Spectrum and spectral density estimation with periodograms and window functions Gerhard Heinzel Max-Planck-Institut f¨urGravitationsphysik, (Albert-Einstein-Institut), Hannover, Germany 1 Noise Bandpass 2V 10 mV/sqrt(Hz) 50−2000 Hz RMS Voltmeter 10 mV/sqrt(Hz) 441 mVrms To ADC y (log.) [V rms] Sine wave 1234 Hz 2Vrms =2.82Vpk 50 1234 2000 f[Hz] This is easy to realize experimentally. This is what we want as result. Abbrev. Name Relation Unit PSD V2/Hz power spectral density LSD LSD = pPSD V/pHz linear spectral density amplitude spectral density PS PS = PSD ENBW V2 power spectrum × LS LS = pPS = LSD pENBW V linear spectrum × amplitude spectrum 2 One practically important relation for a linear spectral density U(f) is its relation to the rms fluctuation e of the quantity U, assumed to be band-limited to the frequency range f1 f f2: ≤ ≤ f v 2 u 2 Urms = u U(f) df ; (1) uZ h i u e tf1 - In GEO we need both PSD and PS - Their relationship is given by the Noise Equivalent BandWidth ENBW. - ENBW depends on the details of the analysis. - If ENBW is not recorded, the results cannot be converted later. - Most software (including MATLAB's pwelch) do not provide ENBW. 3 Some common units U Vpk Vrms t Vpk−pk Vpk Vrms = : (2) p2 signal power signal amplitude ratio[dB] = 10 log = 20 log : (3) 10 reference power 10 reference amplitude dBm: Urms = pP 50 Ω ; (4) × 3 + 0:1 x P = 1 W 10n− × 1 dBmo (5) × 2 Vrms = 2:828 Vpk = 5:657 Vpk pk = 6:02 dBVrms = 126:02 dBµVrms = 19:03 dBm: (6) − 4 A periodogram is just a Discrete Fourier Transform (DFT) of the time series xk: N 1 (1) mk y = − x exp 2πi ; m = 0 :::N 1; (7) m X k k=0 − N − (2) 1 (1) ym = ym ; (8) pN (3) 1 (1) ym = ym : (9) N To become useful, periodograms have to be - modified by window functions and - averaged. Averaging reduces the intrinsically large variance of periodograms of stochastic signals. 5 Doing nothing is equivalent to using a Rectangular window: 5 0 -5 -10 -15 -20 -25 Amplitude [dB] -30 -35 -40 -45 -20 -15 -10 -5 0 5 10 15 20 Frequency offset [bins] 6 A window function to be used with a DFT of length N is defined by a vector of real numbers wj , f g j = 0 :::N 1. It is used by multiplying the time series xj with the window before performing the − DFT, i.e. using x = xj wj as input to the DFT. j0 · All windows studied here have the following symmetry: wj = wN j: (10) − This implies for even lengths N that w0 and wN=2 appear only once, while all other coefficients appear twice. Hence only the N=2 + 1 coefficients w0; : : : ; wN=2 need to be computed and stored. 1 1.2 0.9 w4 1 0.8 w3 w5 0.7 i 0.8 0.6 0.6 0.5 w2 w6 0.4 window value 0.4 window value w 0.3 0.2 0.2 w1 w7 0.1 0 w0 0 0 0.2 0.4 0.6 0.8 1 -5 0 5 10 15 index i/N index i 7 PSLL SLDR NENBW 3 dB BW flatness ROV n Name [dB] [f − ] [bins] [bins] [dB] [%] Rectangular 13.3 1 1.0000 0.8845 3:9224 0.0 − Welch 21.3 2 1.2000 1.1535 2:2248 29.3 − Bartlett 26.5 2 1.3333 1.2736 1:8242 50.0 − Hanning 31.5 3 1.5000 1.4382 1:4236 50.0 − Hamming 42.7 1 1.3628 1.3008 1:7514 50.0 − Nuttall3 46.7 5 1.9444 1.8496 0:8630 64.7 − Nuttall4 60.9 7 2.3100 2.1884 0:6184 70.5 − Nuttall3a 64.2 3 1.7721 1.6828 1:0453 61.2 − Kaiser3 69.6 1 1.7952 1.7025 1:0226 61.9 − Nuttall3b 71.5 1 1.7037 1.6162 1:1352 59.8 − Nuttall4a 82.6 5 2.1253 2.0123 0:7321 68.0 − BH92 92.0 1 2.0044 1.8962 0:8256 66.1 − Nuttall4b 93.3 3 2.0212 1.9122 0:8118 66.3 − Kaiser4 94.4 1 2.0533 1.9417 0:7877 67.0 − Nuttall4c 98.1 1 1.9761 1.8687 0:8506 65.6 − Kaiser5 119.8 1 2.2830 2.1553 0:6403 70.5 − 8 NENBW=2.2830 bins 3.5 0 3 -20 2.5 -40 -60 2 -80 1.5 -100 1 amplitude [dB] -120 0.5 normalized window value -140 0 -160 0 0.2 0.4 0.6 0.8 1 -40 -20 0 20 40 index j/N frequency offset [bins] α PSLL NENBW 3 dB BW zero flatness ROV [dB] [bins] [bins] [bins] [dB] [%] 2.0 45:9 1.4963 1.4270 2.24 1:4527 53.4 2.5 −57:6 1.6519 1.5700 2.69 −1:2010 58.3 3.0 −69:6 1.7952 1.7025 3.16 −1:0226 61.9 3.5 −81:9 1.9284 1.8262 3.64 −0:8900 64.7 4.0 −94:4 2.0533 1.9417 4.12 −0:7877 67.0 4.5 −107:0 2.1712 2.0512 4.61 −0:7064 68.9 5.0 −119:8 2.2830 2.1553 5.10 −0:6403 70.5 5.5 −132:6 2.3898 2.2546 5.59 −0:5854 71.9 6.0 −145:5 2.4920 2.3499 6.08 −0:5392 73.1 6.5 −158:4 2.5902 2.4414 6.58 −0:4998 74.1 7.0 −171:4 2.6848 2.5297 7.07 −0:4657 75.1 − − 9 PSLL SLDR NENBW 3 dB BW flatness ROV n Name [dB] [f − ] [bins] [bins] [dB] [%] SFT3F 31.7 3 3.1681 3.1502 +0:0082 66.7 SFT3M 44.2 1 2.9452 2.9183 0:0115 65.5 FTNI 44.4 1 2.9656 2.9355 +0− :0169 65.6 SFT4F 44.7 5 3.7970 3.7618 +0:0041 75.0 SFT5F 57.3 7 4.3412 4.2910 0:0025 78.5 SFT4M 66.5 1 3.3868 3.3451 −0:0067 72.1 FTHP 70.4 1 3.4279 3.3846 +0− :0096 72.3 HFT70 70.4 1 3.4129 3.3720 0:0065 72.2 FTSRS 76.6 3 3.7702 3.7274 −0:0156 75.4 SFT5M 89.9 1 3.8852 3.8340 +0− :0039 76.0 HFT90D 90.2 3 3.8832 3.8320 0:0039 76.0 HFT95 95.0 1 3.8112 3.7590 +0− :0044 75.6 HFT116D 116.8 3 4.2186 4.1579 0:0028 78.2 HFT144D 144.1 3 4.5386 4.4697 +0− :0021 79.9 HFT169D 169.5 3 4.8347 4.7588 +0:0017 81.2 HFT196D 196.2 3 5.1134 5.0308 +0:0013 82.3 HFT223D 223.0 3 5.3888 5.3000 0:0011 83.3 HFT248D 248.4 3 5.6512 5.5567 +0− :0009 84.1 10 Comparison between Rectangular, Hanning and Flat-top window. 6 0 5 -20 4 -40 3 -60 -80 2 -100 1 amplitude [dB] -120 0 normalized window value -140 -1 -160 0 0.2 0.4 0.6 0.8 1 -40 -20 0 20 40 index i/N frequency offset [bins] 11 Comparison between Rectangular, Hanning and Flat-top window. 0 0.5 0 -50 -0.5 -1 -1.5 -100 -2 -2.5 amplitude [dB] amplitude [dB] -150 -3 -3.5 -200 -4 -10 -5 0 5 10 -0.5 0 0.5 frequency offset [bins] frequency offset [bins] 12 We define the following two sums for normalization purposes: N 1 S = − w ; (11) 1 X j j=0 N 1 S = − w2 : (12) 2 X j j=0 Because we will use S1 and S2 in the normalization of our final results, we can multiply the window values wj with any convenient constant factor. The normalized equivalent noise bandwidth NENBW of the window, expressed in frequency bins, is given by S NENBW = 2 (13) N 2 : (S1) The effective noise bandwidth ENBW is given by fs S ENBW = NENBW = NENBW = 2 (14) fres fs 2 ; · · N (S1) 13 The result of the FFT is a complex vector ym of length N=2+1. We interpret it as a power spectrum, 2 expressed as Vrms, as follows: 2 2 ym PSrms(fm = m fres) = · j 2 j ; m = 0 : : : N=2 ; (15) · S1 If the desired result is a power spectral density (PSD) expressed in V2=Hz, it is obtained by dividing the power spectrum (PS) by the effective noise-equivalent bandwidth ENBW: 2 PSrms(fm) 2 ym PSDrms(fm = m fres) = = · j j ; m = 0 : : : N=2 ; (16) · ENBW fs S2 · LSD = pPSD; (17) LS = pPS: (18) If several spectra/spectral densities are averaged , this averaging must be performed with the power spectra/spectral densities, and the square root, if desired, must be taken only at the end. 14 Overlapping avoids the loss of information when windows are used. Window Window N N N N overlap 15 The recommended overlap is defined as that overlap r where the distance between amplitude flatness AF and overlap correlation OC becomes maximal. rN 1 − wjwj+(1 r)N j=0P − OC(r) = (19) N 1 − 2 wj j=0P 50.0% 1 1 max.