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1. a 0 2000 4000 6000 3Ss000 10000 12000 14000 16000 18000 20000 24000 24000 25600 Figure 6 PCI 4450 Family Spectrum Plot with 997 Hz Input at Full Scale Full Scale 0 dB Using Windows Correctly As mentioned in the Introduction using windows correctly is critical to FFT based measurement This section describes the problem of spectral leakage the characteristics of windows some strategies for choosing windows and the importance of scaling windows Spectral Leakage For an accurate spectral measurement it is not sufficient to use proper signal acquisition techniques to have a nicely scaled single sided spectrum You might encounter spectral leakage Spectral leakage is the result of an assumption in the FFT algorithm that the time record is exactly repeated throughout all time and that signals contained in a time record are thus periodic at intervals that correspond to the length of the time record If the time record has a nonintegral number of cycles this assumption is violated and spectral leakage occurs Another way of looking at this case is that the nonintegral cycle frequency component of the signal does not correspond exactly to one of the spectrum frequency lines There are only two cases in which you can guarantee that an integral number of cycles are always acquired One case is if you are sampling synchronously with respect to the signal you measure and can therefore deliberately take an integral number of cycles A2. Using these functions as building blocks you can create additional measurement functions such as frequency response impulse response coherence amplitude spectrum and phase spectrum FFTs and the Power Spectrum are useful for measuring the frequency content of stationary or transient signals FFTs produce the average frequency content of a signal over the entire time that the signal was acquired For this reason you should use FFTs for stationary signal analysis or in cases where you need only the average energy at each frequency line To measure frequency information that is changing over time use joint time frequency functions such as the Gabor Spectrogram This application note also describes other issues critical to FFT based measurement such as the characteristics of the signal acquisition front end the necessity of using windows the effect of using windows on the measurement and measuring noise versus discrete frequency components National Instruments ni com and LabWindows CVI are trademarks of National Instruments Corporation Product and company names mentioned herein are trademarks or trade names of their respective companies 340555B 01 Copyright 2000 National Instruments Corporation All rights reserved July 2000 Basic Signal Analysis Computations The basic computations for analyzing signals include converting from a two sided power spectrum to a single sided power spectrum adjusting frequency resolution and
3. scale Because of noise level scaling with Af spectra for noise measurement are often displayed in a normalized format called power or amplitude spectral density This normalizes the power or amplitude spectrum to the spectrum that would be measured by a I Hz wide square filter a convention for noise level measurements The level at each frequency line then reads as if it were measured through a Hz filter centered at that frequency line Power spectral density is computed as i 2 Power Spectrum in Power spectral density ___ Power Spectrum in Vis Af x Noise Power Bandwidth of Window 2 2 a N V The units are then in CMS or Hz Hz Amplitude spectral density is computed as Amplitude Spectral Density a L I IN an Af x Noise Power Bandwidth of Window V The units are then in Yrms or Hz Hz The spectral density format is appropriate for random or noise signals but inappropriate for discrete frequency components because the latter theoretically have zero bandwidth FFI Based Network Measurement When you understand how to handle computations with the FFT and power spectra and you understand the influence of windows on the spectrum you can compute several FFT based functions that are extremely useful for network analysis These include the transfer impulse and coherence functions Refer to the Frequency Response and Network Analysis section of this application note for more information about th
4. acquired If N is a power of two LabVIEW uses the efficient FFT algorithm Otherwise LabVIEW actually uses the discrete Fourier transform DFT which takes considerably longer LabWindows CVI requires that N be a factor of two and thus always uses the FFT Typical benchtop instruments use FFTs of 1 024 and 2 048 points So far you have looked at display units of volts peak volts rms and volts rms squared which is equivalent to mean square volts In some spectrum displays the rms qualifier is dropped for Vrms in which case V implies Vrms and V implies Vrms or mean square volts Converting to Logarithmic Units Most often amplitude or power spectra are shown in the logarithmic unit decibels dB Using this unit of measure it is easy to view wide dynamic ranges that is it is easy to see small signal components in the presence of large ones The decibel is a unit of ratio and is computed as follows dB 10log P P where P is the measured power and P is the reference power Use the following equation to compute the ratio in decibels from amplitude values where A is the measured amplitude and A is the reference amplitude When using amplitude or power as the amplitude squared of the same signal the resulting decibel level is exactly the same Multiplying the decibel ratio by two is equivalent to having a squared ratio Therefore you obtain the same decibel level and display regardless of whether you use the amplitude or powe
5. components were multiplied by two to convert from two sided to single sided form you can calculate the rms amplitude spectrum directly from the two sided amplitude spectrum by multiplying the non DC components by the square root of two and discarding the second half of the array The following equations show the entire computation from a two sided FFT to a single sided amplitude spectrum Amplitude spectrum in volts rms 2 e pense for i 1 to gt 1 Magnitude FFT A eisd DO N where i is the frequency line number array index of the FFT of A The magnitude in volts rms gives the rms voltage of each sinusoidal component of the time domain signal To view the phase spectrum in degrees use the following equation Phase spectrum in degrees S e Phase FFT A National Instruments Corporation 5 Application Note 041 The amplitude spectrum is closely related to the power spectrum You can compute the single sided power spectrum by squaring the single sided rms amplitude spectrum Conversely you can compute the amplitude spectrum by taking the square root of the power spectrum The two sided power spectrum is actually computed from the FFT as follows x Power spectrum S f EN where FFT A denotes the complex conjugate of FFT A To form the complex conjugate the imaginary part of FFT A is negated When using the FFT in LabVIEW be aware that the speed of the power spectrum and the FFT computation depend on the number of points
6. graphing the spectrum using the FFT and converting power and amplitude into logarithmic units The power spectrum returns an array that contains the two sided power spectrum of a time domain signal The array values are proportional to the amplitude squared of each frequency component making up the time domain signal A plot of the two sided power spectrum shows negative and positive frequency components at a height a N A 4 where A is the peak amplitude of the sinusoidal component at frequency k The DC component has a height of Ao where Ao is the amplitude of the DC component in the signal Figure 1 shows the power spectrum result from a time domain signal that consists of a 3 Vrms sine wave at 128 Hz a 3 Vrms sine wave at 256 Hz and a DC component of 2 VDC A 3 Vrms sine wave has a peak voltage of 3 0 2 or about 4 2426 V The power spectrum is computed from the basic FFT function Refer to the Computations Using the FFT section later in this application note for an example this formula 200 400 600 800 1000 1200 Hz Figure 1 Two Sided Power Spectrum of Signal Converting from a Two Sided Power Spectrum to a Single Sided Power Spectrum Most real world frequency analysis instruments display only the positive half of the frequency spectrum because the spectrum of a real world signal is symmetrical around DC Thus the negative frequency information is redundant The two sided results from the analysis functions include the pos
7. line indicates no correlation between the response and the stimulus signal A one for a given frequency line indicates that the response energy is 100 percent due to the stimulus signal in other words there is no interference at that frequency For a valid result the coherence function requires an average of two or more readings of the stimulus and response signals For only one reading it registers unity at all frequencies To average the cross power spectrum S p f average it in the complex form then convert to magnitude and phase as described in the Transfer Function section of this application note The auto power spectra S f and Spp f are already in real form and you average them normally Signal Sources for Frequency Response Measurements To achieve a good frequency response measurement significant stimulus energy must be present in the frequency range of interest Two common signals used are the chirp signal and a broadband noise signal The chirp signal is a sinusoid swept from a start frequency to a stop frequency thus generating energy across a given frequency range White and pseudorandom noise have flat broadband frequency spectra that is energy is present at all frequencies It is best not to use windows when analyzing frequency response signals If you generate a chirp stimulus signal at the same rate you acquire the response you can match the acquisition frame size to match the length of the chirp No window is genera
8. main lobe Signal Spectrum Window Spectrum Windowed Signal Spectrum Figure 10 Frequency Characteristics of a Windowed Spectrum Application Note 041 12 www ni com An FFT produces a discrete frequency spectrum The continuous periodic frequency spectrum is sampled by the FFT just as the time domain signal was sampled by the ADC What appears in each frequency line of the FFT is the value of the continuous convolved spectrum at each FFT frequency line This is sometimes referred to as the picket fence effect because the FFT result is analogous to viewing the continuous windowed spectrum through a picket fence with slits at intervals that corresponds to the frequency lines If the frequency components of the original signal match a frequency line exactly as is the case when you acquire an integral number of cycles you see only the main lobe of the spectrum Side lobes do not appear because the spectrum of the window approaches zero at Af intervals on either side of the main lobe Figure 7 illustrates this case If a time record does not contain an integral number of cycles the continuous spectrum of the window is shifted from the main lobe center at a fraction of Af that corresponds to the difference between the frequency component and the FFT line frequencies This shift causes the side lobes to appear in the spectrum In addition there is some amplitude error at the frequency peak as shown in Figure 8 because the main lobe is sampled o
9. than the Af given by the FFT by performing a weighted average of the frequencies around a detected peak in the power spectrum j 3 Y Power i i Af i j 3 Estimated Frequency j 3 y Power i i j 3 where j is the array index of the apparent peak of the frequency of interest and ya J N The span j 3 is reasonable because it represents a spread wider than the main lobes of the windows listed in Table 3 Similarly you can estimate the power in Vrms of a given peak discrete frequency component by summing the power in the bins around the peak computing the area under the peak j 3 3 Power i i j 3 Estimated Power noise power bandwidth of window Notice that this method is valid only for a spectrum made up of discrete frequency components It is not valid for a continuous spectrum Also if two or more frequency peaks are within six lines of each other they contribute to inflating the estimated powers and skewing the actual frequencies You can reduce this effect by decreasing the number of lines spanned by the preceding computations If two peaks are that close they are probably already interfering with one another because of spectral leakage Similarly if you want the total power in a given frequency range sum the power in each bin included in the frequency range and divide by the noise power bandwidth of the windows Application Note 041 16 www ni com Computing Noise Lev
10. to each other spectral resolution is important In this case it is best to choose a window with a very narrow main lobe If the amplitude accuracy of a single frequency component is more important than the exact location of the component in a given frequency bin choose a window with a wide main lobe If the signal spectrum is rather flat or broadband in frequency content use the Uniform window no window In general the Hann window is satisfactory in 95 of cases It has good frequency resolution and reduced spectral leakage The Flat Top window has good amplitude accuracy but because it has a wide main lobe it has poor frequency resolution and more spectral leakage The Flat Top window has a lower maximum side lobe level than the Hann window but the Hann window has a faster roll off rate If you do not know the nature of the signal but you want to apply a window start with the Hann window Figures 7 and 8 contrast the characteristics of the Uniform Hann and Flat Top Windows windows If you are analyzing transient signals such as impact and response signals it is better not to use the spectral windows because these windows attenuate important information at the beginning of the sample block Instead use the Force and Exponential windows A Force window is useful in analyzing shock stimuli because it removes stray signals at the end of the signal The Exponential window is useful for analyzing transient response signals because it damps the en
11. 098 Figure 8 Power Spectrum of 1 Vrms Signal at 256 5 Hz with Uniform Hann and Flat Top Windows National Instruments Corporation 11 Application Note 041 In addition to causing amplitude accuracy errors spectral leakage can obscure adjacent frequency peaks Figure 9 shows the spectrum for two close frequency components when no window is used and when a Hann window is used es S tein i 8 a ee A A S ET N ee O oe a O Figure 9 Spectral Leakage Obscuring Adjacent Frequency Components Window Characteristics To understand how a given window affects the frequency spectrum you need to understand more about the frequency characteristics of windows The windowing of the input data is equivalent to convolving the spectrum of the original signal with the spectrum of the window as shown in Figure 10 Even if you use no window the signal is convolved with a rectangular shaped window of uniform height by the nature of taking a snapshot in time of the input signal This convolution has a sine function characteristic spectrum For this reason no window is often called the Uniform or Rectangular window because there is still a windowing effect An actual plot of a window shows that the frequency characteristic of a window is a continuous spectrum with a main lobe and several side lobes The main lobe is centered at each frequency component of the time domain signal and the side lobes approach zero at intervals on each side of the
12. 3 LabVIEW Function and VI Reference Manual Chapter 42 Analysis Filter Vis In most IIR filter designs and in all of the LabVIEW IIR filters coefficient ay is 1 The output sample at the present sample index i is the sum of scaled present and past inputs x and x when 0 and scaled past outputs _ Because of this IIR filters are also known as recursive filters or autoregressive moving average ARMA filters The response of the general IIR filter to an impulse x 1 and x 0 for all i 0 is called the impulse response of the filter The impulse response of the filter described by equation 42 1 is indeed of infinite length for nonzero coefficients In practical filter applications however the impulse response of stable IIR filters decays to near zero in a finite number of samples IIR filters in LabVIEW contain the following properties e Negative indices resulting from equation 42 1 are assumed to be zero the first time you call the VI e Because the initial filter state is assumed to be zero negative indices a transient proportional to the filter order occurs before the filter reaches a steady state The duration of the transient response or delay for lowpass and highpass filters is equal to the filter order e Delay order e The duration of the transient response for bandpass and bandstop filters is twice the filter order e Delay 2 order You can eliminate this transient response on successiv
13. Elektrisk M tteknik LTH Datorbaserade m tsystem 5 Matdatabehandling 06 08 30 Q7 NATIONAL Application Note 041 p INSTRUMENTS The Fundamentals of FFT Based Signal Analysis and Measurement Michael Cerna and Audrey F Harvey Introduction The Fast Fourier Transform FFT and the power spectrum are powerful tools for analyzing and measuring signals from plug in data acquisition DAQ devices For example you can effectively acquire time domain signals measure the frequency content and convert the results to real world units and displays as shown on traditional benchtop spectrum and network analyzers By using plug in DAQ devices you can build a lower cost measurement system and avoid the communication overhead of working with a stand alone instrument Plus you have the flexibility of configuring your measurement processing to meet your needs To perform FFT based measurement however you must understand the fundamental issues and computations involved This application note serves the following purposes e Describes some of the basic signal analysis computations e Discusses antialiasing and acquisition front ends for FFT based signal analysis e Explains how to use windows correctly e Explains some computations performed on the spectrum and e Shows you how to use FFT based functions for network measurement The basic functions for FFT based signal analysis are the FFT the Power Spectrum and the Cross Power Spectrum
14. Instruments Corporation 42 9 LabVIEW Function and Vi Reference Manual Chapter 42 Analysis Filter Vis that Chebyshev II filters distribute the error in the stopband instead of the passband Elliptic or Cauer Filters Elliptic filters minimize the peak error by distributing it over the passband and the stopband Equi ripples in the passband and the stopband characterize the magnitude response of elliptic filters Compared with the same order Butterworth or Chebyshev filters the elliptic design provides the sharpest transition between the passband and the stopband For this reason elliptic filters are used widely The following graph plots the response of a lowpass elliptic filter Notice that the ripple in both the passband and stopband is constrained by the same maximum tolerable error as specified by ripple amount in dB Also notice the sharp transition edge for even low order elliptic filters Elliptic Response LabVIEW Function and VI Reference Manual 42 10 National Instruments Corporation Bessel Filters Chapter 42 Analysis Filter Vis You can use Bessel filters to reduce nonlinear phase distortion inherent in all IIR filters In higher order filters and those with a steeper rolloff this condition is more pronounced especially in the transition regions of the filters Bessel filters have maximally flat response in both magnitude and phase Furthermore the phase response in the passband of Bessel filters wh
15. andall R B and Tech B Frequency Analysis 3rd Edition Bru l and Kj r September 1979 The Fundamentals of Signal Analysis Application Note 243 Hewlett Packard 1985 340555B 01 Juloo N L ter a Analysis Filter Vis This chapter contains a brief discussion of digital filter theory and describes the VIs that implement IIR FIR and nonlinear filters To access the Analysis Filter palette select Function Analysis Filters The following illustration shows the options that are available on the Filter palette HAnalysis Filters Fit b 2 Filters For examples of how to use the filter VIs see the examples located in examples analysis fltrxmpl 11b Introduction to Digital Filtering Functions Analog filter design is one of the most important areas of electronic design Although analog filter design books featuring simple tested filter designs exist filter design is often reserved for specialists because it requires advanced mathematical knowledge and understanding of the processes involved in the system affecting the filter National Instruments Corporation 42 1 LabVIEW Function and VI Reference Manual Chapter 42 Analysis Filter Vis Modern sampling and digital signal processing tools have made it possible to replace analog filters with digital filters in applications that require flexibility and programmability These applications include audio telecommunications geophysics and me
16. ate a delay with the output sequence o n l delay gt where n is the number of FIR filter coefficients The following graphs plot a typical magnitude and phase response of FIR filters versus normalized frequency LabVIEW Function and VI Reference Manual 42 12 National Instruments Corporation Chapter 42 Analysis Filter VIs The discontinuities in the phase response arise from the discontinuities introduced when you compute the magnitude response using the absolute value Notice that the discontinuities in phase are on the order of pi The phase however is clearly linear See Appendix D References for material that can give you more information on this topic You design FIR filters by approximating a specified desired frequency response of a discrete time system The most common techniques approximate the desired magnitude response while maintaining a linear phase response Designing FIR Filters by Windowing The simplest method for designing linear phase FIR filters is the window design method To design a FIR filter by windowing you start with an ideal frequency response calculate its impulse response and then truncate the impulse response to produce a finite number of coefficients To meet the linear phase constraint by maintain symmetry about the center point of the coefficients The truncation of the ideal impulse response results in the effect known as the Gibbs phenomenon oscillatory behavior near abrupt tra
17. cale and convert to polar form to obtain magnitude and phase The frequency axis is identical to that of the two sided power spectrum The amplitude of the FFT is related to the number of points in the time domain signal Use the following equation to compute the amplitude and phase versus frequency from the FFT Magnitude FFT A real FET A imag FFT A Amplitude spectrum in quantity peak N N l mag ere Ph real FFT A ase spectrum in radians Phase FFT A arctangent real FET A where the arctangent function here returns values of phase between r and 7 a full range of 27 radians Using the rectangular to polar conversion function to convert the complex array FFT A N to its magnitude r and phase is equivalent to using the preceding formulas The two sided amplitude spectrum actually shows half the peak amplitude at the positive and negative frequencies To convert to the single sided form multiply each frequency other than DC by two and discard the second half of the array The units of the single sided amplitude spectrum are then in quantity peak and give the peak amplitude of each sinusoidal component making up the time domain signal For the single sided phase spectrum discard the second half of the array To view the amplitude spectrum in volts or another quantity rms divide the non DC components by the square root of two after converting the spectrum to the single sided form Because the non DC
18. ceding conditions and you cannot obtain its output signals via the convolution operation because a set of coefficients cannot characterize the impulse response of the filter Nonlinear filters provide specific filtering characteristics that are difficult to obtain using linear techniques The median filter is a nonlinear filter that combines lowpass filter characteristics to remove high frequency noise and high frequency characteristics to detect edges Filter VI Descriptions The following Filter VIs are available LabVIEW Function and VI Reference Manual 42 16 National Instruments Corporation
19. d of the signal ensuring that the signal fully decays by the end of the sample block Selecting a window function is not a simple task In fact there is no universal approach for doing so However Table 2 can help you in your initial choice Always compare the performance of different window functions to find the best one for the application Refer to the references at the end of this application note for more information about windows Application Note 041 14 www ni com Table 2 Initial Window Choice Based on Signal Content Unknown content Hann Scaling Windows Windows are useful in reducing spectral leakage when using the FFT for spectral analysis However because windows are multiplied with the acquired time domain signal they introduce distortion effects of their own The windows change the overall amplitude of the signal The windows used to produce the plots in Figures 7 and 8 were scaled by dividing the windowed array by the coherent gain of the window As a result each window yields the same spectrum amplitude result within its accuracy constraints You can think of an FFT as a set of parallel filters each Af in bandwidth Because of the spreading effect of a window each window increases the effective bandwidth of an FFT bin by an amount known as the equivalent noise power bandwidth of the window The power of a given frequency peak is computed by adding the adjacent frequency bins around a peak and is inflated by the band
20. dical monitoring Digital filters have the following advantages over their analog counterparts e They are software programmable e They are stable and predictable They do not drift with temperature or humidity or require precision components e They have a superior performance to cost ratio You can use digital filters in LabVIEW to control parameters such as filter order cutoff frequencies amount of ripple and stopband attenuation The digital filter VIs described in this section follow the virtual instrument philosophy The VIs handle all the design issues computations memory management and actual data filtering internally and are transparent to the user You do not have to be an expert in digital filters or digital filter theory to process the data The following discussion of sampling theory is intended to give you a better understanding of the filter parameters and how they relate to the input parameters The sampling theorem states that you can reconstruct a continuous time signal from discrete equally spaced samples if the sampling frequency is at least twice that of the highest frequency in the time signal Assume you can sample the time signal of interest at Af equally spaced intervals without losing information The Af parameter is the sampling interval You can obtain the sampling rate or sampling frequency f from the sampling interval l Ts Ar which means that according to the sampling theorem the
21. e calls by enabling state memory To enable state memory set the mit cont control of the VI to TRUE continuous filtering LabVIEW Function and VI Reference Manual 42 4 National Instruments Corporation Chapter 42 Analysis Filter Vis Transient _ steady state Original Signal Filtered Signal The number of elements in the filtered sequence equals the number of elements in the input sequence The filter retains the internal filter state values when the filtering completes The advantage of digital IIR filters over finite impulse response FIR filters is that IIR filters usually require fewer coefficients to perform similar filtering operations Thus IIR filters execute much faster and do not require extra memory because they execute in place The disadvantage of IIR filters is that the phase response is nonlinear If the application does not require phase information such as simple signal monitoring IIR filters may be appropriate You should use FIR filters for those applications requiring linear phase responses Cascade Form IIR Filtering Filters implemented using the structure defined by equation 42 2 directly are known as direct form IIR filters Direct form implementations are often sensitive to errors introduced by coefficient quantization and by computational precision limits Additionally a filter designed to be stable can become unstable with increasing coefficient length which is proportional to filt
22. ectrum array for each window at 254 through 258 Hz are shown below the graph Af is 1 Hz Application Note 041 10 www ni com Fower Spectrum of vvindowed Signals dB 0 0 10 0 20 0 30 0 40 0 50 0 60 0 r 0 0 80 0 40 0 l I I l I I I I 1 Hz 240 244 240 ehe hG Fa sj eb ha cre 254 Hz 255Hz 256Hz 257 Hz 258H2 Amplitude Error at 256 Hz dB Uniform 0 0000 0 0000 1 0000 0 0000 0 0000 VYrrms 2 0 0000 Hann 0 0000 0 2500 1 0000 0 2500 0 0000 Vrs 2 0 0000 Flat Top 0 4135 0 9338 1 0000 0 9338 0 4135 Vis 2 0 0000 Figure 7 Power Spectrum of 1 Vrms Signal at 256 Hz with Uniform Hann and Flat Top Windows Figure 8 shows the leakage effects when you acquire 256 5 cycles Notice that at a nonintegral number of cycles the Hann and Flat Top windows introduce much less spectral leakage than the Uniform window Also the amplitude error is better with the Hann and Flat Top windows The Flat Top window demonstrates very good amplitude accuracy but also has a wider spread and higher side lobes than the Hann window Power Spectrum of Windowed Signals dB 0 0 10 0 20 0 30 0 40 0 50 0 60 0 70 0 60 0 40 0 l I I I I I I I 240 cdd 24180 ehe H6 ZKD cb4 eho cre l Hz 254 Hz 255Hz 256Hz 257 Hz 258Hz Amplitude Error at 256 Hz dB Uniform 0 0162 0 0450 0 4053 0 4053 0 0450 ems 3 9224 Hann 0 0006 0 0289 0 7205 0 7205 0 0288 ems 1 4236 Flat Top 0 1591 0 7235 0 9978 0 9978 0 7235 Yims 0 0
23. el and Power Spectral Density The measurement of noise levels depends on the bandwidth of the measurement When looking at the noise floor of a power spectrum you are looking at the narrowband noise level in each FFT bin Thus the noise floor of a given power spectrum depends on the Af of the spectrum which is in turn controlled by the sampling rate and number of points In other words the noise level at each frequency line reads as if it were measured through a Af Hz filter centered at that frequency line Therefore for a given sampling rate doubling the number of points acquired reduces the noise power that appears in each bin by 3 dB Discrete frequency components theoretically have zero bandwidth and therefore do not scale with the number of points or frequency range of the FFT To compute the SNR compare the peak power in the frequencies of interest to the broadband noise level Compute the broadband noise level in Vrms by summing all the power spectrum bins excluding any peaks and the DC component and dividing the sum by the equivalent noise bandwidth of the window For example in Figure 6 the noise floor appears to be more than 120 dB below full scale even though the PCI 4450 Family dynamic range is only 93 dB If you were to sum all the bins excluding DC and any harmonic or other peak components and divide by the noise power bandwidth of the window you used the noise power level compared to full scale would be around 93 dB from full
24. equency multiply the sampling frequency by 0 464 for the 0 1 dB flatness Also the larger the FFT the larger the number of frequency lines A 2 048 point FFT yields twice the number of lines listed above Contrast this with typical benchtop instruments which have 400 or 800 useful lines for a 1 024 point or 2 048 point FFT respectively Dynamic Range Specifications The signal to noise ratio SNR of the PCI 4450 Family boards is 93 dB SNR is defined as 2 n y SNR org V where V and V are the rms amplitudes of the signal and noise respectively A bandwidth is usually given for SNR In this case the bandwidth is the frequency range of the board input which is related to the sampling rate as shown in Figure 5 The 93 dB SNR means that you can detect the frequency components of a signal that is as small as 93 dB below the full scale range of the board This is possible because the total input noise level caused by the acquisition front end is 93 dB below the full scale input range of the board If the signal you monitor is a narrowband signal that is the signal energy is concentrated in a narrow band of frequencies you are able to detect an even lower level signal than 93 dB This is possible because the noise energy of the board is spread out over the entire input frequency range Refer to the Computing Noise Level and Power Spectral Density section later in this application note for more information about narrowband versus b
25. er Vis Windowed FIR Filters You use the filter type parameter of the FIR VIs to select the type of windowed FIR filter you want lowpass highpass bandpass or bandstop The following list gives the two related FIR VIs FIR Windowed Coefficients Generates the windowed or unwindowed coefficients e FIR Windowed Filters Filters the input using windowed or unwindowed coefficients Optimum FIR Filters You can use the Parks McClellan algorithm to design optimum linear phase FIR filter coefficients in the sense that the resulting filter optimally matches the filter specifications for a given number of coefficients The Parks McClellan VI takes as input an array of band descriptions each containing information describing the response you want for the given band The VI outputs the FIR coefficients along with computed ripple which is a measure of the deviation of the resulting filter from the ideal filter specifications Four VIs use the Parks McClellan VI to implement filters whose stopband and passband ripple level are equal Equiripple LowPass Equiripple HighPass Equiripple BandPass and Equiripple BandStop FIR Narrowband Filters You can design narrowband FIR filters using the FIR Narrowband Coefficients VI and then implement the filtering using the FIR Narrowband Filter VI The design and implementation are separate operations because many narrowband filters require lengthy design times while the actual filtering pr
26. er order A less sensitive structure can be obtained by breaking up the direct form transfer function into lower order sections or filter stages The direct National Instruments Corporation 42 5 LabVIEW Function and Vi Reference Manual Chapter 42 Analysis Filter Vis form transfer function of the filter given by equation 42 2 with dp 1 can be written as a ratio of z transforms as follows byt byz t tby yz Orr HZ Tazi dan 0D i 42 2 1 Lr QZ F aa ay _ By factoring equation 42 2 into second order sections the transfer function of the filter becomes a product of second order filter functions N 5 De Oe Foa H z Il Ok Ld 42 3 LA P az a where N N 2 is the largest integer lt N 2 and Na gt Nb N is the number of stages This new filter structure can be described as a cascade of second order filters fom fee Cassed Aler Sages Each individual stage is implemented using the direct form II filter structure because it requires a minimum number of arithmetic operations and a minimum number of delay elements internal filter states Each stage has one input one output and two past internal states s i 1 and Skli 21 If n is the number of samples in the input sequence the filtering operation proceeds as in the following equations Salil Yea Li 1 ay pS pLi 1 Gags pli 2 k 1 2 Ns Vili Oops sli birskli 1 bapsjli 2 k 1 2 Ng
27. ese functions Refer to the Signal Sources for Frequency Response Measurement section for more information about Chirp signals and broadband noise signals Cross Power Spectrum One additional building block is the cross power spectrum The cross power spectrum is not typically used as a direct measurement but is an important building block for other measurements National Instruments Corporation 17 Application Note 041 The two sided cross power spectrum of two time domain signals A and B is computed as x Cross Power Spectrum S f SS The cross power spectrum is in two sided complex form To convert to magnitude and phase use the Rectangular To Polar conversion function To convert to a single sided form use the same method described in the Converting from a Two Sided Power Spectrum to a Single Sided Power Spectrum section of this application note The units of the single sided form are in volts or other quantity rms squared The power spectrum is equivalent to the cross power spectrum when signals A and B are the same signal Therefore the power spectrum is often referred to as the auto power spectrum or the auto spectrum The single sided cross power spectrum yields the product of the rms amplitudes of the two signals A and B and the phase difference between the two signals When you know how to use these basic blocks you can compute other useful functions such as the Frequency Response function Frequency Response and Netw
28. f frequency lines to display The characteristics of the signal acquisition front end affect the measurement The National Instruments PCI 4450 Family dynamic signal acquisition boards and the NI 4551 and NI 4552 dynamic signal analyzers are excellent acquisition front ends for performing FFT based signal analysis measurements These boards use delta sigma modulation technology which yields excellent amplitude flatness high performance antialiasing filters and wide dynamic range as shown in Figure 5 The input channels are also simultaneously sampled for good multichannel measurement performance Application Note 041 8 www ni com At a sampling frequency of 51 2 kHz these boards can perform frequency measurements in the range of DC to 23 75 kHz Amplitude flatness is 0 1 dB maximum from DC to 23 75 kHz Refer to the PCI 445 1 4452 4453 4454 User Manual for more information about these boards Calculating the Measurement Bandwidth or Number of Lines for a Given Sampling Frequency The dynamic signal acquisition boards have antialiasing filters built into the digitizing process In addition the cutoff filter frequency scales with the sampling rate to meet the Nyquist criterion as shown in Figure 5 The fast cutoff of the antialiasing filters on these boards means that the number of useful frequency lines in a 1 024 point FFT based spectrum is 475 lines for 0 1 dB amplitude flatness To calculate the measurement bandwidth for a given sampling fr
29. ff center the spectrum is smeared Figure 11 shows the frequency spectrum characteristics of a window in more detail The side lobe characteristics of the window directly affect the extent to which adjacent frequency components bias leak into adjacent frequency bins The side lobe response of a strong sinusoidal signal can overpower the main lobe response of a nearby weak sinusoidal signal Window Frequency Response Peak Side Lobe Level Side Lobe Roll Off Rate Main Lobe Width Frequency Figure 11 Frequency Response of a Window Another important characteristic of window spectra is main lobe width The frequency resolution of the windowed signal is limited by the width of the main lobe of the window spectrum Therefore the ability to distinguish two closely spaced frequency components increases as the main lobe of the window narrows As the main lobe narrows and spectral resolution improves the window energy spreads into its side lobes and spectral leakage worsens In general then there is a trade off between leakage suppression and spectral resolution Defining Window Characteristics To simplify choosing a window you need to define various characteristics so that you can make comparisons between windows Figure 11 shows the spectrum of a typical window To characterize the main lobe shape the 3 dB and 6 dB main lobe width are defined to be the width of the main lobe in FFT bins or frequency lines where the w
30. highest frequency that the digital system can process is LabVIEW Function and VI Reference Manual 42 2 National Instruments Corporation Chapter 42 Analysis Filter Vis of Inyq J1 The highest frequency the system can process is known as the Nyquist frequency This also applies to digital filters For example if your sampling interval is At 0 001 sec then the sampling frequency is Js 1000 Hz and the highest frequency that the system can process is Jma 900 Hz The following types of filtering operations are based upon filter design techniques e Smoothing windows e Infinite impulse response IIR or recursive digital filters e Finite impulse response FIR or nonrecursive digital filters Nonlinear filters The rest of this chapter presents a brief theoretical background on the IIR FIR and nonlinear techniques and discusses the digital filter VIs corresponding to each technique Refer to Chapter 43 Window VIs for information about the VIs that implement smoothing windows Infinite Impulse Response Filters Infinite impulse response filters IIR are digital filters with impulse responses that can theoretically be infinite in length duration The general difference equation characterizing IIR filters is N 1 N 1 1 y 2 Dix 4 gt sa 42 1 1 j 0 k where N is the number of forward coefficients b and N is the number of reverse coefficients a National Instruments Corporation 42
31. ich have antialiasing filters Note how an input signal at or above half the sampling frequency is severely attenuated Amplitude dB Frequency Sample Rate fs Figure 5 Bandwidth of PCI 4450 Family Input Versus Frequency Normalized to Sampling Rate Limitations of the Acquisition Front End In addition to reducing frequency components greater than half the sampling frequency the acquisition front end you use introduces some bandwidth limitations below half the sampling frequency To eliminate signals at or above half of the sampling rate to less than the measurement range antialiasing filters start to attenuate frequencies at some point below half the sampling rate Because these filters attenuate the highest frequency portion of the spectrum typically you want to limit the plot to the bandwidth you consider valid for the measurement For example in the case of the PCI 4450 Family sample shown in Figure 5 amplitude flatness is maintained to within 0 1 dB at up to 0 464 of the sampling frequency to 20 kHz for all gain settings 1 dB to 95 kHz and then the input gain starts to attenuate The 3 dB point or half power bandwidth of the input occurs at 0 493 of the input spectrum Therefore instead of showing the input spectrum all the way out to half the sampling frequency you may want to show only 0 464 of the input spectrum To do this multiply the number of points acquired by 0 464 respectively to compute the number o
32. ich is the region of interest is nearly linear Like Butterworth filters Bessel filters require high order filters to minimize the error and for this reason are not widely used You can also obtain linear phase response using FIR filter designs The following graphs plot the response of a lowpass Bessel filter Notice that the response is smooth at all frequencies as well as monotonically decreasing in both magnitude and phase Also notice that the phase in the passband is nearly linear Bessel Magnitude Response Order 2 Order 5 Order 10 Order 5 Order 10 National Instruments Corporation 42 1 LabVIEW Function and VI Reference Manual Chapter 42 Analysis Filter Vis Finite Impulse Response Filters Finite impulse response FIR filters are digital filters which have a finite impulse response FIR filters are also known as nonrecursive filters convolution filters or moving average MA filters because you can express the output of an FIR filter as a finite convolution n 1 JE SY hir k 0 where x represents the input sequence to be filtered y represents the output filtered sequence and represents the FIR filter coefficients The following list gives the most important characteristics of FIR filters e They can achieve linear phase because of filter coefficient symmetry in the realization e They are always stable e You can perform the filtering function using the convolution and as such generally associ
33. indow response becomes 0 707 3 dB and 0 5 6 dB respectively of the main lobe peak gain National Instruments Corporation 13 Application Note 041 To characterize the side lobes of the window the maximum side lobe level and side lobe roll off rate are defined The maximum side lobe level is the level in decibels relative to the main lobe peak gain of the maximum side lobe The side lobe roll off rate is the asymptotic decay rate in decibels per decade of frequency of the peaks of the side lobes Table 1 lists the characteristics of several window functions and their effects on spectral leakage and resolution Table 1 Characteristics of Window Functions 3 dB Main Lobe 6 dB Main Lobe Maximum Side Side Lobe Roll Off Window Width bins Width bins Lobe Level dB Rate dB decade Uniform None Hanning Hann Hamming Blackman Harris Exact Blackman Blackman Flat Top Strategies for Choosing Windows Each window has its own characteristics and different windows are used for different applications To choose a spectral window you must guess the signal frequency content If the signal contains strong interfering frequency components distant from the frequency of interest choose a window with a high side lobe roll off rate If there are strong interfering signals near the frequency of interest choose a window with a low maximum side lobe level If the frequency of interest contains two or more signals very near
34. itive half of the spectrum followed by the negative half of the spectrum as shown in Figure 1 In a two sided spectrum half the energy is displayed at the positive frequency and half the energy is displayed at the negative frequency Therefore to convert from a two sided spectrum to a single sided spectrum discard the second half of the array and multiply every point except for DC by two G i S i i 0 DC Application Note 041 2 www ni com where S 4 1 is the two sided power spectrum G44 1 is the single sided power spectrum and N is the length of the two sided power spectrum The remainder of the two sided power spectrum S44 5 through N 1 is discarded The non DC values in the single sided spectrum are then at a height of This is equivalent to where is the root mean square rms amplitude of the sinusoidal component at frequency k Thus the units of a power spectrum are often referred to as quantity squared rms where quantity is the unit of the time domain signal For example the single sided power spectrum of a voltage waveform is in volts rms squared Figure 2 shows the single sided spectrum of the signal whose two sided spectrum Figure shows Figure 2 Single Sided Power Spectrum of Signal in Figure 1 As you can see the level of the non DC frequency components are doubled compared to those in Figure 1 In addition the spectrum stops at half the frequency of that in Figure 1 National Inst
35. lly the best choice for a broadband signal source Because some stimulus signals are not constant in frequency across the time record applying a window may obscure important portions of the transient response National Instruments Corporation 19 Application Note 041 Conclusion There are many issues to consider when analyzing and measuring signals from plug in DAQ devices Unfortunately it is easy to make incorrect spectral measurements Understanding the basic computations involved in FFT based measurement knowing how to prevent antialiasing properly scaling and converting to different units choosing and using windows correctly and learning how to use FFT based functions for network measurement are all critical to the success of analysis and measurement tasks Being equipped with this knowledge and using the tools discussed in this application note can bring you more success with your individual application References Harris Fredric J On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform in Proceedings of the IEEE Vol 66 No 1 January 1978 Audio Frequency Fourier Analyzer AFFA User Guide National Instruments September 1991 Horowitz Paul and Hill Winfield The Art of Electronics 2nd Edition Cambridge University Press 1989 Nuttall Albert H Some Windows with Very Good Sidelobe Behavior IEEE Transactions on Acoustics Speech and Signal Processing Vol 29 No 1 February 1981 R
36. nction and VI Reference Manual 42 8 National Instruments Corporation Chapter 42 Analysis Filter Vis Chebyshey Response Chebyshev II or Inverse Chebyshev Filters Chebyshev II also known as inverse Chebyshev or Type II Chebyshev filters are similar to Chebyshev filters except that Chebyshev II filters distribute the error over the stopband as opposed to the passband and Chebyshev II filters are maximally flat in the passband as opposed to the stopband Chebyshev II filters minimize peak error in the stopband by accounting for the maximum absolute value of the difference between the ideal filter and the filter response you want The frequency response characteristics of Chebyshev II filters are equiripple magnitude response in the stopband monotonically decreasing magnitude response in the passband and a rolloff sharper than Butterworth filters The following graph plots the response of a lowpass Chebyshev II filter Notice that the equiripple response in the stopband is constrained by the maximum tolerable error and that the smooth monotonic rolloff appears in the stopband The advantage of Chebyshev II filters over Butterworth filters is that Chebyshev II filters give a sharper transition between the passband and the stopband with a lower order filter This difference corresponds to a smaller absolute error and higher execution speed One advantage of Chebyshev II filters over regular Chebyshev filters is National
37. ness of the transition proportional to the filter National Instruments Corporation 42 7 LabVIEW Function and VI Reference Manual Chapter 42 Analysis Filter Vis order Higher order Butterworth filters approach the ideal lowpass filter response Butterworth Response Chebyshev Filters Butterworth filters do not always provide a good approximation of the ideal filter response because of the slow rolloff between the passband the portion of interest in the spectrum and the stopband the unwanted portion of the spectrum Chebyshev filters minimize peak error in the passband by accounting for the maximum absolute value of the difference between the ideal filter and the filter response you want the maximum tolerable error in the passband The frequency response characteristics of Chebyshev filters have an equiripple magnitude response in the passband monotonically decreasing magnitude response in the stopband and a sharper rolloff than Butterworth filters The following graph shows the response of a lowpass Chebyshev filter Notice that the equiripple response in the passband is constrained by the maximum tolerable ripple error and that the sharp rolloff appears in the stopband The advantage of Chebyshev filters over Butterworth filters is that Chebyshev filters have a sharper transition between the passband and the stopband with a lower order filter This produces smaller absolute errors and higher execution speeds LabVIEW Fu
38. nother case is if you capture a transient signal that fits entirely into the time record In most cases however you measure an unknown signal that is stationary that is the signal is present before during and after the acquisition In this case you cannot guarantee that you are sampling an integral number of cycles Spectral leakage distorts the measurement in such a way that energy from a given frequency component is spread over adjacent frequency lines or bins You can use windows to minimize the effects of performing an FFT over a nonintegral number of cycles Figure 7 shows the effects of three different windows none Uniform Hanning also commonly known as Hann and Flat Top when an integral number of cycles have been acquired in this figure 256 cycles in a 1 024 point record Notice that the windows have a main lobe around the frequency of interest This main lobe is a frequency domain characteristic of windows The Uniform window has the narrowest lobe and the Hann and Flat Top windows introduce some spreading The Flat Top window has a broader main lobe than the others For an integral number of cycles all windows yield the same peak amplitude reading and have excellent amplitude accuracy Figure 7 also shows the values at frequency lines of 254 Hz through 258 Hz for each window The amplitude error at 256 Hz is 0 dB for each window The graph shows the spectrum values between 240 and 272 Hz The actual values in the resulting sp
39. nsitions cutoff frequencies in the FIR filter frequency response You can reduce the effects of the Gibbs phenomenon by smoothing the truncation of the ideal impulse response using a smoothing window function By tapering the FIR coefficients at each end you can diminish the height of the side lobes in the frequency response The disadvantage to this method however is that the main lobe widens resulting in a wider transition region at the cutoff frequencies The selection of a window function then is similar to the choice between Chebyshev and National Instruments Corporation 42 13 LabVIEW Function and VI Reference Manual Chapter 42 Analysis Filter Vis Butterworth IIR filters in that it is a trade off between side lobe levels near the cutoff frequencies and width of the transition region Designing FIR filters by windowing is simple and computationally inexpensive It is therefore the fastest way to design FIR filters It is not necessarily however the best FIR filter design method Designing Optimum FIR Filters using the Parks McClellan Algorithm The Parks McClellan algorithm offers an optimum FIR filter design technique that attempts to design the best filter possible for a given number of coefficients Such a design reduces the adverse effects at the cutoff frequencies It also offers more control over the approximation errors in different frequency bands control that is not possible with the window method Using the Pa
40. ocess is very fast and efficient Keep this in mind when creating your narrowband filtering diagrams The parameters required for narrowband filter specification are filter type sampling rate passband and stopband frequencies passband ripple linear scale and stopband attenuation decibels For bandpass and bandstop filters passband and stopband frequencies refer to bandwidths and you must specify an additional center frequency parameter You can also design wideband lowpass filters cutoff frequency near Nyquist and wideband highpass filters cutoff frequency near zero using the narrowband filter VIs National Instruments Corporation 42 15 LabVIEW Function and VI Reference Manual Chapter 42 Analysis Filter Vis The following illustration shows how to use the FIR Narrowband Coefficients VI and the FIR Narrowband Filter VI to estimate the response of a narrowband filter to an impulse ripple rp Sampling freq fs passband tpass Stopband fstop center freq fc attenuation db Ar FIR Narrowband Coefficients filter ty pe Nonlinear Filters Smoothing windows IIR filters and FIR filters are linear because they satisfy the superposition and proportionality principles L fax f by t ab tx bLiy where a and b are constants x t and y f are signals L is a linear filtering operation and their inputs and outputs are related via the convolution operation A nonlinear filter does not meet the pre
41. onse Function The impulse response function of a network is the time domain representation of the transfer function of the network It is the output time domain signal generated when an impulse is applied to the input at time t 0 To compute the impulse response of the network take the inverse FFT of the two sided complex transfer function as described in the Transfer Function section of this application note Sag Impulse Response f Inverse FFT Transfer Function H f Inverse FFT asl r S44 f The result is a time domain function To average multiple readings take the inverse FFT of the averaged transfer function Coherence Function The coherence function is often used in conjunction with the transfer function as an indication of the quality of the transfer function measurement and indicates how much of the response energy is correlated to the stimulus energy If there is another signal present in the response either from excessive noise or from another signal the quality of the network response measurement is poor You can use the coherence function to identify both excessive noise and causality that is identify which of the multiple signal sources are contributing to the response signal The coherence function is computed as p Magnitude Averaged S f Coherence Function f GG AA Averaged S f Averaged Spp f The result is a value between zero and one versus frequency A zero for a given frequency
42. ork Analysis Three useful functions for characterizing the frequency response of a network are the transfer impulse response and coherence functions The frequency response of a network is measured by applying a stimulus to the network as shown in Figure 12 and computing the transfer function from the stimulus and response signals Measured Stimulus A Network Applied Stimulus Under Measured Response B Test Figure 12 Configuration for Network Analysis Transfer Function The transfer function gives the gain and phase versus frequency of a network and is typically computed as Cross Power Spectrum Stimulus Response _ SAB f Transfer Function H f p ransfer Function H f Power Spectrum Stimulus S44 f where A is the stimulus signal and B is the response signal The transfer function is in two sided complex form To convert to the frequency response gain magnitude and the frequency response phase use the Rectangular To Polar conversion function To convert to single sided form simply discard the second half of the array You may want to take several transfer function readings and then average them To do so average the cross power spectrum S4 g f by summing it in the complex form then dividing by the number of averages before converting it to magnitude and phase and so forth The power spectrum S44 is already in real form and is averaged normally Application Note 041 18 www ni com Impulse Resp
43. r spectrum As shown in the preceding equations for power and amplitude you must supply a reference for a measure in decibels This reference then corresponds to the 0 dB level Several conventions are used A common convention is to use the reference 1 Vrms for amplitude or 1 Vrms squared for power yielding a unit in dBV or dBVrms In this case I Vrms corresponds to 0 dB Another common form of dB is dBm which corresponds to a reference of I mW into a load of 50 Q for radio frequencies where 0 dB is 0 22 Vrms or 600 Q for audio frequencies where 0 dB is 0 78 Vrms Application Note 041 6 www ni com Antialiasing and Acquisition Front Ends for FFT Based Signal Analysis FFT based measurement requires digitization of a continuous signal According to the Nyquist criterion the sampling frequency F must be at least twice the maximum frequency component in the signal If this criterion is violated a phenomenon known as aliasing occurs Figure 3 shows an adequately sampled signal and an undersampled signal In the undersampled case the result is an aliased signal that appears to be at a lower frequency than the actual signal O Adequately sampled signal Aliased signal due to undersampling Figure 3 Adequate and Inadequate Signal Sampling When the Nyquist criterion is violated frequency components above half the sampling frequency appear as frequency components below half the sampling frequency resulting in an erroneous represen
44. requency To increase the frequency resolution for a given frequency range increase the number of points acquired at the same sampling frequency For example acquiring 2 048 points at 1 024 kHz would have yielded Af 0 5 Hz with frequency range 0 to 511 5 Hz Alternatively if the sampling rate had been 10 24 kHz with 1 024 points Af would have been 10 Hz with frequency range from 0 to 5 11 kHz Computations Using the FFT The power spectrum shows power as the mean squared amplitude at each frequency line but includes no phase information Because the power spectrum loses phase information you may want to use the FFT to view both the frequency and the phase information of a signal The phase information the FFT yields is the phase relative to the start of the time domain signal For this reason you must trigger from the same point in the signal to obtain consistent phase readings A sine wave shows a phase of 90 at the sine wave frequency A cosine shows a 0 phase In many cases your concern is the relative phases between components or the phase difference between two signals acquired simultaneously You can view the phase difference between two signals by using some of the advanced FFT functions Refer to the FFT Based Network Measurement section of this application note for descriptions of these functions Application Note 041 4 www ni com The FFT returns a two sided spectrum in complex form real and imaginary parts which you must s
45. rks McClellan algorithm to design FIR filters is computationally expensive This method however produces optimum FIR filters by applying time consuming iterative techniques Designing Narrowband FIR Filters When you use conventional techniques to design FIR filters with especially narrow bandwidths the resulting filter lengths may be very long FIR filters with long filter lengths often require lengthy design and implementation times and are more susceptible to numerical inaccuracy In some cases conventional filter design techniques such as the Parks McClellan algorithm may fail the design altogether You can use a very efficient algorithm called the Interpolated Finite Impulse Response IFIR filter design technique to design narrowband FIR filters Using this technique produces narrowband filters that require far fewer coefficients and therefore fewer computations than those filters designed by the direct application of the Parks McClellan algorithm LabVIEW also uses this technique to produce wideband lowpass cutoff frequency near Nyquist and highpass filters cutoff frequency near zero For more information about IFIR filter design see Multirate Systems and Filter Banks by P P Vaidyanathan or the paper on interpolated finite impulse response filters by Neuvo et al listed in Appendix D References of this manual LabVIEW Function and VI Reference Manual 42 14 National Instruments Corporation Chapter 42 Analysis Filt
46. roadband levels The spurious free dynamic range of the dynamic signal acquisition boards is 95 dB Besides input noise the acquisition front end may introduce spurious frequencies into a measured spectrum because of harmonic or intermodulation distortion among other things This 95 dB level indicates that any such spurious frequencies are at least 95 dB below the full scale input range of the board The signal to total harmonic distortion THD plus noise ratio which excludes intermodulation distortion is 90 dB from 0 to 20 kHz THD is a measure of the amount of distortion introduced into a signal because of the nonlinear behavior of the acquisition front end This harmonic distortion shows up as harmonic energy added to the spectrum for each of the discrete frequency components present in the input signal The wide dynamic range specifications of these boards is largely due to the 16 bit resolution ADCs Figure 6 shows a typical spectrum plot of the PCI 4450 Family dynamic range with a full scale 997 Hz signal applied You can see that the harmonics of the 997 Hz input signal the noise floor and any other spurious frequencies are below 95 dB In contrast dynamic range specifications for benchtop instruments typically range from 70 dB to 80 dB using 12 bit and 13 bit ADC technology National Instruments Corporation 9 Application Note 041 0 0 sell 40 0 60 0 50 0 100 0 120 0 140 0 160 0 gt
47. ruments Corporation 3 Application Note 041 Adjusting Frequency Resolution and Graphing the Spectrum Figures and 2 show power versus frequency for a time domain signal The frequency range and resolution on the X axis of a spectrum plot depend on the sampling rate and the number of points acquired The number of frequency points or lines in Figure 2 equals NI where N is the number of points in the acquired time domain signal The first frequency line is at 0 Hz that is DC The last frequency line is at Pe aln where F is the frequency at which the acquired time domain signal was sampled The frequency lines occur at Af intervals where Apo T Hy Frequency lines also can be referred to as frequency bins or FFT bins because you can think of an FFT as a set of parallel filters of bandwidth Af centered at each frequency increment from DCt Ms_ fs oN Alternatively you can compute Af as 1 Af f N e At where At is the sampling period Thus N At is the length of the time record that contains the acquired time domain signal The signal in Figures 1 and 2 contains 1 024 points sampled at 1 024 kHz to yield Af 1 Hz and a frequency range from DC to 511 Hz The computations for the frequency axis demonstrate that the sampling frequency determines the frequency range or bandwidth of the spectrum and that for a given sampling frequency the number of points acquired in the time domain signal record determine the resolution f
48. tation of the signal For example a component at frequency 7 lt fo lt F appears as the frequency F fo Figure 4 shows the alias frequencies that appear when the signal with real components at 25 70 160 and 510 Hz is sampled at 100 Hz Alias frequencies appear at 10 30 and 40 Hz Solid Arrows Actual Frequency Dashed A Ali 9 alias ashed Arrows ias 30 Hz F3 alias F4 alias F1 10Hz 25Hz x ne A A fs 2 50 fs 100 Nyquist Frequency Sampling Frequency Figure 4 Alias Frequencies Resulting from Sampling a Signal at 100 Hz That Contains Frequency Components Greater than or Equal to 50 Hz National Instruments Corporation Application Note 041 Before a signal is digitized you can prevent aliasing by using antialiasing filters to attenuate the frequency components at and above half the sampling frequency to a level below the dynamic range of the analog to digital converter ADC For example if the digitizer has a full scale range of 80 dB frequency components at and above half the sampling frequency must be attenuated to 80 dB below full scale These higher frequency components do not interfere with the measurement If you know that the frequency bandwidth of the signal being measured is lower than half the sampling frequency you can choose not to use an antialiasing filter Figure 5 shows the input frequency response of the National Instruments PCI 4450 Family dynamic signal acquisition boards wh
49. width of the window You must take this inflation into account when you perform computations based on the spectrum Refer to the Computations on the Spectrum section for sample computations Table 3 lists the scaling factor or coherent gain the noise power bandwidth and the worst case peak amplitude accuracy caused by off center components for several popular windows Table 3 Correction Factors and Worst Case Amplitude Errors for Windows a ee Se ee Window Coherent Gain Bandwidth Error dB National Instruments Corporation 15 Application Note 041 Computations on the Spectrum When you have the amplitude or power spectrum you can compute several useful characteristics of the input signal such as power and frequency noise level and power spectral density Estimating Power and Frequency The preceding windowing examples demonstrate that if you have a frequency component in between two frequency lines it appears as energy spread among adjacent frequency lines with reduced amplitude The actual peak is between the two frequency lines In Figure 8 the amplitude error at 256 5 Hz is due to the fact that the window is sampled at 0 5 Hz around the center of its main lobe rather than at the center where the amplitude error would be 0 This is the picket fence effect explained in the Window Characteristics section of this application note You can estimate the actual frequency of a discrete frequency component to a greater resolution
50. yli Ynsli for each sample i 0 1 2 n 1 LabVIEW Function and VI Reference Manual 42 6 National Instruments Corporation Chapter 42 Analysis Filter Vis For filters with a single cutoff frequency lowpass and highpass second order filter stages can be designed directly The overall IIR lowpass or highpass filter contains cascaded second order filters For filters with two cutoff frequencies bandpass and bandstop fourth order filter stages are a more natural form The overall IIR bandpass or bandstop filter is cascaded fourth order filters The filtering operation for fourth order stages proceeds as in the following equations Vil DogS eli Oy psp li 1 Oop8pli 2 0338 11 3 bajsili 41 kel 2 N yli Yyslil Notice that in the case of fourth order filter stages N N 1 4 Butterworth Filters A smooth response at all frequencies a nd a monotonic decrease from the specified cutoff frequencies characterize the frequency response of Butterworth filters Butterworth filters are maximally flat the ideal response of unity in the passband and zero in the stopband The half power frequency or the 3 dB down frequency corresponds to the specified cutoff frequencies The following illustration shows the response of a lowpass Butterworth filter The advantage of Butterworth filters is a smooth monotonically decreasing frequency response After you set the cutoff frequency LabVIEW sets the steep
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