Digital Demodulation of aFractional Fringe Interferometer

Tristan Tayag and Christopher BelkTexas Christian University

Fort Worth, TX 76129

DSPS Fest ‘99Houston, TX 77056

August 5, 1999

Acoustic Sensing Application

System Configuration

(Acoustic Sensor)(Acoustic Sensor)

Quadrature Condition

Theoretical Output Signal:Time Domain

0 0 .0 0 5 0 .0 1 0 .0 1 5 0 .0 20

0 .5

1

1 .5

2

Tim e [s e c ]

Ph

oto

de

tec

tor

Cu

rre

nt

[A.U

.]

F ra c tio na l F ring e Inte rfe ro m e te r: Tim e Do m a in

LEGENDLEGEND

Reference:Reference: 50 nm 50 nm 100 Hz 100 HzTarget:Target: 50 nm 50 nm 1 kHz 1 kHzUnity FringeUnity Fringe Contrast Contrast

Theoretical Output Signal:Frequency Domain

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0-1 0 0

-8 0

-6 0

-4 0

-2 0

0

F re q ue nc y [Hz]

Ph

oto

de

tec

tor

Cu

rre

nt

[dB

]

F ra c tio na l F ring e Inte rfe ro m e te r: F re q ue nc y Do m a in

LEGENDLEGEND

Reference:Reference: 50 nm 50 nm 100 Hz 100 HzTarget:Target: 50 nm 50 nm 1 kHz 1 kHzUnity FringeUnity Fringe Contrast Contrast

Discrete Fourier Transform(DFT)

DFT: Parallel Form

Computational Efficiency:Comparative Example

Mathematical Direct FFT Decimation GoertzelOperations DFT Time Frequency Algorithm

Complex Multiplications 8 7 7 0Complex Additions 7 7 3 0Real Multiplications 0 0 0 11Real Additions 0 0 4 19Total Multiplications 32 28 28 11Total Additions 30 28 24 19

TOTAL COMPUTATIONS 62 56 52 30Assumptions: 8-point sequenceAssumptions: 8-point sequence

Calculation of 1 frequency pair Calculation of 1 frequency pair

Goertzel Algorithm

Goertzel Algorithm:Direct Form II

Experimental InterferometerSetup

Experimental Output Signal:Frequency Domain

LEGENDLEGEND

Reference:Reference: 100 Hz 100 HzTarget:Target: 1 kHz 1 kHz

Goertzel Test Configuration

Square Wave: Time & FrequencyRepresentation

Goertzel Frequency Selectivity:658 Hz Square Wave

LEGENDLEGEND

ChCh. 1:. 1: Square Wave Square Wave 658 Hz 658 Hz 2 V 2 V pkpk--pkpkChCh. 2:. 2: 1.25 V DC 1.25 V DC

Goertzel Frequency Selectivity:219 Hz Square Wave

LEGENDLEGEND

ChCh. 1:. 1: Square Wave Square Wave 219 Hz 219 Hz 2 V 2 V pkpk--pkpkChCh. 2:. 2: 0.4 V DC 0.4 V DC

Goertzel Frequency Selectivity:665 Hz Square Wave

LEGENDLEGEND

ChCh. 1:. 1: Square Wave Square Wave 665 Hz 665 Hz 2 V 2 V pkpk--pkpkChCh. 2:. 2: 0 V DC 0 V DC

Goertzel Accuracy:659 Hz Sinusoid

0.07 0.36 0.66 0.95 1.24 1.53 1.83 2.12 2.41 2.71 310

8

6

4

2

0

2

4

6

8

10Goertzel % Error: 3 Vpk-pk AIC Range

Input Voltage [Vpk-pk]

Per

cent

Err

or

%Errori

Vi

Goertzel Linearity:659 Hz Sinusoid

0 0.5 1 1.5 2 2.5 30

2000

4000

6000

8000Goertzel Precision Test (AIC +/- 1.5 V)

Peak-Peak Input Voltage [Volts]

Inpu

t Am

plitu

de [F

loat

ing

Poi

nt]

Goertzeli

yi

Vi

Goertzel Linearity:659 Hz Sinusoid (Low Voltage)

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

5

10

15

20

25

30Goertzel Precision Test (AIC +/- 1.5 V)

Peak-Peak Input Voltage [Volts]

Inpu

t Am

plitu

de [F

loat

ing

Poi

nt]

Goertzeli

yi

Vi

Stabilization Scheme

Demodulation Scheme

Summary

• Constructed a fractional fringe interferometerfor the characterization of microelectromechanicalsystem (MEMS) structures.

• Developed stabilization and demodulationarchitectures based on DSP.

• Implemented the Goertzel algorithm on theTMS320C31 DSK to extract the requisite spectralcomponents for stabilization and demodulation.