ON MEASURING THE EFFECTIVE NOISE BANDWIDTH OF A FILTER
“The noise bandwidth of a non-ideal brick wall filter is defined as the bandwidth of an ideal brick wall filter which will pass the same noise power as the non-ideal filter.”1
Noise Bandwidth can be mathematically defined 2 as:
For convenience, noise bandwidth (“nBW”) is used in this note interchangeably with “effective noise bandwidth”. Note that the noise bandwidth is not the same as the -3 dB bandwidth. For some filters a good approximation of the nBW can be made from the -3 dB bandwidth by a simple multiplier. For example, the nBW of a simple single pole RC filter is approximately , or 1.57 times the -3 dB bandwidth.1,3
The -3 dB bandwidth of a filter can generally be measured by simply applying a constant input amplitude signal to the filter under test and watching with an appropriate measuring instrument for the frequencies where there is a 3 dB drop off on either side of the response curve.
Specifically with respect to making Johnson noise measurements, it can be important to define the noise bandwidth of the filter in use. As noted in our JCan experiment, one method is to assume the validity of the Johnson equation and then to iteratively determine a nBW value by measuring the amplifier gain and then slewing the nBW figure until the experimental resistor noise data (with no bias current) overlays a theoretical Johnson noise curve. In fact, this method turns out to be a relatively easy way to determine the nBW of a filter.
However, during the development of the JCan project, an independent determination of nBW was needed to verify the above method and to help to validate the overall experiment. It quickly became apparent that one way to accomplish the integral was through piecewise integration. That is taking filter response measurements at small frequency steps and then multiplying the delta f times the average of the response at the two frequency ends of the delta f to determine the area of a vertical “sliver” of the filter response curve. Note that the area is determined from the voltage squared for each sliver. The generator can be set so that maximum peak output is 1.000 V, or all of the values can be easily normalized for a measured peak value. Also, where a measurement is limited by, for example, the max frequency response of an AC voltmeter, an approximation can be made for missed area at the very end of the response curve.
One way to measure the piece wise response of a filter is with an automated measurement system, such as GPIB instruments run from National Instruments’ LabView computer program. While this sort of automated test setup is particularly convenient, there is no reason the data cannot be taken by manually setting instruments and writing down the results. Our data acquisition setup used a hp 3325b signal generator and an Agilent 34410A DMM to measure the AC voltage output. The 34410A ACV response is to 300 KHz. We used 100 points since automated collection made the process relatively simple. Where data is being manually recorded, as few as 15 to 20 points can still give good results.
Data from the LabView run was saved to a file suitable for Excel. This was done since many experimenters do not have access to LabView, a relatively expensive package. Otherwise, the entire analysis could have easily been completely done within the LabView environment.
The sample Excel work sheet shows the analysis method. Amplitude response data is entered for each frequency. Then, the frequency difference (delta f) is multiplied by the square of the average voltage over the delta f frequency interval by summing the values at the end points and dividing by 2. The voltage values are normalized by dividing all voltages by the highest value in the response curve. Summing all the piecewise integrations (the small areas of each “sliver”) then gives the area under the entire voltage squared response curve or the nBW of the filter.
2. Jacob Millman and Christos Halkias, “Integrated Electronics: Analog and Digital Circuits and Systems”, Columbia University, 1977, page 402.
3. TI Application Bulletin, Noise Analysis Of Fet Transimpedance Amplifiers, page 4; See also the table in the webpage of footnote 1 .
COPYRIGHT © 2007 JOSEPH M. GELLER