INTERFERENCE ALIGNMENT FOR CHANNEL-ADAPTIVE
WAVEFORM MODULATION
BACKGROUND
Inter-symbol interference (ISI) is a form of distortion of a signal
in which one symbol interferes with subsequent symbols. This is an
unwanted phenomenon as the previous symbols have similar effect as
noise, thus making the communication less reliable. One of the
causes of ISI is multipath propagation in which a wireless signal from
a transmitter reaches the receiver via many different paths. The
causes of this include reflection (i.e., the signal may bounce off
buildings), refraction (such as through the foliage of a tree) and
atmospheric effects such as atmospheric ducting and ionospheric
reflection. Since all of these paths are of different lengths, this results
in the different versions of the signal arriving at different times,
resulting in ISI.
Data communication schemes have handled ISI by a variety of
techniques. One such technique is known as Orthogonal Frequency
Division Multiplexing (OFDM). OFDM uses modulation waveforms that
enable the essential removal of ISI in a frequency-dependent channel.
For example, in OFDM, each transmitted data block is a weighted
superposition of OFDM modulation waveforms. The OFDM modulation
waveforms form an orthonormal basis set over a time period (TS-TG)
where Ts is the length of the OFDM block (also referred to as symbolinterval
of duration Ts) and T is the duration of either a guard
interval or a cyclic prefix, both expressed as a multiple of the sampling
interval. Because ISI does not distort symbols separated by more than
9
the communication channel's delay-spread TD, the guard interval TG
is selected to be greater than or equal to the delay spread TD in OFDM.
In an OFDM block, the weights of the superposition define the data
symbol being transmitted.
At the receiver, in OFDM, each transmitted data block is
demodulated by projecting the received data block onto a basis set of
conjugate OFDM modulation waveforms. Because the OFDM
modulation waveforms are a basis set over the period (TS-TG), the
projections may be performed over the last period (TS-TG) of the OFDM
data blocks. That is, the projections do not need to use prefix
portions of the OFDM data blocks. Because the channel memory is
limited to a time of length TD, an earlier transmitted OFDM block only
produces ISI in the cyclic prefix or guard portion of the next received
OFDM data block. Thus, by ignoring said cyclic prefix or guard
portions of received OFDM data blocks, OFDM produces demodulated
data that is free of distortion due to ISI. OFDM techniques may also
effectively diagonalize the communication channel.
Unfortunately, cyclic prefix and guard portions of OFDM data
blocks consume bandwidth that might otherwise be used to transmit
data. As the communication channel's delay-spread TD approaches
the temporal length of the OFDM data block Ts , the bandwidth TS-TD
remaining for carrying data shrinks to zero. For example, when the
channel delay-spread is equal to the symbol interval, then OFDM is
0% efficient because the redundant cyclic prefix occupies the entire
symbol interval. Increasing the symbol interval Ts would alleviate this
problem, but this results in increased communication delay, which
might not be tolerable depending on the application.
In order to overcome the bandwidth-deficient channel whose
delay-spread approaches the length of the OFDM block, Chen et al.
(U.S. Patent No. 7,653, 120) introduces a Channel Adaptive Waveform
Modulation (CAWM) that generates modulating waveforms from the
channel impulse response itself. When the channel delay-spread is
equal to the symbol interval, CAWM is 50% efficient because the
number of orthogonal data-symbol-bearing waveforms that can be
created is equal to half the symbol-interval. When the delay-spread is
equal to twice the symbol interval, then CAWM is 1/3 (33%) efficient.
This compares to a 0% efficiency of OFDM in both cases.
SUMMARY
Embodiments provide an apparatus and method for interference
alignment for channel-adaptive waveform modulation.
The method includes obtaining at least a part of a first matrix
and a part of a second matrix for the impulse response function of a
communication channel. The part of the first matrix relates to
channel-induced interference between a current data block and a
previously transmitted first data block, and the part of the second
matrix relates to channel-induced interference between the current
data block and a previously transmitted second data block, the
second data block being transmitted before the first data block.
The method further includes designing a set of one or more
linearly independent waveforms based on at least the obtained part of
the first matrix and the obtained part of the second matrix for the
impulse response function such that a first subspace spanned by the
linearly independent waveforms when multiplied by the obtained part
of the first matrix at least partially overlaps a second subspace
spanned by the linearly independent waveforms when multiplied by
the obtained part of the second matrix.
In one embodiment, the designing step designs the set of
linearly independent waveforms such that the first subspace and the
second subspace occupy a same linear space. Further, the designed
set may include a subset of eigenvectors of a product of (1) an inverse
of the first matrix and (2) the second matrix. The subset may
comprise the right eigenvectors of the product.
In another embodiment, the designing step further includes
obtaining eigenvectors and corresponding eigenvalues based on an
eigenvector decomposition of a product of (1) an inverse of the first
matrix and (2) the second matrix, and selecting a subset of the
obtained eigenvectors.
Also, the designing step may further include configuring a
second set of waveforms based on the selected subset, where the
configured second set of waveforms is an orthogonal complement of
the product of the selected subset and either of the first or the second
matrices.
In one embodiment, the first data block immediately precedes
the current data block and the second data block immediately
precedes the first data block.
The method may further include transmitting a set of pilot
signals over the communication channel that is between the
transmitter and the receiver, where the part of the first matrix and the
part of the second matrix for the impulse response function are
obtained responsive to measurements of said pilot signals.
The method further includes, for each one of the data blocks of
the sequence, modulating the waveforms of the designed set to have
amplitudes responsive of a received input data symbol and linearly
superimposing the modulated waveforms to produce each one of the
data blocks.
In one embodiment, the designed set has a number of
waveforms equal to one half of a symbol interval when the delay
spread is twice the symbol interval.
The apparatus includes a transmitter having an array of
modulators, where each modulator is configured to modulate an
amplitude of a corresponding one of linearly independent waveforms
over a sequence of sampling intervals in response to receipt of each of
a sequence of input data symbols, an adder configured to form a
sequence of data blocks, where each data block is a linear
superposition of modulated transmitter waveforms produced by the
modulators responsive to receipt of one of the input data symbols and
the adder is configured to transmit the data blocks via a
communication channel.
The transmitter configures the modulated waveforms in a
manner responsive to a part of a first matrix and a part of a second
matrix for the impulse response function of a communication channel.
The part of the first matrix relates to channel-induced interference
between a current data block and a previously transmitted first data
block, the part of the second matrix relates to channel-induced
interference between the current data block and a previously
transmitted second data block, the second data block being
transmitted before the first data block. The transmitter configures the
modulated waveforms such that a first subspace spanned by the
modulated waveforms when multiplied by the part of the first matrix
at least partially overlaps a second subspace spanned by the
modulated waveforms when multiplied by the part of the second
matrix. .
In one embodiment, the transmitter configures the modulated
waveforms such that the first subspace and the second subspace
occupy a same linear space. Further, the modulated waveforms may
include a subset of eigenvectors of a product of (1) an inverse of the
first matrix and (2) the second matrix. The subset may comprise right
eigenvectors of the product. The transmitter configures the modulated
waveforms by obtaining eigenvectors and corresponding eigenvalues
based on an eigenvector decomposition of a product of (1) an inverse
of the first matrix and (2) the second matrix and selects a subset of
the obtained eigenvectors.
The transmitter may configure the modulated waveforms by
constructing a second set of waveforms based on the selected subset,
where the constructed second set of waveforms is an orthogonal
complement of the product of the selected subset and either of the
first or second matrices.
The apparatus may further include a receiver having an array of
demodulators, where the demodulators project the data blocks onto
conjugate waveforms to produce estimates of a linear combination of
the components of the input data symbols carried by the data blocks
being demodulated.
The transmitter may transmit a set of pilot signals over the
communication channel that is between the transmitter and the
receiver, where the part of the first matrix and the part of the second
matrix for the impulse response function is obtained responsive to
measurements of said pilot signals.
Each modulator may modulate the amplitude of the
corresponding one of linearly independent waveforms to have
amplitudes responsive of a received input data symbol and linearly
superimposing the modulated waveforms to produce each one of the
data blocks.
In one embodiment, the number of modulated waveforms is
equal to one half of a symbol interval when the delay spread is twice
the symbol interval.
BRIEF DESCRIPTION OF THE DRAWINGS
Example embodiments will become more fully understood from
the detailed description given herein below and the accompanying
drawings, wherein like elements are represented by like reference
numerals, which are given by way of illustration only and thus are not
limiting , and wherein:
FIG. 1 illustrates a communication system 10 according to an
embodiment;
FIG. 2 illustrates a data stream 19 that is transmitted over the
channel 13 according to an embodiment;
FIG. 3 illustrates a method for performing interference
alignment for channel-adaptive waveform modulation according to an
embodiment;
FIG. 4 illustrates a method of constructing waveforms and
conjugate waveforms such that inter-symbol interference is removed
according to an embodiment; and
FIG. 5 illustrates a comparison of a number of orthogonal input
waveforms L as a function of channel delay spread T achieved by the
embodiments of the present invention (solid line), channel-adaptive
waveform modulation (dashed line), and the OFMD method (dotted
line).
DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS
Various example embodiments will now be described more fully
with reference to the accompanying drawings in which some example
embodiments are shown. Like numbers refer to like elements
throughout the description of the figures.
It will be understood that, although the terms first, second, etc.
may be used herein to describe various elements, these elements
should not be limited by these terms. These terms are only used to
distinguish one element from another. For example, a first element
could be termed a second element, and, similarly, a second element
could be termed a first element, without departing from the scope of
example embodiments. As used herein, the term "and/or" includes
any and all combinations of one or more of the associated listed items.
The terminology used herein is for the purpose of describing
particular embodiments only and is not intended to be limiting of
example embodiments. As used herein, the singular forms "a," "an"
and "the" are intended to include the plural forms as well, unless the
context clearly indicates otherwise. It will be further understood that
the terms "comprises," "comprising," "includes" and/or "including,"
when used herein, specify the presence of stated features, integers,
steps, operations, elements and/or components, but do not preclude
the presence or addition of one or more other features, integers, steps,
operations, elements, components and/or groups thereof.
It should also be noted that in some alternative
implementations, the functions /acts noted may occur out of the order
noted in the figures. For example, two figures shown in succession
may in fact be executed concurrently or may sometimes be executed
in the reverse order, depending upon the functionality/ acts involved.
Unless otherwise defined, all terms (including technical and
scientific terms) used herein have the same meaning as commonly
understood by one of ordinary skill in the art to which example
embodiments belong. It will be further understood that terms, e.g.,
those defined in commonly used dictionaries, should be interpreted as
having a meaning that is consistent with their meaning in the context
of the relevant art and will not be interpreted in an idealized or overly
formal sense unless expressly so defined herein.
In the following description, illustrative embodiments will be
described with reference to acts and symbolic representations of
operations (e.g., in the form of flowcharts) that may be implemented as
program modules or functional processes that include routines,
programs, objects, components, data structures, etc., that when
executed perform particular tasks or implement particular abstract
data types and may be implemented using existing hardware at
existing network elements. Such existing hardware may include one
or more Central Processing Units (CPUs), digital signal processors
(DSPs), application-specific-integrated-circuits, field programmable
gate arrays (FPGAs) computers or the like machines that once
programmed become particular machines.
It should be borne in mind, however, that all of these and
similar terms are to be associated with the appropriate physical
quantities and are merely convenient labels applied to these quantities.
Unless specifically stated otherwise, or as is apparent from the
discussion, terms such as "obtaining", "designing", "configuring" or
the like, refer to the action and processes of a computer system, or
similar electronic computing device, that manipulates and transforms
data represented as physical, electronic quantities within the
computer system's registers and memories into other data similarly
represented as physical quantities within the computer system
memories or registers or other such information storage, transmission
or display devices.
Below, parts of the description will use a complex baseband
description of the channel and signals as discrete time variables. In
this description, the various signals and channel quantities are
described as complex baseband functions whose values depend on the
sampling interval. Asampling interval t refers to the temporal interval
over which a modulator or demodulator applies one data value to the
signal being modulated or demodulated. The symbol interval refers to
the duration (expressed in terms of the number of sampling intervals)
Ts of one block of symbols. The delay spread TD refers to the length
(expressed again in terms of the number of sampling intervals) of a
communication channel's memory. The embodiments and claims are
meant to cover situations where frequency up-conversion occurs in
the transmitter and frequency down-conversion occurs in the receiver
as well as situations where no such conversions occur.
Embodiments of the present disclosure employ interference
alignment in the context of Channel Adaptive Waveform Modulation
(CAWM), as discussed in U.S. Patent No. 7,653, 120, which is
incorporated by reference in its entirety. Interference alignment in the
CAWM environment ensures that the inter-symbol interference
occupies a relatively smaller subspace than it would otherwise occupy.
For example, multiple interfering symbols are aligned to fall into the
same subspace at the receiver. As a result, an increased number of
waveforms may be used in the symbol interval, improving the
efficiency of the scheme.
FIG. 1 illustrates a communication system 10 according to an
embodiment. The communication system 10 includes a transmitter
1 , a receiver 12, and a frequency-dependent communication channel
13. The transmitter 1 includes a parallel array of L modulators 14,
and an adder 15. In the array, each modulator 14 is configured to
amplitude modulate a received component of an input data symbol
onto a waveform, wherein each waveform corresponds to one of the
modulators 14. For example, the 1-th modulator 14 modulates its
waveform with the 1-th component aiq of the q-th input data symbol
[aiq 2 ] in response to the receipt of the q-th input data
symbol in the transmitter 1 . In the array, each modulator 14
modulates the input data symbol onto its waveform in parallel with
the other modulators 14 of the array. Thus, the array formed by the
modulators 14 will produce a temporally synchronized array of L
modulated waveforms in response to the receipt of L data symbols.
The adder 15 is connected to sum the amplitude modulated
waveforms of the array in a temporally aligned manner to produce a
temporal sequence of output signals, e.g., ... St-i , st , St+i for
transmission over the communication channel 13. Each of the output
signals is a superposition of waveforms modulated at the same
sampling interval.
The communication channel 13 transports the signals from the
transmitter 11 to the receiver 12. The communication channel 13
may be a wireless channel, an optical fiber channel, or a wire line
channel and may be operated in simplex or duplex mode, for example.
FIG. 2 illustrates a data stream 19 that is transmitted over the
channel 13 according to an embodiment. The communication system
10 transmits a data stream 19 over the communication channel 13 as
a sequence of data blocks, e.g., consecutive data blocks (q- ), q, and
(q+1). Each data block spans Ts contiguous, non-overlapping
sampling intervals, and the different data blocks have equal temporal
length. For this reason, it will be convenient to represent the values of
any signal variable over a data block as a Ts - dimensional vector
whose individual components represent the values of the signal
variable at individual sampling intervals. That is, the components of
such a vector group together the values of the signal variable at the
sampling intervals of one data block. For that reason, each
component of such a vector will be labeled by two integer indices. The
first index will represent the position of the corresponding signal
variable in a data block, e.g., an integer in [1, Ts], and the second
integer index will represent the position of the data block in the data
stream. For example, the "k q" component of such a vector will be the
value of the corresponding signal variable during the k-th sampling
interval of the q-th data block, i.e., at sampling interval q-Ts + k.
Referring back to FIG. 1, the transport over the communication
channel 13 transforms each transmitted signal into a corresponding
signal at the receiver 12, e.g., st x t for the signals corresponding to
the sampling interval "t". The transport over the communication
channel 13 effectively convolves output signal, s , by the
communication channel's impulse response, h , and adds a noise, w ,
so that the corresponding signal x received at the receiver 12 for the
sampling interval "t" is given by:
x t = h T t-T + t . Eq. (1)
T=0
In Eq. (1), the integer TD is the delay-spread of the
communication channel 13. The delay-spread TD determines the
number of sampling intervals over which an earlier modulated and
transmitted signal can produce interference in the received signal
corresponding to a later modulated and transmitted signal.
The receiver 12 includes an input 16 and a parallel array of L
demodulators 17. The number L of demodulators 17 is typically equal
to the number of modulators 14. The input port 16 also transmits the
sequence of received signals, i.e., ... x -i, x t , xt+i to the
demodulators 17 of the array in parallel. Each demodulator 17
projects the received signals onto a conjugate waveform corresponding
to the demodulator 17 to produce an estimate, e.g., ylq , of a linear
combination of the components of the input data symbol carried by
the data block being demodulated.
Each individual ylq provides an estimate of the component aiq of
the input data symbol. Thus, the number L of demodulators 17
produces a temporally synchronized array of L estimates in response
to the receiving one data block from the communication channel 13,
e.g., [yiq y q, y q in response to receiving the q-th data block [xi q
X2q , XTsq] .
The waveforms and the conjugate waveforms may
represented by:
Each column of is an independent input waveform, and each
column of is an independent output waveform. When the delay
spread TD is greater than the S n b ol interval Ts, the waveform matrix
, and the corresponding waveform matrix , is based, at least, on
the matrix elements of the part of the impulse response function that
relates to interference between the current data block (q) and the first
previous data block (q-1), and on the matrix elements of the part of
the impulse response function that relates to interference between the
current data block (q) and the second previous data block (q-2), e.g.,
H i and H a s further explained below.
Both the waveforms ( ) and the conjugate waveforms () may be
selected to form orthonormal bases of dimension L over the complex
space of dimension Ts. The ortho normality conditions on the
waveforms and the conjugate waveforms are then described a s follows:
= L L and = I LXL Eq. (3)
Here, I L L is the LxL unit matrix, and the superscript †" denotes
"conjugate transpose". While such orthogonality and/or normality
conditions are not required, they may be advantageous for modulating
data onto the input waveforms and demodulating data form the
output waveforms, a s discussed below.
Each modulator 14 of the parallel array modulates a
corresponding component of the input data symbol onto a preselected
one of the waveforms in parallel with the other modulators 14. For
example, in response to the q-th input data symbol, the k-th
modulator 14 will produce a temporal sequence of output signals
represented by the column vector akq [ v/ k , / 2 k , ..., v/ k ] . Each of the
output signals represents the form of the modulated k-th input
waveform for one of the sampling intervals for one data block.
For each input data symbol, the adder 15 sums the L
modulated input waveforms in a temporally aligned manner. For
example, the adder 15 forms a weighted linear superposition of the
waveforms, e.g., one data block for transmission. In the linear
superposition, the starting sampling intervals of the individual
modulated waveforms are temporally aligned. In response to the
input data S ^nbol aq, the modulating and summing produces an
output data block that m a be represented by a Ts-dimensional
column vector s q. The column vector s q may be written as:
= aq Eq. (4)
In Eq. (4), each term of the sum represents the synchronized
output of a corresponding one of the modulators 14 of Figure 1. The
last form of Eq. (4) writes the output data block s q in terms of a Ts x L
complex matrix representation, , of the set of waveforms of eqs. (2)
and (3).
For each input data symbol, the transmitter 11 transmits the
data block over the communication channel 13 that couples the
transmitter to the receiver 12. The communication channel 13
distorts the data blocks due to its impulse response function and
additive noise.
According to the embodiments, the delay-spread TD of the
communication channel 13 may be greater than each data block or
symbol interval Ts of data blocks, i.e., TD > Ts. For that reason, ISI
may result from not only the immediately adjacent transmitted data
block (q-1), but also may result from the previous data blocks (q-2, q-
3, ...). For ease of exposition, the following discussion assumes that
the delay spread is at most twice the symbol interval, i.e., Ts TD 
2Ts. The present disclosure is, however, not limited to this case, and
the general situation will be discussed below. Under this assumption,
ISI only results from the two previous data blocks (q- 1 and q-2), and
Eq. (1) simplifies when written in data block form so that the q-th
transmitted data block, s q , and the q-th received data block, x q , are
related as follows:
Xq = H o-S q + H S q- + H 2 S q-2 + q Eq. (5a)
= Ho- -aq + H i - -aq i + 2 q - + w q Eq. (5b)
The LxL complex matrices Ho, H and H are formed from the impulse
response function of the communication channel 13 and are given by:
The matrices H i and H 2 produce the inter-data block interference
between the current data block (q) and the previous data block (q-1)
and the inter-data block interference between the current data block
and the previous data block (q-2), respectively. In Eq. (5), the column
vector wq for the additive noise is given by q = [wi q , W , W s q] .
The receiver 12 estimates y q by measuring correlations between
the received data block and the conjugate waveforms. The
measurement of each correlation involves evaluating an inner product
between a received data block and each of the conjugate waveforms.
In particular, the receiver 12 produces for each input data symbol, aq,
an L-dimensional estimate vector, y q, given by:
y q = x q = •a + Hi^ a i + † H aq- + †
q. Eq. (7a)
Here, the last equation results from Eq. (5b) for the channel
transformation of the transmitted data block. is the
interference term between the current data block (q) and the preceding
data block (q-1) and is the interference term between the
current data block (q) and the preceding data block (q-2).
FIG. 3 illustrates a method for performing interference
alignment for channel- adaptive waveform modulation according to an
embodiment. The method may be performed by any type of
transmitter 11 or receiver 12 that is configured for the communication
system 10. The term "device" may encompass the transmitter 11 or
the receiver 12.
In step S2 1, the device obtains at least a part of a first matrix
(Hi) and a part of a second matrix (H2) for the impulse response
function of the communication channel 13 between the transmitter 11
and the receiver 12. The part of the first matrix (Hi) relates to
channel-induced interference between a current data block (q) and a
previously transmitted first data block (q-1). The part of the second
matrix (H2) relates to channel-induced interference between the
current data block (q) and a previously transmitted second data block
(q-2).
However, the device may obtain channel-induced interference
between the current data block and any other two previously
transmitted blocks. In other words, the embodiments of the present
disclosure are not limited to only the interference relating to the
immediately preceding two data blocks. In particular, the
embodiments also encompass the situations where the delay spread
TD is greater than twice the symbol interval Ts. As a result, additional
matrices (H3, H4, ...) may be present, or any number of such matrices.
As such, the device may obtain a part of these matrices (H3, H4, ...) for
the impulse response function, where the part of the matrix H k relates
to channel-induced interference between the current data block (q)
and the data block transmitted k blocks earlier (q-k).
The impulse response may be obtained by transmitting a
sequence of pilot signals over the communication channel 13 between
the transmitter 11 and the receiver 12. Both the transmitter 11 and
the receiver 12 know the sequence of transmitted pilot signals. For
example, these pilot sequences may be programmed into these devices
at their manufacture, installation, or upgrade. The pilot signals are
transmitted on the same communication channel 13 that will be used
to transport data blocks in the communication phase. The pilot
signals may be transmitted along the forward channel from the
transmitter 11 to the receiver 12 in the communication phase. In a
duplex communication system, the pilot signals may alternately be
transmitted along the reverse communication channel 13 provided
that the reverse and forward communications use the same physical
channel and the same carrier frequency, e.g., as in time-division
duplex communications.
Then, the device measures the received pilot signals to evaluate
the impulse response function of the communication channel 13. The
evaluation of the channel's impulse response function involves
comparing received forms of the pilot signals to the transmitted forms
of the same pilot signal. The comparison determines the values of
part or all of the impulse response function, i.e., ht , for different
values of the delay "t" as measured in numbers of sampling intervals.
The comparison determines, at least, the values of hi, h h ,
which define the part of the impulse response function relating to
interference between sampling intervals of adjacent data blocks, i.e.,
H i and H described above. As such, the device obtains a
measurement, at least, of the non-zero H i matrix elements of the
channel's impulse response. Furthermore, the comparison may also
determine the value of h o of Eq. (1), which is not related to such
channel-induced inter-data block interference. In other words, the
above comparison yields the matrix H o .
In S22, the device designs or configures a set of one or more
linearly independent waveforms based on at least the obtained part of
the first matrix (H i and the obtained part of the second matrix (H
for the impulse response function. For example, the device may
design or configure the set of linearly independent waveforms such
that a first subspace spanned by the linearly independent waveforms
when multiplied by the obtained part of the first matrix at least
partially overlaps a second subspace spanned by the linearly
independent waveforms when multiplied by the obtained part of the
second matrix. In other words, the interference to the current block
from the two preceding data blocks (q- 1 and q-2) are aligned to occupy
at least a portion of the same subspace. In one particular
embodiment, the first subspace and the second subspace occupy the
same linear subspace. It is noted, however, that the embodiments
encompass interference alignment for any two not necessarily
consecutive previously transmitted blocks (e.g., q-1 and q-3, or q-2
and q-3, etc.).
The set of waveforms may include the waveforms and the
conjugate waveforms . The waveforms and the conjugate
waveforms are designed such that the inter-block interference terms
of Eq. (7a) vanishes, i.e., •• = 0 and 2• = O. In other words,
all inter-block interference terms of Eq. (7a) vanish. Thus, the
following equation may be used to obtain estimates of the linear
combinations of the components of the input data symbols.
y q = • a q + q Eq. (7b)
Various methods for constructing input and output waveforms
to ensure that the inter-block interference terms of Eq. (7a) vanish are
explained with reference to U.S. Patent No. 7,653, 120. Also, the
embodiments may in addition diagonalize the matrix .
Methods for diagonalizing the matrix - are explained with
reference to U.S. Patent No. 7,653, 120.
As discussed above, according to the embodiments, the
waveforms and the conjugate waveforms are designed such that
the waveforms and the conjugate waveforms , when multiplied by
H i and H^, occu p the same subspace. This is further explained with
reference to FIG. 4.
FIG. 4 illustrates a method of constructing the waveforms and
the conjugate waveforms such that inter-symbol interference is
removed according to embodiments.
In one embodiment, the waveforms are selected to be a subset
[ , . . . . ,UI, . . . ,UL ] of the eigenvectors of the product of the inverse of the
first matrix (i.e., H 1) and the second matrix (H2) of the impulse
response function of the communication channel. In one particular
embodiment, the subset may be the right eigenvectors of H 1 The
matrix H r 1 exists assuming H i is full rank. The corresponding
eigenvalues (di ,...,di,...,dL) may encompass any value. FIG. 4
illustrates the steps of configuring such waveforms.
In step S31, the device obtains eigenvectors and their
corresponding eigenvalues based on an eigenvector decomposition of
the matrix Hi -H2. Embodiments of the present disclosure
encompass any known technique for the decomposition of a matrix
into eigenvalues and eigenvectors such as the Cholesky decomposition
or Hessenberg decomposition, for example.
In step S32, the device configures the waveforms based on the
eigenvectors of the matrix H . For example, in one embodiment,
the device selects a subset of the right eigenvectors for use as the
waveforms .
In step S33, the device configures the conjugate waveforms 
based on the configured waveforms . The total inter- symbol
interference at the receiver is given by the following equation:
H + 2 -2 Eq. (8)
Additional terms corresponding to the matrices H 3 , H 4 , ...may be
present in Eq. (8).
For example, if is selected as a subset of the right
eigenvectors of the matrix H H then the two interference terms
corresponding to Hi and H in Eq. (8) span the same space of at most
dimension L. The two interference terms are hence aligned in the
same subspace. The conjugate waveforms are chosen to be the
projection onto a subspace of dimension L of the orthogonal
complement of this interference subspace. This ensures that the inter
block interference terms of Eq. (7a) vanish, i.e., - = O and
2 • = O.
Because the two interference parts are aligned as described
above, for certain values of the delay spread TD of the communication
channel, an increased number of waveforms L may be transmitted
during any symbol interval Ts .
Referring back to FIG. 3, in step S23, the device transmits a
sequence of data blocks over the channel, where each data block of
the sequence is a weighted linear superposition of the waveforms of
the designed set. In some embodiments, the step of transmitting
includes, for each individual data block, amplitude modulating each
waveform of the designed set responsive of receipt of an input data
symbol and linearly superimposing the modulated waveforms to
produce the individual block.
Similarly, the device (when operating as a receiver) may receive
a sequence of transmitted data blocks at the receiver. For example,
the device estimates y q by measuring correlations between the
received data block and the conjugate waveforms. The measurement
of each correlation involves evaluating an inner product between a
received data block and each of the conjugate waveforms. In
particular, the receiver 12 produces an L-dimensional estimate vector,
y q, for each input data symbol, aq based on Eq. (7a). The conjugate
waveforms are the configured conjugate waveforms , as explained
with reference to FIG. 4.
FIG. 5 illustrates a comparison of a number of orthogonal input
waveforms L achieved by the embodiments (solid line), the channeladaptive
waveform modulation from U.S. Patent No. 7,653, 120
(dashed line), and the OFMD method (dotted line) according to an
embodiment of the present invention.
When comparing the OFDM method to the channel-adaptive
waveform modulation scheme, only when TD/TS is substantially less
than one, the OFDM method performs relatively well. When TD is
greater or equal to Ts, the OFDM method achieves 0% efficiency.
When comparing the embodiments of the present disclosure to the
previous channel-adaptive waveform modulation scheme, the two
schemes have the same performance, when TD is less than or equal to
Ts. However, when TD is greater than 1.5Ts, embodiments of the
present disclosure can yield better performance. In particular, for
TD=2TS, there is a 50% improvement in terms of available orthogonal
input waveforms L.
What is claimed:
1. Amethod for transmitting a sequence of data blocks,
comprising:
obtaining at least a part of a first matrix and a part of a second
matrix for the impulse response function of a communication channel
(13) (S21), the part of the first matrix relating to channel-induced
interference between a current data block and a previously
transmitted first data block, the part of the second matrix relating to
channel-induced interference between the current data block and a
previously transmitted second data block, the second data block being
transmitted before the first data block;
designing a set of one or more linearly independent waveforms
based on at least the obtained part of the first matrix and the obtained
part of the second matrix for the impulse response function such that
a first subspace spanned by the linearly independent waveforms when
multiplied by the obtained part of the first matrix at least partially
overlaps a second subspace spanned by the linearly independent
waveforms when multiplied by the obtained part of the second matrix
(S22); and
transmitting a sequence of the data blocks over the channel
from a transmitter (1 1) (S23), each data block of the sequence being a
weighted linear superposition of the one or more waveforms of the
designed set.
2 . The method of claim 1, wherein the designing step designs the
set of linearly independent waveforms such that the first subspace
and the second subspace occupy a same linear space.
3. The method of claim 1, wherein the designed set includes a
subset of eigenvectors of a product of (1) an inverse of the first matrix
and (2) the second matrix.
4. The method of claim 3, wherein the subset comprises right
eigenvectors of the product.
5. The method of claim 1, wherein the designing step further
includes:
obtaining eigenvectors and corresponding eigenvalues based on
an eigenvector decomposition of a product of (1) an inverse of the first
matrix and (2) the second matrix (S31); and
selecting a subset of the obtained eigenvectors (S32).
6. The method of claim 5, wherein the designing step further
includes:
configuring a second set of waveforms based on the selected
subset (S33), wherein the configured second set of waveforms is an
orthogonal complement of the product of the selected subset and
either of the first or the second matrices.
7 . The method of claim 1, wherein the first data block immediately
precedes the current data block and the second data block
immediately precedes the first data block.
8. The method of claim 1, further comprising:
transmitting a set of pilot signals over the communication
channel that is between the transmitter (1 1) and a receiver (12), the
part of the first matrix and the part of the second matrix for the
impulse response function being obtained responsive to
measurements of said pilot signals.
9. The method of claim 1, further comprising:
for each one of the data blocks of the sequence, modulating the
waveforms of the designed set to have amplitudes responsive of a
received input data symbol and linearly superimposing the modulated
waveforms to produce the each one of the data blocks.
10. The method of claim 1, wherein the designed set has a number
of the waveforms equal to one half of a symbol interval when the delay
spread is twice the symbol interval.
11. An apparatus for communicating data, comprising:
a transmitter (1 1) including
an array of modulators (14), each modulator (14) being
configured to modulate an amplitude of a corresponding one of
linearly independent waveforms over a sequence of sampling
intervals in response to receipt of each of a sequence of input
data symbols; and
an adder (15) configured to form a sequence of data
blocks, each data block being a linear superposition of
modulated transmitter waveforms produced by the modulators
responsive to receipt of one of the input data symbols, the adder
configured to transmit the data blocks via a communication
channel (13); and
wherein the transmitter configures the modulated
waveforms in a manner responsive to a part of a first matrix and
a part of a second matrix for the impulse response function of a
communication channel, the part of the first matrix relating to
channel-induced interference between a current data block and
a previously transmitted first data block, the part of the second
matrix relating to channel-induced interference between the
current data block and a previously transmitted second data
block, the second data block being transmitted before the first
data block,
wherein the transmitter configures the modulated
waveforms such that a first subspace spanned by the
modulated waveforms when multiplied by the part of the first
matrix at least partially overlaps a second subspace spanned by
the modulated waveforms when multiplied by the part of the
second matrix.