An Efficient Antialiasing Filter

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 4, DECEMBER 1985

An Efficient Antialiasing Filter

ROBERT A. LAWTON

Abstract-The application of a solid-state reference filter as an effi-cient antialiasing filter is described. The analytical basis for the effi-ciency of this filter is described and a specific example of measuring a1024-point waveform with an RC filter and the solid-state filter is given.

I. INTRODUCTIONFREQUENCY domain measurements derived fromtime-

domain sampling measurements on broad-band sys-tems such as coaxial attenuators or broad-band antennasystems are subject to aliasing errors when using theFourier transform. These errors are due to the periodicnature of the Fourier transform which results in the fre-quency domain spectrum being folded back on itself asshown in Fig. 1. This folding action modifies the spectrummost drastically in the neighborhood of the Nyquist fre-quency (the reciprocal of twice the sampling interval). Inother words, the measured loss of the attenuator or an-tenna system will be in error most at the Nyquist fre-quency. To reduce these errors one can prefilter the signalto be measured. A low-pass filter is sometimes used toaccomplish this, but this filter will itself introduce errorfor frequencies above its cutoff frequency.The specific type of filter used is, however, quite im-

portant in that it should have as rapid a falloff with fre-quency as possible so as to give a minimal distortion inthe passband of the signal of interest. One definition ofhow much attenuation is enough is achieved when allspectral components at the Nyquist frequency and aboveare less than 1/2 of a least significant bit (LSB) forthe A/D converter used.

II. FiLTER CHOICEAs an example of the importance of the proper choice

of filter, I will now show that a simple RC filter is notadequate for this purpose while a filter with a Gaussianresponse such as encountered with the liquid [1] and solid[2] lossy dielectric waveform filters is a good candidatefilter. The liquid and solid dielectric filters are shown inFigs. 2 and 3, respectively, and their responses to a step-like signal are shown in Figs. 4 and 5, respectively. Thesecurves have the appearance of the integral of a Gaussianand an analysis of the equivalent circuit of the soliddielectric filter confirms the fact that the response to a steplike excitation should be nearly Gaussian [2]. These filtershave been developed to support the rapidly expanding fieldof digital waveform measurement for fast integrated cir-

Manuscript received March 30, 1985; revised June 20, 1985.The author is with the Electromagnetic Fields Division, National Bureau

of Standards, Boulder, CO.

Interaction of the Spectrum amplitude and Its harmonicproducing distortion at the Nyquist (folding) frequency

c0

a)

Tn nfnFrequency

Fig. 1. Frequency domain spectrum folded back on itself.

cuit development with important application in computersand communications.

Let us consider two filters, the RC and Gaussian. Thefrequency domain response of the two filters are shown inFig. 6. Let fc be the corner frequency (that frequencywhere the response is down by 3 dB). Further, let theNyquist frequency be fn and consider a 14-bit A/D con-

verter for which the response atfn must be reduced by 215for less than 1/2 LSB error (this error d(t) is assumed to beuniformly distributed over the time window of the mea-

surement). In the worst case, the filter must do all theattenuation. For example, let the filter transfer function beH(w) where w = 2irf, then

H(wc) = 0.707 = 11/

andH( n) =2 1,

2, where = 2rfc

where wn = 2wxfl.Now for an RC filter

(2- 1/2)/(2- 15) = wuRCI[l + jwc RC]or

1 + (w RC)- = nRC(2 )

which in turn can be expressed as

,1 - cVn2- 14.51c since RC= 1/(

therefore,5=215

Now for a Gaussian filter

H(w) = Ho exp [ _(A2) (C,2)]

where A is an arbitrary parameter. Therefore, at the cor-ner frequency

(l/V/2) Ho = Ho exp [-(A2) (w2)I

U.S. Government work not protected by U.S. copyright

Composite(Distorted)

Spectrum - Spectrum Harmonic ofAmplitude \ SpectrumFunction \ C / Amplitude

Function

I IV -WI

570

LAWTON: ANTIALIASING FILTER

Fig. 2. Liquid dielectric filter.

Fig. 3. Solid dielectric filter.

from which we find that

,o= 0.5 In 2/A.

Similarly at the Nyquist frequency

(2 15) Ho = Ho exp [-(A2) (Cn)]

and

co = A5In 2/A.Therefore,

co,1, = a30 = 5.5

571


HI STOGRAM Dl.!r 0.

L. .

0 P(U) 89

O-.£: IERT = 49. 999?MV,'DIV. YMIN =

HORIZ.= 0.09999 NS/OIU.

THE PULSE PAPAMETERS ARE:BASE MAGNITUDE = 145.1800 MU PULSE START TIMETOP MAGNITUDE = 394.0540 ft! 1ST TRANSITION DURATION -PULSE AMPLITUDE = 249.8740 MUVMAX. UNDERSHOOT s 08039 %MIAX OV?ERSHOOT = 0.986q;.COMMENTS X1B8F TJAL 11/1/cn

Fig. 4. Liquid dielectric filter response to step-like signal.

HISTOGRAM Dlrl= 0.078 VERT.T= 49.9999M1J'OIV. YMIN a

P(V) 7 HORIZ 0.19999 NSODIV.

TIE PLLSE PARAPIETERS ARE

BASE MAGNITUDE = 144.6966MU PULSE START TIME a 0.2 NSTOP MAGNITUDE a 375.9740 MV 1ST TRANSITION DURATION a 06.364 NSPULSE AMPLITUDE a 231.8840 MV

MAX. UNDERSHOOT 0.077 %MAX. OVERSHOOT a 6.000 %COMMWNTS SFOUT = SILICON FILTER OUTPUT, RAL 11 FEB 83

Fig. 5. Solid dielectric filter response to step-like signal.

Comparison of the Attenuation/Distortion Characteristicsof the RC and Gaussian Filters

-10 I

-20- Gaussianco ~~~~~~~~~Filter

a RC Filter0-40

10c

< -60 _

LL_JA

Frequency (Hz)

Fig. 6. Frequency domain responses of RC and Gaussian filters.

which is a much smaller ratio than 215. In other words thecutoff frequency for the Gaussian filter can be much closerto the Nyquist frequency than the RC filter with a resultantdecrease in distortion. A more exact analysis can be madethrough the filter step response rather than the impulseresponse, but the result is the same since the only signif-icant difference is the division of both the filter responseand the error by jco. As an example of the significance ofthis analysis, consider a 1024-point sampled waveform ina time window of 5 ns which has a Nyquist frequency

fn=-At- l/[10-8/1024] - 100 GHz

for which

fc (101) (2- 15) = (1011) (10-4515) - 3 MHz

0.0080 MU

- - - - * -. - . .* . . . I

0.30397 HS0 19643 HS

8. Om8 Mv

..............................- . . . . . . . .

.............................. . ., . . _. . . . . . . . .}o.* * * f- * * * * @. . . / .. . . . . . . / . . . . . . . . . . . . . . .. . .SS . s . . . . . .. . . . . . s . . . . . . . . . . . . . . . . . . .. . B. . . . . .. . /. . . . . ........ ............. s. . > . . . . . ¢

_._. , . _* . * . * .. .. ** .. .. * .. .. ***** . ***_. . . . . . . .. . . . . . . ...... ......... w

.............................

572

........ ...


for an RC filter and

f, = (10")/5.5 = 18 GHz

for a Gaussian filter.Therefore, information in the 3-MHz to 18-GHz fre-

quency band is distorted using the RC filter to eliminatealiasing, but it is not distorted using the Gaussian filter.The particular filters chosen were used for illustrative

purposes only and do not represent the complete set of

efficient antialiasing filters. However, this example illus-trates the need for a careful choice of filter to reduce theeffects of aliasing when using sampled time-domain infor-mation to generate frequency-domain information.

REFERENCES[1] W. D. McCaa, Jr. and N. S. Nahman, "Generation of reference wave-

forms by uniform lossy transmission lines," IEEE Trans. Instrum.Meas., vol. IM-19, pp. 382-390, Nov. 1970.

[2] R. A. Lawton, N. S. Nahman, and J. M. Bigelow, 'A solid state ref-erence waveform standard," IEEE Trans. Instrum. Meas., vol. IM-33.pp. 201-205, Sept. 1984.

Mode-Stirred Chamber Calibration Factor,Relaxation Time, and Scaling Laws

ROBERT E. RICHARDSON, JR.

Abstract-In this paper, single-mode cavity theory relating cavity Q,stored energy, and relaxation time is applied to mode-stirred chamber(MSC) operation.

It is shown that a calibration factor for average power density, de-fined in terms of stored energy density, can be determined solely froma relaxation time measurement, for which a suitable measurement tech-nique is also shown.A possible definition of minimum operating frequency for an MSC

is also defined in terms of parameters that can easily be measured dur-ing normal chamber operation.

Finally, an attempt is made to show how the derived and measuredresults apply to MSC operation and design.

I. CALIBRATION FACTOR DEFINITION IN TERMS OF QAND RELAXATION TIME

1'HE calibration factor (denoted here as M) for a mode-l stirred chamber (MSC) is generally given by a rela-

tionship of the form:

M = (PD>eff (cm)-2PT

(1)

where < PD>eff is the "'effective" power density in thechamber and PT is the net transmitter power into thechamber. For a free-space plane wave, power densitywhich is customarily given by the Poynting Vector, mayalso be given as the product of light speed c and the energy

Manuscript received March 22, 1985; revised July 17, 1985. This workwas supported by NSWC Independent Exploratory Development (IED)Funding.

The author is with the Naval Surface Weapons Center, Dahlgren, VA.

density u:

(~~~c-oE2 jO2\PD = 2-Re E x H* = c +

2 \2 2/

- cU (w/cm2). (2)

For the discussion of MSC operation in this paper wedefine < PD>eff as the product of light speed and averageenergy density in an MSC, or

<PD)eff = Cu (w/cm2). (3)

The average energy density may be measured to withinany desired accuracy by using a suitable number of cali-brated E or H field probes, having minimal perturbation.The Q of a cavity may be written generally as the non-

dimensional parameter

Stored Energy (4)

where PT is net input power to the cavity, and w the radianfrequency. By measuring the average stored energy in theTest MSC and interpreting Q as a composite [1] Q, (3)and (4) may be combined with (1) and (2) to yield

M (<PD>eff CU cQ (Cm-2)

where V is the chamber volume.Furthermore, the Ile decay time (T) of energy stored in

a cavity mode may be written as [2l

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573

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An Efficient Antialiasing Filter