vibration laboratory with a single-stage vibration isolation for microscopy applications”

The construction and the vibrational performance of a low vibration laboratory for microscopy applications comprising a 100 ton floating foundation supported by passive pneumatic isolators (air springs), which rest themselves on a 200 ton solid base plate, are discussed. The optimization of the air spring system leads to a vibration level on the floating floor below that induced by an acceleration of 10 ng for most frequencies. Additional acoustic and electromagnetic isolation is accomplished by a room-in-room concept.

In our publication entitled "Low vibration laboratory with a single-stage vibration isolation for microscopy applications" 1 we have erroneously labeled the measured spectrum N spec of the vibrations as the spectral density N PSD ≡ N v of the vibrations in our laboratory . Due to this, the measured vibration levels (spectral densities) shown in Figs While experts in spectral analysis will be aware of the difference between a spectrum and the spectral density, scientists using spectral analysis only occasionally may not be aware of the difference between both. Thus we discuss, in the following, this difference and how the two can be converted into each other. Moreover, we discuss in which circumstances the one or the other should be used, and how the calibration of a spectrum or a spectral density is experimentally verified.
Nowadays, when spectrum analyzer instruments (with all the calibration steps already included) are less frequently used in favor of analog to digital conversion of the measured signal followed by a subsequent software discrete Fourier transform (DFT), the correct calibration is no more "included" by the spectrum analyzer hardware. Since the discrete Fourier transform just transforms n numbers to n new numbers, the user has to take care about the necessary calibration steps, and errors may occur in this calibration. While this is straightforward in principle, it involves a number of non-trivial details. Here, we also include the description of an experimental calibration procedure which gives an easy cross-check for the correct calibration of the spectrum or spectral density. We realized that the above mentioned information is not easily found in the literature, and thus this information is presented compactly here.
If a continuous signal S(t) is sampled with a sampling frequency f sample , this signal is represented as a discrete time series S(k/f sample ). The discrete Fourier transform (DFT) of a time series of length n is defined aŝ a) Author to whom correspondence should be addressed: b.voigtlaender@ fz-juelich.de with m = 0, · · · , n 1. The power spectral density (termed PSD or N 2 PSD ) is proportional to the absolute square of the discrete Fourier transform (DFT). 2 If we do not consider windowing yet (i.e., consider a rectangular window), 3,4 the single-sided PSD results as Note that other definitions of the DFT than the one in Eq.
(1) result in other factors in Eq. (2). 3 The spectral density N PSD is the square root of the power spectral density N 2 PSD . For a continuous signal, the power spectral density of a signal is related, via Parseval's identity, to the root mean square (RMS) S RMS of the signal as If the signal is a discrete time series, the continuous quantities S(t) and N PSD ( f ) translate to discrete values as respectively, with k = 0, . . ., n 1, and m = 0, . . ., n/2.
The width of the n frequency bins of the DFT is given by In the following, we consider two simple examples for the power spectral density: a constant power spectral density a constant PSD, Eq. (5) results in If, for example, the signal is a voltage, e.g., of RMS amplitude S RMS = 1 V and B = 100 Hz, a spectral density Hz results. In this case, the (power) spectral density is independent of the width of the frequency bins. This case is desirable as the value of the (power) spectral density has a significance independent of the width of the frequency bins, i.e., independent of details of the sampling.
For the case of a tonal (sinusoidal) signal, the situation is different. For the sake of simplicity, we consider cases without spectral leakage present. 4 Then the tonal signal is usually located within a single non-zero frequency bin of width f res at a frequency f t , as shown in Fig. 5(b). In this case, only one term of the sum in Eq. (5) survives and the (power) spectral density of this bin depends (undesirably) on the width of the This means that, for instance, a tonal signal with an RMS amplitude of 1 V results in different values for the (power) spectral density, depending on the width of the frequency bins f res , as also shown in Fig. 5(b) for two different values of the frequency bin width f res and f res , respectively. This dependence of the value of the (power) spectral density on the frequency bin width f res , which depends on the particular length of the time series used for the DFT and the particular sampling rate, is of course undesirable. Thus the value of the (power) spectral density for a tonal signal has no unique significance without the knowledge of some details on the sampling process, such as the sampling rate f sample and the length n of the DFT. Spectrum: A different quantity, the power spectrum N 2 spec or the spectrum N spec , defined as avoids this disadvantage. When inserting Eq. (7), valid for a tonal signal, into Eq. (8), the (power) spectrum of a tonal signal turns out to be independent of the width of the frequency bin as N 2 spec = S 2 RMS .
As Eq. (9) shows, the value of the spectrum is equal to the RMS amplitude of the tonal (sinusoidal) signal, N spec = S RMS (e.g., 1 V). However, undesirably for a signal of constant (power) spectral density, the spectrum N spec depends on the width of the frequency bin f res , as evident when inserting Eq. (6) into Eq. (8), resulting in In conclusion, neither the (power) spectral density nor the (power) spectrum delivers a value which is independent of the frequency bin for a tonal signal as well as for a signal with constant PSD (representative of a broad band signal with a relatively flat PSD). A solution of this dilemma would be to choose the width of the frequency bin f res = 1 Hz so that both the spectral density and the spectrum have the same numeric value (but still different units, e.g., V/

√
Hz and V, respectively). However, if low frequencies approaching 1 Hz and below are of interest, a frequency bin width of 1 Hz is too wide.
So far, we have not considered the windowing in the DFT, 4 which means we have so far implicitly considered a rectangular window function. When applying other window functions in the DFT, a quantity named "normalized equivalent noise bandwidth" (NENBW) can be defined and Eq. (8) is extended with f eff res to and values of NENBW for different windows are shown in Table I. 3 In order to present the complete information of a spectral analysis, both the (power) spectral density and the (power) spectrum have to be presented or one of them and f eff res . In the signal processing from the time series of the signal to the spectral density or spectrum, several proportionality factors are involved due to the use of, for example, either RMS amplitude or peak amplitude, either two-sided spectrum or single-sided spectrum, either natural frequency PSD or angular frequency PSD, or due to different window types, etc. So one has to consider all these factors carefully. Complementarily also an experimental calibration of the spectral density or spectrum is very desirable and will be considered in the following. A tonal signal, e.g., from a signal generator can be used to calibrate the spectrum or the spectral density. According to Eq. (9), the RMS signal amplitude of a tonal signal corresponds directly to the amplitude of the spectrum N spec , independent of the width of a frequency bin. For the calibration of the power spectral density using a tonal signal, the effective width of a frequency bin enters. According to Eq. (7) (extended to f eff res ), the RMS signal amplitude of a tonal signal S 2 RMS has to be divided by f eff res in order to obtain the power spectral density. Alternatively to a calibration with a tonal signal, a signal of constant power spectral density (white noise) and known amplitude can be used for the calibration. Such a signal is provided, for instance, by the Johnson-Nyquist noise (thermal noise) of a resistor R as voltage source of known RMS voltage and constant PSD (white noise spectrum). The bandwidth B in Eq. (12) corresponds to the effective width of a frequency bin of the DFT as B = f eff res . In order to obtain a reasonably large voltage, a resistor of large resistance should be used (R ≥ 500 kΩ) and considered in parallel with the input resistance of the measurement device.
In conclusion, if a case of a spectral analysis includes tonal peaks, as well as (locally) constant regions as a function of frequency, both the spectrum and the spectral density are required in order to deliver quantitative results for tonal peaks and broad band regions. The tonal peaks are represented quantitatively in the spectrum (e.g., in volts), while constant regions are represented quantitatively in the spectral density (e.g., as V/ √ Hz). The spectrum and spectral density can be converted into each other by the proportionality factor f eff res = ( f sample /n) · NENBW.