Digital Low Pass Filters

A test series e.g. a raw file) can be viewed as a superposition of signals plus noise. Since a Fourier transform is additive, the frequency spectrum of a measurement is the sum (superposition) of the signal frequency spectrum and the noise frequency spectrum. If a reflection can be fitted by a Pearson VII profile (Chap. 4.7) and if the maximum of this even function is put in x = 0, 

That is, for a steady signal the share of higher frequencies approximates zero whereas the frequency response of the noise stays more or less constant. Therefore, above a certain frequency only the noise will contribute. If one suppresses this high-frequency part by multiplication with 0, after the reverse transform into the amplitude domain the signal is kept unchanged, only slightly disturbed by the low-frequency part of the noise. 

Instead of the application of a low pass filter in the frequency domain itself (multiplication with 1 in the low-frequency range and above that with 0, i.e. multiplication of the frequency response with a rectangle function), one can use in the amplitude domain (i.e. for the measurement itself) the mathematically fully equivalent convolution with the Fourier transform of that rectangle function. 

Numerically the convolution of a step scan is merely the application of a sliding weighted mean (e.g. like the Savitzky-Golay method). The Fourier transform of the rectangular function has the shape of sin(nv)/(nv) (whereby n is inversely proportional to the width of the rectangle) and unfortunately approaches 0 only very slowly. To make do with a small number of points for a convolution, one must tolerate a compromise and renounce the ideal rectangular shape of the low pass filter (in the frequency domain). 

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In that case, after each newly generated derivative, the noise must be eliminated with a digital low-pass filter (Fig. 3-43). 

The signal before digitization is sufficiently low-pass filtered so that noise is below 1 mV at the digital volt meter (DVM). 

The microhotplate temperature is measured using a polysilicon resistor as temperature sensor (Ry). The resistor is biased with a temperature-independent current source (Free). The voltage drop across the resistor is low-pass filtered and converted to the digital domain using a 10-bit successive-approximation ADC from the analog library of austriamicrosystems (Unterpremstatten, Austria). The inputs of the digi-... 

Sample rates for the majority of digital controllers are adjustable and range from 1 sample every 5 s to 10 samples per second. Some controller designs have fixed sample rates that fall within the same range. Hardwired low-pass filters are usually installed on the analog inputs to the controller to help protect the sampler from aliasing errors. 

The ISFETs are connected to a source-drain follower, and the output of this amplifier is connected to a digital oscilloscope. For presentation purposes, the curves are filtered with a software low-pass filter using a cutoff frequency of 40 Hz for elimination of the 50 Hz main supply interference, which has a top-top amplitude of typically 0.2 mV. 

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