The standard impulse sampler used to convert a continuous signal to a discrete sequence has been well understood since 1924 thanks to Harry Nyquist. Its main weakness is that all the signal information between the sample instants is lost. As a result, the original signal cannot be faithfully reconstructed if it contains energy at frequencies above 1/2 the sampling frequency because of aliasing. These aliased frequencies are at the full amplitude of the high frequency signal. This is normally reduced or suppressed by a band limiting “anti-alias” filter that reduces the signal to negligible levels above the Nyquist frequency (1/2 the sampling frequency). The anti-alias filter introduces several problems. If a sharp cutoff version, such as an elliptic or Chebyschev, is selected to maximize the signal bandwidth the pulse response is very poor with large overshoots and undershoots and significant pulse width distortion. A Bessel (Gaussian) with more linear phase exhibits much less overshoot and distortion but has a considerably lower bandwidth. Another problem is that changes in sampling frequency have to be tracked by proportional changes in the filter cutoff frequency for consistent performance. This involves either tunable analog filters or digital filters with variable clock rates plus the associated clock anti-alias filters and clock feed-through filters. This adds noise, distortion, and DC offset to the signal chain.
An alternative for applications where pulse fidelity, reduced low frequency aliasing, easy frequency tuning, and wide bandwidth are more important than precise amplitude accuracy is an integrating sampler. The idea is to measure the time integral of the signal over the sampling interval and use the value of the integral as the discrete sample value rather than the amplitude of the signal at the end of the sampling interval. The signal is sequentially routed to different integrators, each of which measures the time integral for one sampling interval. While the next integrator is active on the next interval, the previous integrator output is measured by an A-D converter and then reset to ready it for the following interval. The sampler gain is equal to 1 / R C F. The example uses four integrators that sample the signal and a reference in rotation to compensate for gain variations due to components and sampling frequency. For each cycle the signal value is the ratio of the signal channel to the reference channel readings.
Frequency response for a sample clock of 256. Note that amplitudes above 128 are the amplitudes of the aliased spectrum calculated with an FFT of a sampled sine wave of the given frequency. 
The integrating sampler is completely linear without distortion — an input sine wave produces a single frequency spectrum. The sampler has an inherent frequency roll-off with a notch at multiples of the sampling frequency. Note that low frequency aliases, frequencies in the 230 to 255 range, aliasing to 1 to 26, are strongly reduced by 20 dB or more. At the same time the phase response is perfectly linear and there is negligible pulse distortion compared to a conventional filter. The pulse rise time is a single sample interval yielding a very wide bandwidth.
There are applications such as pattern recognition and edge detection where elimination of low frequency aliases and fast accurate pulse response are more important than spectral purity. In these applications an anti-alias filter can be eliminated to improve bandwidth. An example in two dimensions is the Foveon® sensor used in Sigma cameras. Image sensors integrate the light over the area of each pixel during the exposure time. A standard Bayer digital camera sensor has gaps between pixels in each of the red, green, and blue channels. This is functionally equivalent to standard sampling and gives rise to aliasing effects called moire patterns in images. This requires an anti-aliasing filter for Bayer sensors which reduces sharpness and effective resolution for a given pixel pitch if strong enough to eliminate moire. Moire patterns, most noticeable at high amplitude and low spacial frequencies, are especially problematic when they produce color artifacts. The Foveon® sensor, by contrast, does not have a gap between pixels, does not need a filter to eliminate moire patterns, and has higher resolution for a given pixel pitch.
Another inspiration was a project that requiring a variable sampling rate on multiple anti-aliased programmable gain channels. This was for a data acquisition system to detect failure events where pulse bandwidth and fidelity were of primary importance. The anti-alias tracking filters, programmable gain preamps, and associated control hardware required most of the board space.
Integrating sampler gain is inversely proportional to the sampling frequency. As the frequency is varied a tracking reference maintains the relationship between the signal and reference measurements. This supports direct measurement of large, fast signals at low gain with high sampling rates and measurement of low level, slow signals at high gain and low noise with low sampling rates. Depending upon requirements, this could permit the elimination of programmable filters and programmable gain preamplifiers.