Spectral Analysis Using a Deep-MemoryOscilloscope Fast Fourier Transform (FFT)For Use with Infiniium 54830B Series Deep-Memory Oscilloscopes
Application Note 1383-1
Many of todays digital oscillo-scopes include a Fast FourierTransform (FFT) for frequency-domain analysis. This feature isespecially valuable for oscillo-scope users who have limited orno access to a spectrum analyzeryet occasionally need frequency-domain analysis capability. Anintegrated oscilloscope FFT pro-vides a cost effective, space sav-ing alternative to a dedicatedspectrum analyzer. Though thelatter does exhibit better dynamicrange and less distortion, the dig-ital oscilloscope offers severalcompelling benefits.1
The Agilent TechnologiesInfiniium 54800 Series digitaloscilloscopes include FFT func-tions for computing both magni-tude and phase. Several usefulfeatures assist in spectral analy-sis. Controls adjust memorydepth, sampling rate, verticalscale and horizontal scale of theFFT. Automatic measurementsand markers measure spectralpeak frequencies and magnitudesas well as deltas between peaks.The Infiniium Help system pro-vides extensive information on
FFT theory and application.Several features designed prima-rily for time-domain analysis arealso useful for the FFT. Displaytraces can be annotated andsaved to a file. The oscilloscopeconfiguration can be saved andrecalled as a setup file. Func-tions can be chained together toperform complex tasks, such ascomputing the average, maxi-mum or minimum of several FFTspectrums. Measurement statis-tics are available for computingthe mean and standard deviationof a measurement over severalacquisitions. With all this capa-bility, the oscilloscope FFT pro-vides a very convenient tool forspectral analysis.
The deep-memory Infiniium54830B family of oscilloscopesenable an increase in the recordlength of the FFT, which in turnimproves the frequency spec-trum estimate. Longer recordlengths provide finer frequencyresolution and better dynamicrange. By upgrading the proces-sor speed and improving the effi-ciency of the FFT algorithm, theInfiniium deep-memory oscillo-scopes can perform FFTs on longrecords very quickly.
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
FFT Fundamentals . . . . . . . . . . . . . . . . . . 2Discrete Fourier Transform . . . . . . . . 2Sampling Effects . . . . . . . . . . . . . . . . . 4Spectral Leakage and Windowing . . 6
Practical Considerations . . . . . . . . . . . . 8Frequency Span and Resolution . . . . 8Dynamic Range . . . . . . . . . . . . . . . . . 10Averaging . . . . . . . . . . . . . . . . . . . . . . 13Window Selection. . . . . . . . . . . . . . . 15Equivalent Time Sampling . . . . . . . . 16
Application . . . . . . . . . . . . . . . . . . . . . . . 17Characterizing an AM Signal. . . . . . 17Selecting Sampling Rate andMemory Depth. . . . . . . . . . . . . . . . . . 18Scaling Time and Frequency . . . . . . 18Resulting FFT Spectrum . . . . . . . . . . 19
Support, Services, and Assistance . . . . . . . . . . . . . . . . . . . . 20
This application note begins with a discussion of FFT funda-mentals and highlights thosecharacteristics that are importantfor understanding FFT-basedspectral analysis. It goes on toexplain practical considerationsfor using an oscilloscope FFT for spectral analysis and techniques that can be used toimprove dynamic range and accuracy. Finally, it concludeswith a specific application thatillustrates the benefits of a deep-memory oscilloscope FFT formaking high-frequency resolutionspectral measurements.
Discrete Fourier Transform
To better understand the limita-tions of using an oscilloscope FFTfor spectral analysis, it is impor-tant to understand some funda-mental properties of the DiscreteFourier Transform (DFT) and theeffects of sampling. A brief reviewis covered here. The reader isreferred to other sources formore detailed information.
The DFT represents discrete samples of the continuous
Fourier transform of a finitelength sequence.2 The DFT for asequence, x(nT), is given by:3
whereN = number of samplesF = spacing of frequency
domain samplesT = sample period in the
The FFT is simply an efficientalgorithm for computing the DFT.Actually, there are several FFTalgorithms. Infiniium uses aradix-2 FFT algorithm for com-puting the DFT. A radix-2 FFT iscomputed on a number of pointsequal to a power of 2. The effi-ciency of the FFT is oftenexpressed in terms of the numberof complex multiplications. Thenumber of complex multiplica-tions for a radix-2 FFT can beshown to be Nlog2N. This is a bigimprovement over the number ofcomputations for a DFT, which isapproximately N2. For example,for a one-million-point sequence,the FFT takes 0.002% of the DFTcomputation time.2
The spacing of the frequency-domain samples or bins in theDFT is given by the followingequation:
whereFs = sampling frequency
Thus, the frequency resolutioncan be improved by increasing Nor decreasing Fs.
The DFT is symmetrical aboutN/2. The magnitude of the DFT isan even symmetric function andthe phase is an odd symmetricfunction. Infiniium plots only thefirst half of the FFT points sinceno additional information is pro-vided by the remaining points.The frequency of a particular FFTpoint, k, is:
The maximum frequency plottedis Fs/2, where k = N/2.
X(kF) = x(nT)e-j2kFntN-1
n = 0 F = = 1
Fk = kFsN
Discrete Fourier Transform continued
The DFT is a complex exponentialfrom which both magnitude andphase can be computed. Infiniiumhas functions for computing boththe magnitude and phase. Thephase is computed in degrees.The magnitude is computed indBm.3 The voltage form for dBm is:
The reference voltage, VREF, isdefined as the voltage that pro-duces 1 milliwatt of power into50 . For example, if 1 volt dc isconnected to Infiniium, then themagnitude of the FFT result at0 Hz frequency will be approxi-mately 13 dBm:
P(dBm) = 20log(VRMS / VREF)
0.001 watts * 50 = 0.2236 volts
20log(1.0 / 0.2236) = 13.0 dBm
The Infiniium waveform recordlength captured is generally not apower of 2. However, the radix-2FFT requires a number of pointsequal to a power of 2. Infiniiumhandles this by padding the endof the waveform with zeros to get to the next power of 2sequence length.
Although zero padding increasesthe number of points, it does notchange the shape of X(F). It sim-ply extends the number of pointsin the DFT. For example, if zeropadding extends the sequencelength by a factor of 2, then everyother point of the DFT of thezero-padded sequence has thesame value as the DFT of theunpadded sequence. Thus, zeropadding has the same effect asinterpolation: it fills in pointsbetween frequency samples, giv-ing a better visual image of thecontinuous Fourier transform.
The Fourier transform of an ana-log signal, xa(t), is defined by:
To obtain a digital representa-tion, x(n), of an analog signal,xa(t), a digital oscilloscope samples the signal at uniformintervals, T:
Equation 6 assumes the samplingprocess is ideal such that there isno voltage quantization or otherdistortion. The Fourier transformof this ideal discrete timesequence is:2
Equation 7 shows the relation-ship between the Fourier trans-form, Xa(F), of the continuous signal, and the Fourier transform,X(F), of the discrete timesequence. X(F) is the sum of aninfinite number of amplitude-scaled, frequency-scaled, andtranslated versions of Xa(F).
Figure 1 (a) shows the Fouriertransform, Xa(F), of a continuoussignal. Figure 1 (b) and (c) showthe Fourier transforms, X(F), oftwo discrete-time signalsobtained by periodic sampling.Notice in (b) and (c) that X(F) isperiodic with period 1/T or Fs.Also notice that if Fo is greaterthan Fs/2, as shown by (b), theperiodic repetitions of the con-tinuous-time transform overlap,and it is not possible to recover
Xa(F) from X(F). When overlapoccurs, high-frequency compo-nents of Xa(F) fold into lower fre-quencies in X(F). This effect iscalled aliasing. The frequency atwhich aliasing occurs, Fs/2, iscalled the folding frequency. Themaximum frequency in Xa(F) iscalled the Nyquist frequency. The Nyquist rate is the minimumsampling rate required to pre-vent aliasing and is twice theNyquist frequency.2
-Fs -Fs/2 2FsFsFs/2
-2Fs -Fs -Fs/2 2FsFsFs/2
X(F) Unknown in Overlap Regions
Figure 1. Fourier transform of (a) continuous signal, (b) discrete-timesignal with overlap, and (c) discrete-time signal without overlap
Xa(F) = xa(t)e-j2Ftdt
x(n) = xa(t) |t=nT
X(F) = Xa[ (FT + k)]k = -
Equation 8 can be used to com-pute the alias frequency, F', fromthe original frequency, F:
Sampling Effects continued
It is insightful to consider aliasingfrom a time-domain perspective.Assume a sine wave signal is sam-pled uniformly. If there are lessthan two samples per period,then it is not possible to deter-mine the frequency of the contin-uous sine wave signal from thesampled version. In this case, thesampled version will appear to bea sine wave at a lower frequency.
If the sampling rate is at leasttwice the highest frequency con-tent in the analog signal, no alias-ing will occur, and the DFT of thesampled sequence will provide agood estimate of the Fouriertransform of the analog signal