Reconstruction filter
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In a mixed-signal system (analog and digital), a reconstruction filter (or anti-imaging filter) is used to construct a smooth analogue signal from the output of a digital to analogue converter (DAC) or other sampled data output device.
Whilst in theory a DAC gives a series of impulses, in practice, the output of a DAC is more typically a series of stair-steps. The low pass reconstruction filter smooths the stair step (removes the harmonics above the Nyquist limit) to (re)construct the analogue signal corresponding to the digital time sequence.
The sampling theorem describes why the input of an ADC requires a low-pass analog electronic filter, called the anti-aliasing filter. For the same reason, the output of a DAC requires a low-pass analog filter, called a reconstruction filter. Ideally, both filters should be brick-wall filters, constant phase delay in the pass-band with constant flat frequency response, and zero response from the Nyquist frequency. This is given by a filter with a 'sinc' impulse response.
Practical filters have non-flat frequency or phase response in the pass band and incomplete suppression of the signal elsewhere, as a sinc waveform has an infinite response to a signal, in both the positive and negative time directions, which is impossible to perform in real time.
In systems that have both, the anti-aliasing filter and a reconstruction filter may be of identical design. For example, both the input and the output for audio equipment is sampled at 44.1 kHz. Both audio filters block as much as possible above 22 kHz and pass as much as possible below 20 kHz. Typically both filters are active op-amp filters, with exactly the same selection of resistors and capacitors.
Reconstruction filters are also used when "reconstructing" a waveform or an image from a collection of wavelet coefficients. In medical imaging, a common technique is to use a number of 2D X-ray photos or MRI scans to "reconstruct" a 3D image.
