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MIT 12 215 - Lecture Notes

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12.215 Modern NavigationSummary of Last classToday’s ClassHistory of EDMGeodimeterLater model GeodimeterHistory of EDMTelluometerModern versionsTheory of EDMTime of flight measurementTime-of-flight measurementPulse characteristicsPulse characteristicsEM PropagationEM ReceiverDifference measurement (stays constant with time and depends on distance)Higher frequency. Phase difference still says something about distance but how to know number of cycles?Phase measurement of distanceResolving ambiguitiesAmbiguity exampleFrequency shiftingFrequency shiftingApplication areasIssues with EDMSummary of class12.215 Modern NavigationThomas Herring11/13/2006 12.215 Modern Naviation L15 2Summary of Last class• Finish up some aspects of estimation– Propagation of variances for derived quantities– Sequential estimation– Error ellipses• Discuss correlations: Basic technique used to make GPS measurements.– Correlation of random signals with lag and noise added (varying amounts of noise)– Effects of length of series correlated– Effects of clipping (ex. 1-bit clipping)11/13/2006 12.215 Modern Naviation L15 3Today’s Class• Electronic Distance Measurement (EDM)• History• Methods:– Theory: Propagating electromagnetic signals – Timing signal delays– Use of phase measurements– Application areas (other than GPS)11/13/2006 12.215 Modern Naviation L15 4History of EDM• Development of this type of technology started during World War II with the development of RADAR (Radio Detection and Ranging)• Radars returned the distance to an object (and later versions the speed of the object through the Doppler shift) by timing the length of time the from the transmission of a pulse to its return. Accuracy was set by timing resolution (1μsec=300meters)• In 1949, Dr. Erik Bergstrand of Sweden introduced the Geodimeter (Geodetic Distance Measurement) that used light (550 nm wavelength) to measure geodetic quality distances (instrument weighed 100kg)11/13/2006 12.215 Modern Naviation L15 5Geodimeter• First units circa 1959 (50 kg each for measurement unit and optics11/13/2006 12.215 Modern Naviation L15 6Later model Geodimeter• Example of a latter model Geodimeter (circa 1966)Front and back views11/13/2006 12.215 Modern Naviation L15 7History of EDM• Distance range was about 10km during daylight and 25km at night.• Greater range during daytime was achieved by using radio waves, and in Dr. T. L. Wadley, South Africa introduced the Telluometer in 1957.• Instrument used X-band radio waves (~10GHz)• Receive and transmit ends looked similar (receiver actually re-transmitted the signal) (The Geodimeter used one or more corner cube reflectors.)• Distances up to 50 km could be measured in daylight with this instrument and later models.11/13/2006 12.215 Modern Naviation L15 8Telluometer• Example of circa 1962 model.Back and front of instrument (9 kg with case)1970’s version (1.7kg)11/13/2006 12.215 Modern Naviation L15 9Modern versions• These types of measurements are now directly built into the telescope assemblies of theodolites and you can see these on most construction sites. The angles are now also read electronically (compared to glass optical circles).• Modern example (circa 2000)Corner cube reflector, Infrared light source used11/13/2006 12.215 Modern Naviation L15 10Theory of EDM• EDM is based on the idea that light (and radio waves) travel at a finite velocity and by measuring how long a signal takes travel back and forth between two points and knowing the speed of light, the distance can be measured.• However, very few instruments actually make a time-of-flight measurement. Most instruments use a phase measurement (actually as series of such measurements). We will see shortly why.• Start with time-of-flight because concept it is simple then move to phase (GPS actually uses both measurement types with some interesting twists).11/13/2006 12.215 Modern Naviation L15 11Time of flight measurement• In time of flight measurement, a pulse is transmitted and the time to the return is measured.• Measurement can be made a number of ways:– Leading edge detection (signal level passes a threshold)– Centroid measurement (assume pulse shape of return)– “Matched filter”: outgoing pulse correlated with return pulse• Accuracy of measurement depends on duration of pulse11/13/2006 12.215 Modern Naviation L15 12Time-of-flight measurement• If a “box-car” is transmitted (i.e., a rectangular pulse), correlation with another box car, will generate a triangular correlation function.• The width of this function is twice the pulse length. A narrower pulse; the more precise the measurement.• However a perfect box-car is impossible to generate because of the instantaneous rise time.• Nature of pulse is accessed by Fourier transform of time-domain signal (i.e., its frequency content is determined).11/13/2006 12.215 Modern Naviation L15 13Pulse characteristics• The Fourier transform of a box car of height C and duration T seconds is:• The function on the left is called the sinc function• Notice that the width of the sinc function is 1/T (between zeros) and that its amplitude decays as 1/f• The equivalent width of a “pulse” is thought of as 1/(frequency range) [called bandwidth]Ce−i2πft−T /2T /2∫dt = 4CTsin(4πTf)4πTf11/13/2006 12.215 Modern Naviation L15 14Pulse characteristics• Very narrow pulses, need a large frequency bandwidth and broad pulses require a small bandwidth (consider internet data transfer rates)• In real systems bandwidth is limited by losses in the system that attenuate signals away from the center of the transmission frequencies (e.g., antennas only work around a certain frequency band).• One of the advantages of optical frequencies is that since the frequency is so high (3x106GHz compared to GPS at ~1GHz (109Hz)11/13/2006 12.215 Modern Naviation L15 15EM Propagation• Theory of propagation of EM (and interaction with antennas) is set by Maxwell’s equations. We will not cover this area except to note that the solution to Maxwell’s equations for a signal propagating in uniform, isotropic medium can be written as:• Where E is the electric field, t is time, x is a position vector and k is the wave vector (vector in direction of propagation divided by wavelength λ=v/f r E (x,t)=r E oe−i2π(ft−k⋅x)11/13/2006 12.215 Modern Naviation L15 16EM Receiver• All an EM receiver does is sample the E field


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