perfectreconstruction

Perfect reconstruction refers to the property of a signal-processing system that can recover the original continuous-time or discrete-time signal from its transformed representation without any loss of information, assuming ideal conditions. It is a central goal in multi-rate signal processing, sampling theory, and frame/wavelet design.

In multi-rate filter banks, a common setting uses analysis and synthesis filter banks to decompose and reassemble a signal. Perfect reconstruction occurs when the combined effect of analysis and synthesis yields a system that is equivalent to a pure delay and potentially a scale factor, with aliasing canceled. Mathematically, in polyphase form, if the product of the synthesis and analysis polyphase matrices equals a constant times a delay times the identity, P(z)Q(z) = c z^{-k} I, the original signal can be reconstructed exactly from the subband signals.

In sampling and reconstruction, perfect reconstruction means that sampling at a sufficient rate and applying the correct interpolation (e.g., ideal low-pass filtering with sinc response) can recover the original continuous-time signal exactly, ignoring practical limitations.

In practice, achieving perfect reconstruction requires ideal components; in real systems, quantization, finite filter length, and numerical precision introduce errors. Designers seek approximate perfect reconstruction with small distortion, or enforce perfect reconstruction within a chosen class of signals.