lecroy 93XXC-OM-E21 电路图.pdf

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1、 Xn is the resulting complex frequency-domain waveform; We j N = 2/ ; and N is the number of points in xk and Xn. The generalized FFT algorithm, as implemented here, works on N, which need not be a power of 2. 4. The resulting complex vector Xn is divided by the coherent gain of the window function,

2、 in order to compensate for the loss of the signal energy due to windowing. This compensation provides accurate amplitude values for isolated spectrum peaks. 5. The real part of Xn is symmetric around the Nyquist frequency, i.e. Rn = RN-n , while the imaginary part is asymmetric, i.e. In = IN-n . th

3、e energy at frequency 0 is completely contained in the 0 term. The first half of the spectrum (Re, Im), from 0 to the Nyquist frequency is kept for further processing and doubled in amplitude: Rn = 2 ? Rn0 n N/2 In = 2 ? In0 n Mmin angle = 0Mn Mmin . &? $SSHQGL?& Where Mmin is the minimum magnitude,

4、 fixed at about 0.001 of the full scale at any gain setting, below which the angle is not well defined. The dBm Power Spectrum: dBm PS M M M M n ref n ref = = 1020 10 2 2 10 loglog where Mref = 0.316 V (that is, 0 dBm is defined as a sine wave of 0.316 V peak or 0.224 V RMS, giving 1.0 mW into 50).

5、The dBm Power Spectrum is the same as dBm Magnitude, as suggested in the above formula. dBm Power Density: ()dBmPDdBmPSENBWf=10 10 log where ENBW is the equivalent noise bandwidth of the filter corresponding to the selected window, and f is the current frequency resolution (bin width). 7. The FFT Po

6、wer Average takes the complex frequency- domain data Rn and In for each spectrum generated in Step 5, and computes the square of the magnitude: Mn2 = Rn2 + In2, then sums Mn2 and counts the accumulated spectra. The total is normalized by the number of spectra and converted to the selected result typ

7、e using the same formulae as are used for the Fourier Transform. &? ?)7?*ORVVDU *ORVVDU Defines the terms frequently used in FFT spectrum analysis and relates them to the oscilloscope. $OLDVLQJIf the input signal to a sampling acquisition system contains components whose frequency is greater than th

8、e Nyquist frequency (half the sampling frequency), there will be less than two samples per signal period. The result is that the contribution of these components to the sampled waveform is indistinguishable from that of components below the Nyquist frequency. This is aliasing. The timebase and trans

9、form-size should be selected so that the resulting Nyquist frequency is higher than the highest significant component in the time-domain record. &RKHUHQW?*DLQThe normalized coherent gain of a filter corresponding to each window function is 1.0 (0 dB) for a rectangular window and less than 1.0 for ot

10、her windows. It defines the loss of signal energy due to the multiplication by the window function. This loss is compensated in the oscilloscope. This table lists the values for the implemented windows. :LQGRZ?)UHTXHQF?RPDLQ?3DUDPHWHUV Window Type Highest Side Lobe (dB) Scallop Loss (dB) ENBW (bins)

11、 Coherent Gain (dB) Rectangular133.921.0 0.0 von Hann321.421.5 6.02 Hamming431.78 1.37 5.35 Flat Top440.01 2.9611.05 BlackmanHarris671.13 1.71 7.53 &? $SSHQGL?& (1%:?Equivalent Noise BandWidth (ENBW) is the bandwidth of a rectangular filter (same gain at the center frequency), equivalent to a filter

12、 associated with each frequency bin, which would collect the same power from a white noise signal. In the table on the previous page, the ENBW is listed for each window function implemented and is given in bins. )LOWHUVComputing an N-point FFT is equivalent to passing the time- domain input signal t

13、hrough N/2 filters and plotting their outputs against the frequency. The spacing of filters is f = 1/T while the bandwidth depends on the window function used (see Frequency bins). )UHTXHQF?ELQVThe FFT algorithm takes a discrete source waveform, defined over N points, and computes N complex Fourier

14、coefficients, which are interpreted as harmonic components of the input signal. For a real source waveform (imaginary part equals 0), there are only N/2 independent harmonic components. An FFT corresponds to analyzing the input signal with a bank of N/2 filters, all having the same shape and width,

15、and centered at N/2 discrete frequencies. Each filter collects the signal energy that falls into the immediate neighborhood of its center frequency, and thus it can be said that there are N/2 “frequency bins”. The distance in hertz between the center frequencies of two neighboring bins is always: f

16、= 1/T, where T is the duration of the time-domain record in seconds. The width of the main lobe of the filter centered at each bin depends on the window function used. The rectangular window has a nominal width at 1.0 bin. Other windows have wider main lobes (see table). )UHTXHQF?5DQJHThe range of f

17、requencies computed and displayed is 0 Hz (displayed at the left-hand edge of the screen) to the Nyquist frequency (at the rightmost edge of the trace). &? ?)7?*ORVVDU )UHTXHQF?5HVROXWLRQIn a simple sense, the frequency resolution is equal to the bin width f. That is, if the input signal changes its

18、 frequency by f, the corresponding spectrum peak will be displaced by f. For smaller changes of frequency, only the shape of the peak will change. However, the effective frequency resolution (i.e. the ability to resolve two signals whose frequencies are almost the same) is further limited by the use

19、 of window functions. The ENBW value of all windows other than the rectangular is greater than f and the bin width. The table on page C17 lists the ENBW values for the implemented windows. /HDNDJHIn the power spectrum of a sine wave with an integral number of periods in the (rectangular) time window

20、 (i.e. the source frequency equals one of the bin frequencies), the spectrum contains a sharp component whose value accurately reflects the source waveforms amplitude. For intermediate input frequencies this spectral component has a lower and broader peak. The broadening of the base of the peak, str

21、etching out into many neighboring bins is termed leakage. It is due to the relatively high side lobes of the filter associated with each frequency bin. The filter side lobes and the resulting leakage are reduced when one of the available window functions is applied. The best reduction is provided by

22、 the BlackmanHarris and Flat Top windows. However, this reduction is offset by a broadening of the main lobe of the filter. 1XPEHU?RI?3RLQWVFFT is computed over the number of points (Transform Size) whose upper bounds are the source number of points, and by the maximum number of points selected in t

23、he menu. FFT generates spectra of N/2 output points. 1TXLVW?)UHTXHQFThe Nyquist frequency is equal to one half of the effective sampling frequency (after the decimation): f N/2. 3LFNHW?)HQFH?(IIHFWIf a sine wave has a whole number of periods in the time domain record, the power spectrum obtained wit

24、h a rectangular window will have a sharp peak, corresponding exactly to the frequency and amplitude of the sine wave. Otherwise the spectrum peak with a rectangular window will be lower and broader. The highest point in the power spectrum can be 3.92 dB lower (1.57 times) when the source frequency i

25、s halfway between two &? $SSHQGL?& discrete bin frequencies. This variation of the spectrum magnitude is called the picket fence effect (the loss is called the scallop loss). All window functions compensate this loss to some extent, but the best compensation is obtained with the Flat Top window. 3RZ

26、HU?6SHFWUXPThe power spectrum (V2) is the square of the magnitude spectrum. The power spectrum is displayed on the dBm scale, with 0 dBm corresponding to: Vref2 = (0.316 Vpeak)2, where Vref is the peak value of the sinusoidal voltage, which is equivalent to 1 mW into 50 . 3RZHU?HQVLW?6SHFWUXP The po

27、wer density spectrum (V2/Hz) is the power spectrum divided by the equivalent noise bandwidth of the filter in hertz. The power density spectrum is displayed on the dBm scale, with 0 dBm corresponding to (Vref2/Hz). 6DPSOLQJ?)UHTXHQFThe time-domain records are acquired at sampling frequencies depende

28、nt on the selected time base. Before the FFT computation, the time-domain record may be decimated. If the selected maximum number of points is lower than the source number of points, the effective sampling frequency is reduced. The effective sampling frequency equals twice the Nyquist frequency. 6FD

29、OORS?/RVVLoss associated with the picket fence effect. :LQGRZ?)XQFWLRQVAll available window functions belong to the sum of cosines family with one to three non-zero cosine terms: Wa k N mkN km m m M = = = 0 1 2 0cos p , where:M = 3is the maximum number of terms, am are the coefficients of the terms,

30、 N is the number of points of the decimated source waveform, and k is the time index. &? ?)7?*ORVVDU The following table lists the coefficients am. The window functions seen in the time domain are symmetric around the point k = N/2. &RHIILFLHQWV?2I?:LQGRZ?)XQFWLRQV Window Typea0a1a2 Rectangular1.00.

31、00.0 von Hann0.50.50.0 Hamming0.540.460.0 Flat-Top0.2810.5210.198 Blackman-Harris0.4230.4970.079 $SSHQGL?&?5HIHUHQFHVBergland, G.D., A Guided Tour of the Fast Fourier Transform, IEEE Spectrum, July 1969, pp. 4152. A general introduction to FFT theory and applications. Brigham, E.O., The Fast Fourier

32、 Transform, Prentice Hall, Inc., Englewood Cliffs, N.J., 1974. Theory, applications and implementation of FFT. Includes discussion of FFT algorithms for N not a power of 2. Harris, F.J., On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform, Proceedings of the IEEE, vol. 66, No. 1, January 1978, pp. 5183. Classic paper on window functions and their figures of merit, with many examples of windows. Ramirez, R.W., The FFT Fundamentals and Concepts, Prentice Hall, Inc., Englewood Cliffs, N.J., 1985. Practice-oriented, many examples of applications.