thus truncating the series yields a low-pass filtered version of the series: Notably, the middle approximation (2-term) differs. From the frequency domain perspective, this is a better approximation, but from the time domain perspective it has drawbacks – it exhibits undershoot – one of the values is negative, though the original series is non-negative everywhere – and ringing, where the right side is non-zero, unlike in the wavelet transform. On the other hand, the Fourier approximation correctly shows a peak, and all points are within of their correct value, though all points have error. The wavelet approximation, by contrast, places a peak on the left half, but has no peak at the first point, and while it is exactly correct for half the values (reflecting location), it has an error of for the other values. This illustrates the kinds of trade-offs between these transforms, and how in some respects the DWT provides preferable behavior, particularly for the modeling of transients. Definition One level of the transform The DWT of a signal is calculated by passing it through a series of filters. First the samples are passed through a low-pass filter with impulse response resulting in a convolution of the two: The signal is also decomposed simultaneously using a high-pass filter . The outputs give the detail coefficients (from the high-pass filter) and approximation coefficients (from the low-pass). It is important that the two filters are related to each other and they are known as a quadrature mirror filter. However, since half the frequencies of the signal have now been removed, half the samples can be discarded according to Nyquist's rule. The filter output of the low-pass filter in the diagram above is then subsampled by 2 and further processed by passing it again through a new low-pass filter and a high- pass filter with half the cut-off frequency of the previous one, i.e.: This decomposition has halved the time resolution since only half of each filter output characterises the signal. However, each output has half the frequency band of the input, so the frequency resolution has been doubled. With the subsampling operator the above summation can be written more concisely. However computing a complete convolution with subsequent downsampling would waste computation time. The Lifting scheme is an optimization where these two computations are interleaved. Cascading and filter banks This decomposition is repeated to further increase the frequency resolution and the