  Echo protocols provide a means to investigate the arrow of time in
macroscopic processes. Starting from a nonequilibrium state, the many-body
quantum system under study is evolved for a certain period of time $\tau$.
Thereafter, an (effective) time reversal is performed that would -- if
implemented perfectly -- take the system back to the initial state after
another time period $\tau$. Typical examples are nuclear magnetic resonance
imaging and polarization echo experiments. The presence of small, uncontrolled
inaccuracies during the backward propagation results in deviations of the "echo
signal" from the original evolution, and can be exploited to quantify the
instability of nonequilibrium states and the irreversibility of the dynamics.
We derive an analytic prediction for the typical dependence of this echo signal
for macroscopic observables on the magnitude of the inaccuracies and on the
duration $\tau$ of the process, and verify it in numerical examples.
