Spectrum_of_Quantization_Noise_WBWC_03_02_Part_II

Spectrum of Quantization Noise WesBrodsky Wireless Communication

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What is the Power Spectrum of Quantization Noise?

Part II

Contents 2.2. BaseBand Inputs ............................................................................................................... 2

2.3 Discussion of Results ..................................................................................................... 18

3. DAC Case ............................................................................................................................. 19

3.1 BandLimited MSK output .................................................................................................. 19

3.2 BaseBand MSK output ....................................................................................................... 19

3.3 Discussion of Results .......................................................................................................... 22

4. Summary and Conclusions ................................................................................................... 22

5. Abbreviations Used ............................................................................................................... 22

6. References ............................................................................................................................. 23


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2.2. BaseBand Inputs

For the BaseBand input cases, the ADC sample rate was 550 MHz. The inputs were 1-tone

(Figure 32), 2-tones (Figure 33), MSK (Figure 34) and OFDM (Figure 35).

Figure 32 Base-Band 1 Tone Input

Figure 33 Base Band 2 Tone Input


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Figure 34 Base Band MSK Input

Figure 35 Base Band OFDM Input


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Figure 36; a, b, c: 1 Tone, 0 dBrmsFS


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Figure 37; a, b, c: 1 Tone, -3 dBrmsFS


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Figure 38; a, b, c: 1 Tone, -43 dBrmsFS


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Figure 39; a, b, c: 2 Tones, -3 dBrmsFS


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Figure 42; a, b, c: MSK, 0 dBrmsFS


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Figure 43; a, b, c: MSK -3 dBrmsFS


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Figure 44; a, b, c: MSK -4 dBrmsFS


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Figure 45; a, b, c: MSK, -43 dBrmsFS (LSB = 0.0078 V)


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Figure 46; a, b, c: OFDM, -9.2 dBrmsFS


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2.3 Discussion of Results A spectrum will be defined as “uniform enough” if equation 1 approximately holds for any

reasonable filter that might follow the ADC, at any center frequency. A “reasonable filter” is one

that is not too narrow. For example, the outputs with 1- and 2- tones in have spikes, and a

narrowband filter would either pass or reject these spikes depending on its center frequency. But,

it is doubtful that a filter this narrow would ever be used, and a reasonable wide filter would pass

many spikes. If the output of this filter remained constant wherever its center frequency was, the

spectrum would be “uniform enough”.

Looking over all the figures, it is possible to make some observations:

1) Even a teensy-weensy bit of clipping (overflow) in the ADC will cause the quantization

noise spectrum to not be uniform enough. For example, look at Figures 47 and 48. In the

time waveform, Figure 47a, there is one peak which clips at the maximum ADC input of

+1 V. The spectrum of the quantization noise, Figure 47c, is not uniform. But, Figure 48

is with the input 1.04 dB lower than in Figure 47. There is no clipping, and the

quantization noise spectrum in Figure 48c is uniform. This is in agreement with the

results of [14].

2) For all inputs, if there is no clipping and the input is above approximately 28 dBrmsLSB,

the spectrum will be “uniform enough”.

3) For the baseband MSK input, version .01 of this white paper inadvertently had an integer

number (29) ADC samples per MSK symbol. This apparently caused the error spectrum

to be very different from uniform. The later version(s) have 29.1875 ADC samples per

MSK symbol. The error spectra, while not perfectly uniform, looks uniform enough as

long as there is no clipping and the input is well above an LSB.

Some overall conclusions are given in Section 4.


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3. DAC Case For the DAC (Figure 50) case the error was determined the same as the ADC case.

Figure 50 Determination of DAC Quantization Error

The quantization noise spectrum is the spectrum of the quantization Error. For DAC

applications, usually the DAC clock and the modulation clock are derived from the same source,

so there is an integer relationship between them.

3.1 BandLimited MSK output Due to time constraints, the BandLimited MSK DAC was not simulated for this version. This section

is a placeholder for when there is more time.

3.2 BaseBand MSK output For the BaseBand DAC case, the model is shown in Figure 51; and consists of just an 8-bit

quantizer, since the data already comes in at the DAC sample rate of 550 MHz. Each quantizer

output is repeated 13 times, at 13 times the DAC sample rate, to simulate a continuous-time

signal. In order to complete this report in a reasonable time, only an MSK signal was simulated.

Figure 51 BaseBand DAC Model

Since the input to the DAC is controlled digital, only 0 dBpeakFS was simulated.

Digital

Signal In

Quantized

Signal Out

Unquantized

Signal Out Quantization

Error

DAC Model

17 August 2014

Signal

Generated

at DAC

sample rate

Repeat each

sample by 13

to simulate

Continous

Time

Ideal

Uniform

Quantizer

NE Bits

Analog

Signal

Out


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Figure 52 MSK Unquantized and Quantized. The "Molière" pattern is due to the graphics sample

Figure 53 Time zoom-in MSK Sampled, Unquantized and Sampled + Quantized


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Figure 54 Spectra of Unquantized and Quantized MSK

Figure 55 Spectrum of BaseBand MSK Quantization Error


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3.3 Discussion of Results For the BaseBand MSK DAC output, the Quantization Noise is not uniform, even when there is

no DAC overflow and the entire dynamic range is used. Usually, for DAC applications, there is

an integer relationship between the DAC clock rate and the MSK symbol rate. The MSK symbol

rate was 13 times the DAC clock rate.

4. Summary and Conclusions A large number of different cases was examined. The following conclusions can be obtained:

Quantization noise is “uniform enough” (Equation 1 approximately holds for any reasonable

filter that might follow the ADC, at any center frequency; where a “reasonable filter” is one that

is not too narrow) if all the following conditions hold:

1) There is absolutely no clipping (overflow).

2) The signal +28 dBrmsLSB or higher.

3) The data converter clock rate is not an integer multiple of the modulation rate.

Observation (1) is important when OFDM, which has a large Peak-to-Average ratio is used, and

it might be optimum to allow some level of clipping [14].

5. Abbreviations Used Abbreviation Meaning

ADC Analog-to-Digital Converter

dBrmsFS rms voltage of waveform referred to peak of Data Converter response

dBpeakFS peak voltage of waveform referred to peak of Data Converter response

DAC Digital-to-Analog Converter

FS Full Scale

I/Q Inphase & Quadrature

LSB Least Significant Bit

MSK Minimum-Shift Keying

OFDM Orthogonal Frequency Division Multiplexing

PAPR Peak-to-Average Power Ratio;

power in the real peak divided by the power in the real average

rms Root-Mean-Square


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6. References 6.1. Introduction and Motivation

[1] Brodsky, Wesley; Should I and Q Combining and Separation be done Digitally or

Analogly?; WesBrodsky Wireless Communications Document WBWC.01; 2014.

[2] Brodsky, Wesley; How Should Data Converters be Modeled for System

Simulations?; WesBrodsky Wireless Communications Document WBWC.02; 2014.

6.2. Quantization noise with or without clipping effects

6.2.1 ADC & DAC

[3] Maloberti, Franco; Data Converters; Springer Publishing; 2007

6.2.2 ADC Specific

[4] Lever, K. V.; Cattermol, K.W., "Quantising noise spectra," Electrical Engineers,

Proceedings of the Institution of , vol.121, no.9, pp.945,954, September 1974

Lever, K.V.; Cattermole, K.W., "Erratum: Quantising noise spectra," Electrical

Engineers, Proceedings of the Institution of, vol.122, no.3, pp.272, March 1975

[5] Gersho, A, "Principles of quantization," Circuits and Systems, IEEE Transactions on,

vol.25, no.7, pp.427, 436, Jul 1978

[6] Gersho, A, "Quantization," Communications Society Magazine, IEEE, vol.15, no.5,

pp.16, 16, September 1977

[7] Schuchman, L., "Dither Signals and Their Effect on Quantization Noise,"

Communication Technology, IEEE Transactions on, vol.12, no.4, pp.162, 165,

December 1964

[8] Walden, R.H., "Analog-to-Digital Converters and Associated IC Technologies,"

Compound Semiconductor Integrated Circuits Symposium, 2008. CSIC '08. IEEE, vol.,

no., pp.1, 2, 12-15 Oct. 2008

[9] Walden, R.H., "Performance trends for analog to digital converters,"

Communications Magazine, IEEE, vol.37, no.2, pp.96, 101, Feb 1999

[10] Walden, R.H., "Analog-to-digital converter technology comparison," Gallium

Arsenide Integrated Circuit (GaAs IC) Symposium, 1994, Technical Digest 1994., 16th

Annual , vol., no., pp.217,219, 16-19 Oct. 1994

[11] Walden, R.H., "Analog-to-digital converter survey and analysis," Selected Areas in

Communications, IEEE Journal on, vol.17, no.4, pp.539, 550, Apr 1999

[12] Morgan, D.R., "Finite limiting effects for a band-limited Gaussian random process

with applications to A/D conversion," Acoustics, Speech and Signal Processing, IEEE

Transactions on , vol.36, no.7, pp.1011,1016, Jul 1988

[13] Chow, P.E.-K., "Performance in waveform quantization," Communications, IEEE

Transactions on, vol.40, no.11, pp.1737, 1745, Nov 1992

[14] Dardari, D., "Exact analysis of joint clipping and quantization effects in high speed

WLAN receivers," Communications, 2003. ICC '03. IEEE International Conference on,

vol.5, no., pp.3487, 3492 vol.5, 11-15 May 2003


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[15] Gray, R.M., "Quantization noise spectra," Information Theory, IEEE Transactions

on, vol.36, no.6, pp.1220,1244, Nov 1990

[16] Echard, J.; Watt, M.L., "The quantization noise spectrum of a sinusoid in colored

noise," Signal Processing, IEEE Transactions on , vol.39, no.8, pp.1780,1787, Aug 1991

[17] He Jing; Li Gang; Xu Xibin; Yao Yan, "Estimation for the quantization noise

spectrum of linear digital filter," Communication Technology Proceedings, 2000. WCC -

ICCT 2000. International Conference on, vol.1, no., pp.184, 187 vol.1, 2000

[18] Bennett, W.R., "Spectra of quantized signals," Bell System Technical Journal, The,

vol.27, no.3, pp.446, 472, July 1948

[19] Mohamed, E.M., "Low complexity channel estimation technique for MIMO-

Constant Envelope Modulation," Wireless Technology and Applications (ISWTA), 2013

IEEE Symposium on, vol., no., pp.97, 102, 22-25 Sept. 2013

[20] Clavier, A G.; Panter, P. F.; Grieg, D.D., "Distortion in a Pulse Count Modulation

System," American Institute of Electrical Engineers, Transactions of the, vol.66, no.1,

pp.989, 1005, Jan. 1947

6.2.3 DAC Specific

[21] Ling, W.A, "Shaping Quantization Noise and Clipping Distortion in Direct-

Detection Discrete Multitone," Lightwave Technology, Journal of, vol.32, no.9, pp.1750,

1758, May1, 2014

6.3. Clipping effects only; ADC Only

[22] Mazo, J.E., "Asymptotic distortion spectrum of clipped, DC-biased, Gaussian noise

[optical communication]," Communications, IEEE Transactions on, vol.40, no.8,

pp.1339, 1344, Aug 1992

[23] Dakhli, M.C.; Zayani, R.; Bouallegue, R., "A theoretical characterization and

compensation of nonlinear distortion effects and performance analysis using polynomial

model in MIMO OFDM systems under Rayleigh fading channel," Computers and

Communications (ISCC), 2013 IEEE Symposium on , vol., no., pp.000583,000587, 7-10

July 2013

[24] Dardari, D.; Tralli, V.; Vaccari, A, "A theoretical characterization of nonlinear

distortion effects in OFDM systems," Communications, IEEE Transactions on, vol.48,

no.10, pp.1755, 1764, Oct 2000

[25] Giannetti, F.; Lottici, V.; Stupia, I, "Theoretical Characterization of Nonlinear

Distortion Noise in MC-CDMA Transmissions," Personal, Indoor and Mobile Radio

Communications, 2006 IEEE 17th International Symposium on , vol., no., pp.1,5, 11-14

Sept. 2006

[26] Van Vleck, J.H.; Middleton, D., "The spectrum of clipped noise," Proceedings of

the IEEE, vol.54, no.1, pp.2, 19, Jan. 1966


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6.4. Other relevant mathematical treatments

[27] Ermolova, N.Y.; Haggman, S.-G., "An extension of Bussgang's theory to complex-

valued signals," Signal Processing Symposium, 2004, NORSIG 2004. Proceedings of the

6th Nordic, vol., no., pp.45, 48, 11-11 June 2004

[28] Requicha, Aristides A G, "Expected Values of Functions of Quantized Random

Variables," Communications, IEEE Transactions on , vol.21, no.7, pp.850,854, Jul 1973

[29] Pirskanen, J.; Renfors, M., "Quantization and jitter requirements in multimode

mobile terminals," Communications, 2001. ICC 2001, IEEE International Conference

on, vol.4, no., pp.1182, 1186 vol.4, 2001

[30] Irons, Fred H.; Riley, K.J.; Hummels, D.M.; Friel, G.A, "The noise power ratio-

theory and ADC testing," Instrumentation and Measurement, IEEE Transactions on ,

vol.49, no.3, pp.659,665, Jun 2000

[31] Widrow, B., "A Study of Rough Amplitude Quantization by Means of Nyquist

Sampling Theory," Circuit Theory, IRE Transactions on, vol.3, no.4, pp.266, 276, Dec

1956

[32] Rowe, H.E., "Memoryless nonlinearities with Gaussian inputs: Elementary results,"

Bell System Technical Journal, The, vol.61, no.7, pp.1519, 1525, Sept. 1982

[33] VanTrees, Harry L; Detection, Estimation, and Modulation Theory, Part III,

Radar/Sonar Signal Processing and Gaussian Signals in Noise; John Wiley and Sons;

1971. Appendix: “Complex Representation of Bandpass Signals, Systems, and

Processes”


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Advertising

Section

247 High Street

Medford, MA 02155

781 866 9816

[email protected]

ADVERTISING SECTION

mailto:[email protected]


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WesBrodsky Wireless Communication provides consulting for

communications and RF systems; including the fields of analog and digital signal

processing, RF/microwave design, antennas, and propagation.

The principal of WesBrodsky Wireless Communication, Wesley G.

Brodsky, has 40 years experience. He received a Bachelor’s Degree from New

York University and a Master’s Degree from the Massachusetts Institute of

Technology, both in Electrical Engineering. He holds two patents and has

published several papers. Wes contributes to system simulations analysis,

specifications and proposals. He can provide the initial design of overall system

being modeled and system optimization via variation of system parameters. The

emphasis is on Microwave and Radio Frequency Hardware Systems Design, with

functional design, modeling, and simulation of Digital portions of systems to

evaluate interactions between analog and digital hardware.

*The IEEE Communications Society has declared Wes an “IEEE Wireless

Communication Professional®” (WCP). This certification is awarded to those who

have demonstrated abilities in RF, propagation and antennas; access technologies;

network and service architecture; network management and security; facilities’

infrastructure; agreements, standards, policies and regulations; and fundamental

knowledge.

(See http://www.ieee-wcet.org/specifications.html for more information on the

scope of WCP certification.)

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http://www.ieee-wcet.org/specifications.html


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A complete resume of Wesley Brodsky

follows:

Wesley G. Brodsky (WCP*)

247 High Street

Medford, MA 02155

781 866 9816

[email protected]

Present Position: WesBrodsky Wireless Communication; Consultant. Expert in

communications systems, millimeter-wave, microwave, and RF applications.

Communication systems experience in physical layer receiver/transmit algorithms and

modem development, noise and interference suppression, and advanced modulation

techniques. U.S. Citizen.

*WCP: Certified an “IEEE Wireless Communication Professional®” by the IEEE

Communications Society. This certification is awarded to those who have demonstrated abilities

in RF engineering, RF propagation, antennas; network access technologies; and network

architecture. Certification requires knowledge of the Physical, Link, and Network Layers.

Skill Overview:

Design: Experience with linear and mixed signal board-level design for

communications and radar, including design, simulation, and layout, mostly using ADS. The

designs used RF (amplifiers, filters, and mixers), linear baseband, ADC, DAC, and FPGAs. The

actual FPGA “programming” was done by others.

System Modeling and Simulation – Have used both ADS and MATLAB to

perform link budget analysis and performance trade off along signal chain. Performed analysis

and simulation (using ADS Ptolemy) of diversity for OFDM troposcatter system. Performed

analysis and simulation of Maximum-Ratio Combining of Phased Array Antennas for

communications link (including the model for the Phased Arrays) in MATLAB. Wes has done

modeling and simulation of digital and analog portions of communications and radar systems

using Agilent’s Advanced Design System (ADS) Ptolemy. The simulations included detailed

models of the RF portions of the systems, including effects of amplitude, group delay variation,

mixer and amplifier intermodulation products, nonlinear amplifiers and mixers, and A/D and

D/A Converters. Functional design, modeling, and simulation of Digital portions of systems to

evaluate interactions between analog and digital hardware; including the effects of finite bit-

precision and clocking.

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Test: Wes has developed test programs for mixed signal and digital. Included

generation of test vectors for testing VHDL models of digital signal processing. Familiar with

Microwave, Millimeter Wave, and analog test equipment such as signal generators, vector signal

sources, oscilloscopes, spectrum analyzers, vector signal analyzers, and vector network

analyzers. Have been the person who decides what test equipment to buy and arranges to buy it

on some projects.

Software: Wes has experience in MATLAB and SPICE (both of which Wes has

on my personal computers); and also with Agilent ADS.

MATLAB Toolboxes Wes owns and is experienced in are:

o RF Toolbox

o Communications System Toolbox

o DSP System Toolbox

o Fixed-Point Toolbox

o Optimization Toolbox

o Phased Array System Toolbox

o Signal Processing Toolbox

o Statistics Toolbox

Education:

o B.S. (Electrical Engineering); New York University;

o M.S. (Electrical Engineering); Massachusetts Institute of Technology where I

took graduate-level courses in Detection, Estimation, and Modulation; Probability and Random

Processes; Microwave Design; and Antennas. Thesis on design of 118 GHz mixer.

o Wes has recently taken the following in-person seminars, as well as many

webinars;

"Implementing Measurement and Analysis Techniques with MATLAB" given by

Tektronix and The Mathworks.

"Mathematical Modeling with MATLAB", and "Crest Factor Reduction Theory

and Design Practices for FPGAs"; and well as many MATLAB webinars; all given by The

Mathworks.

“Introduction to LabView”, given by National Instruments.

“Combine the Power of Digital and RF Test Tools” given by Agilent.

“Developing Measurement and Analysis Systems using MATLAB” given Agilent

and The Mathworks.

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Details of Experience:

Present Position: WesBrodsky Wireless Communication; Consultant. Expert in

communications systems, millimeter-wave, microwave, and RF applications.

Communication systems experience in physical layer receiver/transmit algorithms and

modem development, noise and interference suppression, and advanced modulation

techniques.

Prior Experience:

Raytheon Company; October 1983 to November 2010: last position was Engineering

Fellow, a position for the top 4% of engineers.

Design, analysis, and development of Civilian Communications Systems, Military

Communications Systems, and Air Traffic Control Radar.

Civilian Communications:

-Design, simulation, and analysis of Modulator and Demodulator for Wireless Local Area

Network plug-in PCMICA card for original, Frequency Hopped IEEE 802.11 link.

-A member of IEEE 802.11 Committee, which formulated the original Wi-Fi (802.11a

and 802.11b) Standards; 1998 to 1999.

-Design, simulation, and analysis of QPSK modulator for commercial Satellite

Communications ground station.

-Proposal for Maine Communications Network Proposal. Responsible for the

demonstration and explanation of requirements compliance. Evaluation of use of a

satellite link for a case where a terrestrial microwave link would not support the desired

bandwidth.

-System Design and Proposal for LA-RICS (Los Angeles Regional Interoperable

Communications System; a wireless voice and data communications system that will

support first responders and local mission-critical personnel.) Participated in review of

potential vendors and visited their facilities. Suggested method of Intermodulation

Analysis for sites.

Military Communications:

-Overview, use of ADS: System and high-level Hardware Design of modulators and

demodulators. Modeling and simulation of digital and analog portions using Agilent’s

Advanced Design System (ADS) Ptolemy. The simulations included modeling of

Microwave front-end components (Low-Noise Amplifiers, Mixers, Non-Linear Power

Amplifiers, Filters, VCOs, and Synthesizers). Bit Error Rates were evaluated for several

of the applications. Models were from several MHz to 45 GHz.

-Overview; use of MATLAB for system modeling and simulation:

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-Tracking using pseudo-monopulse receiver system for satellite ground station, including

modeling of a microwave directional coupler.

-Tracking of moving target for Data Link, using a Phased Array Antenna.

- Performance of Ionospheric Gradient Monitor for aircraft approach and landing system

using single-frequency GPS

- Analyzed Probability-of-Intercept for satellite communications.

M.I.T Lincoln Laboratory: August 1977 to October 1983: Design of RF and analog

hardware for adaptive antenna and Identification Friend-or-Foe Systems. Analysis of

finite-dynamic range effects on Spread-Spectrum Communications Systems.

Varian Associates: August 1976 to August 1977: Design and test of high-power

microwave control devices.

U.S. Air Force Cambridge Research Laboratory: June 1974 to August 1976:

Measurement and analysis of millimeter-wave propagation through the troposphere.

Design of millimeter-wave radiometers.

Patents:

-Curtis; Robert G., Brodsky; Wesley G., U.S. Patent 5,963,599, “Truncated maximum

likelihood sequence estimator,” October 5, 1999.

-Washakowski; Steven, Brodsky; Wesley G., U.S. Patent 7,346,125, “Method and device

for pulse shaping QPSK signals,” March 18, 2008.

Papers published openly:

-Bickford, W. J.; Brodsky, W. G.; RF Subsystem Design for Microwave Communication

Receivers. IEEE MILCOM 1987.

-Brodsky, W.; Book Review: Digital communications and spread spectrum systems,

Communications Magazine, IEEE, Volume: 25, Issue: 9, 1987, Page(s): 62 – 63.

-Horowitz, L.L.; Blatt, H.; Brodsky, W.G.; Senne, K.D.; Controlling Adaptive Antenna

Arrays with the Sample Matrix Inversion Algorithm, Aerospace and Electronic Systems,

IEEE Transactions on, Volume: AES-1, Issue: 6, 1979, Page(s): 840 – 848.

Copyright ©2014 WesBrodsky Wireless Communication. All rights reserved. To request a copy of this document; eMail: [email protected] specifically requesting a copy.


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