Bounds of biorthogonal decompositions and a lossless modification of signal-to-noise ratio

There are signal processing applications which can benefit from a modified signal-to-noise ratio in the transform domain. In the applications, e.g. a digital watermarking of images, it is the additive information to the signal which could be of interest in an enhanced form. While the value of the signal-to-noise ratio in transform domain relative to the value of the ratio in the original domain is unchanged if an expansion onto an orthogonal basis is used, in expansions onto biorthogonal and nonorthogonal bases the value of the signal-to-noise ratio in transform domain is determined by the characteristics of the signal and noise. The biorthogonal decomposition can therefore improve or deteriorate the SNR depending on the characteristics of the noise. An understanding of the signal processing applications such as de-noising of signals in transform domain, or a wavelet based signal compression in transmission through noisy channels, involves the knowledge of the bounds of the employed decomposition. The inequality which characterizes biorthogonal decompositions using the, bounds is discussed