We consider the problem of detecting additive structured correlated perturbations affecting the measurement outputs of a system whose state is estimated by a Kalman filter. We advocate the time series of the gradients of the loglikelihood with respect to the output measurements as an indicator, notably through its fast Fourier transform (FFT). This provides a novel unifying method to detect structured perturbations, namely small sinusoidal perturbations with unknown frequency, and slowly growing errors, such as a ramp, or more generally any known incipient profile with unknown starting time. The method allows for identification of their parameters too, i.e., frequency, and starting time of the ramp. Thanks to recent results on backpropagation in Kalman filters, and the use of the FFT, the method remains numerically tractable even for large datasets, as demonstrated by simulations.