Understanding Data Converters’ Frequency Domain …

1 DATEL, Inc., Mansfield, MA 02048 (USA) Tel: (508)339-3000, (800)233-2765 Fax: (508)339-6356 Email: Internet: Application Note AN-4By Bob Leonard, DATEL, data Converters Frequency Domain SpecificationsInnovation and Excellence in Precision data AcquisitionTime DomainFrequency DomainFrequencySpectral ComponentsSignal AnalysisComposite Analog Signal(Steady State Instantaneous Algebraic sum of all Spectral Components)TimeAmplitudeTIME Domain VS. Frequency DOMAINDATEL was founded in 1970 as a producer of high performance Analog-to-Digital Converters and data Acquisitionproducts. Our administrative offices, engineering, modular and subsystem production facilities and hybrid productionfacilities qualified to MIL-STD-1772, are housed in our 180,000 square foot facility in Mansfield, Domain data collected from an A/D converter is mapped into the Frequency Domain using the FastFourier Transform (FFT)

2 Note AN-4 data Acquisition BoardsVMEM ultibusPC/ATPanel ProductsDigital Panel MetersProcess MonitorsPrintersCalibratorsData Conversion ComponentsA/DsAmplifiersSampling A/DsFiltersSample-HoldsHDASsD/AsMultiple xersPower ProductsDC-DC ConvertersAC-DC ConvertersFourier Transform for A/D sThe Fourier Transform is intended to operate on continuous waveform data from to + .Discrete Fourier Transform (DFT) is used on sampled A/Ddata. The ideal continuous waveform from to + has beenreplaced with sampled points on a waveform for a limitedtime Fast Fourier Transform (FFT) Algorithm is used inimplementing the Discrete Fourier Transform due to theFFT S mathematical Fast Fourier Transform is a mathematically efficientalgorithm to supplant the Discrete Fourier Transform.

3 Likewisethe DFT is used to supplant the Continuous Fourier Transformfor time sampled data . In principle, most aspects of the CFTtransfer to the DFT and FFT, however there are nuances thatdemand data points in the time Domain produce N/2 points (ampli-tude and phase) in the Frequency : = 2 fX(f) = x(t) e dt j t Where: N = Total # of points in the recordn f = Finite # of Frequency points t = Sampling intervalk = IntegerX (n f)= X(k t)e tN 1K=0D j2 n fk t# OF POINTS FFT IMPROVEMENTNN2 /N LOG2 N128 32512 VERSUS DFT PROCESSINGThe Fast Fourier Transform (FFT) requires NLog2 N opera-tions (multiplication & addition) while the Discrete FourierTransform (DFT) requires N2 Fourier TransformWeaknesses/CuresAliasing/Nyquis t SamplingLeakage/WindowingPicket-Fence effect /# of FFT PointsSuccessful application of the FFT demands an appreciationof three basic limitations.

4 Aliasing, leakage and of the Nyquist Sampling rate, utilizing windowsto weight the non-infinite sequenced data and choosing anappropriate number of FFT points provide the appropriatesolutions to these non-ideal FFT plot of an actual Sampling A/D illustrates the inputfrequency, harmonics generated and the noise floor. A single-tonefrequency of MHz was the input for this 12-bit, 10 MHz 10 20 30 40 50 60 70 80 90 Signal2nd Harmonic3rd HarmonicFREQUENCY (MHz)SIGNALAMPLITUDE(dB)3 Application Note AN-4 Signal Alias FrequencySampling PulsesAn inadequate sampling rate has the effect of producing an alias Frequency in the recovered signal.

5 Sampling at a rate less than twice per cycleresults in an alias which is significantly different from the original Frequency . ALIAS Frequency CAUSED BY INADEQUATE SAMPLING RATEV0fcffs-fcfsfs + fcFrequency Folding(b) Sampled Signal Spectrum(a) Continuous Signal SpectrumV0fcfs/2 Frequency SPECTRA DEMONSTRATING THE SAMPLING THEORUM4 Application Note AN-4 Frequency Domain SpecificationsSignal-to-Noise Ratio & Distortion (Sinad)Signal-to-Noise Ratio without DistortionTotal Harmonic DistortionIn-Band HarmonicsSpurious Free Dynamic Range (SFDR)two-Tone Intermodulation DistortionNoise Power Ratio (NPR)Effective BitsSome key Frequency Domain specifications for Sampling A/Dconverters are listed.

6 Understanding how these are definedand under what conditions is as important as knowing the FFTpitfalls and SpectraDemonstrating The Sampling TherorumThe Nyquist Sampling Theorum requires that a continuousbandwidth-limited analog signal, with Frequency componentsout to fc, must be sampled at a rate fs which is a minimum the sampling Frequency fs is not high enough, part of thespectrum centered about fs will fold over into the original signalspectrum ( Frequency folding). Frequency folding can beeliminated in two ways: first by using a high enough samplingrate and second by filtering the signal before sampling to limitits bandwidth to fs/2.

7 0151 SignalSampling Rate3rd Harmonic ( ) Frequency (MHz)2nd Harmonic ( )Alias of ( )Alias of (300 KHz)Original Signal SpectrumExample:Analyze where the 2nd harmonic of a MHz signal will alias when digitizedwith a 10 MHz sampling =10 MHzFin = 2nd HARMONIC = MHzFs/2 = 5 MHzsubstituting into the formula Fin = K(Fs/2) + Fenables the determination that: K = 1 (ODD) and DF = MHz therefore aliasfrequency = (Fs/2) F = 300 KHzFREQUENCY FOLDING AND ALIASINGF requency Folding and AliasingProcedure:Analyze the input Frequency as: Fin = K(Fs/2) + FWhere:Fin = Input FrequencyFs = Sampling RateK = Odd or even integer(multiple of half the sampling rate)(need to determine K when substituting into the formula) F = Differential change in Frequency needed toequate the formula(need to determine F when substituting into the formula)If K is odd: Alias Frequency = (Fs/2) FIf K is even: Alias Frequency = FThe aliasing formulas are useful in determining where aharmonic will alias back into the signal spectrum.

8 Conversely, aharmonic or spurious Frequency can suggest possible frequen-cies that caused them. An example could be a system clockoperating at a much higher Frequency appearing as an alias inthe signal initial disconcertment over two unknowns and one formulais reduced once familiarity with the substitution process Note AN-4 Sample -To -Hold TransientSpecified Error bandHold-Mode Settling TimeOutputHold CommandStart Convert PulseKEY SAMPLE-HOLD SPECIFICATIONSTime Switching Time Delay(Aperture Delay)Sample CommandAcquisition TimeCapacitor ChargingFinal Value VoltageOvershootSpecified Error BandFScVFrequency Folding and AliasingUtilizing the alias formulas, a 2nd harmonic ( MHz)

9 Of MHz signal Frequency when sampled at 10 MHz willappear as an alias Frequency of 300 KHz on an FFT , a 3rd harmonic of the MHz signal ( MHz)would then yield K = 2 (even) and DF = MHz. The aliasfrequency would appear as DF, or MHz, on the FFT of Change =VVt01Dt=tAPERTURE UNCERTAINTYA perture UncertaintyThe actual voltage digitized by the data converter dependson the input signal slew rate and the aperture time. The aperture time is application and architecture aperture time, Dt, could be the conversion time of theA/D converter used without a Sample-Hold. The aperture time could also be the aperture delay time specification of aSample-Hold.

10 Applications with repetitive sampling, such asthose utilizing the FFT, can use the aperture uncertainty (tu)specification. The aperture delay is just phase informationwhich is not required in identifying the Frequency components. Classic Sample-HoldThe sampling process begins for many applications with theSample-Hold (S/H) in front of the Analog-to-Digital ConverterThe classic open loop follower sample-hold architecture hasa buffer in front of the switch to quicken capacitor charging andgives the S/H a high input impedance. Adding a buffer behindthe hold capacitor reduces capacitor charge bleeding andoutput InputAnalog OutputCHControlX1X1 CLASSIC SAMPLE-HOLDKey Sample -Hold SpecificationsAmong the key specifications for Sample-Hold are Acquisi-tion Time and Hold Mode Settling Time of a S/H starts with the sample commandand end when the voltage on the hold capacitor enters andstays in the error band.