BACKGROUND OF THE INVENTION
This invention relates to the detection and correction
of I/Q crosstalk in complex quadraturemodulated signals, and to the removal of
unwanted complex conjugate components from a signal.
Quadrature modulation is commonly used to transmit two
signal components on a single carrier. The two components comprise an inphase component
'I' and a quadraturephase component 'Q'. These signals have to be demodulated at
the receiving end, and this involves downconverting the signals for the frequency
of the transmission carrier. Quadrature downconversion to a zero intermediate frequency
(IF) is a commonlyused technique in RF (radio frequency) systems since it simplifies
filtering, and allows compact tunable systems to be realised. However, an RF system
employing a quadrature downconvertor (also known as an I/Q downconvertor) suffers
to some degree from I/Q crosstalk in its output. That is, part of the original 'I'
signal ends up in the 'Q' channel and vice versa. Sometimes this is not a
problem, for example if the crosstalk represents a simple time delay which is dealt
with in timing recovery stages; mathematically, that case corresponds to a simple
(frequencydependent) rotation of the I/Q axes.
A more troublesome type of crosstalk is the case where
a positive baseband (zero IF) frequency +&ohgr;_{0} is partially mapped
to a negative frequency &ohgr;_{0}. Mathematically, this case is modelled
by an unwanted additive complex conjugate of the original signal. It is this
type of crosstalk the present proposal is designed to address, with the added capability
of dealing with the case where the added conjugate term is distributed in time.
The mapping &ohgr;_{0} to +&ohgr;_{0} is normally present as
well in this case. Circumventing such crosstalk by use of a lowIF involves the
use of bulky imagechannel filtering. Such filters, as well as being large, are
relatively expensive and far from perfect. The present invention has as an object
the detection and preferably reduction of such crosstalk, thereby allowing the use
of I/Q downconversion in performancecritical systems.
Reference may be made to
United States Patent Application 2005/0089120
which describes a frequency shift keying demodulator system that provides
programmable tunable spectral shaping; to
United States Patent 6,765,623
which describes the measurement of correlation between I and Q components;
and to
United States Patent 4,581,586
which describes the demodulation of quadrature phase shift keyed signals
in a manner that seeks to reduce crosstalk between I and Q channels.
SUMMARY OF THE INVENTION
The invention in its various aspects is defined in the
independent claims below to which reference may now be made. Advantageous features
are set forth in the appendent claims.
Preferred embodiments of the invention are described below
by way of example with reference to the drawings. In each of these embodiments,
I/Q crosstalk is reduced in a complex signal. A complex signal containing inphase
I and quadraturephase Q components is received at an input, and its complex conjugate
is formed. This complex conjugate signal is in general delayed to provide one or
more complex conjugate signals with different delays. In a very simple ('onetap')
system no delay element need be used. These signals are then weighted in accordance
with respective weighting factors. The thusweighted signals are combined with each
other and with the input signal or a signal derived therefrom in the sense such
as to reduce I/Q crosstalk. The weighting factors are generated by providing the
input signal or a signal derived therefrom as a first signal, and providing the
input signal and/or a signal or signals derived therefrom as one or more second
signals. Where there are plural second signals they have different delays corresponding
to the delays of the complex conjugate signals. The complex nonconjugate correlation
of the first signal and the or each of the second signals is then generated to provide
one or more complex correlation signals, and each of the complex correlation signals
is accumulated and multiplied by a convergence factor, to provide a weighting factor
for a respective first signal.
The preferred use of delays allows for effective cancellation
even if the crosstalk is frequencyselective. Furthermore, although the following
discussion is directed primarily to the correction of downconversion errors, any
similar I/Q errors in the transmitted signal can also be corrected.
It is assumed that there are no intentional conjugate
terms in the desired signals, and that the signals are noise like. This is true,
for instance, for DVBT and DAB COFDM signals. It is also true for OFDM signals
used in other applications such as in mobile telephony.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will now be described in more detail by way
of example with reference to the accompanying drawings, in which:

Fig. 1 is a block diagram of a first crosstalk canceller embodying the invention;

Fig. 2 is a block diagram of a second crosstalk canceller embodying the invention
based on Fig. 1 but extended to detect 'twosided' crosstalk terms; and

Fig. 3 is a block diagram of a third crosstalk canceller embodying the invention
also based on Fig. 1 and including a secondary LMS equalizer to compensate for certain
errors introduced in the crosstalk canceller.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
Various preferred embodiments of the invention will now
be described by way of example with reference to the drawings. The embodiments may
each be implemented in discrete hardware components, but may alternatively be implemented
in software in for example a digital signal processor (DSP), in which case the figures
should be regarded as flow charts. In each case the structure will first be described,
followed by the manner of its operation.
First Embodiment  Fig. 1
The first preferred embodiment of the invention is illustrated
in the block circuit diagram of Fig. 1. The figure illustrates a crosstalk canceller
10 which has an input 12 for receiving a complex baseband input (I/P) signal, that
is a signal that is quadrature modulated with inphase (I) and quadraturephase
(Q) components. It is assumed that the input signal suffers from I/Q crosstalk.
To the input 12 is connected the noninverting input 16 of a combining circuit in
the form of a subtractor 14, which subtracts from the input signal a correction
signal, the generation of which will be described. The output of the subtractor
14 is applied to an output 18 and constitutes the output (O/P) of the canceller
10.
Also connected to the input 12 is a complex conjugate circuit
20 which generates the complex conjugate of the input signal. That is, it inverts
the phase of the input signal, in known manner. If the input signal is designated
X, then the complex conjugate is conventionally represented X*. Thus the complex
conjugate circuit 20 is marked with an asterisk (*) in Fig. 1. The output of the
complex conjugate circuit is then applied to a tapped delay line 22 consisting of
one or more seriallyconnected delays 22_{1} ... 22_{N}. In the
example there are two delays (i.e. N = 2), giving three taps, but there may be just
one delay, or even no delays, or there may be more than two, typically up to about
twelve, depending on the application. The number of delays is not critical to the
principle of operation. The duration of the delays is selected in relation to the
intended signal. In a sampled system, the delays are multiples of the sampling interval,
in the sequence 0,1,2 etc., though in principle other spacings could be used. It
is desirable for convergence that the sampling interval be associated with a Nyquist
bandwidth which just encompasses the signal.
The outputs of the N+1 taps, that is the output of the
complex conjugate circuit 20 and each of the delays 22_{N}, are complex
conjugate signals which are applied to respective multipliers 24_{0} ...
24_{N} where they are each multiplied by a tap weight or weighting factor
in the form of a respective complex number. The outputs of the N multipliers are
then combined to produce the sum of the signals produced by the multipliers, in
the manner of a transversal filter. As illustrated, the summing is achieved by a
series of N twoinput adder circuits 26_{0} ... 26_{N1}. The final
output of the adder circuits 26 is applied as the correction signal to the subtractive
input 28 of the subtractor 14 to be combined with the input signal in the sense
such as to reduce I/Q crosstalk in the input signal. The output of the subtractor
14 constitutes the output of the canceller.
The arrangement for generating the tap weights applied
to the multipliers 24_{0} ... 24_{N} will now be described. The
tap weight generation circuit or detecting circuit generally illustrated at 30 has
an input 32 coupled to receive the corrected signal from the subtractor 14 which
is applied to the output 18. To the input 32 is connected a tapped delay line 34
comprising N delays 34_{1} ... 34_{N} corresponding in delay length
and in number to the number of delays in the delay line 22. The tapped delay line
thus provides N+1 signals, namely the signal at the input 32 and the signal at the
output of each of the delays 34_{1} ... 34_{N}.
The circuit 30 also includes N+1 complex correlators 36_{0}
... 36_{N}. In what follows all references to 'correlator' are in fact nonconjugate
correlators, i.e. they do not, as is usual, take the complex conjugate of one input
before performing the complex multiplication operation. A first input to each correlator
36 is connected to the input 32 of circuit 30 to receive a first signal provided
thereby which comprises the input signal or a signal derived therefrom, here
via the subtractor 14. The second input to each correlator 36 is connected
to a respective one of the taps in the tapped delay line 34 to receive a second
signal provided thereby which comprises the input signal or a signal derived therefrom,
here via the subtractor 14 and the delay line 34. Thus the first correlator
36_{0} is, in fact, a nonconjugate autocorrelator. The output of each correlator
36_{0} ... 36_{N} is a nonconjugate complexnumber correlation
signal which is then applied to a respective one of N+1 multipliers 38_{0}
... 38_{N}. These multipliers multiply the complex correlation output of
the respective correlator by a constant but selectable convergence or update factor.
This will typically be less than 0.1 and may, for example, be down to 0.00001, depending
on the speed of convergence required. Typical values can be 0.001 or 0.00005.
Finally in the tap weight generation circuit 30, N+1 integration
or accumulation circuits 40_{0} ... 40_{N} are each connected to
a respective one of the convergence factor multipliers 38_{0} ... 38_{N}.
These are of simple and in itself wellknown structure. The structure of the integration
circuit 40_{2} is shown in Fig. 1, and is seen to consist of a twoinput
adder 42 and a delay 44 of the same delay length as the delays in the delay lines
22 and 34. One input to the adder 42 is connected to the output of the respective
convergence factor multiplier 38, and the delay 44 is coupled from the output to
the other input of the adder 42. Thus the outputs of the multipliers 38 are steadily
accumulated for what will, in due course, be a large number of iterations to increase
the accuracy of the correction and track changes in the crosstalk in the input signal.
The other integration circuits 40 are similarly constructed. The outputs of the
integration circuits 40_{0} ... 40_{N} are then applied as the coefficient
or weighting factor inputs to the multipliers 24_{0} ... 24_{N}
respectively, as previously noted. The multipliers 38 and integration circuits 40
may be arranged in the other order.
Thus, in operation, a complex baseband signal suffering
from complexconjugate I/Q crosstalk is received at the input 12. It is split into
two paths. The main path passes to the subtractor 14, where the correction signal
is subtracted from the input signal; the resultant signal from the subtractor 14
forms the output of the crosstalk canceller. The input signal is also applied through
the complex conjugate circuit 20, the output of which feeds the delay line 22. The
output of each tap from the delay line is multiplied by the complexnumber tap weight,
and summed to produce the correction signal. The values of the tap weights are steered
by the outputs of the complex correlators 36 fed from the canceller output.
The action of correlator 36_{0} is to ignore the
wanted component, which will give a zero nonconjugate autocorrelation output, and
to isolate the unwanted complex conjugate component. This is then used to allow
a proportion of the complexconjugate generated in circuit 20 to be subtracted from
the input signal in subtractor 14, and the operation then repeats in an interactive
manner. The system will converge to a steady state in which the amount of the complex
conjugate in the output of the subtractor is minimised or indeed eliminated.
The other correlators 36_{1} ... 36_{N}
operate similarly but with the corresponding delays. Each correlator effectively
searches for complexconjugate I/Q crosstalk in the output, at the corresponding
delay, and steers the particular tap weight to minimize it.
As noted above, each of the delay lines 22, 34 contains
two delays. If fewer delays are required, the unwanted ones (and associated components)
are eliminated. If further delays are used, the components 22_{i}, 24_{i},
26_{i}, 34_{i}, 36_{i}, 38_{i}, and 40_{i}
are repeated down the page, as indicated by the downward arrows 48 on Fig. 1.
The components which generate the weighting factors applied
to the multipliers 24 will be seen to operate as a crosstalk detector, and can be
used independently to that end, with the weighting factor signals being appropriately
combined in any desired manner. The components which generate the correction signal
28 are similarly seen to constitute a correction signal generator, even if used
without the subtractor 14.
Theory  background
The techniques described are of value because we have appreciated
that many commonlyencountered I/Q impairments can be regarded as equivalent to
complex conjugate crosstalk. In particular, this can be shown to be true for:
 baseband crosstalk, where a proportion of I is added into Q and vice versa
 quadrature error, where the I and Q axes are not exactly orthogonal; note that
this can occur at either the upconversion or downconversion ends of a transmission
system
 I/Q gain inequality, where the gains applied to the I component and Q component
are different.
Under certain conditions, this crosstalk can be minimised
by the use of a system embodying the invention. These conditions are that:
 the frequency spectrum of the signal is at least approximately noiselike
 the signal does not contain any wanted complex conjugate components
 the signal is bandlimited (i.e. of finite effective band without significant
outofband components).
The conditions are in particular met by orthogonal frequencydivision
multiplex (OFDM) signals. OFDM signals are used in many modern transmission signals
in the fields of digital broadcasting, telephoning, and broadband communications.
Theory  mathematical analysis
The steadystate situation will be considered in detail.
It can be shown that the method approximates to a standard least mean square (LMS)
system so far as the crosstalk terms are concerned, and standard LMS convergence
theory can then be applied to show that convergence will take place, at least for
practical low levels of crosstalk. Thus it is sufficient to consider the steadystate
situation.
Looking at the steadystate case, we start by assuming
a white noiselike signal Y(n) having frequency components n is corrupted
by frequencyselective complexconjugate crosstalk to form a new signal
X(n).
That is:
$$X\left(n\right)=Y\left(n\right)+{\displaystyle \sum _{k=0}^{K1}}{\mathit{\&ggr;}}_{k}{Y}^{*}\left(n,,k\right)$$
The second term on the righthand side of equation (1)
represents the complexconjugate crosstalk. The coefficients &ggr;
_{k}
in this term form an FIR (finite impulse response) filter acting on the signal
conjugate Y* to model the frequency selectivity. A total of K coefficients
are included, i.e. 0 to K.
Referring to Fig. 1, the requirement for the correlators
36 to be in steadystate can be expressed as:
$$\mathbf{E}\left\{Z,\left(n\right),,Z,,\left(n,,j\right)\right\}=\mathrm{0................}j\in \mathrm{0...}N1$$
where E denotes statistical expectation, N is the number of taps on
the delay line, and Z(n) is the output signal.
Considering the inputside delay line 22 and associated
taps (h_{j}
say), together with the input X(n), allows us to write
Z(n) as:
$$Z\left(n\right)=X\left(n\right)+{\displaystyle \sum _{j=0}^{N1}}{h}_{j}{X}^{*}\left(n,,j\right)$$
Hence:
$$Z\left(n\right)=Y\left(n\right)+{\displaystyle \sum _{k=0}^{K1}}{\mathit{\&ggr;}}_{k}{Y}^{*}\left(n,,k\right){\displaystyle \sum _{j=0}^{N1}}{h}_{j}\left({Y}^{*},,\left(n,,k\right),+,{\displaystyle \sum _{k=0}^{K1}},{\mathit{\&ggr;}}_{k}^{*},,Y,,\left(n,,j,,k\right)\right)$$
Second order small quantities can be ignored;
h and y are considered firstorder small quantities. With this approximation,
and choosing N=K, simplifies the expression to:
$$Z\left(n\right)=Y\left(n\right)+{\displaystyle \sum _{k=0}^{K1}}\left({\mathit{\&ggr;}}_{k},,{h}_{k}\right){Y}^{*}\left(n,,k\right)$$
Now, if the correlator bank reaches steadystate with
h_{k} = &ggr;_{k}
for all k, then the output Z(n) will be free of crosstalk terms
to a degree determined by the secondorder terms which have been ignored.
To show this is indeed the case, apply the condition expressed
in (2) to Z(n) as defined in (5):
$$\mathbf{E}\left\{Z,\left(n\right),,Z,,\left(n,,j\right)\right\}=\mathbf{E}\left\{[,Y,\left(n\right),+,{\displaystyle \sum _{k=0}^{K1}},\left({\mathit{\&ggr;}}_{k},,{h}_{k}\right),,{Y}^{*},,\left(n,,k\right),],[,Y,,\left(n,,j\right),+,{\displaystyle \sum _{k=0}^{K1}},\left({\mathit{\&ggr;}}_{k},,{h}_{k}\right),,{Y}^{*},,\left(n,,j,,k\right),]\right\}$$
which must be equal to zero for all j.
Since Y is whitenoiselike, its autocorrelation
is impulsive, and the only expectation terms which return a nonzero value are of
the form E{Y(l)Y*(m)} with lm. In particular,
terms of the form E{Y(l)Y(m)} are always zero.
This allows (6) to be simplified to:
$$\mathbf{E}\left\{Z,\left(n\right),,Z,,\left(n,,j\right)\right\}=\mathbf{E}\left\{[,Y,\left(n\right),{\displaystyle \sum _{k=0}^{K1}},\left({\mathit{\&ggr;}}_{k},,{h}_{k}\right),,{Y}^{*},,\left(n,,j,,k\right),+,Y,,\left(n,,j\right),{\displaystyle \sum _{k=0}^{K1}},\left({\mathit{\&ggr;}}_{k},,{h}_{k}\right),,{Y}^{*},,\left(n,,k\right),]\right\}$$
The first term of the righthand side of this expectation
only contributes if j=k=0. This contribution is zero if:
$${h}_{0}={\mathit{\&ggr;}}_{0}$$
The second term has potential contributions at all vales
of k within the range 0 to K1, again requiring j=k. These terms are
only all zero if:
$${h}_{k}={\mathit{\&ggr;}}_{k}\mathrm{..............................}k\in \mathrm{0...}K1$$
Hence if (9) is satisfied, (6) equates to zero, the correlators
are in steadystate, and (5) reduces to:
$$Z\left(n\right)=Y\left(n\right)$$
as is required. (The fact that the index0 term has two contributors to the expectation
calculation points to the fact that that correlator has twice the gain of the others.
This can be compensated for by dividing the associated convergence constant µ
by two, though this is only optional in practice.)
The above mathematical discussion has assumed a white noiselike
signal. In fact we believe that the system will still work for most practical nonwhite
signals, though it may leave a slightly greater residual error and take longer to
converge.
The system has been described and illustrated in relation
to quadrature downconversion to zero IF, that is to baseband where the centre frequency
of the transmitted band is downconverted to zero. There are however applications
which use a socalled 'low IF', which is an IF which is within the bandwidth of
the transmitted signal or is close to it, though not actually at zero. The system
is also applicable to such downconverted signals.
Second Embodiment  Fig. 2
The arrangement of Fig. 1 only uses samples prior to a
given sample to provide the correction for that sample. However the subsequent samples
are equally important. The embodiment of Fig. 2 uses both prior and subsequent samples.
The arrangement of Fig. 2 is largely the same as Fig. 1
and thus only the differences will be described to avoid unnecessary repetition.
The canceller 50 of Fig. 2 includes additional delays 52 and 54. Each of these delays
provides a delay of about half the overall delay of the delay lines 22, 34. The
first additional delay is between the input 12 and the noninverting input 16 to
the subtractor 14. The signal from the input to the first delay line 22 is not delayed.
The second additional delay 54 is between the output of the subtractor 14 and the
output 18 of the canceller, on the one hand, and those inputs to the complex correlators
that are not connected to the delay line 34, on the other. Thus the correlators
36 are coupled to input 32a of the tap weight generation circuit 30 which is connected
to the delay 54, and the delay line 34 is coupled to an input 32b connected directly
to the output of subtractor 14. Thus the correction signal at any instant is derived
from portions e.g. samples of the input signal both prior to and subsequent to that
instant.
The algorithm described above is generalised to deal with
'twosided' crosstalk terms as follows, that is to say, equation (1) becomes:
$$X\left(n\right)=Y\left(n\right)+{\displaystyle \sum _{k=K}^{K}}{\mathit{\&ggr;}}_{k}{Y}^{*}\left(n,,k\right)$$
The total number of terms in the expansion is now 2K+1,
and they are evenly spread over positive and negative values of delay. At first
sight this may appear noncausal (impossible), but it can happen that the nonconjugate
term Y(n) exists timeadvanced elsewhere in the system and generates the
crosstalk in such a way that the situation modelled by equation (11) is realised.
The cancellation system of Fig. 2 deals with this case
by the use of the two extra delays 52, 54 incorporated in the system as shown. Choosing
N=K allows the twosided crosstalk to be dealt with in the same way as previously
described for singlesided crosstalk in relation to Fig. 1..
Third Embodiment  Fig. 3
Referring back to equation (4) it will be recalled that
certain secondorder terms were ignored. These are terms in Y(n), not Y*(n). This
implies that a small frequency response error will have been introduced on the wanted
output. it is believed that this will be negligible in most cases, but if critical
a secondary LMS (least mean square) equalizer can be included to restore the signal
frequency response to virtually its original characteristic.
This is illustrated in Fig. 3 where the crosstalk canceller
60 includes an LMS equalizer 62 which is connected between the output of the subtractor
14 and the output 18 of the canceller. The LMS equalizer is itself of wellknown
construction and has a plurality of q taps. The output of the equalizer 62 constitutes
the output 18 of the canceller and is applied to the inverting input of a subtractor
64. A delay 66 of delay substantially equal to half the delay in the LMS equalizer
62 is coupled to the input 12 and provides a signal to the noninverting input of
the subtractor 64. The Y(n) terms at the output of the subtractor 14 and the output
of the delay 66 should be the same. If they are not, the equalizer with the subtractor
loop tends to reduce the difference.
Although three separate embodiments have been described
it will be appreciated that their features may be used in combinations other than
those specifically shown and described. In particular, the twosided variant of
Fig. 2 may be used with the supplemental LMS equalizer of Fig. 3.
Experimental simulation results
A threetap simulation was performed to check that the
system works as intended. A complex white noise source was used as the 'wanted'
signal and deliberately corrupted with delayed conjugate terms. In this simulation,
the run length was 2000000 iterations; the convergence factor µ was 0.00005
for all the taps; the wanted signal power was defined as 1.0; and the subsidiary
LMS convergence factor in the LMS equalizer was 0.0001
In the nomenclature of equation (1), the coefficients &ggr;
_{k}
were arbitrarily chosen as:
$${\mathit{\&ggr;}}_{0}=\mathrm{\&bgr;}\left(\mathrm{0.8},\mathrm{+},\mathrm{j},,\mathrm{0.05}\right)$$
$${\mathit{\&ggr;}}_{1}=\mathrm{\&bgr;}\left(0.5,\mathrm{+},\mathrm{j},,\mathrm{0.05}\right)$$
$${\mathit{\&ggr;}}_{2}=\mathrm{\&bgr;}\left(\mathrm{0.3},,\mathrm{j},,\mathrm{0.1225}\right)$$
All higherindex terms were zero. The sum of the squared
magnitudes of the above terms is unity if &bgr; =1. In practice &bgr; was set
to be either 0.1 (representing 20dB I/Q crosstalk) or 0.5 (representing 6dB).
Two parameters were measured on each run. The first was
the final crosstalk reduction achieved in dB with respect to the initial crosstalk
level, and secondly the power of the difference signal formed by subtracting the
original uncorrupted signal from the final output.
Run 1: Crosstalk level 20dB
Corrected crosstalk level: 0.00085 + j0.00073 (39dB on
uncorrected level)
Power of difference signal: 37dB (with respect to unity wanted output power)
Run 2a: Crosstalk level 6dB
Corrected crosstalk level: 0.0059 + j0.00052 (38.5dB on
uncorrected level)
Power of difference signal: 10dB (with respect to unity wanted output power)
Run 2b: Crosstalk level 6dB; additional 24tap LMS equalizer to correct frequency
response
Corrected crosstalk level: 0.0059 + j0.00052 (38.5dB on
uncorrected level)
Power of difference signal: 36dB (with respect to unity wanted output power)
In the case of run 2b, the difference signal was taken between the input and output
signals, where the former had been delayed by half the LMS length (i.e. 12).
It will be seen that the system is effective in very significantly
reducing the unwanted complex conjugate crosstalk, with a typical crosstalk rejection
in excess of 30dB. The system works well in its simplest form for levels of incoming
I/Q crosstalk of 20dB down on the main input signal. The system can be readily extended
to correct incidental frequency response errors which occur when the crosstalk level
is higher by using the supplemental LMS equaliser 62 as in Fig. 3. In this case
levels of crosstalk up to 6dB can be comfortably accommodated. The systems described
are of wide application and can be used in broadcasting applications, including
DVBT (digital video broadcasting  terrestrial) receivers, transposers, radio cameras,
and onchannel repeaters, and DAB (digital audio  or sound  broadcasting) receivers.