ONO SOKKI - CF-5200 FAQ - Relationship between Time Axis Waveforms and Spectrum

	Q	A
1	Reading time axis waveforms	The X axis displays 2048 points of data AD-converted with a sampling frequency of 2.56 times the frequency range. A waveform with the X axis display time T = 2048 / (frequency range) / 2.56 s is displayed. The FFT spectrum is displayed from this 2048 points of sampled data. The Y axis of the spectrum shows the amplitude voltage (rms value) for each input waveform frequency. (see Chapter 2·7-1 item 2 Spectrum in Instruction Manual of CF-5210/5220)
2	Relationship between time axis waveforms and power spectra What is a Fourier expansion	The 2048 points of data AD-converted with a sampling frequency of 2.56 times the frequency range is displayed on the time axis, Fourier-expanded (Fourier-transformed), to find the Fourier spectrum (pair of sine and cosine coefficients). In other words, the sine and cosine coefficients, the Fourier coefficient, and the Fourier spectrum are found in that order to express the Fourier series. Fourier series is f (t) = A0 + Σ｛Ainωi + Bicosωi｝ωi:2π x (each frequency f) Spectrum can be conceptually expressed with the cos coefficient value Bi as the real part (Re) on the X axis, and the sine coefficient value Ai as the imaginary part (Im) on the Y axis. This value is referred to as the Fourier spectrum. It is displayed on the screen after displaying the power spectrum by setting the REAL and IMAG buttons to ON. Power spectrum Vr is the size of the Fourier spectrum. Vi = √｛Re^２ + Im^２｝Values in brackets subject to square root operation^. Phase is found with; tan Θ = Im / Re or Ai sin ωi + Bi cos ωi = Vi cos (ωi + Θ) From these conversions, f (t) can be considered as a synthesis of cosine waveform for each frequency ωi (in the CF5200 Series, Vi is the initial value, and is displayed as the rms value) With the Y axis is Lin, Vi is displayed as is, however if the Y axis is Log, Pi = 10 LOG (Vi)^２ is computed and displayed. Since Vi² is used, the power spectrum generally indicates Log display data. Furthermore, the FFT is displayed at a resolution of 1/800 across the frequency range for the sample of 2048 points. On the power spectrum display screen, therefore, 801 data items are displayed across the X axis 0Hz – frequency range.
3	(1) What happens when dBVr is converted to voltage? (2) What happens when the amplitude is read from the time axis waveform and converted to dB?	(1) dBVr = 10 LOG｛Vrms/(reference)｝^２ = 20 LOG(Vrms) The reference is the rms value 1Vrms Vrms = 10dbVr / 20 (2) Since time waveforms are syntheses of various frequencies, care is required when computing dB from waveform peaks. If P-P = 1.39 mV, 0.695 mV is obtained for the half-amplitude (0-P). Thus, as with (1), dBV = 20 LOG (0.695×10^-3). The reference is therefore a half-amplitude of 1V. Note that the frequency is unclear.
4	What is the relationship between number of sampling points and frequency resolution?	When data length is 2048 points, frequency resolution is 1/800. When data length is 1024 points, frequency resolution is 1/400. The relationship between data length and resolution is 2048 / 2.56 = 800 1024 / 2.56 = 400
5	Basic FFT operation	See Chapter 2 Overview of the instruction manual for details of the time axis waveform, power spectrum, frequency range, and Y axis units, etc.
6	The time axis waveform P-P value and the order spectrum overall P-P value do not match.	The time waveform peak is a synthesis of various frequency components. Fundamentally, the overall value of the FFT results and the time waveform peak do not match.
7	Time axis waveforms, power spectrum, and overall values	See Chapter 3·5-2 Antialiasing filter, and the chapter 5·3 Calibration function on calibration with a sound level meter in Instruction Manual of CF-5210/5220. The overall value is the product sum of power spectra for each frequency (Y axis value) subjected to FFT analysis. Matches the mean-square on the time axis. Since a low-pass (antialiasing) filter is applied, components beyond the frequency range are also eliminated.
8	What is the AC coupling cut-off frequency?	0.5Hz/-3dB Almost no damping at 1Hz.
9	The FFT mean is used, however what is the measurement time per cycle?	Computation time is approximately 25ms, and display time approximately 400ms. Varies with setup conditions (e.g. data length, overlap, number of display screens). Increase average number of cycles and measure measurement time, and compute time per cycle to verify. See amount of overlap and non-display functions.
10	What is the overall value? What is the partial overall value?	Overall (OA) value is the product sum of all frequency power values (Y axis values) subject to the FFT. Since a low-pass filter is applied, components beyond the frequency range are eliminated. Care is required. First, when the data length is 2048 (default setting), the frequency range is divided by 800, and a spectrum of 0Hz + 800 = 801 is displayed on the screen. With i = 0 – 800 Vi = i th spectrum voltage amplitude, the individual spectra displayed on the screen are as follows. (1) With Log display (when Y-axis displays dB) Pi = 10 LOG (Vi)^２ (2) With Lin display (when Y-axis displays Vr) √ (Vi)^２ = Vi Overall computations are as follows. (3) When Y-axis displays Log For P0: DC component Y-axis value (dB), Pi: i th Y-axis value (dB) (OA) = 10 LOG｛(10^P0/10+--+10^Ｐi/10+10^P800/10)Hf｝ Hf: Window adjustment 1 when rectangular 2/3 when hanning (Example) With Pi = 30, 50, --- 40 (dB) (read value when Y-axis displays dB) (OA) = 10 LOG｛(10^30/10+10^50/10+--+10^40/10) Hf｝ (4) When Y axis displays Lin If V0: Y-axis value of DC component and Vi: I th Y axis value (Vr), (OA) LIN = √｛(V0^２+--+Vi^２+V800^２) Hf｝ Note: Values in brackets subject to square root operation. (Example) With 0.1, 0.3, ---0.4 (V), (OA) LIN = √｛(0.1^２+0.3^２+ - - - +0.4^２) Hf｝ Pi and Vi become partial overalls when computed within desired range. When considered in terms of a time region, they match the mean-square of 2048 data items.