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Appendix F: Effective Noise Bandwidth of Analog RC Filters and the Selection of Filter Parameters to Optimize Signal-to-Noise Ratio
Engineering calculations involving noise and signal-to-noise ratio need to use the effective noise bandwidth (ENB) in order to calculate noise properly. Often the conventional (–3-dB voltage, –6-dB power) bandwidth is used, leading to erroneous results. When the difference between –3-dB bandwidth and ENB is recognized, it is often oversimplified by attempting to relate ENB to the –3-dB bandwidth. Table F.1 shows several of these relationships found in the open and corporate literature. These discrepancies, while not extremely serious, are disconcerting, particularly for the two-section filter, which is readily realizable with a single operational amplifier and a handful of R and C components. So I set off to find out which values are correct. This was accomplished by means of simple spreadsheet analysis and BASIC computer programs to do the necessary integrations. There are many multiple-pole filter types in the literature. This appendix is limited to simple Butterworth RC filters (maximally flat-frequency response), where each section has the same R and C and therefore the same cutoff frequency. The primary emphasis is limited to “white” noise with a uniform power spectral density. We conclude with a simple bandwidth optimization to maximize the SNR of a single-frequency signal in the presence of white noise.
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