A Novel FMCW Radar Altimeter Design Capable of Achieving Fine Range

Accuracy Needed for Autonomous UAV Operation

Theresa Stumpf and Reid Crowe

University of Kansas

2335 Irving Hill Road

Lawrence, KS 66045-7612

http://cresis.ku.edu

Technical Report

CReSIS TR 156

2013

2

A Novel FMCW Radar Altimeter Design Capable

of Achieving Fine Range Accuracy Needed for

Autonomous UAV Operation By Theresa Stumpf and Reid Crowe We are studying the use of FMCW radars for various applications in geophysical surveys. In

the Fall of 2012, we developed a concept for an FMCW radar altimeter that used two

transmit chirps at different start frequencies in order extract very fine accuracy range

estimates from by inverting the difference in phase of the dechirped digitized beat signals.

Given recent interest in the use of dual-frequency radars for improving range accuracy, we

decided to publish this work in order to facilitate future study of this technique.

Table of Contents 1 Problem Statement ............................................................................................................. 5

2 Proposed Solution ............................................................................................................... 5

3 System Description and Analysis ........................................................................................ 5

3.1 Transmitter Section ...................................................................................................... 5

3.1.1 Waveform Generator and Transmit Waveform Characteristics ............................. 5

3.1.2 RF Gain and Filter ................................................................................................ 6

3.1.3 LO Generation ...................................................................................................... 6

3.1.4 SAW Filter as a Delay Element ............................................................................. 6

3.2 Receiver Section .......................................................................................................... 7

3.3 Antenna Characteristics ............................................................................................... 8

3.4 A/D Requirements........................................................................................................ 9

4 Signal Processing ............................................................................................................... 9

4.1 Signal Model ................................................................................................................ 9

4.2 Accounting for Doppler Shift Due to Maximum Vertical Velocity Component...............10

4.3 Mathematical Description of Algorithm for Inverting Phase Difference to Extract Range

11

4.4 Converting Range to Altitude ......................................................................................13

4.5 Processing Flow and Estimated Execution Time .........................................................13

4.6 Simulated Performance ..............................................................................................14

5 Conclusions .......................................................................................................................14

References ...............................................................................................................................16

Appendix A: Figures .................................................................................................................17

Figure 1: System Block Diagram ..........................................................................................17

Figure 2: Timing Used In Choosing PRF ..............................................................................18

Figure 3: Geometry Assumed For Converting Range Into Altitude ........................................18

4

Figure 4: Simulated Performance Over 0.2 to 9.2 Meters Above Surface in Presence of

Worst Case Doppler Shift and Thermal Noise, Assuming Dielectric Contrast of 2 .................19

Figure 5: Simulated Performance Over 6 to 120 Meters Above Surface In the Presence of

Worst Case Doppler Shift and Thermal Noise, Assuming Dielectric Contrast of 2 .................20

Figure 6: Simulated Performance Over 0.2 to 9.2 Meters Above Surface in Presence of Worst

Case Doppler Shift and Thermal Noise, Assuming Dielectric Contrast of 20 .........................21

Figure 7: Simulated Performance Over 6 to 120 Meters Above Surface in Presence of Worst

Case Doppler Shift and Thermal Noise, Assuming Dielectric Contrast of 20 .........................22

Appendix B: Tables ..................................................................................................................23

Table 1: Transmit Waveform Characteristics by Altimeter Setting .........................................23

Table 2: Processing Steps and Associated DSP With Estimated Machine Cycles ................24

Appendix C: Calculations .........................................................................................................25

Worst Case SNR Calculation .................................................................................................25

Worst Case RX Power ...........................................................................................................27

Appendix D: Simulator Code ....................................................................................................29

Appendix E: Datasheets for Parts Not in Project Description ....................................................38

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1 Problem Statement For this project, we are asked to design an FMCW radar altimeter to support landing events of a

Yak-54 UAV. The specifications require a system that is capable of providing altitude

measurements with 3% accuracy when the aircraft is within 10 meters of the surface. For

altitudes greater than 10 meters, height-above-surface measurements are required to be

accurate to within 1 meter. The major challenge of this design is achieving the range fine

accuracy required for autonomously landing the UAV.

2 Proposed Solution We propose an FMCW radar altimeter that transmits two consecutive chirps that are

characterized by the same sweep time, bandwidth and chirp rate but with slightly different start

frequencies. The use of dual chirps allows us to achieve the fine range accuracy required for

this application. The modulation of our transmit chirps is a sawtooth type. To provide altitude

information of the UAV from zero to 120 meters above the surface, we recommend a low

altitude regime that can provide altitude data from the surface up to 9.2 meters and a high

altitude regime that can operate from 6.2 to 120 meters above the surface. The start

frequencies of waveform 1 and waveform 2 are offset by 8.885 MHz in the low altitude regime

and by 1.1125 MHz in the high altitude regime. By using two chirps with slightly different start

frequencies, we can invert the difference in phase of the dechirped digitized beat signals to

estimate the height of the UAV above the surface.

3 System Description and Analysis The block diagram for the FMCW altimeter is located in Appendix A.

3.1 Transmitter Section

3.1.1 Waveform Generator and Transmit Waveform Characteristics

• Transmit Waveform Modes: Low Altitude (0.2 m < h < 9.2 m)

High Altitude (6 m < h < 120 m)

• Transmit Power: -3.3 dBm (radiated)

• Waveform Type: Sawtooth

• Sweep Time: 30 µs

• Bandwidth: 26 MHz (33.875 MHz)

• Center Frequencies: Waveform 1 – 1440.9 MHz (1444.84 MHz)

Waveform 2 – 1449.785 MHz (1445.9625 MHz)

• PRF: 31.65 kHz

6

In the transmit section of the FMCW altimeter system, the recommended waveform generator,

Analog Devices ADF4158 Waveform Generating Frequency Synthesizer, was implemented.

The design uses sawtooth waveforms with characteristics summarized in Table 1, located in

Appendix B. Additionally the timing assumed in choosing the pulse repetition interval is

provided in Figure 2, located in Appendix A.

The ADF4158 outputs a control voltage, used by a VCO. The VCO used for this application is

the Z~Communications V602ME40-LF. When the output of the VCO is fed back into the

AF4158, as shown in Figure 6.2-1, a Phase-Locked Loop is formed creating a signal that is

coherent to the reference clock.

3.1.2 RF Gain and Filter

From the VCO, the 0 dBm signal is then amplified by a RFMD NBB-312 amplifier, in order to

achieve the required LO drive power for the mixer out of the coupled port of the directional

coupler. The NBB-312 is a surface mount, lightweight, amplifier that provides 12.5 dB of gain

with a P1dB compression point of +15.8 dBm; more than enough for this application.

The SAW filter used throughout this design is the TriQuint 856928. The insertion loss is

typically 1.25 dB, but the altimeter design utilizes the worst case specification of 2.5 dB. By

placing the filter at this point in the design, both the RF and LO signals are filtered.

3.1.3 LO Generation

After the filter, the Mini-Circuits BDCA-10-25 10 dB Bi-Directional Coupler is used to couple the

LO signal from the transmitted signal. This device was selected for its size, and light weight (0.3

grams). The insertion loss at the frequency of interest is 1 dB, with a coupling of 10 dB. The

forward coupled port sends the LO signal to the mixer to produce the beat signal from the

received RF and LO. The reverse coupled port is terminated in 50 ohms as it is not needed for

this application.

3.1.4 SAW Filter as a Delay Element

A unique property of the SAW filter is the amount of delay that it introduces over other filter

types. In a SAW filter electrical signals are converted to acoustic waves within the device. The

result is a propagation velocity much lower than the speed of light that ranges from ~3000 to

~4000 m/s, depending on the material [1]. The two SAW filters that follow the coupler are used

to introduce delay in the signal path. At close ranges, the beat frequency is too close to DC to

be of any use. Options to increase the beat frequency are increasing bandwidth, decreasing the

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repetition period or increase the round-trip time to target. Increasing the bandwidth and

decreasing the repetition period proved to not be practical options, leaving increasing the round-

trip time. To accomplish the round trip time, without changing the range of the aircraft to

ground, a delay element must be added to the transmit and/or receive path. Coax and optical

delay lines were considered, but these proved to be lossy, heavy or possibly add too much load

on the power supply. Cascaded saw filters were placed in the transmit chain to minimize losses

in the receiver front end, minimizing the effect on the noise figure.

Attenuation was added to the transmit section before the antenna to reduce the transmitted

power. This helps to reduce the amount of power that is coupled into the receiver. The 10 dB

attenuation value was selected as it provides the most reduction in transmit power, while

maintaining a worst case SNR of 16 dB, 6 dB higher than the minimum requirements. SNR

calculations for each test case are shown in Table 6.3-1and Table 6.3-2, located in the

Appendix. The SNR is calculated by first calculating the noise power, PN, as shown in equation

3.1

PN = kT0ΔfF [W] (3.1)

Where k is Boltzmann constant, T0 is room temperature 290 K, F is the noise figure which is

given later in this section, and Δf is defined by equation 3.2

Δf= 1TR

[Hz] (3.2)

In equation 3.2 TR is the length of the frequency sweep in seconds. After the noise power is

calculated, the SNR can be found by taking the received power over the noise power. Return

power is calculated assuming a specular surface. The dielectric contrast used for the returned

power, Pr, calculations ranged from ϵr = 2 to 20. The expression shown in Equation 3.3 gives

the returned power for a specular surface.

Pr = PtG2λ2Γsp

(8πR)2 (3.3)

Where Pt is the transmitted power, G is the antenna gain, l is the wavelength, Γsp is the

reflectivity of the specular surface and R is the range of the transmitter to the specular surface.

3.2 Receiver Section On the receive side, the SAW filter introduces an additional 14 ns delay and filters signals

outside the band of interest. The added delay, including the coax sections and cascaded SAW

8

filters in the transmit section, moves the beat frequency away from DC when the aircraft is near

the ground.

To produce the beat signal, the Analog Devices AD8342 Active Receive Mixer is used. The

mixer provides 3.7 dB of gain, but has a noise figure of 12.2 dB. The noise figure is

representative of the noise power added to the system by the device. For passive components,

such as the SAW filter, the noise figure is the loss of the device. For passive devices, such as

the SAW filter, the noise figure is the loss of the device.

Following the mixer is the Analog Devices AD8283 LNA, PGA, AAF with ADC. As the name

implies, this device provides a low-noise amplifier, programmable gain amplifier, and

programmable low-pass antialiasing filter along with the 12-Bit 80MSPS ADC. The analog

section of the AD8283 can provide a gain of 16, 22, 28 and 34dB and has a noise figure of 12.7

dB. The AD8283 operates off of the same 10 MHz reference clock used by the AF4158, but is

lowered to 4 MHz using a fractional PLL. This keeps the system coherent while reducing the

number of samples that need to be processed by the DSP section.

The cascaded gain of the receive section is from 16.87 dB to 34.87 dB. The total noise figure of

the receiver is 16.65 dB. Active components should not exceed their 1 dB compression point. If

exceeded, the output from that component will become distorted. In addition, the maximum

peak to peak voltage of the ADC must not be exceeded; otherwise the digitized signal will be

distorted. The 1dB compression point for each component was checked for the worst case to

ensure that the received signal would not become distorted within the receiver. The test

conditions were -6.33 dBm of transmit power over a specular surface with a reflection coefficient

of 0.403, antenna gain of 5.15 dBi, and a range of 0.2 meters. For these levels, at the highest

gain setting, the ADC is 0.86 dBm away from compressing. The instantaneous dynamic range

for the receiver is 76 dB.

3.3 Antenna Characteristics

• Type: Half-wave dipoles

• Gain: 5.15 dBi (with ground plane)

• Polarization: Linear

• Mounting Location: Fuselage

The antennas for the FMCW radar are linearly polarized 𝜆2 dipoles, spaced 𝜆

4 from a ground

plane. This provides 5.15 dBi of gain for each antenna. The antennas are 10.4 cm long and

9

are placed 20.8 cm apart at the center to allow for 𝜆2 spacing from the ends. Antennas are

mounted inside the fuselage of the aircraft to minimize drag and protect the antennas. Since

the aircraft is made from balsa wood and covered with MonoKote, a plastic material, the

structure should have little effect on the antenna. The antennas are oriented so the nulls in the

radiation pattern face each other. This orientation will minimize coupling.

3.4 A/D Requirements

• Sampling Frequency

fs = 4 MHz

• Fast Time Vector Length

N = 120 samples

Our selection of sampling frequency was simulation-driven. We assumed the worst case

Doppler shift given a vertical velocity component of 3 m/s and a minimum dielectric contrast of 2

(leading to a reflectivity of approximately -15 dB). We then tested our ability to retrieve accurate

altitude data in the presence of thermal noise as the sampling frequency decreased from the

maximum value of 80 MS/s towards our theoretical minimum sampling frequency of 1.8 MHz

(needed to satisfy Nyquist for a maximum beat frequency of 903.3 kHz at 120 meter altitude).

We chose 4 MHz because this value kept the error within the necessary bounds imposed by the

accuracy requirements with some guard band.

4 Signal Processing

4.1 Signal Model Our design uses two different transmit waveforms on successive sweeps to get around the

constraint imposed by the preselector’s bandwidth. These two chirps use the same chirp rate,

k, but require different start frequencies, denoted fc1 and fc2. The dechirped, digitized beat

signals of waveform 1 and waveform 2 are both vectors of real values, with dimensions 1x120

that are denoted as sIF1(n) and sIF2(n) respectively. Let T represent the time required for a

signal launched from the antenna to travel to the surface and reflect back to the receiving

antenna. Suppose T remains approximately constant over the time it takes the altimeter to

transmit, receive, dechirp and digitize waveforms 1 and 2. Then in the presence of thermal

noise, sIF1(n) and sIF2(n) may be modeled as

𝒔𝑰𝑭𝟏(𝑛) = 𝐴1cos(2𝜋𝑓𝐵𝑛 + 2𝜋𝑓𝑐1𝑇 − 𝜋𝑘𝑇2) + 𝒏𝟏(𝑛)

10

𝒔𝑰𝑭𝟐(𝑛) = 𝐴2cos(2𝜋(𝑓𝐵±𝑓𝐷)𝑛 + 2𝜋𝑓𝑐2𝑇 − 𝜋𝑘𝑇2) + 𝒏𝟐(𝑛)

where

A1,2 = amplitude of digitized beat signal

fB = beat frequency for the surface echo

fD = relative Doppler shift realized between waveform 1 and waveform 2

n = discrete time index

n1,2(n) = additive noise vector with dimensions 1x120

We model the contribution of noise as a real, stationary, zero-mean, Gaussian random process

with variance, σn equal to the thermal noise power. We evaluate the thermal noise power as

𝑃𝑛 = 𝑘𝑇0∆𝑓𝐹 (𝑊𝑎𝑡𝑡𝑠)

where

k = 1.3807x10-23 (Watts / (seconds * kelvin)) is Boltzmann’s Constant

T0 = 290 kelvin is the assumed ambient temperature

Δf = 33 MHz is the resolution bandwidth

F = 47 is the receiver noise figure expressed as a linear quantity

4.2 Accounting for Doppler Shift Due to Maximum Vertical Velocity Component Assuming a maximum vertical velocity, vr-max of 3 m/s, we can calculate a worst-case Doppler

shift as follows

𝑓𝐷−𝑚𝑎𝑥 = 2𝑣𝑟−𝑚𝑎𝑥

𝜆= 2

3(𝑚𝑠 )0.21(𝑚)

≈ 30 𝐻𝑧

When the relative Doppler shift between the echoes received from waveform 1 and waveform 2

is sufficiently less than the smallest beat frequency, we represent the digitized beat signals as

𝒔𝑰𝑭𝟏(𝑛) = 𝐴1cos(2𝜋𝑓𝐵𝑛 + 2𝜋𝑓𝑐1𝑇 − 𝜋𝑘𝑇2) + 𝒏𝟏(𝑛)

𝒔𝑰𝑭𝟐(𝑛) = 𝐴2cos(2𝜋𝑓𝐵𝑛 + 2𝜋𝑓𝑐2𝑇 − 𝜋𝑘𝑇2) + 𝒏𝟐(𝑛), 𝑓𝐵 ≫ 𝑓𝐷

To ensure that the smallest anticipated beat frequency was not sensitive to Doppler shifts, we

chose a sweep time, τ, that would provide a resolution bandwidth that was less than the lowest

beat frequency and much greater than fD-max. For a range of 0.2 meters and system delay of 50

11

ns, we calculated a minimum beat frequency of approximately 46 kHz. A 30 μs sweep provides

a resolution bandwidth of 33.3 MHz and ensures that the minimum beat frequency would not be

mapped to the DC bin when subject to the worst case Doppler shift. Furthermore a 30 Hz

Doppler shift should not be resolvable given a 33.3 MHz resolution in the frequency domain.

4.3 Mathematical Description of Algorithm for Inverting Phase Difference to Extract Range The range dependent phases of our digitized beat signals are denoted as Ø1(T) and Ø2(T)

respectively for a propagation delay T and are expressed mathematically as follows

∅1(𝑇) = 2𝜋𝑓𝐵𝑛 + 2𝜋(𝑓𝑐1𝑇 − 0.5𝑘𝑇2)𝑚

∅2(𝑇) = 2𝜋𝑓𝐵𝑛 + 2𝜋(𝑓𝑐2𝑇 − 0.5𝑘𝑇2)𝑚

Note that when m = 1, the difference of these terms leads to an unambiguous estimate of range:

∅2(𝑇)− ∅1(𝑇) = 2𝜋(𝑓𝑐2 − 𝑓𝑐1)𝑇 =4𝜋(𝑓𝑐2 − 𝑓𝑐1)𝑅

𝑐

Our objective is to evaluate this phase difference in order to provide a more accurate estimate

or R.

Assume the representation below for our digitized beat signals, sIF1(n) and sIF2(n):

𝒔𝑰𝑭𝟏(𝑛) = 𝐴1cos(2𝜋𝑓𝐵𝑛 + 𝛽1) + 𝒏𝟏(𝑛), 𝑤ℎ𝑒𝑟𝑒 𝛽1 = 2𝜋𝑓𝑐1𝑇 − 𝜋𝑘𝑇2

𝒔𝑰𝑭𝟐(𝑛) = 𝐴2cos(2𝜋𝑓𝐵𝑛 + 𝛽2) + 𝒏𝟐(𝑛), 𝑤ℎ𝑒𝑟𝑒 𝛽2 = 2𝜋𝑓𝑐2𝑇 − 𝜋𝑘𝑇2

Let SIF1(ω) and SIF2(ω) represent the spectra of sIF1(n) and sIF2(n). An estimate of the beat

frequency, 𝑓𝐵, due to the surface echo may be obtained by finding the frequency corresponding

to the maximum peak of either signal’s magnitude spectrum.

Let sref(n) be a unity magnitude reference function defined as a complex exponential with

fundamental frequency equal to the estimated beat frequency:

𝒔𝒓𝒆𝒇(𝑛) = 𝑒𝑗2𝜋�̂�𝐵𝑛

When the error between the true beat frequency and the estimated beat frequency is much less

than the value of the true beat frequency, the reference is modeled as

12

𝒔𝒓𝒆𝒇(𝑛) = 𝑒𝑗2𝜋𝑓𝐵𝑛

Let 𝒔�𝟏(𝑛) and 𝒔�𝟐(𝑛) denote the complex signals created by mixing each beat signal with the

reference. These signals lead to a complex exponential whose phase contains the range-

dependent term of interest. We show this derivation for 𝒔�𝟏(𝑛) and apply the result to 𝒔�𝟐(𝑛).

Note that the noise vector is neglected in the derivation because we are interested in the

deterministic portion of the signals. Noise was not however neglected in simulation.

𝒔�𝟏(𝑛) = 𝒔𝒓𝒆𝒇(𝑛) ∙ 𝒔𝑰𝑭𝟏(𝑛)

= 𝑒𝑗2𝜋𝑓𝐵𝑛 ∙ (𝐴1 cos(2𝜋𝑓𝐵𝑛 + 𝛽1))

= [cos(2𝜋𝑓𝐵𝑛) + 𝑗 sin(2𝜋𝑓𝐵𝑛)] ∙ [𝐴1cos(2𝜋𝑓𝐵𝑛 + 𝛽1)]

= cos(2𝜋𝑓𝐵𝑛) ∙ 𝐴1 cos(2𝜋𝑓𝐵𝑛 + 𝛽1) + 𝑗 sin(2𝜋𝑓𝐵𝑛) ∙ 𝐴1cos(2𝜋𝑓𝐵𝑛 + 𝛽1)

=𝐴12

[cos(2𝜋𝑓𝐵𝑛 + 2𝜋𝑓𝐵𝑛 + 𝛽1) + cos(2𝜋𝑓𝐵𝑛 − 2𝜋𝑓𝐵𝑛 + 𝛽1) + 𝑗 sin(2𝜋𝑓𝐵𝑛 + 2𝜋𝑓𝐵𝑛 + 𝛽1)

+ 𝑗 sin(2𝜋𝑓𝐵𝑛 − 2𝜋𝑓𝐵𝑛 + 𝛽1)]

=𝐴12

[cos(2𝜋(2𝑓𝐵)𝑛 + 𝛽1) + cos(𝛽1) + 𝑗 sin(2𝜋(2𝑓𝐵)𝑛 + 𝛽1) + 𝑗 sin(𝛽1)]

=𝐴12

[cos(𝛽1) + 𝑗 sin(𝛽1)] +𝐴12

[cos(2𝜋(2𝑓𝐵)𝑛 + 𝛽1) + 𝑗 sin(2𝜋(2𝑓𝐵)𝑛 + 𝛽1)]

The first term in this expression represents a DC component. The AC component on the left

hand side is a complex exponential with frequency 2fB offset in phase from the reference by β1

radians. We can isolate β1 by removing the DC component and evaluating the phase of the AC

term in either the time domain or the frequency domain. The final zero-mean result is given by

𝒔�𝟏′ (𝑛) =𝐴12

[cos(2𝜋(2𝑓𝐵)𝑛 + 𝛽1) + 𝑗 sin(2𝜋(2𝑓𝐵)𝑛 + 𝛽1)]

=𝐴12𝑒𝑗(2𝜋(2𝑓𝐵)𝑛+𝛽1)

The corresponding spectrum of this complex signal in the first and second Nyquist zones

(0 ≤ 𝑓 ≤ 𝑓𝑠) is

𝑺�𝟏′ (𝜔) = 𝐴1 ∙ 𝜋 ∙ 𝑒𝑗𝛽1 ∙ 𝛿(𝜔 − 2𝜔𝐵) = 𝐴1 ∙ 𝜋 ∙ 𝛿(𝜔 − 2𝜔𝐵) ∙ (cos𝛽1 + 𝑗 sin𝛽1)

13

We can obtain β1 by evaluating the phase spectrum at its peak value (located theoretically at

2ωB)

∡𝑺�𝟏′ (𝜔 = 2𝜔𝐵) = tan−1 �ℐ𝓂�𝑺�𝟏′ (𝜔 = 2𝜔𝐵)�ℛℯ�𝑺�𝟏′ (𝜔 = 2𝜔𝐵)�

� = tan−1 �sin𝛽1cos𝛽1

� = tan−1(tan𝛽1) = 𝛽1

Following the same approach, we can determine β2. Then our phase difference used for

inversion is

∅2(𝑇)− ∅1(𝑇) = 𝛽2 − 𝛽1 =4𝜋(𝑓𝑐2 − 𝑓𝑐1)𝑅

𝑐

We are able to estimate ∅1(𝑇) and ∅2(𝑇) directly from the complex time vectors 𝒔�𝟏′ (𝑛) and 𝒔�𝟐′ (𝑛)

respectively. In the interest of reducing computational cost, we chose to estimate the phase

difference in the time domain to get around having to do two additional FFTs.

4.4 Converting Range to Altitude The geometry assumed for determining a value for altitude from range is presented in the

Figure 3, located in Appendix B. From range dependent phase difference derived in the

previous section, we compute the value of R, correct for system delay and solve for the

corresponding altitude, h as

ℎ = �𝑅2 − �𝑏2�2

where

b = separation distance between phase centers of the transmit and receive antennas

measured in meters

4.5 Processing Flow and Estimated Execution Time

• Total Number of Machine Cycles: 290,724 cycles

• Estimated Execution Time: 290.7 µs

Table 2 located in Appendix B outlines the processing flow and associated machine cycles for

each calculation. We start at the point that the vectors sIF1(n) and sIF2(n) (120 samples of real

values) are available in the DSP. Where the number of a machine cycles for a given

calculation was unknown, a value was assumed. Assumed values are denoted with a star in

Table 2.

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We require the following be stored in memory and available to the DSP:

• Discrete frequency vector (length 8,192) representing the frequencies from 0 to fs/2 with

a frequency step factor of 33.3 MHz (1/ τ).

• Discrete time vector (length 120) from 0 to τ with time step equal to the sampling period

(250 ns).

Though the execution time presented here is only an estimate, we believe that this approach

can satisfy the required refresh rate over the entire altitude envelope.

4.6 Simulated Performance To verify the algorithm, we developed MATLAB code to simulate inverting phase differences

between two beat signals in order extract range. This code is provided in Appendix D. The

user must chose low altitude or high altitude mode. We used the equation for received power

from a specular surface (presented in equation 3.3) to scale the amplitude of the beat signal for

a specific range and dielectric contrast. Simulator options allow the user to choose to add

worst-case Doppler shift to the simulation and thermal noise.

Results are presented in Figures 4 through 7 of Appendix A. Figures 4 and 5 show simulated

altimeter performance in low and high altitude modes respectively, assuming a dielectric

contrast of 2. These results were generated assuming the presence of thermal noise and worst-

case Doppler shift. The figures show measured altitude versus true altitude along with

corresponding measurement error. For convenience, we plot the required accuracy bounds on

top of the error to demonstrate that the algorithm met the specifications. Figures 6 and 7 show

simulated altimeter performance in low and high altitude modes respectively, assuming a

dielectric contrast of 20. These results were generated assuming the presence of thermal noise

and worst-case Doppler shift.

The simulator results suggest that the processing technique is capable of realizing 3% accuracy

when the UAV is less than 10 meters away from the surface. Altitude measurements easily

achieve the 1 meter accuracy required when the aircraft is more than 10 meters away from the

surface.

5 Conclusions We have presented a design for a lightweight FMCW radar altimeter capable of achieving the

fine range accuracy in real-time required to support autonomous landing of a UAV. The system

can provide meaningful data up to 120 meters above the surface with better than 1 meter

15

accuracy. The range accuracy is achieved by inverting the phase differences between two

distinct beat signals. We demonstrated this with simulation of the beat signals and found that

the performance of the algorithm was extremely sensitive to variations in the sweep time as well

as variations in the start frequencies of waveforms 1 and 2. Since the altitude estimates rely on

measurements of the phase, future work should involve a detailed analysis of timing and

frequency stability in the system. Additionally, we recommend that additional simulations

account for the effects of aircraft motion on altitude measurements.

16

References

Appendix A: Figures

Figure 1: System Block Diagram

Figure 2: Timing Used In Choosing PRF

Figure 3: Geometry Assumed For Converting Range Into Altitude

Figure 4: Simulated Performance Over 0.2 to 9.2 Meters Above Surface in Presence of Worst Case Doppler Shift and Thermal Noise, Assuming Dielectric Contrast of 2

20

Figure 5: Simulated Performance Over 6 to 120 Meters Above Surface In the Presence of Worst Case Doppler Shift and Thermal Noise, Assuming Dielectric Contrast of 2

21

Figure 6: Simulated Performance Over 0.2 to 9.2 Meters Above Surface in Presence of Worst Case Doppler Shift and Thermal Noise, Assuming Dielectric Contrast of 20

22

Figure 7: Simulated Performance Over 6 to 120 Meters Above Surface in Presence of Worst Case Doppler Shift and Thermal Noise, Assuming Dielectric Contrast of 20

Appendix B: Tables

Table 1: Transmit Waveform Characteristics by Altimeter Setting

Setting Waveform # Altitude [m] Start Freq. Stop Freq. Sweep Time PRF

Low 1

0.2 < h <9.2 1427.9 1454.015

30uS 31.64 kHz 2 1436.785 1462.9

High 1

6 < h <120 1427.9 1461.775

2 1429.025 1462.9

24

Table 2: Processing Steps and Associated DSP With Estimated Machine Cycles

Step in Algorithm Action in DSP Cycles

• Calculate, SIF2(ω), the spectrum of

sIF2(n)

16,384 point FFT 270,495

• Keep first 8,192 samples Move block of memory 2,063

• Detect surface echo in SIF2(ω) Find index of maximum of 8,192

length vector

2,602

• Estimate beat frequency Evaluate stored frequency vector

at index from above

1*

• Create complex reference function,

sref(n)

Unknown 500 *

• Calculate 𝒔�𝟏(𝑛) and 𝒔�𝟐(𝑛) by

multiplying sIF1(n) and sIF2(n) by sref(n)

Vector multiply x 2 306

• Calculate mean of 𝒔�𝟏(𝑛) and 𝒔�𝟐(𝑛) Vector dot product and scalar

multiply by 1/120

100

• Calculate 𝒔�𝟏′ (𝑛) and 𝒔�𝟐′ (𝑛) by removing

mean from 𝒔�𝟏(𝑛) and 𝒔�𝟐(𝑛)

Scalar negate and sum of vector

and scalar (x 2)

40 *

• Compute range-dependent phase

vectors, ∅𝟏(𝑇) and ∅𝟐(𝑇), from 𝒔�𝟏′ (𝑛)

and 𝒔�𝟐′ (𝑛) respectively

atan2(im(…)/real(…)) 14,916 *

• Calculate ∆∅ by taking the difference

between 120 length vectors ∅𝟏(𝑇) and

∅𝟐(𝑇)

Vector negate and vector sum 142 *

• Calculate the mean of ⟨∆∅⟩ Vector dot product and multiply

by 1/120

100

• Calculate range and correct for system

delay

Scalar multiplies and additions 0 *

• Compute altitude Vector addition and square root 1 *

Total Machine Cycles

290,724 * Number of machine cycles unknown. Starred values represent an estimated number of machine cycles.

Appendix C: Calculations

Worst Case SNR Calculation Worst Case SNR Er=2 R=120m

RX Chain

Component Coax SAW Filter Mixer LNA/PGA/

AAF 16dB LNA/PGA/AAF 22dB

LNA/PGA/AAF 28dB

LNA/PGA/AAF 34dB

ADC 16dB

ADC 22dB

ADC 28dB

ADC 34dB

Gain (dB) -0.33 -2.50 3.70 16.00 22.00 28.00 34.00 Gain 0.93 0.56 2.34 39.81 158.49 630.96 2511.89 NF (dB) 0.33 2.50 12.20 12.70 12.70 12.70 12.70 NF 1.08 1.78 16.60 18.62 18.62 18.62 18.62 P1dBin (dBm) 8.30 -6.20 -12.20 -12.20 -24.20 10.00 10.00 10.00 10.00 Pin (dBm) -92.65 -92.98 -95.48 -91.78 -91.78 -91.78 -91.78 -75.78 -69.78 -63.78 -57.78 Pout (dBm) -92.98 -95.48 -91.78 -75.78 -69.78 -63.78 -57.78 NFtotal 46.26 NFtotal (dB) 16.65 Te (K) 1.05E+04

26

TX Chain

Component Amp SAW Filter Coupler SAW Filter SAW Filter Atten. Coax

Gain (dB) 12.5 -2.5 -1 -2.5 -2.5 -12 -0.33 Pin (dBm) 0 Pout (dBm) -8.33

Pout (mW) 0.14689 T0 (K) 290 Peirp (dBm) -3.18 k 1.38E-23 Peirp (mW) 0.48084 fc 1.45E+09 c 3.00E+08 Range (m) 120 lambda (m) 2.08E-01

Pr (mW) 5.4E-10 Antenna

Gain 5.15E+00 Pr (dBm) -92.65 E1 (Air) 1

E2 (Specular

Surface) 2 Pn (mW) 6.17E-12 Gamma_sp 0.0294 Pn (dBm) -112.09 Bandwidth 3.50E+07 tau 3.00E-05 SNR (dB) 19.4433 DelFreq 3.33E+04

27

Worst Case RX Power Worst Case RX Power Er=20 R=0.2m

RX Chain

Component Coax SAW Filter Mixer LNA/PGA/AAF

16dB LNA/PGA/AAF

22dB LNA/PGA/AAF

28dB LNA/PGA/AAF

34dB ADC 16dB

ADC 22dB

ADC 28dB

ADC 34dB

Gain (dB) -0.33 -2.50 3.70 16.00 22.00 28.00 34.00 Gain 0.93 0.56 2.34 39.81 158.49 630.96 2511.89 NF (dB) 0.33 2.50 12.20 12.70 12.70 12.70 12.70 NF 1.08 1.78 16.60 18.62 18.62 18.62 18.62 P1dBin (dBm) 8.30 -6.20 -12.20 -12.20 -24.20 10.00 10.00 10.00 10.00 Pin (dBm) -25.73 -26.06 -28.56 -24.86 -24.86 -24.86 -24.86 -8.86 -2.86 3.14 9.14 Pout (dBm) -26.06 -28.56 -24.86 -8.86 -2.86 3.14 9.14 NFtotal 46.26 NFtotal (dB) 16.65 Te (K) 1.05E+04

28

TX Chain

Component Amp SAW Filter Coupler SAW Filter SAW Filter Atten. Coax

Gain (dB) 12.5 -2.5 -1 -2.5 -2.5 -12 -0.33 Pin (dBm) 0 Pout (dBm) -8.33

Pout (mW) 0.14689 T0 (K) 290 Peirp (dBm) -3.18 k 1.38E-23 Peirp (mW) 0.48084 fc 1.45E+09 c 3.00E+08 Range (m) 0.2 lambda (m) 2.08E-01

Pr (mW) 2.67E-03 Antenna Gain

5.15E+00 Pr (dBm) -25.73 E1 (Air) 1

E2 (Specular Surface)

20 Pn (mW) 6.17E-12 Gamma_sp 0.4026 Pn (dBm) -112.09 Bandwidth 3.50E+07 tau 3.00E-05 SNR (dB) 86.3662 DelFreq 3.33E+04

Appendix D: Simulator Code % close all clear % ------------------------------------------------------------------------- % Set flag to choose either low altitude simulation (0.2 < range < 9.2) % or high altitude (6 < range < 120) % ------------------------------------------------------------------------- param.simulate.low_altitude.en = 1; param.simulate.high_altitude.en = 0; param.simulate.discrete_low_alt.en = 0; % USED FOR DEBUG % ------------------------------------------------------------------------- % Assign variable radar configuration parameters % ------------------------------------------------------------------------- param.system_config.tau = 30e-6; % chirp sweep time (s) param.system_config.fs = 10e6; % sampling frequency of IF wfs (Hz) param.system_config.G_antenna = 5.51; % antenna gain (dBi) param.system_config.theta_HP = 87; % antenna 3dB beamwidth (degrees) param.system_config.td_bpf = 14e-9; % prop delay through SAW filt (s) param.system_config.td_coax = 11e-9; % prop delay through coax (s) param.system_config.vr_max = 3; % max radial velocity of YAK (m/s) param.system_config.P_tx_dBm = -3.3; % transmit power (dBm) param.system_config.NF = 16.7; % receiver noise figure (dB) param.system_config.baseline = 0.1; % separation baseline between % transmit and receive phase % centers (meters) % Assign offset frequency based on low or high alt simulation if param.simulate.low_altitude.en param.system_config.d_freq = 8.885e6; end if param.simulate.high_altitude.en param.system_config.d_freq = 1.125e6; end % ------------------------------------------------------------------------- % Assign variable target characteristics % ------------------------------------------------------------------------- if param.simulate.low_altitude.en param.target_config.range = [0.2:0.2:9.2]; end if param.simulate.high_altitude.en param.target_config.range = [6:0.2:120]; end if param.simulate.discrete_low_alt.en param.target_config.range = 0.2; param.system_config.d_freq = 8.885e6; end

30

param.target_config.er_surf = 2; % relative permittivity of % surface % ------------------------------------------------------------------------- % Setup simulation options % ------------------------------------------------------------------------- param.proc.debug = 0; param.proc.simulate_IF.en = 1; param.proc.add_WGN.en = 1; param.proc.add_fd_max.en = 1; param.proc.plot_IF_time_domain.en = 0; param.proc.plot_IF_freq_domain.en = 0; param.proc.zero_pad.en = 1; param.proc.zero_pad.NFFT = 2^14; param.proc.IQ_mode.en = 1; param.proc.IQ_mode.atan2.en = 1; param.proc.plot_IQ_freq_domain.en = 0; param.proc.plot_SNR.en = 1; % ------------------------------------------------------------------------- % Declare speed of light in a vacuum and Boltzmann's Constant % ------------------------------------------------------------------------- c = 2.9979e+08; % speed of light (m/s) K = 1.3807e-23; % Boltzmann's Constant (Watts/(s*kelvin)) % ------------------------------------------------------------------------- % Calculate system delay % ------------------------------------------------------------------------- td_bpf = param.system_config.td_bpf; td_coax = param.system_config.td_coax; t_system_delay = 2*(td_bpf + td_coax); % system delay (seconds) R_system_delay = (c*t_system_delay)/2; % range equivalent system % delay (meters) % ------------------------------------------------------------------------- % Calculate theoretical limit of unambiguous range that may be extracted % using dual chirp approach % ------------------------------------------------------------------------- d_freq = param.system_config.d_freq; R_max = c/(2*pi*d_freq); % Theoretical maximum unambiguous range for % low altitude scenario fprintf('\n\n\n\n\n\n') fprintf('-----------------------------------------------------------\n') fprintf('The theoretical maximum range that can be extracted using \n') fprintf('dual-chirp approach for the selected values of tau and \n') fprintf('d_freq is \n\nR_max = %2.4f\n',R_max) fprintf('-----------------------------------------------------------\n') fprintf('\n\n\n') % ------------------------------------------------------------------------- % Calculate Fresnel reflection coefficient

31

% ------------------------------------------------------------------------- er_surf = param.target_config.er_surf; er_air = 1; % relative dielectric constant of air diel_contrast = er_surf/er_air; % dielectric contrast at air surface % interface Gamma_sp = abs((1-sqrt(diel_contrast))/(1+sqrt(diel_contrast)))^2; % ------------------------------------------------------------------------- % Setup given constraints % ------------------------------------------------------------------------- fc = 1445.4e6; % carrier frequency (Hz) f_L = 1427.9e6; % lower edge of SAW filter passband (Hz) f_U = 1462.9e6; % upper edge of SAW filter passband (Hz) % ------------------------------------------------------------------------- % Calculate maximum Doppler shift for an assumed maximum radial velocity of % YAK % ------------------------------------------------------------------------- vr_max = param.system_config.vr_max; % maximum radial (vertical) lambda = c/fc; % wavelength (m) fd_max = (2*vr_max)/lambda; % maximum Doppler shift (Hz) % ------------------------------------------------------------------------- % Calclulate chirp bandwidth, start frequencies for waveforms 1 and 2, and % chirp rate % ------------------------------------------------------------------------- tau = param.system_config.tau; BW_chirp = f_U - f_L - d_freq; % bandwidth of transmit chirp (Hz) k = BW_chirp/tau; % chirp rate (Hz/s) fc1 = f_L; % waveform 1 start freq (Hz) fc2 = fc1 + d_freq; % waveform 2 start freq (Hz) f_stop1 = fc1 + BW_chirp; % stop frequency of waveform 1 (Hz) f_stop2 = fc2 + BW_chirp; % stop frequency of waveform 2 (Hz) % ------------------------------------------------------------------------- % Setup dB quantities for radar range equation % ------------------------------------------------------------------------- G = param.system_config.G_antenna; lambda_dBsm = 10*log10(lambda); Gamma_sp_dB = 10*log10(Gamma_sp); % ------------------------------------------------------------------------- % Calculate thermal noise power % ------------------------------------------------------------------------- if param.proc.add_WGN.en T0 = 290; % ambient temperature (kelvin) F = 10^(param.system_config.NF/10); % receiver noise figure

32

bw = 1/tau; % bandwith after mixer (Hz) Pn_lin = K*T0*bw*F; % thermal noise power (Watts) end for range_idx = 1:length(param.target_config.range) if length(param.target_config.range) > 1 param.proc.plot_IF_time_domain.en = 0; param.proc.plot_IF_freq_domain.en = 0; param.proc.plot_IQ_freq_domain.en = 0; end % --------------------------------------------------------------------- % Calculate received signal delay % --------------------------------------------------------------------- R = param.target_config.range(range_idx); t_prop = (2*R)/c; T_rx = t_prop + t_system_delay; % --------------------------------------------------------------------- % Convert simulated range to altitude % --------------------------------------------------------------------- b = param.system_config.baseline; h = sqrt(R^2-(b/2)^2); % --------------------------------------------------------------------- % Calculate amplitude of receive chirp using radar range equation % --------------------------------------------------------------------- P_tx_dBm = param.system_config.P_tx_dBm; spread_loss_dB = 10*log10(8*pi*R); PC_gain = 10*log10(BW_chirp*tau); P_rx_dBm = P_tx_dBm + 2*G + 2*lambda_dBsm + PC_gain ... + Gamma_sp_dB - 2*spread_loss_dB; P_rx_dBW = P_rx_dBm - 30; % received signal power (dBW) P_rx_lin = 10^(P_rx_dBW/10); % received signal power (Watts) A = sqrt(P_rx_lin); % amplitude (Volts) if param.proc.debug A = sqrt(P_tx_lin); end if param.proc.simulate_IF.en % --------------------------------------------------------------------- % Create digitized IF signals % --------------------------------------------------------------------- fs = param.system_config.fs; Ts = 1/fs; % sampling period of A/D (secs/sample) t_AD = 0:Ts:tau; % discrete time vector (samples) f_b = k*T_rx; % beat frequency (Hz) f_b1 = f_b;

33

% Add worst case doppler offset to the beat frequency of waveform 2 to % simulate error in estimated range due to vertical component o if param.proc.add_fd_max.en f_b2 = f_b + fd_max; else f_b2 = f_b; end % Create digitized beat signals of waveform 1 and waveform 2 s_IF1 = A *... cos(2*pi*(f_b1.*t_AD + fc1*T_rx - 0.5*k*T_rx^2)); s_IF2 = A *... cos(2*pi*(f_b2.*t_AD + fc2*T_rx - 0.5*k*T_rx^2)); % Add thermal noise to IF signals if param.proc.add_WGN.en noise_1 = sqrt(Pn_lin).*randn(1,length(s_IF1)); s_IF1 = s_IF1 + noise_1; noise_2 = sqrt(Pn_lin).*randn(1,length(s_IF2)); s_IF2 = s_IF2 + noise_2; end % DEBUG ONLY % Plot waveform 1 and waveform 2 beat signals if param.proc.debug || param.proc.plot_IF_time_domain.en figure;plot(t_AD.*1e6,real(s_IF1).*1e3) hold on grid on plot(t_AD.*1e6, real(s_IF2).*1e3,'r') % ylim([-500 500]) xlabel('time (us)','FontSize',12) ylabel('IF Signal Amplitude (mV)','FontSize',12) title_string = sprintf('Dechirped Waveforms, R = %2.4f (meters), f_B = %3.4f (kHz)',R,f_b*10^-3); title(title_string,'FontSize',12,'FontWeight','Demi') legend('Wf 1','Wf 2','Location','NorthEast') end % Specify number of points in FFT if param.proc.zero_pad.en NFFT = param.proc.zero_pad.NFFT; M = NFFT - length(s_IF1); else M = 0; NFFT = length(s_IF1); end % DEBUG ONLY % Compute spectra of wf 1 and wf2 and plot if param.proc.debug || param.proc.plot_IF_freq_domain.en

34

S_IF1 = fft([zeros(1,M) s_IF1]); S_IF2 = fft([zeros(1,M) s_IF2]); freq = fs.*linspace(0,1,NFFT); df = 1/tau; figure;subplot(2,1,1);plot(freq.*1e-6,10*log10((abs(S_IF1).^2)./(NFFT*df))); xlim([0 1]) grid on hold on plot(freq.*1e-6,10*log10((abs(S_IF2).^2)./(NFFT*df)),'r') xlabel('Frequency (MHz)','FontSize',12) ylabel('Power With Respect to 1 Ohm (dBW/Hz)','FontSize',12) title('Beat Signal Frequency Response (Magnitude)',... 'FontSize',12,'FontWeight','Demi') legend('Wf 1','Wf 2','Location','NorthEast') subplot(2,1,2);plot(freq.*1e-6,angle(S_IF1)) xlim([0 1]) grid on hold on plot(freq.*1e-6,angle(S_IF2),'r') xlabel('Frequency (MHz)','FontSize',12) ylabel('Phase (radians)','FontSize',12) title('Beat Signal Frequency Response (Phase)',... 'FontSize',12,'FontWeight','Demi') legend('Wf 1','Wf 2','Location','NorthEast') end % ===================================================================== % Range Extraction Algorithm % ===================================================================== if param.proc.IQ_mode.en % Compute FFT of waveform 2 and use the max function to find the % index corresponding to the peak of wf 2's spectrum. % ------------------------------------------------------------------- S_IF2 = fft([zeros(1,M) s_IF2]); [~,idx2] = max(S_IF2(1:floor(NFFT/2))); % Create a frequency vector and evaluate it at the index % corresponding to the max of wf 2's spectrum. This provides an % estimate of the beat frequency. % ------------------------------------------------------------------- freq = fs.*linspace(0,1,NFFT); f_b_est = freq(idx2); % Estimated beat frequency % Create reference beat signals % ------------------------------------------------------------------- s_ref = exp(1i*2*pi*f_b_est.*t_AD); % Multiply wf1 and wf2 by complex envelope, s_ref. % -------------------------------------------------------------------

35

s_IF1_IQ = s_ref.*(s_IF1); s_IF2_IQ = s_ref.*(s_IF2); % The spectra of these complex signals contain a term at DC and a % term at twice the beat frequency. The phase of the 2f_b terms will % be 2*pi*(fc1*T - 0.5*k*T^2) and 2*pi*(fc2*T - 0.5*k*T^2) % respectively. The difference in phases is 2*pi*(fc2 - fc1)*T. We % wish to isolate the 2f_b terms in order to extract range from the % phase difference. We can do this by removing the DC components. % This is realized by subtracting from each signal its respective % mean. % ------------------------------------------------------------------- s_IF1_IQ = s_IF1_IQ - mean(s_IF1_IQ(:)); s_IF2_IQ = s_IF2_IQ - mean(s_IF2_IQ(:)); % Evaluate phase of each complex time series, evaluate their % difference and take the mean to provide the estimate of % 2*pi*(fc2-fc1)*T % ------------------------------------------------------------------- if param.proc.IQ_mode.atan2.en arg1_IQ = atan2(imag(s_IF1_IQ),real(s_IF1_IQ)); arg2_IQ = atan2(imag(s_IF2_IQ),real(s_IF2_IQ)); delta_phi = mean(unwrap(arg2_IQ - arg1_IQ)); if delta_phi < 0 delta_phi = 2*pi - abs(delta_phi); end end end end % Convert extracted phase into a time delay and measured range T_meas = delta_phi/(2*pi*d_freq); R_meas = (c*T_meas)/2; % Convert measured delay into actual delay by compensating for system % delay. Use this delay to calculate correc T_corrected = T_meas - t_system_delay; R_corrected = (c*T_corrected)/2; range_error = R_corrected - R; % Convert range to an altitude h_IQ_mode = sqrt(R_corrected^2 - (b/2)^2); h_error = h_IQ_mode - h; fprintf('\n\n\n') fprintf('-----------------------------------------------------------\n')

36

fprintf('The estimated height above surface is: %2.4f meters\n',h_IQ_mode) fprintf('The true height above surface is: %2.4f meters\n',h) fprintf('The error for this measurement is: %3.2f meters\n',h_error) fprintf('\n') fprintf('-----------------------------------------------------------\n') fprintf('\n\n\n') range_IQ_mode(range_idx) = R_corrected; range_error_IQ_mode(range_idx) = range_error; true_range(range_idx) = R; true_altitude(range_idx) = h; altitude_IQ_mode(range_idx) = h_IQ_mode; altitude_error_IQ_mode(range_idx) = h_error; if param.proc.plot_IQ_freq_domain.en S_IF1_IQ = fft([zeros(1,M) s_IF1_IQ]); S_IF2_IQ = fft([zeros(1,M) s_IF2_IQ]); df = 1/tau; figure;subplot(2,1,1);plot(freq.*1e-6,10*log10((abs((S_IF1_IQ)).^2)./(NFFT*df))) hold on plot(freq.*1e-6,10*log10((abs((S_IF2_IQ)).^2)./(NFFT*df)),'r') xlim([0 1]) ylim([-140 -80]) grid on xlabel('Frequency (MHz)','FontSize',12) ylabel('Power With Respect to 1 Ohm (dBW/Hz)','FontSize',12) title('Modulated Beat Signal Frequency Response (Magnitude)',... 'FontSize',12,'FontWeight','Demi') legend('Wf1','Wf2','Location','NorthEast') subplot(2,1,2);plot(freq.*1e-6,angle(S_IF1_IQ)) hold on plot(freq.*1e-6,angle(S_IF2_IQ),'r') xlim([0 1]) grid on hold on xlabel('Frequency (MHz)','FontSize',12) ylabel('Phase (radians)','FontSize',12) title('Modulated Beat Signal Frequency Response (Phase)',... 'FontSize',12,'FontWeight','Demi') legend('Wf1','Wf2','Location','NorthEast') end end if length(param.target_config.range) > 1

37

figure;subplot(2,1,1);plot(true_altitude,altitude_IQ_mode,'xk') hold on grid on plot(true_altitude,true_altitude,'--c') xlabel('Simulated Altitude (meters)','FontSize',12) ylabel('Estimated Altitude (meters)','FontSize',12) title_string_1 = sprintf('Surface Tracking Performance Over %2.1f to %2.1f Meter Interval',true_altitude(1),true_altitude(end)); title_string_2 = sprintf('Sweep Time = %2.1f us, fc2-fc1 = %2.4f MHz',tau*1e6,d_freq*1e-6); title({title_string_1;title_string_2},'FontSize',12,'FontWeight','Demi') legend('IQ mode','True Altitude','Location','NorthWest') subplot(2,1,2);stem(true_altitude,altitude_error_IQ_mode) grid on hold on if param.simulate.low_altitude.en threshold_plus = .03.*true_altitude; threshold_minus = -1*.03.*true_altitude; end if param.simulate.high_altitude.en idx = find(true_altitude > 10, 1); r = length(true_altitude(idx + 1:end)); threshold_plus = [0.03*true_altitude(1:idx) ones(1,r)]; threshold_minus = [-1*0.03*true_altitude(1:idx) -1*ones(1,r)]; end plot(true_altitude,threshold_plus,'--r') plot(true_altitude,threshold_minus,'--r') xlabel('Simulated Altitude (meters)','FontSize',12) ylabel('Measurement Error (meters)','FontSize',12) title_string_1 = sprintf('Measured Altitude Error Over %2.1f to %2.1f Meter Interval',true_altitude(1),true_altitude(end)); title_string_2 = sprintf('Assumed Dielectric Contrast of %2d',er_surf); title({title_string_1;title_string_2},'FontSize',12,'FontWeight','Demi') legend('Altitude Estimation Error','Required Accuracy Bound','Location','NorthWest') if param.simulate.low_altitude.en end if param.simulate.high_altitude.en ylim([-1.5 1.5]) xlim([6 120]) end end

38

Appendix E: Datasheets for Parts Not in Project Description RFMD NBB-312 Cascadeable Broadband Amplifier

39

40

41

42

43

44

45

46

47

Mini-Circuits BDCA-10-25 Bi-Directional Coupler

48

Mini-Circuits 141-35SM+ Coaxial Cable

49

50