In a mixed-signal system (analog and digital), a reconstruction filter, sometimes called an anti-imaging filter, is used to construct a smooth analog signal from a digital input, as in the case of a digital to analog converter (DAC) or other sampled data output device.
Sampled data reconstruction filters
The sampling theorem describes why the input of an ADC requires a low-pass analog electronic filter, called the anti-aliasing filter: the sampled input signal must be bandlimited to prevent aliasing (here meaning waves of higher frequency being recorded as a lower frequency).
For the same reason, the output of a DAC requires a low-pass analog filter, called a reconstruction filter - because the output signal must be bandlimited, to prevent imaging (meaning Fourier coefficients being reconstructed as spurious high-frequency 'mirrors'). This is an implementation of the Whittaker–Shannon interpolation formula.
Ideally, both filters should be brickwall filters, constant phase delay in the pass-band with constant flat frequency response, and zero response from the Nyquist frequency. This can be achieved by a filter with a 'sinc' impulse response.
Implementation
While in theory a DAC outputs a series of discrete Dirac impulses, in practice, a real DAC outputs pulses with finite bandwidth and width. Both idealized Dirac pulses, zero-order held steps and other output pulses, if unfiltered, would contain spurious high-frequency replicas, "or images" of the original bandlimited signal. Thus, the reconstruction filter smooths the waveform to remove image frequencies (copies) above the Nyquist limit. In doing so, it reconstructs the continuous time signal (whether originally sampled, or modelled by digital logic) corresponding to the digital time sequence.
Practical filters have non-flat frequency or phase response in the pass band and incomplete suppression of the signal elsewhere. The ideal 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 – as it would require infinite delay. Consequently, real reconstruction filters typically either allow some energy above the Nyquist rate, attenuate some in-band frequencies, or both. For this reason, oversampling may be used to ensure that frequencies of interest are accurately reproduced without excess energy being emitted out of band.
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 may be sampled at 44.1 kHz. In this case, both audio filters block as much as possible above 22 kHz and pass as much as possible below 20 kHz.
Alternatively, a system may have no reconstruction filter and simply tolerate some energy being wasted reproducing higher frequency images of the primary signal spectrum.
Image processing
In image processing, digital reconstruction filters are used both to recreate images from samples as in medical imaging and for resampling.
A number of comparisons have been made, by various criteria; one observation is that reconstruction can be improved if the derivative of the signal is also known, in addition to the amplitude,
- nearest-neighbor interpolation, with kernel the box filter – for downsampling, this corresponding to averaging;
- bilinear interpolation, with kernel the tent filter;
- bicubic interpolation, with kernel a cubic spline – this latter has a free parameter, with each value of the parameter yielding a different interpolation filter.
These are in increasing order of stopband suppression (anti-aliasing), and decreasing speed
For reconstruction purposes, a variety of kernels are used, many of which can be interpreted as approximating the sinc function, some proprietary, for which opinions are mixed. Evaluation is often subjective, with reactions being varied, and some arguing that at realistic resampling ratios, there is little difference between them, as compared with bicubic, though for higher resampling ratios behavior is more varied.
Wavelet reconstruction filters
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.
- Reconstruction algorithm
- Iterative reconstruction
See also
- Signal processing
- Signal reconstruction