  Solving evolutionary equations in a parallel-in-time manner is an attractive
topic and many algorithms are proposed in recent two decades. The algorithm
based on the block $\alpha$-circulant preconditioning technique has shown
promising advantages, especially for wave propagation problems. By fast Fourier
transform for factorizing the involved circulant matrices, the preconditioned
iteration can be computed efficiently via the so-called diagonalization
technique, which yields a direct parallel implementation across all time
levels. In recent years, considerable efforts have been devoted to exploring
the convergence of the preconditioned iteration by studying the spectral radius
of the iteration matrix, and this leads to many case-by-case studies depending
on the used time-integrator. In this paper, we propose a unified convergence
analysis for the algorithm applied to $u'+Au=f$, where
$\sigma(A)\subset\mathbb{C}^+$ with $\sigma(A)$ being the spectrum of
$A\in\mathbb{C}^{m\times m}$. For any one-step method (such as the Runge-Kutta
methods) with stability function $\mathcal{R}(z)$, we prove that the decay rate
of the global error is bounded by $\alpha/(1-\alpha)$, provided the method is
stable, i.e., $\max_{\lambda\in\sigma(A)}|\mathcal{R}(\Delta t\lambda)|\leq1$.
For any linear multistep method, such a bound becomes $c\alpha/(1-c\alpha)$,
where $c\geq1$ is a constant specified by the multistep method itself. Our
proof only relies on the stability of the time-integrator and the estimate is
independent of the step size $\Delta t$ and the spectrum $\sigma(A)$.
