Papoulise, A. (1984). Probability, Random Variables and Stochastic Processes, McGraw-Hill, NY
