Cascaded Integrator Comb Filter

A cascaded integrator comb filter , abbreviated to CIC filter (English for cascaded integrator differentiator filter ) is a time and value discrete digital filter in digital signal processing that is used for sample rate conversion between different sample rates . They belong to the group of multi-rate filters. This type of filter was developed by Eugene Hogenauer in 1981 .

function

CIC filters are a special type of FIR filter and can be implemented very advantageously in digital hardware such as FPGAs ( Field Programmable Gate Array ), since all the coefficients of filter “1” and therefore no computationally intensive multiplications are necessary. Only simple addition stages and memories, so-called taps , are used. CIC filters are used to convert digital signals between areas with different sampling rates, which have a fixed sampling rate ratio to one another, and to avoid the mirror spectra that occur or to avoid aliasing .

Since all coefficients in the filter are equal to “1”, however, the adaptation of the filter properties is only possible to a very limited extent. The variants of the filter are limited to the number of individual integration or differentiation stages connected in series, which describe the steepness of the filter, and for the differentiation stages to a different number of delay stages (taps) that influence the bandwidth in very rough steps. In practice, therefore, combinations of CIC filters with other filters are often used in order to obtain the desired transmission properties as the sum of the individual filters.

variants

The CIC filters are divided into interpolation filters and decimation filters.

Interpolation filters: These are used to convert a discrete signal sequence from a low sampling rate to a higher sampling rate. The signal values that result between the input samples due to the higher output rate are interpolated . The mirror spectra that inevitably arise in the output area during the upconversion are suppressed by the CIC filter.

Decimation filters: These filters are used to switch from a high to a low sampling rate. All signal components of the input signal that are above half the output sampling rate must be suppressed by the filter in order to avoid aliasing.

Filter structure

The structure of a CIC filter as used as an interpolator. The schematic switch in the middle of the picture represents the actual sample rate converter, the blocks on the left and right are the filter.

A CIC interpolation filter is shown in the illustration opposite. On the left you can see the individual differentiation stages, implemented as comb filters , which can vary in number. Each level represents a high pass , the frequency response of which increases by approx. 6 dB per octave. Only one storage register (z ⁻¹ ) per stage is shown. An increase to two storage registers (z ⁻² ) would halve the bandwidth. The actual sampling rate converter can then be seen in the center of the image, which in this case converts to a higher sampling rate. During the conversion, intermediate values are replaced by the value 0 . The levels on the right are the integration levels ( low-pass filter ), also different in number depending on the application. These stages perform the signal interpolation for the output sequence with the high sampling rate.

A CIC decimation filter only differs in that the order of the integration levels and the differentiation levels is exchanged.

Transfer function

The transfer function H (z) of a CIC filter, based on the side with the high sampling rate f _s , is:

{\ displaystyle H (z) = {\ frac {(1-z ^ {- R \ cdot M}) ^ {N}} {(1-z ^ {- 1}) ^ {N}}} = \ left [\ sum _ {k = 0} ^ {R \ cdot M-1} z ^ {- k} \ right] ^ {N}}

The parameters mean:

R = decimation or interpolation factor - ratio of the two sampling rates to one another.

M = number of storage registers ( taps ) per stage. This parameter is typically 1, sometimes also 2.

N = number of levels

Individual evidence

↑ Eugene B. Hogenauer: An economical class of digital filters for decimation and interpolation . In: IEEE Transactions on Acoustics, Speech and Signal Processing . 2nd Edition. No. April 29 , 1981, ISSN 0096-3518 , pp. 155-162 , doi : 10.1109 / TASSP.1981.1163535 .

Web links

CIC Filter Introduction

[hogen1-1] Eugene B. Hogenauer: An economical class of digital filters for decimation and interpolation . In: IEEE Transactions on Acoustics, Speech and Signal Processing . 2nd Edition. No. April 29 , 1981, ISSN 0096-3518 , pp. 155-162 , doi : 10.1109 / TASSP.1981.1163535 .