that the filter is the mirrored signal model. This ensures that the operation to be applied in order to find the optimum is indeed the convolution between the observed sequence and the matched filter . The filtered sequence assumes its maximum at the position where the observed sequence best matches (in a least-squares sense) the signal model . Implications The matched filter may be derived in a variety of ways, but as a special case of a least-squares procedure it may also be interpreted as a maximum likelihood method in the context of a (coloured) Gaussian noise model and the associated Whittle likelihood. If the transmitted signal possessed no unknown parameters (like time-of-arrival, amplitude,...), then the matched filter would, according to the Neyman–Pearson lemma, minimize the error probability. However, since the exact signal generally is determined by unknown parameters that effectively are estimated (or fitted) in the filtering process, the matched filter constitutes a generalized maximum likelihood (test-) statistic. The filtered time series may then be interpreted as (proportional to) the profile likelihood, the maximized conditional likelihood as a function of the time parameter. This implies in particular that the error probability (in the sense of Neyman and Pearson, i.e., concerning maximization of the detection probability for a given false-alarm probability) is not necessarily optimal. What is commonly referred to as the Signal-to-noise ratio (SNR), which is supposed to be maximized by a matched filter, in this context corresponds to , where is the (conditionally) maximized likelihood ratio. The construction of the matched filter is based on a known noise spectrum. In reality, however, the noise spectrum is usually estimated from data and hence only known up to a limited precision. For the case of an uncertain spectrum, the matched filter may be generalized to a more robust iterative procedure with favourable properties also in non-Gaussian noise. Frequency-domain interpretation When viewed in the frequency domain, it is evident that the matched filter applies the greatest weighting to spectral components exhibiting the greatest signal-to-noise ratio (i.e., large weight where noise is relatively low, and vice versa). In general this requires a non-flat frequency response, but the associated "distortion" is no cause for concern in situations such as radar and digital communications, where the original waveform is known and the objective is the detection of this signal against the background noise. On the technical side, the matched filter is a weighted least-squares