  We introduce a wavelet-based model of local stationarity. This model enlarges
the class of locally stationary wavelet processes and contains processes whose
spectral density function may change very suddenly in time. A notion of
time-varying wavelet spectrum is uniquely defined as a wavelet-type transform
of the autocovariance function with respect to so-called autocorrelation
wavelets. This leads to a natural representation of the autocovariance which is
localized on scales. We propose a pointwise adaptive estimator of the
time-varying spectrum. The behavior of the estimator studied in homogeneous and
inhomogeneous regions of the wavelet spectrum.
