Understanding Noisy Signal

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Signal conditioning includes selectable anti-alias l-tering in addition to basic facilities such as multipleinput ranges. Dynamic signals often are AC coupled,which eliminates DC offset and drift problems. ADSA also may contain a signal source that you canuse to excite the system under test.

Because even a small amount of energy at thewrong frequency is undesirable, a DSA’s dynamicrange is an important specication. The term dy-namic range refers only to the ratio of the largest tosmallest signals handled and has no other connectionto dynamic systems.

A high-resolution ADC preserves the dynamicrange of the analog input circuitry better than onewith less resolution and is absolutely necessary if the DSA is to have a low noise oor. The highestcommonly available resolution is 24 b, and manyDSAs have this resolution. In addition to supportinga large dynamic range, high resolution also is moreforgiving of a suboptimal gain setting.

Unfortunately, most high-resolution ADCs are of

the delta-sigma type and include a very high-order digital lter. The extremely sharp lter cutoff in thefrequency domain creates artifacts in transient time-domain signals. Obviously, these are not part of the

real signal and add to the dataacquisition system errors.

If the signal bandwidth pre-sented to the ADC has beenreduced by an analog anti-aliaslter, the effect can be avoided.However, it is economicallyadvantageous to use one xed-frequency analog anti-aliaslter ahead of a delta-sigmaADC running at a high samplerate. At lower sample rates, fur-ther ltering occurs via digitaldecimation with the attendanttime-domain artifacts. The side-bar accompanying this articledescribes a digital method thateliminates the artifacts, and apatent has been applied for thistechnique.

Beyond basic characteristics,

DSA feature sets and their implementations vary widely.

DYNAMIC SIGNAL ANALYZERS

Understanding Noisy Signalsby Tom Lecklider, Senior Technical Editor

A dynamic signal analyzer (DSA)is used to investigate signals rep-resenting forces and motions. In

this context, the word dynamic has a precise mean-ing distinct from the usual synonyms such as spiritedor energetic. Dynamics is the branch of mechanicsconcerned with the motions of material bodiesunder the action of given forces. It follows that adynamic system is a group of interacting mechanicalcomponents and forces.

Dynamic systems clearly have a time element as-sociated with them, although many effects are better

understood by their frequency spectrum.For example, noises within a car may beof the squeak or rattle variety. Becausethis type of problem is so common,especially in dashboard assemblies, it’sbecome a standard procedure for automanufacturers to test for buzz, squeak,and rattle. Similarly, there’s also noise,

vibration, and harshness testing.

DSAs sample their input signals in the time do-main but provide comprehensive fast Fourier trans-form (FFT)–based frequency-domain capabilities.

With a good DSA,analyzing the sound

of one hand clappingis not a problem.

Figure 1. Autocorrelation of a Waveform With Positive and Negative Ramps



Some vendors, such as National Instruments (NI), providethe constituent parts of a DSA in the form of data acquisi-tion boards and data processing software. The type 446x and447x products have from two to eight input channels, a 3-dBbandwidth up to about 80 kHz, and built-in anti-alias ltering.

You need to use the company’s Sound and Vibration Toolkitsoftware to perform signal analysis.In contrast, the 16-b NI 455x data acquisition boards have

a built-in DSP. This means that spectral analysis is performedin hardware much more quickly than with a software-basedapproach. The board interface is via the dedicated NI-DSAinstrument driver, and you must select the algorithms thatwill run on the DSP.

The usual advantages for modular, software-based instru-ments apply to this type of approach: a high degree of ex-ibility and the capability of higher performance with a faster PC. Against these positive factors, the user must do much moreinstrument development work, although having commonly

used algorithms bundled as a software toolkit helps.Several PC-based products include a separate data acquisi-

tion chassis with DSP processing. Instrument interface func-tions and display of analysis results are handled by the PC, butthe combination of PC and hardware behaves as an integratedturnkey solution.

Other manufacturers package their DSAs as stand-aloneinstruments similar to spectrum analyzers or DSOs. The func-tionality may be limited to that originally shipped, or you mayhave the option of downloadable eld upgrades and customanalysis routines. Upgradeability is a capability worth look-ing for because you probably will be using a newly purchasedDSA for many years.

Physical phenomena usually are below 100 kHz in frequencyand don’t change a lot from year to year, so a mature instru-ment design may not be a bad thing: The manufacturer shouldhave corrected any early production problems, for example.Also, to keep up with the competition, the instrument softwarewill have been periodically enhanced with new features andperhaps more efcient algorithms.

Frequency-Domain Tools

FFT Most DSAs use the FFT to produce a frequency-domain

representation of data acquired in the time domain. For a block of N = 2 n data points, the corresponding single-sided spectrumwill contain N/2 discrete frequency lines spaced F s /N apart,where F s is the sampling frequency. The rst line is at DC andthe last at F s /2 - F s /N. For example, if 2,048 points are acquiredat a 1,024-Hz rate, the FFT will produce 1,024 lines spaced0.5 Hz apart extending from DC to 511.5 Hz.

Although many DSA users simply may accept the FFT-basedfrequency-domain display presented to them, there are manyreasons to maintain a healthy skepticism. The FFT assumesthat the continuous function of time that has been sampledhas a period corresponding to the length of the captured datablock. This coherence condition is satised if the samples areall unique and an integer number of cycles of the periodicsignal t within the N samples.


A transient signal that decays to zero before the N th sampleis a special case that meets these conditions. Another suitablesituation may be created when you can precisely control thesample rate and input signal frequency. In this case, you canachieve coherent sampling if N•F in = B•F s where F in is the

input signal frequency and B an odd prime number smaller than N.For example, if B = 1, exactly one cycle of the input signal

will be captured and F s = N•F in.The samples all will representdifferent points on the waveform.

To reduce the ratio of F s:F in, a much larger value of B couldbe used. If B = 211 and N = 4,096, for a 1-kHz input signal, thesampling rate Fs must be 19.412322 kHz. In other words, therewill be 19.412322 samples per cycle of F in, and the samples willrepresent different points on each of the 211 cycles.

Satisfying the coherence criteria causes all the energy as-sociated with each spectral line to be correctly represented.In general, sampling is not coherent perhaps because youdon’t have precise control of the sampling rate but more oftenbecause the exact input frequency is unknown. In this case,the energy from one spectral line will become spread acrossseveral adjacent lines, an effect called spectral leakage.

To reduce leakage, windowing is required. If an input signalthat is not periodic with respect to the sampled data block isconvolved with a suitable function in the time domain, it can bemade to equal zero at the beginning and end of the data block.There are many different windowing functions, and choosingwhich to use depends on many factors.

The point of the discussion is not to get into the details of Blackman, Hann, or Hamming windows but rather to explain

that the spectrum produced by the FFT depends on much morethan just the input signal. The difference between coherent andnoncoherent sampling is large, and test results can be mislead-ing if the FFT is indiscriminately applied.

Frequency-Domain FunctionsWhatever DSA you may have, it probably includes analysis

functions such as linear, cross, and power spectrum; power spectral density; autocorrelation; cross correlation; histogram;probability density functions (PDF); and cumulative distribu-tion functions (CDF). What are these things and what are theyused for?

When the FFT function is applied to a series of N time-

domain values, it returns the real and imaginary componentsfor N/2 positive and N/2 negative frequency bins. This is adouble-sided FFT. To convert to a single-sided spectrum, ndthe magnitude at each frequency:

Phase can be expressed as phase = arctan [imaginary(FFT)/ real(FFT)] where the arctan function needs to operate over theentire range of ± π radians.

Starting with the line after the DC line, double the amplitudefor all the positive frequency lines and discard the negativefrequency lines. The single-sided spectrum represents the peak amplitudes of cosine functions at each frequency bin and withindividual phase offsets.



The power spectrum can be computedby multiplying the complex value of theFFT by its conjugate: power is propor-tional to [real(FFT)+imaginary(FFT)j] •[real(FFT)–imaginary(FFT)j]. Because

the inner products cancel when complexconjugates are multiplied

(x+jy) • (x-jy) = x2+y2 = magnitude 2

More generally, if the FFT of onetime-domain signal Q is multiplied bythe complex conjugate of the FFT of an-other signal R, a cross power spectrumresults. Here the inner products do notcancel, so the phase of each line will not

be the same as either of the phases of the corresponding lines in Q or R. Thepower spectrum is the same as the crosspower spectrum but with signals Q andR identical. Because of this, the power

spectrum also is called the auto power spectrum or the auto spectrum.Similarly, autocorrelation is cross cor-

relation performed on a series of valuesand a copy of that series: It’s a measureof how well parts of a signal resembleother parts of the same signal. Cross cor-relation measures how well parts of onesignal resemble parts of another.

The cross correlation of two sets of N samples A[a 1, a2,…a N] and B[b 1,b2,…b N] at an offset of m samples isdened as

For a real series of N samples and for values of m from -N to +N, cross or au-

tocorrelation gives a resulting waveformtwice as long as the original sampledsignal.

Figure 1 shows an example of autocorrelation. The red and green

waveforms represent the same databut shifted vertically away from zeroto reduce overlap of the curves andhorizontally to demonstrate the slid-ing action of correlation. This relativehorizontal position of the two curvescorresponds to the zero value at the leftedge of the blue autocorrelation curvewhere m = -N.

Summing the products of red andgreen points having the same horizontalvalue gives a nonzero result as the redwaveform is progressively moved tothe right with an increasing value of m. The blue peak about one divisionfrom the left edge corresponds to thered curve being shifted one division tothe right so that its last ramp sectionand the green curve’s rst ramp section

Test results can be

misleading if the FFT is

indiscriminately applied.




coincide. As that shifting took place,the blue autocorrelation sum increasedcorresponding to the increasing degreeof red-green overlap.

The largest peak occurs after N

shifts, at m = 0, as it does for any au-tocorrelation operation. If the signalis periodic, such as a sine wave, theremay be other peaks as large, but theautocorrelation value cannot ever belarger than when m = 0 and the curveis exactly aligned with itself. Further shifting causes the degree of overlapto reduce until, after an additional Nshifts, the red curve will be completelyoff the right edge of the graph. Thisposition corresponds to m = +N andthe last blue value of zero.

Figure 2a shows the FFT of the sameset of N samples and Figure 2b the sumof several frequency components thatapproximates the original waveform.

Histograms, PDFs, and CDFs displaythe relationships among the values of thesamples within a group. For example, ahistogram generated from the set of Nsamples shown in Figure 1 is displayedin Figure 3 . It’s intuitive from the wave-form in Figure 1 that the value zero oc-curs most often, and this is clearly shownin Figure 3. Also, because a single cycleof the waveform contains two positiveramps and only one negative ramp, posi-tive values occur twice as often.

PDFs and CDFs generally are usedto help understand noise-like signalsrather than the well-dened waveformin Figure 1. For example, the distribu-tion of power at different frequenciesdescribes a vibration test and often ispresented as a power spectrum density(PSD) display. Of course, a swept sine

source could be used, which wouldbe very predictable. More often, andespecially in pneumatic actuator drivenHALT/HASS chambers, vibration ap-pears to be random but with a prede-termined distribution.

A PSD is a density function with unitsof power per hertz, usually expressed asg2 /Hz. A CDF is the integral of a PDF.Integrating the area under a PSD curvedisplays the percentage of the totalpower that is applied below a certain

frequency. For example, in a normaldistribution, 50% of the power existsFigure 3. Histogram of the Ramp Waveform

Figure 2a. FFT of the Figure 1 Waveform

Figure 2b. Partial Reconstruction of Figure 1 Waveform by Summing FFT Lines



Data acquisition systems apply anti-alias ltering before the

signal is converted from analog to digital. In almost all systems,

the sampling rate can vary greatly, so the cutofffrequency should adapt accordingly.

It is very difcult and sometimes even impossibleto adapt the cutoff frequency of a high-order analoglter. So digital decimation often is used to vary thesampling rate. With this approach, the hardwaresampling rate is xed at a relatively high value, andthe cutoff frequency of the analog lter does notneed to adapt.

The sharp decimation lter causes the step responseto have signicant overshoot as shown in Figure 4 .The ripple can easily be as high as 10% of the stepamplitude, causing measurement accuracy issues.

A Bessel lter has about the best step responsein the analog domain because of its constant groupdelay or linear phase and slow transition band. Asimple extension of this concept would be to usea digital Bessel lter. However, because a digitalrepresentation of a Bessel lter does not preserve a linearphase relationship, the resulting performance is not as goodas an analog Bessel lter.

Since linear phase characteristics are preserved with an FIRlter, it has the potential to overcome certain limitations of a

for frequencies less than the mean because the distribution issymmetrical.

In general, distributions used in real vibration testing con-centrate energy within dened frequency bands to simulate thestresses experienced in actual use. In the case of a pneumatic

actuator-driven vibration table, signicant energy simply is notproduced outside of a frequency band from a few hundred toa few thousand hertz.

SummaryIt should be clear from the many specialized functions

DSAs provide that it’s difcult to accomplish detailed vi-

bration analysis with basic data acquisition tools. If you’reinvestigating vibration, you need a DSA or at least the ap-propriate analysis and display software. Manufacturers of DSAs tend to use standard mathematical terms, although thenames of the instruments themselves vary. The DSA name is

common, but the product also is called a spectrum analyzer,which emphasizes the frequency-domain aspect of many of the analysis functions.

digital Bessel lter. However, an FIR lter usually has a very sharp

transition band. Indeed, in the past, engineers have strived tomake the transition band as sharp as possible, but toeliminate transient ripples, we need a slow transitionband.

An FIR lter with a few taps creates the desiredeffect. Figure 5 represents the step response with afour-tap FIR ripple elimination lter, showing that theovershoot has been eliminated. The result is as goodas an analog Bessel lter.

The ripple elimination lter overcomes the poor stepresponse of a delta-sigma ADC. Delta-sigma ADCs withripple elimination lters offer the following benetsrelative to acquisition systems using Bessel lters:• Better Phase Match Between ChannelsBecause the analog lter is xed at a relatively highfrequency, the phase distortion by capacitor error willbe decreased, improving the channel phase match.• Wider Bandwidth With Same Sampling RateIf the frequency-domain property is emphasized, the

cutoff frequency of the decimation lter can be set at near0.45 Fs.• More Versatile

A powerful digital lter can be applied after sampling accordingto the needs of different applications.

Ripple Elimination in Delta-Sigma Systemsby Gordon Chen, LDS-Dactron, LDS Test and Measurement

Figure 5. Delta-Sigma Transient Response With Added Linear-Phase FIR Filter

Figure 4. Large Overshoot Caused by High-Order Digital Filter

Reprinted from EE-Evaluation Engineering , August 2005Copyright © 2005 by Nelson Publishing Inc. • www.evaluationengineering.com

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