EX. 2013 - IPR2021-00375
`Qualcomm Incorporated v. UNM Rainforest Innovations
`
`

`

`1.
`
`I have been retained by DiMuro Ginsberg, P.C., as an independent
`
`technical expert in the Inter Partes Review between UNM Rainforest Innovations
`
`(“UNM”), and Qualcomm Incorporated (“Qualcomm”), PTAB Case No. IPR2021-
`
`00375 involving U.S. Patent No. 8,265,096 (“the ’096 patent”).
`
`2.
`
`I am being compensated for my work as a technical expert at my customary
`
`rate of $675 per hour. My compensation does not in any way depend on the outcome
`
`of this review, and I have no personal interest in the outcome of this review.
`
`I.
`
`QUALIFICATIONS
`
`3.
`
`I am an expert in wireless technology and other telecommunications areas.
`
`I am an Emeritus Professor of Engineering and Applied Science at The George
`
`Washington University, where I have been a member of the faculty since September
`
`1991. In addition, I have served as a consultant for a number of companies in the
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`wireless communications industry in various technology areas. I have also served
`
`on numerous committees and as a reviewer and editor for several journals,
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`conferences, and organizations.
`
`4.
`
`I received my Diploma of Engineering, Master of Science, and Doctor of
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`Science degrees in Electrical Engineering from the University of Belgrade in
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`Yugoslavia in 1981, 1986, and 1989, respectively. The primary focus of my Doctor
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`of Science studies was on Code Division Multiple Access (CDMA) and spread
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`spectrum communications technologies.
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`5.
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`In 1991, I joined The George Washington University as an Assistant
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`Professor and was promoted to Associate Professor and Professor in 1997 and 2000,
`
`respectively. From 2001 to 2004, I served as Chairman of the Electrical and
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`Computer Engineering Department at The George Washington University and
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`retired in 2015. During my tenure at The George Washington University, I have
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`taught many different courses on communications theory and networks, wireless
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`communications, CDMA, and I have been a course director for a number of courses
`
`in communications. I have supervised students mostly in the areas of wireless
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`communications
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`and
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`CDMA
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`(including
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`IS-95,
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`CDMA2000,
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`WCDMA/HSDPA/HSUPA) and OFDM/LTE and have been a thesis director for a
`
`number of Doctor of Science candidates, who now have successful careers in
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`academia, industry, and government.
`
`6. My research in the areas I just mentioned has been supported by the
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`communications industry and various government agencies, such as the Advanced
`
`Research Project Agency (ARPA), the National Science Foundation (NSF), and the
`
`National Security Agency (NSA). Much of this research concerns communications
`
`theory, performance evaluation, modeling of wireless networks, multi-user
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`detection, adaptive antenna arrays, and ad-hoc networks.
`
`7.
`
`I have authored and co-authored a number of journal and conference
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`papers, contributed to various books, and co-authored a textbook on CDMA, entitled
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`

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`The CDMA2000 System for Mobile Communications, Prentice Hall, 2004. I have
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`also served as a co-editor of a book on wireless communications, entitled
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`Multiaccess, Mobility, and Teletraffic in Wireless Communications, Volume III,
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`Kluwer Academic Publishers, Norwell, Massachusetts, 1998. My CV includes a
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`detailed listing of my publications. See Attachment A.
`
`8.
`
`I have also received awards for my work. For example, in 1995, I received
`
`the prestigious National Science Foundation Faculty Early CAREER Development
`
`Award. The award is given annually by the NSF to a select number of young
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`professors nationwide to promote excellence in teaching and research.
`
`9.
`
`I have served as a consultant for numerous companies in the wireless
`
`communications industry in technology areas, including the areas of 2G/3G/4G
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`mobile technologies, Wireless LANs, new generation broadcast systems, advanced
`
`mobile satellite systems and other aspects of modern communication systems. I
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`have also taught academic courses as well as short courses for the industry and
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`government on various aspects of communications in the areas of 2G, 2.5G, 3G, and
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`4G cellular standards, such as CDMA2000 1xRTT, CDMA2000 Evolution Data
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`Optimized (EVDO), Wideband Code Division Multiple Access (WCDMA), and
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`LTE.
`
`10.
`
`I am a Senior Member of the IEEE and was an Associate Editor for IEEE
`
`Communications Letters and the Journal on Communications and Networks. I have
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`served as a member of technical program committees, as a session organizer for
`
`many technical conferences and workshops, and as a reviewer of technical papers
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`for many journals and conferences.
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`11.
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`I am a co-inventor on about 20 patents; some representative examples
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`include: U.S. Patent No. 6,523,147, entitled “Method and Apparatus for Forward
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`Error Correction Coding for an AM In-Band On-Channel Digital Audio
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`Broadcasting System,” U.S. Patent No. 8,595,590 B1, entitled “Systems and
`
`Methods for Encoding and Decoding Check-Irregular Non-Systematic IRA Codes,”
`
`and applications including “Joint Source-Channel Decoding with Source Sequence
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`Augmentation,” U.S. 20140153654 A1, June 5, 2014, “Systems and Methods for
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`Advanced Iterative Decoding and Channel Estimation of Concatenated Coding
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`Systems,” U.S. 20140153625 A1, June 5, 2014, “Advanced Decoding of
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`High/Medium/Low Density Parity Check Codes,” PCT/US13/72883, and
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`International Application Number PCT/CA01/01488, entitled “Multi-User Detector
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`For Direct Sequence - Code Division Multiple Access (DS/CDMA) Channels.”
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`12. Over the last several years, I have evaluated many, on the order of
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`hundreds, of patents that are essential or potentially essential to wireless standards
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`for various clients. These evaluations typically include, for example, analyzing
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`whether the patent claims read on the relevant standard, considering the importance
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`of the technological inventions claimed, analyzing how such claimed inventions
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`compare to other similar patents in the field, searching for and reviewing potential
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`prior art, reviewing and analyzing the prosecution histories of patents relevant to
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`potential claim construction, infringement, or related issues, reviewing and
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`analyzing the working group documents related to the relevant standard in relation
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`to the claimed invention, and considering whether there are available alternatives to
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`the claimed inventions.
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`13. A copy of my CV is attached as Attachment A.
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`II. BASES OF OPINIONS
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`14. The basis and reasoning of my opinions include my education, training,
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`and experience as an engineer, including my decades of experience in
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`telecommunications. In the course of conducting my analysis and forming my
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`opinions, I have considered the materials listed below:
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`a. U.S. Patent No. 8,265,096 to Yan-Xiu Zheng, et al., filed July 7, 2008
`
`and issued on Sep. 11, 2012 and its file history;
`
`b. Qualcomm’s Petition for Inter Partes Review, PTAB Case No.
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`IPR2021-00375, and all its relevant technical exhibits;
`
`c. Declaration of Dr. Sumit Roy (“Roy”) supporting Qualcomm’s
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`petition;
`
`d. U.S. Pub. No. 2009/0067377 A1 (“Talukdar”);
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`e. U.S. Pub. No. 2007/0155387 A1 (“Li”);
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`f. U.S. Pub. No. 2007/0104174 A1 (“Nystrom”);
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`g. Certain materials from STC.UNM v. Apple Inc., No. 1-20-cv-00351
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`(W.D. Tex. Apr. 9, 2020), including the Claim Construction Order,
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`ECF No. 69;
`
`h. and any other materials referenced herein.
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`15. My opinions in this declaration are based on the understanding of a person
`
`of ordinary skill in the art at the time of the invention of the claims in the ’096 patent.
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`16.
`
`In assessing the level of skill of a person of ordinary skill in the art, I have
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`considered the type of problems encountered in the art, the prior solutions to those
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`problems found in the prior art references, the rapidity with which innovations are
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`made, the sophistication of the technology, the level of education of active workers
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`in the field, and my own experience working with those of skill in the art at the time
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`of the invention.
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`17. A person of ordinary skill in the art (“POSITA”) at the time of the filing
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`of the ’096 patent would typically have at least a MS Degree in Computer
`
`Engineering or Electrical Engineering, or equivalent work experience, along with at
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`least 1 year of experience related specifically to wireless communications, including
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`MIMO and OFDM.
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`18.
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`I am very familiar with this level of skill. In the course of my decades of
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`experience, I have supervised and worked with engineers in this field having at least
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`the level of skill identified above.
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`III. TECHNICAL ANALYSIS
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`19. A POSITA as of July 2007 would have known that TSYM = TGI + TDFT
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`= N/Fs + K/Fs, where TDFT is the IDFT/DFT period, TGI is the length of the cyclic
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`prefix (also called guard interval), N is the number of carriers.
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`20. OFDM dates back as a concept from the 1970s. At that time analog
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`implementations were envisioned. The first paper suggesting the discrete Fourier
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`transform for OFDM was published in 1971. Ex. 1 (Weinstein and Ebert, Data
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`Transmission by Frequency-Division Multiplexing Using the Discrete Fourier
`
`Transform, IEEE Transactions On Communication Technology, Vol. Com-19, No.
`
`5, October 1971) (“Weinstein”).
`
`21. According to the 1971 paper, the time variable t is between 0 and N Δt:
`
`See Weinstein at 2.
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`Id.
`22. Therefore, a POSITA in 1971 would have known that the symbol period
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`is the product of the sampling period Δt, and the number of samples N, TDFT = N
`
`
`
`Δt.
`
`23. This is confirmed from a notable 1985 paper:
`
`
`
`See Ex. 2 (Leonard J. Cimini, Analysis and Simulation of a Digital Mobile
`Channel Using Orthogonal Frequency Division Multiplexing, 33 IEEE
`Transactions on Communications 665 (Jul. 1985)) (“Cimini”) at pg. 2.
`In other words, this 1985 paper also shows that “the signaling interval”, in
`24.
`
`other words the symbol period, is TDFT = N Δt.
`
`25. The Cimini paper also establishes the well-known result that Δt =1/Fs.
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`26. Further, the 1985 paper also shows a modulo extension (which present-day
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`systems refer to as a cyclic prefix or guard interval). At the bottom of Fig. 8 of
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`Cimini, the length of a block is “now N+l long”. In the Cimini system, l is the length
`
`of the guard interval, which in our notation is K.
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`27. Consequently, a block is now N+K long. Including this extension, the
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`length of a block is TSYM = TGI + TDFT.
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`Cimini at pg. 8.
`28. Another prior art system is the 1999 standard IEEE 802.11a.
`
`
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`
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`See IEEE 802.11a at p. 9.
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`29.
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`IEEE 802.11 specifies that the number of carriers N is 64, N=64. These
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`carriers are shown in the figure below, labeled from 0 to 63.
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`
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`See IEEE 802.11a-1999 at page 12.
`30. A POSITA in 1999 would have known the 1971 and 1985 papers and
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`would have been able to understand Table 79 of the standard, shown above.
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`According to Table 79 the IFFT/FFT period is 3.2 µs. Therefore TDFT = 3.2 µs =
`
`64 Δt = 64 /Fs.
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`31. The conclusions are that Δt = 50 ns and Fs = 1/ Δt = 20 MHz.
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`32. According to Table 79 above, the guard interval is TDFT /4. Therefore,
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`the guard interval in samples is K = 64/4 = 16 samples. The duration of the guard
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`interval is 800 ns.
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`33. A POSITA would have been able to calculate the symbol interval as
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`TSYM = TGI + TDFT = 4 µs. Table 79 itself contains an entry for the symbol period
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`as 4 µs.
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`34.
`
`In conclusion, a POSITA as of at least 1999 would have been able to
`
`calculate the symbol period of an OFDM system as TSYM = TGI + TDFT = N/Fs +
`
`K/Fs = (N+K)/Fs.
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`35. Further, A POSITA would understand that OFDM systems are sensitive to
`
`frequency errors and Doppler shifts and that Intercarrier interference in OFDM
`
`increased with Doppler shift. Thus, in a system with higher mobility intercarrier
`
`spacing should be increased, or equivalently, OFDM symbol duration should be
`
`decreased. A POSITA would have known this in 2007.
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`36. Specifically, as shown in Figure 5 (shown below) in Ex. 4 (Armstrong,
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`Grant, and Povey, Polynomial Cancellation Coding Of OFDM To Reduce
`
`Intercarrier Interference Due To Doppler Spread), signal to inter-carrier
`
`interference (ICI) ratio rapidly decreases, adversely impacting the performance) as
`
`the Doppler shift due to higher mobility increases. T is equal to 1/carrier
`
`separation, i.e. 1/Fs, thus as the Doppler shift fd increases, the normalized Doppler
`
`shift fdT, for a fixed T, increases and, thus, signal to ICI ratio decrease. Thus, to
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`mitigate adverse Doppler effects at higher mobility, symbol duration T must be
`
`decreased which also results in a larger subcarrier frequency spacing Fs, i.e.
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`

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`decrease fdT, making the system more tolerant to Doppler. In some very high-
`
`speed scenarios, it might also be needed to implement very complex ICI
`
`cancelation schemes in addition, as in Ex. 4.
`
`
`37. As explained above, a POSITA would understand that the symbol duration
`
`in a high mobility 802.1m system needs to be shorter than in the legacy system
`
`802.1e, e.g., L times, or, equivalently, the inter-carrier spacing needs to be larger L
`
`times. This is confirmed in slide 3/9 (EX2002 (’096 Provisional) at 3) where it is
`
`stated that subcarriers bandwidth (i.e., spacing) in a legacy system 16.e is B, while
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`in 16.m system it is B*L, i.e., L times larger. Therefore, a POSITA would
`
`understand that the number of subcarriers N, and therefore the number of samples in
`
`the cyclic prefix, K, in both systems are the same in the provisional disclosure,
`
`taking into account the arrangement in the example L=3 in the provisional
`
`application at 3/9. Thus, it also follows that Ts = (N+K)/3B is 3 times shorter than
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`TsL = (NL+KL)/B. However, this example in the provisional should not be read as
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`limiting, as a POSITA would understand that there are other possible arrangements
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`such that Ts is shorter than TsL while the number of subcarriers is not necessarily the
`
`same.
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`IV. CONCLUSION
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`38. My opinions are subject to change or revision. I may acquire additional
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`opinions regarding arguments Petitioner or its expert may present, information I may
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`receive in the future, or additional work I may perform. With this in mind, based on
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`the analysis I have conducted and for the reasons set forth herein, I have
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`preliminarily reached the conclusions and opinions in this declaration.
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`39.
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`In the event of a hearing or trial in this matter, I may create and/or use
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`various exhibits for the purposes of demonstrating my testimony as discussed in this
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`declaration. I have not yet created, directed the creation of, or selected the particular
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`demonstrative exhibits for this purpose.
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`EXHIBIT 1
`EXHIBIT 1
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`628
`
`Data Transmission by Frequency-Division Multiplexing
`Using the Discrete Fourier Transform
`
`
`
`
`
` IEEE TRANSACTIONS
`
`O N COMMUNICATION TECHNOLOGY, VOL. COM-19, NO. 5 , OCTOBER 1971
`
`S. B. WEINSTEIN, MEMBER, IEEE, AND PAUL M. EBERT, MEMBER, IEEE
`
`t l \ A r\ A * r
`
`W W W W U : as COS(IOllt/T)
`T
`
`+I 1
`I
`
`. a 0
`
`T
`
`(C)
`Fig. 1. Comparison of waveforms in serial and parallel dat.a
`transmission systems. (a) Serial stream of six binary digits.
`(b) Typical appearance
`transmission.
`of baseband serial
`to
`(c) Typical appearance
`of waveforms that are summed
`create para,llel data signals.
`
`commtdcation system
`lourief transform data
`AZisfracf-The
`i6 a realization of freqbency-division multiplexing (FDM) in which
`discrete Fourier transforms are computed as part of the modulation
`and demodulation processes. In addition to eliminating the banks
`of subcarrier oscillators and coherefit demodulators usually required
`in FDM systems, a completely digital implementation can be built
`around a special-purpose computer, performing the
`fast Fourier
`transform. In this paper, the system is described and the effects of
`linear channel distoriion
`are investigated. Signal design criteria
`and equalization algorithms are derived and explained. A differential
`phase modulation scheme is presented that obviates any equaliza-
`tion.
`
`I. INTRODUCTION
`
`ATA ARE usually sent as a serial pulse train, but
`there has
`long been
`interest in frequency-divi-
`sion mult.iplexing with
`overlapping subchannels
`as a means of avoiding equalization, combating impulsive
`noise, and making fuller use of the available bandwidth.
`These “parallel da.ta” systems, in which emh member of
`a sequence of N digits modulates a subcarrier, have been
`studied in [2] and [4]. Multitone systems are widely used
`and have proved to be effective in [3], [SI, and [9]. Fig.
`1 compares the transn~issions of a serial and a parallel
`system.
`For a large number
`of channels, the arrays, of sinu-
`soidal generators and coherent demodulators required in
`parallel systems become unreasonably expensive and com-
`plex. However, it can be shown [l] that a multitone data
`signal is effect,ively the Fourier trans.form of the origina.1
`serial data train, and that the bank of coherent deniodu-
`lators is effectively an inverse Fourier trans’form genera-
`tor. This point
`of view suggests
`a completely digita’l
`modem built around a special-purpose computer per-
`forming the
`fast Fourier transform (FFT). Fourier
`transform techniques, although not necessarily the signal
`format described
`in this, paper, have been incorporated
`into several military d a h communication systems
`[5]-
`[71.
`a small fraction
`Because each subchannel covers only
`the original bandwidth, equalizat’ion is potentially
`of
`for very
`simpler than for a serial system. In particular,
`narrow subchannels, soundings made at the centers of the
`
`subchanneis may be used in simple transformations
`the receiver output data .to produce, excellent estimates
`of the original data. Further, a simple equalization algo-
`rithm will minimize mean-square distortion on each sub-
`channel, and differential encoding
`of t’he original data
`may make it
`possible to avoid equalization altogether.
`MULTIPLEXING AS A DISCRETE
`11. FREQUENCY-DIVISION
`TRANSFORMATION
`‘data sequence (d,,, d l , . . . , d,-
`Consider a
`each d, is a complex number.d,, = a,, + jb,.
`If a discrete Fourier transform
`(DFT) is performed
`a vector S =
`{2dn}n,0N-1, the result is
`on the vector
`(S,,, S1, . . . , S,,- 1) of iV complex numbers, with
`2dne-i(2*nm/N) - - 2 dne-j2=fntm,
`N-1
`8, =
`m = 0 , 1, . . . , N - 1,
`
`of
`
`where
`
`(1)
`
`N - 1
`
`n = O
`
`n-0
`
`of
`Pa.per approved by the Data Communications Committee
`Technology Group for publication
`the IEEE Communication
`after presentation a.t the 19871 IEEE International Conference on
`Communications, Montreal,
`Que., Canada, June
`14-16. Manu-
`script received January 15, 1971 ; revised March 29, 1971.
`The authors are with the Advanced Data Communicat.ions
`N. J .
`Department, Bell Telephone Laboratories, Holmdel,
`
`where
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on December 05,2021 at 23:26:52 UTC from IEEE Xplore. Restrictions apply.
`
`

`

`AN5
`
`
`
`WEINSTEIN
`
`EBERT: TRANSMISSION BY FREQUENCY-DIVISION MULTIPLEXING
`
`- :(REAL PART
`
`DFT
`
`-: ONLY)
`
`SOURCE
`-
`dn'an+i&.
`n=O,I, ..., N-l
`
`629
`
`.
`
`LOW-
`PASS
`FILTER
`
`y ( t )
`. :
`
`(3)
`
`Y , = 2
`
`N - 1
`
`,,=O
`
`'
`A
`
`t, = m At
`and A t is an arbitrarily chosen interval. The real part
`of the vector X has components
`(a, cos 2nj,t, + 6, sin 2nfntm),
`, N - 1.
`m = 0, 1,
`(4)
`If these components are apjdied to
`a low-pass filter a t
`is obtained that closely ap-
`time intervals, At, a signal
`proximates the frequency-division multiplexed signal
`
`y(t) = 2
`
`N - 1
`
`% = n
`
`(a, cos 2 ~ f , t + b, sin 2nf,t),
`0 _< t _< N At.
`(Fi)
`A hlock diagram of the communication system in which
`y ( t ) is the trammitted signal appears in Fig. 2.
`Demodulation at. t.he receiver is carried out via a dis-
`crete Fourier transformatid
`of a vector of samples of
`the received signal. Because only the
`rea.1 part of the
`Fourier transform has been transmitted, it is necessary to
`sample twice as fast as expected, i.e., a t intervals A t / 2 .
`DFT
`When there is no channel distortion, the receiver
`operates on the 2N samples
`(u,, cos 2~ + b,sin __
`2 m k
`2N
`K = 0, 1, . * . , . 2 N - 1,
`(6)
`where definitions (2) a n i (3) have been substituted into
`(4). The DFT yields
`
`Y , = y(k 9) = 2
`
`+
`
`2N--1
`
`=1
`
`J2%,
`a, - jb,,
`irrelevant,
`where the equality
`
`l = O
`1 = I, 2, . . * , N - 1
`1 > N - 1,
`
`(7)
`
`a[ and b, are
`has been employed. The original data
`available (except for
`1 = 0 ) as the real and imaginary
`conlponents, respectively, of z l , as indicated in Fig. 2. A
`synchronizing signal is reciuired, but one or several chan-
`nels of the transnlitted signal can readily
`be utilized for
`this purpose.
`Because the sinusoidal components of the parallel data
`signal y ( t ) are truncated
`in time, the power density
`spectrum of y ( t ) consis'ts of [sin (f) / f ] "shaped spectra,
`as sketched in Fig. 3. Nevertheless, the data on the dif-
`ferent subchannels can
`be completely separated by the
`DFT operation of (7). This will not be exactly true
`when linear channel distortion affects the received
`sig-
`nal, hut it will be shown later that a modest reduction in
`transmission rate eliminates most. interferences.
`
`DISTORTIONLESS CHANNEL
`
`Fig. 2. Fourier transform communication system
`channel distortion.
`
`in absence of
`
`' n-2
`
`!WI
`
`'n
`
`fn+ I
`
`'n+2
`
`Fig. 3. Power dtnsity spectra of subchannel components of y(t)
`
`BY USE OF CHANNEL SOUNDINGS
`111. EQUALIZATION
`Except for the added linear channel distortion and
`final equalizer, the Fourier transform data communica-
`tion systeni shown in Fig. 4 is identical to that of Fig. 2.
`Ideally, the discrete Fourier transformation in the re-
`ceiver should be replaced by another linear transforma-
`1111, which minimizes the error in the
`tion, derived in
`receiver output. However, it is preferable,
`if possible, to
`retain the DFT with its "fast" implementations and
`carry out suboptimal but adequate correctional transfor-
`mations at th2 receiver out<put. The system
`of Fig. 4
`performs this approximate equalization.
`Consider the waveform a t t h e receiver input,
`
`(9)
`r(t) = Y ( t ) * W ,
`deliotcs convolution. This waveform
`where the asterisk
`is a collection of truncated sinusoids modified by a. linear
`filter. If the sinusoid cos 2Tf,,t were not truncated, then
`the result of passing it through a channel with transfer
`H , cos(2~f,t + +,,) , where
`function H ( f ) would be
`
`y ( t ) [see (5) 3
`The sinusoids in the transmitted signal
`are truncated to the interval
`(0, N A t ) , so that the nth
`a [sin N T ( ~ - f n ) A t ] /
`subchannel must accommodate
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on December 05,2021 at 23:26:52 UTC from IEEE Xplore. Restrictions apply.
`
`

`

`630
`
`SOURCE
`dn*an+ibn.
`
`n=O,l, ..., N-I
`
`OFT
`
`:(REAL PART
`:ONLY1
`
`- L o w -
`e
`PASS
`FILTER
`
`Y(t)
`:
`
`I
`
`'
`
`. ", .. -
`i
`SAMPLE
`.
`AT
`INTERVALS-
`kAt/2
`communicati.on system
`Fig. 4. Fourier trmsform
`linear channel distortion and final equalization.
`
`-
`
`DFT
`
`*
`EOUALIZATION
`TRANSFORMATION
`
`A
`
`f
`7 bn
`including
`
`[hTr(f - fn)At] spectrum instead of the impulse at
`fn,
`which would correspond
`to a pure sinusoid. However, if
`l/(NAt) is small conlpared with the total transmission
`bandwidth, then H ( f ) does not change significantly over
`express,ion for the
`the subchannel and an approximate
`received signal r ( t ) is
`
`r(t)
`
`2
`
`N-1
`
`n = 1
`
`N- 1
`
`= 2
`
`Hn[an COS (27rfat + 4 n )
`+ b, sin (W,t + 4Jl + 2H0an
`H,[(a, cos 4, + b, sin &) cos 27rf,t
`n= 1 + ( b , cos 4" - a, sin +J sin 27rfnt]
`+ 2H0ao,
`0 5 t 5 N A,?.
`(11)
`As indicated in Fig. 4, r ( t ) is sampled at times k: (At/2) ,
`k = 0, 1, . .
`2N - 1, and the samples { r k } are applied
`to a discrete Fourier transformer. The output
`of
`the
`DFT is
`
`e ,
`
`
`
`El
`
`2N-1
`
`.o
`
`2Hoao,
`
`1 =.o
`H,[at cos 4, + b, sin - H , [ b , cos 4,
`- a, sin +,I,
`1 = 1, 2, .. . , N - 1
`1 > N - 1 .
`irreievant,
`Estimates of nl and b, are obt,ained from the computa-
`ions
`
`(12)
`
`IEEE TRANSACTIONS O N COMMUNICATION TECHNOLOGY, OCTOBER 1971
`2 X 2 transformation
`Equations (13a-c) describe a
`to be performed on each of the DFT outputs z I , I = 1, 2,
`*, N - 1. For a reasonably large
`N and a typical
`of H ( f ) by
`communication channel, the approximation
`a constant over each subchannel, which leads-to (13a-c) ,
`may be adequate. However, linear rather than constant
`approximations to the amplitude and phase of the chan-
`nel transfer function as it affects each subchannel wave-
`form are much closer to reality. The following section
`examines the consequences of these approximations. It
`is shown that the truncated subchannel sinusoids are de-
`layed by differing amounts, and that distortion is
`con-
`centra-ted at the on-off
`transitions of these waveforms.
`Further, the magnitude of the distortion is, proportional
`to the abruptness
`of the transitions. Hence
`a "guard
`space," consisting of a modest increase in the signai dur-
`ation together with a smoothing of the on-off transitions,
`will eliminate most interference among channels and
`be-
`tween adjacent transmission blocks. The individual
`channels can then hc equalized in accord with (13a-c).
`IV. APPROXIMATE ANALYSIS
`OF THE EFFECTS OF
`CHANNEL DISTORTION
`y ( t ) as given by (5) exists
`The transmitted signal
`only on the interval (0, NAt), so that each suhchannel
`must, as noted earlier, accommodate a sin f/f type spec-
`trum. As suggested in t,he last section, let this spectrum
`he narrowed by increasing the signal duration
`t o some
`T > N A t and requiring gradual rather than abrupt roll-
`offs of the transmitted waveform. S'pecifically, the trans-
`mitted signal will be redefined :IS
`
`o l t < T
`
`6, = - [Re (2,) cos 4, + Im (z,) sin + 1 ]
`
`1
`H ,
`
`, N - 1.
`1 = 1, 2, e
`Tn complex notation, the appropriate computation is
`&, - j6, = w,z,,
`
`.
`
`.
`
`
`
`10 elsewhere.
`
`The "window function" g, ( t ) is. sketched in Fig. 5.
`When y ( t ) is pa.ssed through.the channel filter with
`impulse response h ( t ) , the received signal is
`
`(15)
`
`(13a)
`
`(13b)
`
`where
`
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`
`

`

`WEINSTEIN AND EBERT: TRANSMISSION
`
`BY FREQUENCY-DIVISION MULTIPLEXING
`
`63 1
`
`--io1
`
`I
`
`i
`
`TiZOT
`
`Fig. 5 . Window g.(t) multiplying all subchannel sinusoids in
`transmitted signal.
`
`where
`
`p ( t ) = 2h(t) * g&) cos 2af,t
`qb'n)(t) = 2h(t) * g,(t) sin 2nfnt.
`(17)
`In Appendix I, linear approximations to t.he amplitude
`and phase of H ( f ) around f = *f,, result in the follow-
`ing approximate expression for qa(n) ( t ) .
`qn(n)(t) E 2Hn COS Y27rfnt + +.]ga(t - PJ
`
`where (HN, is the channel sounding
`a t frequency f n ,
`and LY, and Pn are the slopes at
`f = fn of the linear
`of H ( f ) , as
`approximations to amplitude and phase
`shown in Fig. 6. A similar expression resulte for q b ( n ) ( t ) .
`of (18) is the 7zth
`The first term on the right-hand side
`cosine element in the transmitted signal
`(14), except
`( H a , &) and
`that it is modified by a channel sounding
`Pa. Interblock interfer-
`subjected to an envelope delay
`ence can result
`if a delayed sinusoid from
`a previous
`block impinges on the current sampling period. The sec-
`ond term is distortion arising from the amplitude varia-
`tions of H (f) , and it is a potential source of interchannel
`T is large enough
`interference. Suppose, however, that
`so that
`
`T > (2N - 1) y + max (PJ - min ( A ) ,
`
`At
`
`n
`n
`7. Then for all n there exists a t,ime
`as pictured in Fig.
`to > max /3, such that
`
`(19)
`
`Fig. 6. Linear approximations
`
`in relation to spectra G,,(f - fn) and Ga(f + 1.n).
`
`to amplitude and phase of H ( f )
`
`I
`
`r Z N SAMPLES-
`
`Fig. 7 . Shifted versions
`of g,(t) corresponding to subchannels
`wit.h minimum and maximum delay and locations
`of samples
`t,,,, = min, fin, L a x = rnax,%
`taken by receiver. Here
`6".
`
`N - 1
`
`r(t) E 2
`
`Hn[a, cos (2nfnt + 4,J + bn sin (27rfnt + 4JI
`n = 1 + 2Hoao,
`t o 5 t 5 t o + (2N - 1) 5. (23)
`
`At
`
`of definition, ( 2 3 ) is
`Except for the shifted domain
`identical t,o ( l l ) , which led to ( 1 3 ) for retrieval of the
`data. It can be shown
`that initiating sampling of r ( t )
`a t t = t,) instead of a t t = 0 is equivalent t.o incrementing
`each phase +[ by f l t,, rad in the equalization equations
`( 1 3 ) .
`Intuitively, r ( t ) reduces to ( 2 3 ) because linear dis-
`tortion delays different spectral components by different
`anlounts and responds to abrupt transitions with "ring-
`ing." If the subchannel waveforms, are examined during
`an interval when they are all present, and
`if all the on-
`off transitions lie well
`outside this interval, then the
`waveforms will look like the collection
`of sinusoids de-
`scribed by (5). The linear approximations to amplitude
`and phase of H ( f ) restrict the ringing from the transi-
`tion periods
`t,o those periods themselves.
`In practice,
`the ringing can be expected to die out very rapidly out-
`side of the transition periods. This use o f a guard space
`is a common technique, as, for example, described in
`V
`I
`.
`
`Ti. ALGORITHM FOR MIXIMIZING MEAN-SQUARE
`DISTORTION
`of the
`Under the assumption supported by the results
`last section that interchannel interference is negligible, a
`simple algorithm can be devised for determination of the
`
`t o 5 t I (2N - 1) + t o .
`At
`d
`- g,(t - pn) = 0,
`(20)
`dt
`(to, to + ( 2 N - 1) ( A t ]
`Fig. 7 shows where the interval
`2 ) ) is located with respect to the minimum and maxi-
`mum values of the time shift &. Thus,
`q.'")(t) "= 2H, cos (27rf"t + +"),
`
`By a similar derivation,
`
`~ ~ ( " ' ( 2 )
`
`2H, sin (2af,t + 4,J,
`t o 5 t i t o + -2--
`( 2 N - 1) At .
`
`(22)
`
`Therefore, substituting
`for n = O),
`
`( 2 1 ) and ( 2 2 ) into (16) (except
`
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`
`

`

`632
`parameters cos +!/HI and sin + J H , on each channel,
`which minimize mean-square distortion when
`used in
`the transformation of (13a-c). Because of propagation
`delay and the' presence of noise, these parameters may
`a sounding of the Ith sub-
`not exactly correspond to
`carrier channel. The w e of an automatic equalization
`procedure based on the minimum mean-square distortion
`algorithm leads to accurate demodulation without hav-
`ing to make
`precise channel soundings. Further, the
`algorithm can work adaptively after an initial coarse
`adjustment.
`The receiver produces estimates d l and 6, according to
`the formulas
`d , = TI, R e 2, + T,, Im 2,
`6, = T,, R e 2, - T,, I m Z , ,
`(24)
`that the T coefficients
`which resemble (13a-c), except
`are to
`be chosen
`to nlinilnize the estimation err0r.l
`Mean-square distortion is defined by
`
`= E
`
`N - 1
`
`[(TI,. R e 2, + T,, . I m Z , - a,)'
`2=1 + (T,,.Re 2, - T,, . I m 2, - b,)'],
`
`(25)
`
`where
`
`It is shown in Appendix I1 that E is a convex function of
`the vector F, where
`
`Thus a steepest descent algorithm is sure to converge to
`the vector
`yielding the minimum mean-square
`dis-
`tortion.
`The l,,,th components of the gradient of E with respect
`t o T are
`
`(VE),, = __ " - - E[2e,, R e 2, - 2elb I m Z,]
`
`The steepest descent algorithm makes changes at the end
`of each bloc