NASA/TM- 1999-206584

Design and Calibration of an Airborne

Multichannel Swept-Tuned Spectrum

Analyzer

Philip J. Hamory

Dryden Flight Research Center

Edwards, California

John K. Diamond

Langley Research Center

Hampton, Virginia

Arild Bertelrud

Analytical Services and Materials, Inc.

Edwards, California

October 1999

https://ntrs.nasa.gov/search.jsp?R=20000004531 2018-05-14T14:05:08+00:00Z
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NASA/TM- 1999-206584

Design and Calibration of an Airborne

Multichannel Swept-Tuned Spectrum

Analyzer

Philip J. Hamory

Dryden Flight Research Center

Edwards, California

John K. Diamond

Langley Research Center

Hampton, Virginia

Arild Bertelrud

Analytical Services and Materials, Inc.

Edwards, California

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, California 93523-0273

October 1999
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5285 Port Royal Road

Springfield, VA 22161-2171

(703) 487-4650
DESIGN AND CALIBRATION OF AN AIRBORNE MULTICHANNEL

SWEPT-TUNED SPECTRUM ANALYZER

Philip J. Hamory

National Aeronautics and Space Administration

Dryden Flight Research Center

Edwards, California

John K. Diamond

National Aeronautics and Space Administration

Langley Research Center

Hampton, Virginia

Arild Bertelrud

Analytical Services and Materials, Inc.

Edwards, California

ABSTRACT

This paper describes the design and calibration of a four-channel, airborne, swept-tuned spectrum analyzer

used in two hypersonic flight experiments for characterizing dynamic data up to 25 kHz. Built mainly from

commercially available analog function modules, the analyzer proved useful for an application with

limited telemetry bandwidth, physical weight and volume, and electrical power. The authors discuss

considerations that affect the frequency and amplitude calibrations, limitations of the design, and example

flight data.

KEY WORDS

Spectrum analysis, Signal processing, Multiple input calibration, Multivariable calibration, Analog

circuits
INTRODUCTION

In 1994 and 1998 NASA flew two hypersonic aerodynamic experiments called FX-A and FX-1,

respectively, aboard the Pegasus ®* launch vehicle as part of a project called PHYSX (for Pegasus ®

HYperSonic eXperiments) (ref. 1). The requirements for these experiments included dynamic data up to

25 kHz to characterize the vibration environment and the state of the boundary layer. Real-time

transmission of time histories of the dynamic data via a pulse code modulation telemetry encoder would

require sample rates of at least 50,000 samples per second (sps). The telemetry bandwidth necessary to

support those sample rates was not available, so two airborne spectrum analyzers were developed: one

based on the swept-tuned receiver, and one based on digital signal processing (DSP) technology. The

frequency content of the data was of most interest, and transforming the data into the frequency domain

with an onboard spectrum analyzer offered the opportunity to trade off frequency resolution for a reduced

bandwidth.

Work on developing the DSP-based analyzer has been reported previously (refs. 2 and 3), but work on the

swept-tuned analyzer has not yet been reported. (The design of a related analyzer was reported in ref. 4.)

This paper describes the design and calibration of the swept-tuned analyzer used on FX-A and FX- 1. The

design discussion includes the underlying signal processing technique, the circuit elements, packaging,

and limitations. A calculation showing the tradeoff between frequency resolution and telemetry bandwidth

is also presented. The calibration discussion includes the process and considerations that affect the

frequency and amplitude calibration coefficients. Finally, FX-A flight data are presented to show results

obtained from actual flight conditions. Use of trade names or names of manufacturers in this document

does not constitute an official endorsement of such products or manufacturers, either expressed or implied,

by the National Aeronautics and Space Administration.

DESIGN

Introduction to the Design

The fundamental technique used in this design is the same as that of laboratory spectrum analyzers.

Reference 5 is an excellent presentation on spectrum analysis basics, and the discussion in this paper is

organized and written based upon that presentation. Reference 6 also contains a section (ref. 6(a)) on the

subject.

Spectrum analyzers are typically used for radio frequency (RF) measurements. Because the required

measurement bandwidth for FX-A and FX-1 extended only slightly beyond the audio range, commercially

available analog function modules were used rather than RF components.

Another difference between this analyzer and a laboratory spectrum analyzer is that this analyzer has no

display unit. Rather, an airborne telemetry encoder samples its amplitude and frequency analog outputs.

By plotting the amplitude readings against the frequency readings, engineers on the ground obtain the

display of data. Calibrations are applied digitally on the ground.

*Pegasus ® is a registered trademark of Orbital Sciences Corporation, Fairfield, Virginia.
The Design Elements

A spectrum is a range of frequencies, and a spectrum analyzer is a device that breaks up a signal into its

frequency components. The most common type of spectrum analyzer is the swept-tuned receiver, the most

widely accepted, general-purpose tool for frequency domain measurements. The technique most widely

used is superheterodyne. Heterodyne means to mix--that is, to translate frequency--and super refers to

frequencies above the audio range. Basically, these analyzers use a bandpass filter as a window to view a

limited part of the signal's spectrum, and this window appears to sweep across the spectrum and identify

all the frequency components present.

It is difficult, however, to build a filter that tunes over a wide range. An easier implementation is to use a

tunable local oscillator (LO) and keep the bandpass filter fixed. In this implementation, the input signal

sweeps past the fixed filter. For now, assume that this resembles how an AM radio works except that a

radio dial controls the tuning and the radio has a speaker instead of a display. The concept will become

more clear later in this paper' s discussion of the circuit elements of the analyzer.

The major components in this spectrum analyzer, as figure 1 shows, are the input filter, analog multiplier,

intermediate frequency (IF) filter, root-mean-square (RMS) to direct current (DC) converter, local

oscillator, and sweep generator. The analog multiplier, IF filter, RMS-to-DC converter, and voltage-tofrequency converter in the local oscillator are the analog function modules referred to earlier. Before

discussing how the components work together, this paper describes the fundamentals of each one

individually.

The analog multiplier (ref. 7) plays the same role as the mixer in an RF spectrum analyzer; that is, it

converts a signal from one frequency to another. From modulation theory, it is known that when two

signals are multiplied together, the product contains frequencies that are the sum and difference of the two

[ Analysis channel 1 (amplitude reading) I

I I

I

I

I

Input

signal#1 1 I

I

Input filter

(250 Hz - 25 kHz)

/ \

Analog IF filter RMS-to-DC I

multiplier (fc = 40 kHz, Q = 50) converter I

 telemetry

I I li system

To the analog

multipliers of < i

the other

analysis channels

....

LO

(40 - 65 kHz)

Sweep voltage channel (frequency reading)

[

Sweep generator

To

___ telemetry

system

990126

Figure 1. PHYSX swept-tuned spectrum analyzer (only one of four analysis channels shown;

frequencies shown are for FX- 1).
signals. The following trigonometric identity, involving frequencies F 1 and F2, and time t, provides a

glimpse into that fact:

cos(2rCFlt)COS(2rcF2t ) = 0.5 { cos(2rc[F 1 + F2]t ) + cos(2rc[F 1 -F2]t ) } (1)

In this spectrum analyzer, the difference frequency is of interest. The multiplier has converted the input

signal to an IF that the analyzer can now detect.

The IF filter is a narrow bandpass filter used as the window for detecting signals. In this design, eight

second-order analog filter stages (ref. 8) are cascaded together. Each stage is identical, having a center

frequency (fc) of 40 kHz and a quality factor (Q) of 15. The entire filter has a total Q of 50 yielding a

resolution bandwidth of 800 Hz.

The analyzer must convert the IF signal to a baseband signal that the telemetry encoder can sample. This

is accomplished with an envelope detector. The output of the narrow bandpass filter is AC coupled to the

detector, and the output from this detector represents the amplitude of the spectrum. An RMS-to-DC

converter (ref. 9) was selected as the detector because detecting the power of the signal was of interest. In

general, a true power measurement can be made only by averaging the square of the actual voltage

waveform (ref. 6(b)). Although the RMS-to-DC converter actually gives the square root of the true power,

the square root was accepted to benefit from the quality and compactness of the commercially available

converter. The averaging time of the RMS-to-DC converter was set to the minimum time described in

reference 9. For FX-1, this time was 12.5 microseconds (bts).

The local oscillator is the element that, in effect, tunes the analyzer, and this element was constructed out

of three parts: a voltage-to-frequency converter, a digital sine-wave generator, and a bandpass filter.

A sine-wave generator can be constructed out of digital circuitry by hooking up a shift register as a

walking-ring counter and by resistively summing the register outputs to produce a staircase waveform.

When passed through a bandpass filter to remove the DC component and high-order frequency

components, the staircase waveform produces a low-distortion (less than 1 percent), sinusoidal output with

stable amplitude. For this application, a 5-stage shift register was used to make a 10-stage walking-ring

counter (ref. 10). A voltage-to-frequency converter (ref. 11) drove the counter's clock and was in turn

driven by the sweep generator.

The sweep generator is a precision, temperature-stable, triangle waveform generator (ref. 12) that tunes

the LO to change its frequency linearly. The telemetry encoder samples this voltage and the results

represent the frequency component of the signal being analyzed. Most swept-tuned spectrum analyzers use

a ramp generator and, therefore, sweep across the spectrum unidirectionally. Even though a bidirectional

sweep makes the calibration process more involved, the triangle wave was chosen for circuit simplicity.

The input filter is a wideband anti-aliasing filter. The filter consists of a third-order Chebyshev high-pass

filter with a frequency passband beginning at 250 Hz in series with a fourth-order Chebyshev low-pass

filter with a frequency passband ending at 25 kHz. Voltage-controlled, voltage-source (VCVS) hardware

implementation was used for these filters. Reference 6(c) has a suitable procedure for designing equalcomponent VCVS filters; however, the unity-gain VCVS filter was chosen primarily because it has fewer

components. Component values were computed using filter design software developed in house. Figure 2,

obtained by circuit simulation, shows the nominal frequency response of the input filter.
Gain

1.25

1.00

.75

.50

.25

0

10 100 1,000 10,000

Frequency, Hz

Figure 2. Nominal frequency response of input filter.

100,000

990127

How It All Works Together

Suppose a sine-wave, whose amplitude is 1.2 V peak-to-peak (Vpp) and whose frequency fsig is 10 kHz,

is applied to the input. Suppose also that the telemetry system is set up to sample the analysis channel and

the sweep voltage channel at the same rate. Figure 3 simulates what the calibrated output of the analyzer

would look like if the points in the spectrum were 62.5 Hz apart. To visualize how this figure is

constructed, imagine the sweep generator increasing in voltage causing the LO frequency (fLo) to sweep

upward. As soon as the difference frequency fLo - fsig comes into the skirt of the IF filter, the signal

starts to appear at the output. When the difference frequency is at the center frequency (40 kHz), the full

amplitude of the signal passes through. As the sweep generator increases voltage further, the difference

frequency leaves the filter skirt, and the amplitude reading goes to zero.

A swept-tuned analyzer only produces a correct spectrum when the sweep rate is fast enough to capture the

spectrum before it changes. On the other hand, the response time of the IF filter (about 1 millisecond (ms)

in this design) limits how high the sweep rate can go. Swept-tuned analyzers are, therefore, more suited for

signals whose frequency content is stationary rather than transient during the sweep time. For more details

on this issue, see references 5 and 6(a).

Suppose the period of the sweep generator triangle wave is 200 ms (as it was in this application) so that

each sweep is completed in 100 ms. This arrangement gives a sweep rate of 25 kHz/100 ms or 250 Hz/ms.

To acquire points that are 62.5 Hz apart, samples are needed every 250 gs, which corresponds to 4000 sps.

Therefore, to acquire data from the four analysis channels and the sweep voltage channel, the aggregate

sample rate needs to be 20 ksps. In contrast, if the time history data are transmitted and the spectrum is

computed digitally, a sample rate of at least 50 ksps is needed to have frequency content up to 25 kHz.
Signal

amplitude,

Vpp

1.2

.9

.6

.3

.................................................................................................. Q_ ..............................................................................................

o i

°io

o i o

o

o

o

o

o

o

o

o

o o

Q .......... ..................................................................

0

i

i

0 0

8 9 10 11 12 13

Signal frequency, kHz

990128

Figure 3. Simulated spectrum for 1.2 Vpp, 10 kHz sine wave (upsweep only).

For four input signals, the aggregate sample rate is 200 ksps. Using the sweep time (100 ms) for the record

length of a discrete Fourier transform on the time history data, the frequency resolution equals the sample

rate (50 ksps) divided by the number of samples in the record (5000 samples), that is, 10 Hz (ref. 13). Thus,

in this example, a factor of 6.25 in frequency resolution is lost, but a factor of 10 in required telemetry

system bandwidth is saved by using the airborne spectrum analyzer.

Packaging

The sweep generator and LO were placed on one circuit board measuring 6 in. × 3.75 in. × 0.5 in. (the last

dimension includes part height), and the four analysis channels were placed on a second circuit board of

the same size to make a four-channel spectrum analyzer whose total power consumption was 5.4 Watts

(W) and whose weight was 0.43 lb without the chassis or 2.5 lb with the chassis. The small size, weight,

and power consumption enabled the analyzer to be incorporated in experiments aboard the Pegasus®

launch vehicle.

CALIBRATION

Frequency Calibration Considerations

As discussed earlier, the sweep-generator voltage determines the signal frequency that passes through the

IF filters. Although one set of calibration coefficients for the sweep voltage would appear to apply to all
four input signals,this isnot thecase.Theanalysischannelshavenonnegligibleandunequalpropagation

delays.Different coefficients,therefore,applyto thesweepvoltagefor eachinput signal.Sweepdirection

alsomatters,ascanbeseenmoreclearlyby studyingfigure 4.

Figure4 depictsthe triangular sweepvoltagein the time domain.Let Vsw ° be the sweep voltage that

causes frequency F o to pass through the IF filters. Let to and t o' be the times that Vsw ° is reached during

the upsweep and downsweep, respectively. If tpd I is the propagation delay through analysis channel 1,

then the amplitude readings for channel 1 are available at times t I = t o + tpd I for the upsweep and

t 1' = t 0' + tpd I for the downsweep. If the amplitude readings and the sweep voltage are sampled at

those times, then voltages Vsw I and Vsw 1' rather than Vsw ° become associated with F o . Likewise, if the

propagation delays through the other analysis channels are not equal to tpd 1, as was the case in this

implementation, then each analysis channel has different sweep voltages that become associated with F o .

Thus a total of eight sets of coefficients (two sets per channel) are required to determine frequency in this

four-channel analyzer.

Vswl

Sweep

voltage Vswo

Vsw 1' - _ _ _

I I

to tl t o' t I '

Time

990129

Figure 4. Sweep voltage as a function of time.

For a given channel and sweep direction, the relationship between input signal frequency and sweep

voltage was linear. If the variable f represents the sweep-voltage reading, that is the frequency reading,

then equation 2 can be written as follows:

F = Po + Plf (2)

where Po and p i are the calibration coefficients.

Amplitude Calibration Considerations

Taking another look at figure 2, the reader will notice that the passband gain is not fiat but sloped. In fact,

in this implementation, the slope was greater than shown because of a limited selection of component
valuesandtolerances.This slopemadetheamplitudecalibrationmoreinvolved.Hadthe passbandgain

beenflat, thecalibrationprobablywouldhavebeengivenby equation3,

V = a 0 + alv (3)

where V is the signal voltage in volts RMS, v is the amplitude reading in counts, and a o and a I are

constants. Because it is not flat, the gain coefficient a I is a function of frequency. In figure 2, a I increases

with frequency except for the upper transition region where a I decreases as frequency increases. To

characterize a I over the entire range requires an equation that has both the frequency reading f and

• , 1 , , ,

the reverse of the frequency reading f . By trial and error at was found that equation 4 produced the best

fit for the calibration points used.

al = bo + bl f 1 + b2 f 2 + b3 f + b4f2 (4)

The overall fit was improved by letting a 0 be a function of frequency as well. See equation 5.

ao = Co + Cl f 1 + c2 f + c3f2 + c4f3 (5)

Combining equations 3, 4, and 5, equation 6 results.

V = c 0 + elf 1 + c2 f + c3f2 + c4f3 + bov + blVf 1 + bzvf 2 + b3vf + b4vf2 (6)

Here, too, sweep direction effects required different coefficients for the upsweep than for the downsweep.

Calibration Process and Results

The analyzer inputs had a range of 0 to 1.6 Vpp from 250 Hz to 25 kHz. The analyzer was calibrated with

24 sine waves defined from a combination of 4 voltages and 6 frequencies. The sine waves were produced

by a function generator that was connected to all four inputs simultaneously. For each sine wave the

telemetry stream was recorded and the readings separated by sweep direction. Then, for each channel and

sweep direction, the sine-wave voltage V and the frequency F was associated with the peak amplitude

reading Vp and the corresponding sweep voltage reading fp. (See figure 3 again to observe how the

spectrum has a distinct peak that can be related to the input signal voltage and frequency.)

To get an accurate peak reading, data were recorded for many seconds. Because the sweep generator was

asynchronous to the telemetry system, each sweep produced additional points that filled in the spectrum.

The input voltage and frequency pair {V, F} and output amplitude and frequency reading pair {Vp,fp}

can be referred to as a calibration point. The 24 sine waves therefore produced 24 calibration points. The

four voltages used were 0.4, 0.8, 1.2, and 1.6 Vpp. The six frequencies were 1, 5, 10, 15, 20, and 24 kHz.

Six other calibration points were desired as well; namely, zero volts at the same six frequencies. To obtain

these points, the signal inputs were grounded and the amplitude readings recorded. Because the resulting

spectrum was flat, no fp could be obtained directly. The fp, however, was obtained from the nonzero

calibration voltages because fp was the same for a particular frequency whether the input voltage was 0.4,

0.8, 1.2, or 1.6 Vpp.
With these6 additionalpoints,atotalof 30calibrationpointsweregathered.In retrospect,oneortwomore

frequenciesbetween20 and24 kHz shouldalsohavebeenaddedto identify wherethe filter transition

regionactuallybegan.

The calibrationcoefficientswerecalculatedusing the methodof least-squares(ref. 14).The appendix

showsthemethodappliedto theamplitudecalibrationcoefficients.Thecurvefit errorsare+0.6 percent

full scale for the amplitude and +0.1 percent full scale for the frequency.

Figure 5(a) shows superimposed uncalibrated readings obtained from one analysis channel and the sweep

voltage channel given six of the calibration sine waves. Each frequency has two peaks because of the

propagation delay discussed and illustrated in figure 4. The unequal heights of the spectra are evidence of

the frequency-dependent gain discussed earlier.

Figure 5(b) is the result of applying calibration equations 2 and 6 with appropriate coefficients to the data

in figure 5(a). The calibration equalizes the amplitudes and centers the spectra. For this figure, sweep

direction was determined by comparing the value of one frequency reading with the value of the previous

frequency reading. This scheme works adequately for most of the spectrum. At the extremes of the

spectrum, however, the readings appear to belong to one sweep direction when, because of propagation

delays, they really belong to the other. This fact explains why some of the results at 1 and 24 kHz appear

incorrect.

Amplitude,

counts

2500

l

24) (20) (15) (10)

, I | •

; ii ill 1

i i

0 1000 2000

(5) (1)1

l: It!

:!l_i it

iIli 'I!

3000 4000

1.2

Amplitude, .6

Vpp

0

Frequency, counts

990130

(a) Uncalibrated readings.(Number in parentheses is

the frequency in kHz.)

Figure 5. Spectra for 1.2 Vpp sine waves at six frequencies.

it :. i!

i ii ii il ilt

• , : , i _ i I

t

1_0 ' , 0 15 20 25

Frequency, kHz 990131

(b) Calibrated results.

FLIGHT DATA

On the FX-A experiment, the analyzer processed accelerometer data. The objective was to characterize

vehicle vibration because, if the vibration level was too high at certain frequencies, subsequent boundarylayer experiments would not be worth performing.
In figure 6,two vibrationspectraaresuperimposed.Thedashedline representsthevibrationlevel during

captivecarry flight, that is, beforetherocketenginewas ignited;the solid line representsthevibration

level 19secafterlaunchathigh angleof attack.

.50

Amplitude,

Vrms .25

0 2 4 6 8 10

Frequency, kHz

12

990132

Figure 6. Accelerometer spectra before and after rocket ignition.

Each spectrum comes from one sweep, and the points are 100 Hz apart. Clearly the vibration level is higher

after the rocket has been ignited than before. The vibration level drops off above 4 kHz--safely below the

frequencies of interest for the FX- 1 hypersonic boundary-layer transition experiment.

SUMMARY

This paper described the design and calibration of an airborne multichannel swept-tuned spectrum

analyzer for characterization of dynamic data up to 25 kHz. The need for the analyzer was driven primarily

by telemetry bandwidth limitations. After the data are transformed on board into the frequency domain,

frequency resolution can be traded for reduced telemetry bandwidth. In the example presented, a factor of

6.25 in frequency resolution was sacrificed to reduce telemetry bandwidth by a factor of 10.

The analyzer design was based upon the widely used superheterodyne technique, and the frequency range

of interest made the use of commercially available analog function modules possible. A four-channel

analyzer was constructed in a small, lightweight, low-power package suitable for flight aboard the

Pegasus® launch vehicle. The main limitation of the design was that the frequency content of the input

signal remain stationary when the analyzer sweeps across the spectrum. Propagation delays in the analyzer

were shown to require a set of calibration coefficients for each sweep direction. Frequency-dependent
coefficientswereusedfortheamplitudecalibrationto dealwiththeslopedpassbandof thewidebandinput

filter.

Finally,FX-A flight dataillustratehowthespectrumanalyzerisused,andgaveconfidenceto proceedwith

FX-1.

REFERENCES

. Bertelrud, Arild, et al., "Pegasus Wing-Glove Experiment to Document Crossflow TransitionDevelopment and Current Status," AIAA-98-1522, 8th AIAA International Space Planes and

Hypersonic Systems and Technologies Conference, Norfolk, Virginia, April 1998.

2. Graves, Sharon, et al., "High Frequency Data Acquisition Using Digital Signal Processing,"

AIAA-94-2134, 7th Biennial AIAA Flight Test Conference, Colorado Springs, Colorado, June 1994.

3. Bertlrud, Arild, et al., "Development of a System for Transition Characterization," AIAA-93-3465-

CP, 1 lth AIAA Applied Aerodynamics Conference, Monterey, California, August 1993.

. Diamond, John K., "Miniature PCM Compatible Wideband Spectral Analyzer for Hypersonic Flight

Research," ISA-88-0732, Instrumentation Society of America 34th International Instrumentation

Symposium, Albuquerque, New Mexico, May 1988.

5. Brown, Christie, "Spectrum Analysis Basics," 1997 Back to Basics Seminar, Hewlett-Packard

Company, Santa Rosa, California, 1997, (HP Publication Number 5965-7008E).

. Horowitz, Paul, and Winfield Hill, The Art of Electronics, Second Edition, Cambridge University

Press, New York, New York, 1989: (a) "Measurements & Signal Processing," pp. 1035-1038;

(b) "High Frequency & High-Speed Techniques," p. 890; (c) "Active Filters and Oscillators,"

pp. 273-276.

7. "Internally Trimmed Precision IC Multiplier: AD534," Special Linear Reference Manual, Analog

Devices, Inc., Norwood, Massachusetts, 1992, pp. 2-15 to 2-22.

8. "4th- and 8th-Order Continuous-Time Active Filters: MAX274/MAX275," 1993 New Releases Data

Book, Volume II, Maxim Integrated Products, Sunnyvale, California, 1993, pp. 6-3 to 6-29.

9. "High Precision, Wideband RMS-to-DC Converter: AD637," Special Linear Reference Manual,

Analog Devices, Inc., Norwood, Massachusetts, 1992, pp. 4-23 to 4-30.

10. Lancaster, Don, revised by Howard M. Berlin, "Counters and Shift Registers," CMOS Cookbook,

Second Edition, Howard W. Sams & Co., Indianapolis, Indiana, 1988, pp. 379-383.

11. "Voltage-to-Frequency and Frequency-to-Voltage Converter: AD650," Data Converter Reference

Manual, Volume II, Analog Devices, Inc., Norwood, Massachusetts, 1992, pp. 3-15 to 3-26.

12. "Dual Current Source]Current Sink: REF200," Burr-Brown IC Data Book--Linear Products,

Burr-Brown Corporation, Tucson, Arizona, 1995, p. 8.59 (figure 12, "Precision Triangle Wave Form

Generator").
13.Ludeman,LonnieC.,"The DiscreteFourierTransform,"Fundamentals of Digital Signal Processing,

John Wiley & Sons, New York, New York, 1986, p. 287.

14. Dally, James W., et al., "Statistical Methods in Experimental Measures," Instrumentation for

Engineering Measurements, John Wiley & Sons, Inc., New York, New York, 1984, pp. 534-539.

15. Strang, Gilbert, "Orthogonal Projections and Least Squares," Linear Algebra and Its Applications,

Second Edition, Academic Press, New York, New York, 1980, p. 138.

16. "Matrices and Linear Algebra," Using MATLAB, The MathWorks, Inc., Natick, Massachusetts,

December 1996, pp. 4-13, 4-22, and 4-23.

APPENDIXmSOLVING FOR THE CALIBRATION COEFFICIENTS

Using subscripts 1 through 30 to denote the 30 calibration points, 30 equations in the form of equation 6

are obtained. The key to solving for the set of coefficients that has minimum error in the least-squares sense

is to organize the terms of the equations into vectors and a matrix; namely,

V1

V31

fPl 1

1 fP3o 1

2 3 1 2 2

f Pl f Pl f Pl vpl Vplf Pl Vplf Pl Vplf Pl Vplf Pl

2 3 1 2

f P3o f P3o f P3o Vp3o Vp3of P3o Vp3of P3o Vp3of P3o Vp3of P3o 2

c(

c_

bc

b_

If vector V 1 through V3o is represented by b, vector [Co..cabo..b 4] by x, and the matrix by A, then the

system of equations is in the form of Ax = b. This structure is common in linear algebra and has a

standard solution; namely, x = (ArA) lAth, where A r is the transpose of A and (ArA) 1AT is the

pseudoinverse of A (ref. 15). The solution looks complicated, but mathematics software packages that

handle matrices (ref. 16) are commercially available to compute the solution•
REPORT DOCUMENTATION PAGE FormApproved

OMB No. 0704-0188

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VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

2. REPORT DATE

October 1999

1. AGENCY USE ONLY (Leave blank)

4.TITLE AND SUBTITLE

Design and Calibration

Spectrum Analyzer

3. REPORTTYPE AND DATES COVERED

Technical Memorandum

5. FUNDING NUMBERS

of an Airborne Multichannel Swept-Tuned

6. AUTHOR(S)

Phillip J. Hamory, John K. Diamond, andArild Be_elrud

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Dryden Flight Research Center

RO. Box 273

Edwards, California 93523-0273

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU 529-60-24-00-17-00-ANL

8. PERFORMING ORGANIZATION

REPORT NUMBER

H-2361

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NAS A/TM- 1999 -206584

11. SUPPLEMENTARY NOTES

Presented at the 35th Annual International Telmetering Conference (ITC) "Telemetry: Meeting The 21st

Century Challenge," Las Vegas, Nevada, October 25-28, 1999.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified--Unlimited

Subject Category 06

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

This paper describes the design and calibration of a four-channel, airborne, swept-tuned spectrum analyzer

used in two hypersonic flight experiments for characterizing dynamic data up to 25 kHz. Built mainly from

commercially available analog function modules, the analyzer proved useful for an application with limited

telemetry bandwidth, physical weight and volume, and electrical power. The authors discuss considerations

that affect the frequency and amplitude calibrations, limitations of the design, and example flight data.

14. SUBJECTTERMS

Spectrum analysis,

Multivariable calibration, Analog circuits

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION

OF REPORT OFTHIS PAGE

Unclassified Unclassified

NSN 7540-01-280-5500

Signal processing, Multiple input calibration,

19. SECURITY CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

17

16. PRICE CODE

A03

20. LIMITATION OF ABSTRACT

Unlimited

Standard Form 298 (Rev. 2-89)

Prescribed byANSI Std. Z39-18

298-102