MULTI-CARRIER
`
`TECHNOLOGIES
`
`FOR WIRELESS
`
`COMMUNICATION
`
`Carl R. Nassar
`
`B. Natarajan
`Z. Wu
`
`D. Wiegandt
`SA. Zekavat
`
`S. Shattil
`
`L‘
`
`w K
`
`luwer Academic Publishers
`
`MTel., Exhibit 2003, ARRIS v. MTel., Page 1, IPR2016-00765
`
`

`

`MULTI-CARRIER TECHNOLOGIES
`FOR WIRELESS COMMUNICATION
`
`MTel., Exhibit 2003, ARRIS v. MTel., Page 2, IPR2016-00765
`
`

`

`MULTI-CARRIER TECHNOLOGIES
`FOR WIRELESS COMMUNICATION
`
`by
`
`Carl R. Nassar, B. Natarajan, Z. Wu
`
`D. Wiegandt, S. A. Zekavat
`
`Colorado State University
`
`S. Shattil
`
`Idris Communications
`
`KLUWER ACADEMIC PUBLISHERS
`NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
`
`MTel., Exhibit 2003, ARRIS v. MTel., Page 3, IPR2016-00765
`
`

`

`eBook ISBN:
`Print ISBN:
`
`0-306-47308-9
`0-792-37618-8
`
`©2002 Kluwer Academic Publishers
`New York, Boston, Dordrecht, London, Moscow
`
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`
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`
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`
`MTel., Exhibit 2003, ARRIS v. MTel., Page 4, IPR2016-00765
`
`

`

`Chapter 6
`HIGH-THROUGHPUT
`HIGH-PERFORMANCE,
`OFDM WITH LOW PAPR VIA CARRIER
`INTERFEROMETRY PHASE CODING
`
`6.1 Introduction
`
`Experimentation with parallel data transmission techniques began as
`early as the 1950’s [1], and in the mid 1960’s a multitude of work was
`emerging on the topic of Frequency Division Multiplexing, or FDM [2]. The
`basic premise for FDM was to avoid the hazards of the frequency selective
`fading channel by dividing the band into many smaller bands. Specifically,
`serial-to-parallel conversion of the incoming information bits, and
`transmission of each bit upon its own unique carrier, created a data rate per
`carrier that was a factor of N smaller than the original data rate. Hence, the
`bandwidth per carrier was only
`of the overall system bandwidth. As a
`result, each transmitted bit (one per carrier) experienced a flat fade.
`
`When the ability to avoid the frequency selective fading channel first
`became possible, the overall bandwidth efficiency was low. Weinstein and
`Ebert introduced the discrete Fourier transform (DFT) to FDM in 1971 [3],
`and through this addition to the modulation/demodulation process made it
`possible to orthogonally overlap the smaller bands. This gave way to
`Orthogonal Frequency Division Multiplexing (OFDM).
`
`Since its first-introduction some four decades ago, advances in digital
`signal processing, specifically the Fast Fourier Transform (FFT), have led to
`OFDM’s growing popularity. Applications to date include variable rate
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`126
`modems [4], wideband data communications over mobile radio FM channels
`[5], high data rate subscriber lines [6], digital terrestrial TV broadcasting [7],
`fixed wireless [8], and wireless ATM [9]. More recently, OFDM has emerged
`as the standard in a number of high data rate applications: Digital television
`broadcasting (such as the digital ATV terrestrial broadcasting [10] and
`European DAB and DVB-T [11]), and numerous wireless local area networks
`(most notably IEEE 802.11 operating at 5 GHz [12] and ETSI BRAN’s
`HYPERLAN 2 standards [13]).
`
`While the excitement for OFDM continues to grow, and even as
`OFDM emerges as a possible “platform” technology, it is not without its
`drawbacks: problematic bit loss arises due to deep fades, throughput loss
`results due to “performance-aiding” coding, and peak-to-average power ratio
`dilemmas have led to questions regarding implementation.
`
`In the following sections, we introduce a novel OFDM architecture,
`one which enables OFDM to overcome its limiations. This architecture,
`referred to as Carrier Interferometry OFDM (or CI/OFDM for short), utilizes
`frequency diversity to increase OFDM performance without bandwidth
`expansion and without decreased data
`rate.
`Specifically, Carrier
`Interferometry OFDM (1) simultaneously modulates each information bit
`onto all carriers, and (2) assigns a unique phase code set to the carriers of
`each bit to assure orthogonality between bits. This creates a frequency
`diversity benefit for each bit, and leads to high performances. Moreover, in
`addition to the dramatic performance gains that are possible through the use
`of CI/OFDM, Pseudo-Orthogonal Carrier Interferometry (PO-CI) codes can
`be applied to the CI/OFDM systems. Here, (1) each bit is still simultaneously
`sent over all N carriers, but now (2) each bit’s N carriers are assigned pseudo-
`orthogonal CI codes, making them nearly orthogonal to other bits at the
`transmitter. By applying pseudo-orthogonal codes to the N carriers of each
`bit, we can assign 2N bits onto the N carriers, doubling the number of
`information bits on the original number of carriers, without bandwidth
`expansion.
`It will also be shown in section 6.7, that CI/OFDM is capable of
`not only enhancing probability of error performance and doubling throughput,
`but also acts to eliminate the peak-to-average power ratio (PAPR) problem
`inherent in traditional OFDM systems.
`
`Since typical OFDM systems already employ coding to overcome
`channel degradations, we also present
`the application of coding to the
`proposed CI/OFDM system, leading to further performance enhancement. In
`the resulting CI/COFDM systems, a time interleaving is incorporated to create
`a time diversity benefit alongside the channel coding gain and the frequency
`diversity gain inherent in CI/OFDM.
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`127
`Sections 6.2 and 6.3 describe the transmitter and receiver models
`respectively for CI/OFDM and PO-CI/OFDM. The remaining sections of this
`chapter provide additional information regarding CI and the performance
`benefits available through its use.
`
`6.2 Novel CI Codes and OFDM Transmitter Structures
`
`In today’s OFDM, incoming information bits are mapped into
`transmit data symbols corresponding to Quadrature Amplitude Modulation
`(QAM) or Phase Shift Keying (PSK) symbols. To simplify the discussion of
`this chapter, Binary Phase Shift Keying (BPSK) will be assumed as the
`mapping (i.e., incoming information bits consisting of 0’s and 1’s are mapped
`to –1 and +1 respectively). After mapping, the OFDM transmit operation is
`shown in Figure 6.1 (a). Here, N symbols are serial-to-parallel converted and
`sent simultaneously over N orthogonal carriers [14]. The data rate per carrier
`is a factor of N smaller than the original data rate, and hence the bandwidth
`per carrier is only
`of the overall system bandwidth. As a result, each
`transmitted bit (one per carrier) experiences a flat fade. This translates into
`simple receiver design and a system that drastically reduces inter-symbol
`interference and avoids multipath in a frequency selective channel.
`
`However, there is a very real disadvantage in this OFDM architecture.
`Since each carrier experiences a flat fade and reaches the receiver with a
`different amplitude, it is possible, even likely, that some of the N data
`symbols are completely lost due to deep fades. To account for this, Coded
`OFDM (COFDM) has been introduced (e.g., [15][16][17][18]). Here,
`incoming information bits are channel coded prior to serial-to-parallel
`coder, each bit is effectively sent over n frequency
`conversion. In a rate
`carriers, introducing a frequency diversity benefit and channel coding gain,
`which overcomes the fading degradation. The draw back is of course a
`lowered throughput (by a factor of n).
`
`The following illustrates the incorporation of the CI phase codes to
`the OFDM transmitter. This will enable full utilization of the frequency
`diversity available in the channel.
`
`6.2.1 CI/OFDM & CI/COFDM
`
`A typical OFDM transmitter is shown in Figure 6.1(a), and the novel
`CI/OFDM transmitter is depicted in Figures 6.1(b) and 6.1(c). In both OFDM
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`128
`and CI/OFDM, input bits are serial to parallel converted. However, unlike
`OFDM, where each bit is modulated onto its own carrier, in CI/OFDM each
`bit is modulated onto all of the N carriers. To separate bits located on
`identical carriers, we introduce a phase offset to each of bit k’s carriers.
`Specifically,
`is the phase offset applied to the
`carrier for bit k
`(Figure 6.1(c)).
`The
`set of phases applied to bit k’s
`carriers,
`is known as the spreading code for bit k. Careful
`lead to spreading codes that ensure orthogonality
`will
`selection of
`among the N transmitted bits, even though bits occupy the same carriers at the
`same time. This notion is very similar to that of MC-CDMA systems [19],
`where N users occupy all N carriers at the same time, but are separated by
`spreading codes corresponding to phase offsets.
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`129
`
`The spreading codes used in CI/OFDM, referred to as CI codes, correspond to
`those used to create user orthogonality in CI/MC-CDMA (Chapter 3): the
`spreading
` code
`for
`user
`where
`Therefore, the spreading codes for
`in the CI/OFDM system utilizes the following code-
`
`bits
`defined phase offsets:
`
`The transmitted signal for the
`
`bit in a CI/OFDM system is:
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`refers to the
`
`130
`bit and is assumed to be +1 or –1 with equal
`where (1)
`is the bit
`rate)
`to assure
`and
`probability; (2)
`is the phase offset used to
`orthogonality among carriers; (3)
`generate bit k’s spreading code, and ensures orthogonality among the N bits;
`and (4)
`ensures a bit energy of unity. Now, over the entire OFDM
`block of N bits, the transmitted signal in CI/OFDM is:
`
`As mentioned, channel coding is incorporated into most traditional
`OFDM architectures, leading to coded OFDM (COFDM). In typical COFDM
`systems, prior to the serial to parallel conversion of Figure l(a), each l input
`) are channel coded to n output bits (typically
`bits (typically
`Then, in the same serial-to-parallel manner, each bit is transmitted on its own
`information bits sent on N carriers. In this way, l
`carrier for a total of
`information bits are effectively sent on n carriers, enabling frequency
`diversity benefits at a cost of decreased throughput.
`In our CI/COFDM
`system, each set of l input bits are similarly coded to n output bits (e.g. 1 bit to
`2 bits). Now, since CI/OFDM already sends each bit on all N carriers
`(exploiting the full frequency diversity benefit), each set of n coded bits are
`sufficiently time interleaved to add an nfold time diversity benefit. Figure
`6.2 illustrates this interleaving methodology.
`
`convolutional
`Referring to Figure 6.2, it can be seen that for a rate
`coder, one bit is input and two coded bits are output, creating “coded output
`bit 1” (denoted in Figure 6.2 as A) and “coded output bit 2” (denoted in
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`131
`Figure 6.2 as B). These coded output bits are then time interleaved onto two
`different CI/OFDM symbols such that one CI/OFDM symbol contains N
`“coded output 1 bits” and another CI/OFDM symbol contains the second N
`“coded output 2 bits.” In this way, CI/COFDM has the same degree of
`redundancy (i.e. same throughput) as COFDM, but instead of the redundant
`bits being transmitted on the carriers at the same time, they are time
`interleaved. CI/COFDM then, offers the same full frequency diversity benefit
`of CI/OFDM, and adds an nfold time diversity benefit, all with the same
`throughput of a COFDM system.
`
`6.2.2 Addition of Pseudo-Orthogonality to CI/OFDM & CI/COFDM
`
`CI/OFDM, as presented to date, represents a powerful alternative
`implementation for OFDM that enables significant gains in performance (via
`enhanced diversity gains). However, the benefits of CI/OFDM are not limited
`to performance, as CI/OFDM also creates a doubling in throughput. This
`benefit is demonstrated in this subsection, where we refer to the CI/OFDM
`implementation that doubles throughput as PO-CI/OFDM (pseudo-orthogonal
`CI/OFDM).
`
`In PO-CI/OFDM, we transmit 2N data symbols on N carriers; rather
`than the usual N symbols on N carriers. Specifically, a data stream with twice
`the usual OFDM throughput is serial to parallel converted into 2N data
`streams. Each parallel data stream has the same data rate of a traditional per-
`carrier OFDM data stream. Next, just as in CI/OFDM, each bit is modulated
`onto all of the N carriers. To separate bit k from the (2N -1) other bits located
`on identical carriers, we again introduce a phase offset to each of bit k’s
`carrier for bit k is assigned phase offset
`carriers. Specifically, the
`In
`is applied to bit k’s
`other words, the spreading code
`carriers, and
`is referred to as the code-defining phase offset. By
`careful selection of the code-defining phase offset, the 2N bits can be
`supported on N orthogonal carriers in a manner that makes them highly (but
`pseudo) orthogonal. Specifically, we support the first bits on the N carriers by
`using the usual CI/OFDM code-defining phase offsets (equation (6.1)), i.e.,
`
`To the next set of N bits (bits N, N+1,...,
`spreading codes with code-defining phase offsets
`
`2N-1), we assign
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`132
`
`The second set of bits are hence assigned code-defining phase offsets that
`allow them to be orthogonal to one another, but pseudo-orthogonal to the first
`set. We select
`such that we minimize the amount of inter-bit interference
`at the transmitter. The intuitive solution is to select
` asthis creates
`a second set of code-defining phase offsets equidistant from the original set.
`This has been proven mathematically to minimize the inter-bit interference
`(analogous to the derivation in Chapter 3). Hence, the following is used as
`the second set of code-defining phase offsets:
`
`That is, for CI/OFDM systems incorporating K = 2N bits on N carriers,
`referenced as bit 0 to bit 2N-1, each bit is applied to all N carriers and
`assigned spreading code
`
`where
`
`Again, this means the first N bits are orthogonal amongst themselves, the
`second N bits are orthogonal amongst themselves, but both sets of N bits (for
`a total of 2N bits) are pseudo-orthogonal to each other.
`
`The transmitted signal forthe
`
` bit in PO-CI/OFDM is therefore:
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`133
`and the PO-CI/OFDM transmitted signal considering the entire OFDM block
`of 2N bits is thus:
`
`Channel coding, common in OFDM, can also be applied to PO-
`CI/OFDM, leading to PO-CI/COFDM. In PO-CI/COFDM, each l input bits
`are channel coded to n output bits (typically n = 2) prior to
`(typically
`the seriako-parallel conversion. COFDM transmits each of the n bits on a
`information bits sent on N carriers. (This
`unique carrier, for a total of
`introduces frequency diversity and channel coding to OFDM.) PO-
`information bits on N
`CI/COFDM, on the other hand, transmits
`carriers, and time interleaves each set of n coded bits to create a time diversity
`benefit in addition to the channel coding gain. Figure 6.3 illustrates the time
`interleaving in the PO-CI/COFDM architecture.
`
`convolutional coder,
`Referring to Figure 6.3, and again for a rate
`one bit is input and two coded bits are output, creating “coded output bit 1”
`and “coded output bit 2.”
`
`In Figure 6.3, for the first set of N “coded output 1” bits and the first set of N
`“coded output 2” bits are denoted as A and B respectively, and the second set
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`134
`of N “coded output 1” bits and N “coded output 2” bits are denoted C and D
`respectively. These blocks of coded output bits are time interleaved onto two
`PO-CI/OFDM symbols such that one PO-CI/OFDM symbol contains 2N
`“coded output 1 bits” and another PO-CI/OFDM symbol contains the second
`N “coded output 2 bits.” When utilizing this strategy, PO-CI/COFDM, offers
`the same full frequency diversity benefit of PO-CI/OFDM and adds an n-fold
`time diversity benefit (in addition to the channel coding gain).
`
`6.3 Novel OFDM Receiver Structures
`
`In the Carrier Interferometry implementation of OFDM, one major
`benefit is the receiver’s ability to fully exploit the channel frequency diversity
`and, as a result, significantly enhance performance. The novel CI/OFDM
`receivers that achieve large performance gains are the topic of this subsection.
`
`The received signal, assuming the sent signal s(t) in (6.3) or (6.10), is
`mathematically characterized by the following equation:
`
`Here, K = N if s(t) is based on equation (6.3) and K = 2N if s(t) corresponds
`to equation (6.10);
`and
`are the fade parameter and phase offset,
`respectively, introduced into the
`carrier by the frequency selective Rayleigh
`fading channel; and n(t) is additive white Gaussian noise (AWGN). We will
`assume perfect phase synchronization, for reasons of simplicity in
`presentation.
`
`The CI/OFDM receiver is depicted in Figure 6.4 for detection of the
`bit. Here, r(t) is separated into its N orthogonal carriers, and the
`bit's
`is removed from carrier i.
`phase offset
`
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`

`

`This lead to the decision vector
`
`where
`
`135
`
`The second term represents the existence of the (K – 1) other bits on the
`bit; that is, it represents inter-bit interference.
`In an AWGN channel, this
`term, when combined across carriers (i.e. after performing
`sums
`to zero due to the orthogonality between bits created by the appropriate choice
`of
`. (In the case where bits are pseudo-orthogonal and not orthogonal,
`i.e., when K = 2N,
`this term is minimized in an AWGN channel via the
`combining of
`In the frequency selective channel, however, a
`
`fails to minimize the presence of the
`simple combining across
`interference term, due to the presence of the carrier dependent fade,
`. In
`frequency selective channels, a different combining strategy will be employed
`in the CI/OFDM receiver, to rebuild our bit from the newly created
`While numerous combining techniques are possible, it has been shown in the
`MC-CDMA literature (e.g., [20]) that minimum mean square error combining
`(MMSEC) offers the best performance. This MMSEC minimizes the inter-bit
`interference and noise while best exploiting the frequency diversity benefits.
`This combining corresponds to the following decision variable:
`
`For uncoded CI/OFDM and PO-CI/OFDM, the variable C enters a hard
`decision device which outputs
`.
`In the cases of CI/COFDM and PO-
`CI/COFDM, the decision variable C enters a deinterleaver, followed by a soft
`decision decoding Viterbi Algorithm (VA) employing the Euclidean distance
`metric.
`
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`

`136
`
`6.4 Channel Modeling
`In order to compare OFDM, CI/OFDM and PO-CI/OFDM,
`appropriate channel modeling must be employed. The following elaborates
`on the channel models.
`
`Extensive work has been done on the modeling of wireless channels.
`These models, typically based on measurement data, emulate realistic
`environments for transmission. Here we focus on the indoor channel models,
`since many of today’s OFDM systems are intended for this environment.
`These models characterize environments such as the small office/home office
`(SOHO), large office, and warehouse type structures. Each of these is
`characterized by a specific delay spread and path model. As discussed in the
`literature (e.g., [21]), root mean squared (rms) values of delay spread
`vary from 20-50 ns for small office/home offices (SOHO) and from 50-100 ns
`for large office buildings.
`
`The specific models used for simulation are based on the UMTS
`indoor office and large office models [22]. Specifically, rms delay spreads for
`these environments correspond to 35 ns and 100 ns respectively (as specified
`by the UMTS channel model for indoor test environments [22]).
`
`The 35 ns and 100 ns delay spreads correspond to a 2.8-fold and an
`8.125-fold frequency diversity (respectively), over the entire bandwidth.
`(This assumes a bandwidth consistent with the IEEE 802.11a standard) We
`also utilize an “average” channel model with a 4-fold frequency diversity over
`the entire bandwidth.
`With this 3 to 8-fold frequency diversity, the channel fades
`equation (6.11) are correlated according to [23]:
`
`in
`
`fade and carrier
`is the correlation between carrier
`where
`is the frequency separation between these two carriers, and
`channel’s coherence bandwidth.
`
`fade,
`is the
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`137
`
`6.5 Performance Results
`Throughout this chapter, we have discussed the promise of CI/OFDM
`and PO-CI/OFDM in terms of increased performance and throughput. In this
`section, we demonstrate these benefits via performances and throughput
`curves. Figure 6.5 illustrates the bit error rate (BER) versus signal to noise
`ratio for OFDM, COFDM, CI/OFDM, and CI/COFDM. Each system
`transmits N = 32 bits over N = 32 carriers. We also assume a channel with a
`4-fold frequency diversity. In cases of OFDM and CI/OFDM, the N = 32
`transmit bits all correspond to information bearing bits; in COFDM and
`CI/COFDM, only 16 of these N = 32 bits are information bearing (the rest are
`redundancy bits).
`
`Referring to Figure 6.5, the CI/OFDM system offers 10 dB
`performance gain over OFDM at a BER of
`This gain is due to the
`frequency diversity benefit inherent in the CI/OFDM system. It is apparent
`that the interbit interference due to the second term in (6.12) (reduced by the
`combining in (6.13)) is more than compensated for by the gain achieved via
`frequency diversity (sending the same bit over the N =32 carriers). The
`performance gain is even larger at lower BER’s: for example, at BER of
`an 18 dB gain is available.
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`138
`
`For the coded systems, we have implemented the coder as a rate
`convolutional coder with a constraint length of 3, and utilized a soft decision
`decoding Viterbi Algorithm.
`Referring once again to Figure 6.5,
`it is
`observed that the traditional COFDM system gains approximately 14 dB over
`OFDM at BER of
`The substantial benefits of channel coding, in terms of
`both coding gain and frequency diversity benefit, are apparent, but the cost is
`high -- in this case a factor of 2 degradation in throughput. Without any
`coding, and hence without loss in throughput, CI/OFDM offers 10 of
`COFDM’s 14 dB gain. Moreover, for only a 4 dB performance loss relative
`to COFDM, CI/OFDM is available without the complexity of a soft decision
`decoding VA at its receiver.
`
`When the identical rate convolutional coding scheme is applied to
`the CI/OFDM system, creating CI/COFDM, 16 dB gain is achieved over
`OFDM, and a 2 dB gain is available over COFDM at BER of
`By BER =
`23 dB gains are observed in relation to OFDM and 3 dB gains are
`achieved relative to COFDM. The performance benefits of CI/OFDM are
`observed because not only is the full frequency diversity exploited, but in
`addition, (1) a time diversity benefit is achieved in bit interleaving the channel
`coded bits, and (2) convolutional decoding using a VA offers well-
`documented benefits.
`
`Figures 6.6 and 6.7 illustrate the bit error rate (BER) versus signal to
`noise ratio for OFDM, COFDM, CI/OFDM, CI/COFDM, PO-CI/OFDM and
`PO-CI/COFDM. In all cases, N = 32 carriers are employed, and the coding
`applied to the coded systems is rate with constraint length 3.
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`

`139
`
`The OFDM, COFDM, CI/OFDM and CI/COFDM systems all transmit N= 32
`bits over the N = 32 carriers. In the coded cases only 16 of every 32 bits are
`information bearing. In PO-CI/OFDM, by application of pseudo-orthogonal
`codes to each bit, 2N = 64 bits are sent over N = 32 carriers. In the coded
`cases, 32 of the 64 bits are information bearing. To emulate realistic wireless
`environments, we assume a 4-fold frequency diversity over the entire
`bandwidth.
`
`Referring to Figure 6.6, we see that 64-bit, 32-carrier PO-CI/OFDM
`loses 2 dB relative to 32-bit, 32-carrier CI/OFDM at a BER of
`and that
`the 32-information bit, 32-carrier PO-CI/COFDM system loses 2 dB relative
`to the 16-information bit, 32-carrier CI/COFDM system at a BER of
`These losses in performance demonstrate the impact of inter-bit interference
`created by the pseudo-orthogonal spreading codes assigned to the bits.
`Degradation in performance is a cost paid for the doubling of the throughput.
`However, CI/OFDM is known to significantly outperform OFDM (Figure
`6.5), and the losses in PO-CI/OFDM (relative to CI/OFDM) are small enough
`that PO-CI/OFDM will still outperform OFDM.
`
`Referring to Figure 6.7, the increased capacity 64-bit, 32-carrier PO-
`CI/OFDM system offers 8 dB of gain over a 32-bit, 32-carrier OFDM system
`at a BER of
`While it loses 6 dB relative to the 16-information bit, 32-
`
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`

`140
`carrier COFDM system, PO-CI/OFDM has four times the throughput relative
`to COFDM over the same 32 carriers (and a less complex receiver design).
`
`Also in Figure 6.7, the 32-information bit, 32-carrier PO-CI/COFDM
`system demonstrates essentially the same performance as the 16-information
`bit, 32-carrier COFDM system, and hence the same gain of 14 dB over typical
`OFDM at a BER of
`This means that the inter-bit interference in PO-
`CI/COFDM, even with pseudo-orthogonal codes applied to the bits, is more
`then compensated for by the gain achieved via the full frequency diversity
`(sending the same bit over the N = 32 carriers), the time diversity benefit
`(induced
`in time
`interleaving the channel coded bits), and the VA
`convolutional decoding. These benefits allow our 32-information bit, 32-
`carrier PO-CI/COFDM system to perform as well as its 16-information bit,
`32-carrier COFDM counterpart. Hence, PO-CI/COFDM achieves the
`performance of COFDM, with the same throughput as in OFDM. We achieve
`the best of both worlds. The cost, of course, is transmitter and receiver
`complexity.
`
`6.6 Peak to Average Power Ratio Considerations
`
`Of great concern in OFDM systems is high peak-to-average power
`ratios (PAPR). Specifically, in OFDM and COFDM, high peaks in power (up
`to N times the average) are observed, a consequence of using independently
`modulated carriers. This, in turn, leads to inefficient operation of the transmit
`power amplifier.
`A number of solutions to OFDM’s peak-to-average power ratio
`(PAPR) problem have been proposed in the literature (i.e., block coding [24],
`partial transmit sequences [25], selective mapping [26], and clipping [27]).
`While reducing the PAPR, these schemes typically increase the complexity of
`the OFDM system.
`
`The proposed Carrier-Interferometry OFDM (CI/OFDM) system
`demonstrates a low PAPR. That is, in CI/OFDM, PAPR is simply not an
`issue. Specifically, the phase codes applied to the N carriers result in one bit’s
`power reaching a maximum when the powers of the remaining N-1 bits are at
`a minimum. Therefore, a stable transmit envelope is observed, and, the PAPR
`is small.
`
`Pseudo-Orthogonal Carrier-Interferometry OFDM (PO-CI/OFDM)
`demonstrates even lower PAPR values than those in CI/OFDM. Specifically,
`as in CI/OFDM, when one bit’s power reaches a maximum, the powers of the
`remaining 2N-1 bits are at a minimum; and now, because there are twice as
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`141
`many bits per N carriers (in PO-CI/OFDM relative to CI/OFDM), an even
`better averaging of the power across the OFDM symbol is observed.
`
`6.6.1 PAPR in OFDM and CI/OFDM
`
`PAPR is defined as the peak power per OFDM symbol versus the
`average power per OFDM symbol, i.e.,
`
`The average power in CI/OFDM (and OFDM) is:
`
`where
`
`is the power on one carrier, i.e.,
`
`incoming
`The OFDM method of serial-to-parallel converting
`information bits and transmitting each bit on its own unique carrier leads to
`the potential for high peak power. This is a result of a possible in-phase,
`coherent addition of all the carriers. In this worst case (WC) senario, where
`the N carriers combine coherently, OFDM’s peak power is equal to:
`
`In CI/OFDM, as discussed in section 6.2.1, all bits are transmitted
`simultaneously over all carriers, and an appropriate selection of phase offsets
`makes bits separable at the receiver. However, these phase offsets have a
`second benefit: they reduce the peak power. Specifically, they ensure that
`when one bit’s carriers add coherently, other bit’s carriers do not add
`coherently. Therefore,
`is much less than
`That is,
`considering worst case scenarios:
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`142
`
`and
`
`Figure 6.8 shows PAPR levels across 10,000 transmit symbols for
`both OFDM (black) and CI/OFDM (gray), each with N=32 carriers. As seen
`in Figure 6.8, spurious peaks with PAPR > 7.5 are quite common in OFDM
`transmissions (arising 2.5% of the time), and even peaks of 15 < PAPR < 20
`result at select transmission times. CI/OFDM, on the other hand, displays no
`peaks with PAPR > 6.5, and displays PAPR < 5 at almost all times. On
`average, OFDM demonstrates a PAPR of 3.79 while Cl/OFDM’s PAPR is
`3.41.
`
`Figure 6.9 demonstrates the standard deviation of the PAPR as a
`function of increasing number of carriers. With N = 32 carriers, OFDM’s
`PAPR demonstrates a standard deviation of 1.23 (a variance of 1.5), while
`CI/OFDM’s standard deviation is only 0.665 (a variance of 0.442).
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`143
`
`Now, referring to Figure 6.10, 98% of the CI/OFDM transmissions
`demonstrate PAPR < 5, and all transmissions (100%) demonstrate PAPR <
`6.5. Meanwhile, only 88% of the OFDM transmissions demonstrate PAPR <
`5, and it is not until y = 32 that Pr(PAPR = y) = 100%. Clearly, the PAPR
`values in CI/OFDM will allow amplifiers at the transmit side to operate with
`much greater power efficiency.
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`
`6.6.2 PAPR in PO-CI/OFDM
`
`Figure 6. 11 illustrates PAPR levels across 10,000 transmit symbols
`for 32-bit, 32-carrier OFDM (black) and 64-bit, 32-carrier PO-CI/OFDM
`(gray). Referring to Figure 6.11, OFDM’s PAPR can be characterized as
`erratic, displaying a mean PAPR of 3.79, and consistently reaching levels
`exceeding 6 (5% of the time), with some PAPR values exceeding 15 and even
`20. PO-CI/OFDM, on the other hand, displays no PAPR value above 4.4 and
`stays close to its mean PAPR level of 2.5.
`
`Figure 6.12 demonstrates the standard deviation of the PAPR as a
`function of increasing number of carriers. As the number of carriers
`increases, the standard deviation of OFDM’s PAPR also increases, but the
`opposite is true in PO-CI/OFDM: in PO-CI/OFDM, the standard deviation of
`the PAPR decreases with increasing number of carriers. For the 32-bit, 32-
`carrier OFDM and 64-bit, 32-carrier PO-CI/OFDM systems shown in Figure
`6.11, OFDM’s PAPR demonstrates a standard deviation of 1.23 (a variance of
`1.5), while PO-CI/OFDM’s standard deviation is only 0.355 (a variance of
`0.125).
`
`When compared to an OFDM system that has had the clipping
`algorithm of [28] applied, similar results are none-the-less observed. Figure
`6.13 displays the PAPR levels across 10,000 transmit symbols for a 32-bit,
`32-carrier OFDM system with clipping (in black), and the 64-bit, 32-carrier
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`145
`PO-CI/OFDM system (in gray). Here, a Clipping Ratio (CR), (defined in
`[27]), of 1.4 was implemented.
`
`Referring to Figure 6.13, the clipping algorithm greatly reduces the
`number of times the PAPR exceeds a level of 5, but spurious levels are still
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`146
`prevalent. The mean and standard deviation of OFDM’s PAPR, with
`clipping, are reduced to 2.412 and 1.053 respectively. However, these are still
`far worse than of PO-CI/OFDM’s PAPR values, where the mean is effectively
`the same but the standard deviation is only 0.355.
`
`Figure 6.14 plots the pdf (probability density function) of the PAPR
`for OFDM, OFDM with clipping, and PO-CI/OFDM.
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`147
`see how clipping effectively
`to Figure 6.14, we
`Referring
`concentrates the PAPR levels about the mean, but does little to contain the
`spurious peaks. This can be directly attributed to the in-band distortion
`caused by clipping.
`
`Figure 6.15 plots the cumulative distribution function (CDF) of the
`PAPR. Clipping improves the PAPR statistics (relative to OFDM), but it is
`not until y = 22.5 that Pr(PAPR = y) = 100%, which is a result similar to that
`of unclipped OFDM.
`PO-CI/OFDM, on the other hand, demonstrates
`Pr(PAPR < y) = 100% when y = 4.4.
`
`6.7 Conclusions
`
`In this chapter, Carrier Interferometry and Pseudo-Orthogonal Carrier
`Interferometry