GB2150719A - Hilbert transform correction system - Google Patents

Abstract

A Hilbert transform correction system for modifying two input signals which should be in phase quadrature thereby to produce two corresponding output signals which are in phase quadrature, comprising weighting signal producing means 3 fed with the two input signals X1,X2 so as to produce two weighting signals, W1,W2, two mutlipliers 6, 7 each fed with one of the two input signals and each fed also with one of the weighting signals so as to produce two product signals 8,9 and an adder 10 fed with the other of the two input signals and one of the product signals, thereby to produce a sum signal Y2 which corresponds to one of the two output signals, the other Y1 of the two output signals comprising the other of the product signals. <IMAGE>

Description

SPECIFICATION Hilbert transform correction system This invention relates to Hilbert transform correction systems.

A Hilbert transform applied to a waveform produces a quadrature phase change in all the frequency components of the waveform. Phase quadrature related signals at baseband are required in various signal processing systems such as quadrature demodulators and complex adaptive antenna array processing systems. In order to produce quadrature related signals, a Hilbert transform is applied and for example a signal a cosA is transformed by a Hilbert transform to a signal a sinA whereby two quadrature related signals are made available, (where a and A are, in general, time-varying). This process can be carried out digitally but is costly in computation. Alternatively, a good Hilbert transform can be performed very simply in analogue form at RF or IF, and then the two waveforms are down-converted to baseband for A/D conversion.Apparatus such as hybrid devices for performing the transform at IF are well known but due to a number of factors, such as phase or amplitude imbalance in the down-conversion process, ideal Hilbert transforms at baseband can not easily be produced, with the result that a true phase quadrature relationship (with amplitude balance) essential for optimising complex signal processing performance may not be produced at all times.

It is an object of the present invention to provide a digital Hilbert transform correction system effective for correcting spurious overall phase shifts (or amplitude imbalance; whereby a phase quadrature relationship is maintained at substantially all times between two signals. This can be implemented very economically by comparison with a digital Hilbert transform process.

According to the present invention a Hilbert transform correction system for modifying two input signals which should be in phase quadrature thereby to produce two corresponding output signals which are in phase quadrature, comprises weighting signal producing means fed with the two input signals so as to produce two weighting signals, two multipliers each fed with one of the two input signals and each fed also with one of the weighting signals so as to produce two product signals and an adder fed with the other of the two input signals and one of the product signals, thereby to produce a sum signal which corresponds to one of the two output signals, the other of the two output signals comprising the other of the two product signals.

The adder may be fed with the other of the two input signals via a first delay device and the other of the two product signals may be fed via a second delay device to provide the other of the two output signals, whereby signal delays contributed by the system are compensated for.

The first delay device may be arranged to contribute a delay corresponding to the delay which is contributed by the multiplier which feeds the second delay device and the second delay device may be arranged to contribute the delay which corresponds to the delay which is contributed by the adder.

The weighting signal producing means may comprise a first squarer and averager to which the said one input signal is fed to provide a first averaged square signal, a second squarer and averager to which the said other input signal is fed to provide a second averaged square signal, a multiplier and averager responsive to the two input signals for producing an averaged product signal, a first divider operative to divide the averaged product signal by the first averaged square signal to produce a first quotient signal which constitutes one of the two weighting signals, a second divider operative to divide the second averaged square signal by the first averaged square signal to produce a second quotient signal, a further squarer operative to square the first quotient signal to produce a quotient squared signal and a data processor responsive to the quotient squared signal and to the second quotient signal for producing the other of the two weighting signals.

The data processor may be operative to take the square root of the difference between the second quotient signal and the squared quotient signal thereby to produce the said other of the two weighting signals.

Alternatively the data processor may be arranged to be operative for halving one plus the difference between the second quotient signal and the squared quotient signal thereby to produce the said other of the two weighting signals.

The system may be a digital system and the two input signals may accordingly be digital signals which are processed digitally throughout the system.

The two input signals may be base band signals produced by a pair of base band mixers which are nominally phase quadrature related, the base band mixers being fed with quadrature related local oscillator signals.

One embodiment of the invention will now be described solely by way of example with reference to the accompanying drawings in which: Figure 1 is a block schematic diagram of a Hilbert transform correction system; Figure 2 is a block schematic diagram of a part of the correction system shown in Fig. 1; and Figure 3 is a block schematic diagram showing one application of a Hilbert transform correction system.

Referring now to Fig. 1, a Hilbert transform correction system is fed via two input lines 1 and 2 with two input signals x, and x2 which are nominally phase quadrature related. Under ideal conditions if x1 corresponded to a signal a cosA then x2 would be a sinA. Due however to imperfections in Hilbert transform devices the signals x, and x2 may not be precisely the same and the difference may be represented by an amplitude difference k and a phase difference B and so therefore if x1 = a cosA, then x2 = kasin(A + B). We assume k and B are virtually constant over the signal band width and are slowly varying in time, compared with the signal modulation functions a and A. It follows that the weighting signals w1 and w2 will also be slowly varying in time.

In order to correct for these deviations from the ideal conditions, the signals xl and x2 which are applied to the input lines 1 and 2 respectively are fed to a weighting signal generator 3 which produces on lines 4 and 5 two weighting signals w1 and w2 respectively. The input signal on the line 1 is fed also to a pair of multipliers 6 and 7 the multiplier 6 being fed with the weighting signal w1 and the multiplier 7 being fed with the weighting signal w2. The multiplier 7 is operative to produce a product signal on an output line 8 and the multiplier 6 is operative to produce a product signal on an output line 9. The product signal on the output line 8 is fed to an adder 10 which is fed also with the input signal x2 via a first delay device 11.The product signal on the line 9 is fed via a second delay device 1 2 to produce on a line 1 3 an output signal y1 which is one of two output signals which are in phase quadrature, the other output signal y2 being produced by the adder 10 on an output line 14. As will hereinafter be shown, the signal y1 = ka cosB cosA and the signal y2 = ka cosB sinA and thus the two signals Y1 and Y2 are in phase quadrature, and of equal mean power.

The weighting signal generator 3 of Fig. 1 may be constructed as shown in Fig. 2. Referring now to Fig. 2 the weighting signal generator 3 comprises a first squarer 1 5 to which the input signal x1 is fed on the line 1 and a second squarer 1 6 to which the input signal x2 is fed on the line 2. The input signals x1 and x2 are also fed to a multiplier 1 7. The squarer 1 5 is arranged to feed an averager 18, the squarer 1 6 is arranged to feed an averager 19, and the multiplier 1 7 is arranged to feed an averager 20.Output signals from the averagers 1 8 and 20 are fed to a divider 21 and output signals from the averagers 18 and 19 are fed to a further divider 22. The output signal from the divider 21 constitutes (after change of sign) the weighting signal w2 which is present on the line 5. (The sign change can be applied at any convenient point between multiplier 1 7 and multiplier 7-it is simply a matter of changing the sign of an input to any of the devices 1 7, 20, 21 or 7). The weighting signal w2 is fed to a squarer 23 to produce a squared signal which is fed via a line 24 to a data processor 25 fed from the divider 22 on a line 26.The data processor 25 serves to provide the weighting signal w1 on the line 4 and may either be arranged to calculate k2-w22 where k2 corresponds to the output signal from the divider 22 or alternatively the data processor 25 may be arranged to give I(1 - k2 - w22).

A Hilbert transform correction system as just before described with reference to Fig. 1 and Fig. 2 may be used in a system as shown in Fig. 3. Referring now to Fig. 3 a base band phase quadrature demodulator system comprises an in-phase power splitter 41 to which received signals are fed on a line 42 and which serves to produce two output signals on lines 27 and 28 which are in-phase and fed to a pair of mixers 29 and 30 respectively. The mixers 29 and 30 are fed from a Hilbert transform network 31 on lines 32 and 33 with phase quadrature related local oscillator signals which correspond to the carrier frequency of the signals on line 42 whereby base band demodulated signals are provided on lines 34 and 35 at the output of the mixers 29 and 30 respectively.The signals at base band on the lines 34 and 35 are fed via A to D converters 36 and 37 to a Hilbert transform correction system 38 of the kind hereinbefore described, to produce phase corrected quadrature related output signals y1 and y2 on the lines 39 and 40 respectively.

This system will compensate for amplitude imbalances which could arise in the power dividers 41 and 31, the mixers 29 and 30 or the A/D converters 36 and 37, and for phase errors arising from mismatched cable lengths, in particular cables 27, 28, 32 and 33.

In order to give a better understanding of the invention the following mathematical analysis will now be considered.

Consider an IF or RF signal given by a cos(wot + A) where w0 is the carrier angular frequency and a and A are time-varying but with bandwidth small compared with wO, i.e. the signal is fractionally narrow band. After complex down-conversion with a local oscillator at we two signals a cosA and a sinA are obtained in an ideal case. These may be considered as the real and imaginary components of the complex signal a exp iA. The term a sinA is the Hilbert transform of a cosA, and the quality of the down-conversion, and the cancellation performance achievable (in an interference cancellation application) depends on the accuracy of the Hilbert transform, which may actually be performed in an IF circuit. There may also be common amplitude and phase factors applying to both channels, but these do not affect the quality of the Hilbert transform, of concern here.

With errors present we can express the outputs as a cosA and ka sin(A + B) where generally k = 1 and B is small, for small errors. Common amplitude and phase factors have been taken out to make the real component equal to the ideal value. We assume the factors k and B are the same for all components of the spectrum of a exp iA, because these errors originate at IF where the signal is narrowband. Two properties required of two signals which are Hilbert transforms are that their mean powers should be equal and that they should be precisely in phase quadrature for all frequency components. This will occur only if k = 1 and B = O. If we expand sin(A + B): sin(A + B) = sinA cosB + cosA sinB we see that it contains a component proportional to cosA, which, in principle, can be removed by adding a suitable multiple of the waveform cosA, if available.In fact cosA is available from the other baseband sub-channel, and after cancellation of this component we are left with sinA cosB which is strictly orthogonal to cosA but not equal to it in power. We can either multiply this component by secB or the cos component by cosB to obtain two orthogonal components with the same mean power, as required. The second of these options is perhaps simpler, requiring less delay compensation.

The required correction system is shown in Fig. 1 and Fig. 2. Given x1 = a cosA and x2 = ka sin(A + B) we have x,x2 = ka2sin(A + B)cosA = aka2(sin(2A + B) + sin B) and x1x2 = aka2 sin B where a2 is the power in the ideal complex signal, and we assume the mean value of sin(2A + B), or rather a2sin(2A + B) is zero. The overbar indicates a time average. Similarly we find X,2 = a2 so that w2 = - x1x2/x12 = - k sin B The output is y2 = x2 + w2x1 = kasin(A + B) - ksinB acosA kacosB sinA The output y2 is now correctly in quadrature with the input x, but differs in amplitude.We must either divide y2 by kcosB or multiply xl by kcosB to obtain equal power outputs. This second alternative is implemented by the circuit in Fig. 2.

We note that (k cosB)2 = k2 ~ k2sin2B = k2 ~ w22 and that k2 is the ratio of the input powers, i.e.

k2 = x22/xl2 as x22 = Tk2a2 To avoid taking the square root, if kcosB is near 1 we can put kcosB = 1 + d where d is small, so that (1 + d)2-1 + 2d and so 1 + d-+(1 + (1 + d)2) or kcosB""'+(1 + k2 - w22) The full weighting signal generator required to obtain amplitude and phase correction system is shown in Fig. 2.

The computation required is: (1) form three products and accumulate: x,2, xlx2 and x22 (2) form w2 = - x1x2/x12 (3) form Y2 = x2 + w2x1 (4) form w1 = x22/x12 - w22 or +(1 + x22/x12 - w22) (5) form y1 = wlxl The mean products in (1) need not be formed from every signal sample. Samples could be taken in at any convenient rate, assuming the input statistics are stationary, or virtually so, as long as sufficient samples are taken to form sound averages and assuming k and B are steady.

The calculations in (2) and (4) can be formed quite slowly, if the errors k and B are slowly changing, for example due to temperature drifts. The oniy rapid calculations required are in (3) and (5) as these must be made on each signal sample. The delay device 1 2 included in the path appertaining to y1 is to compensate for the delay in the adder 10 in the other signal path, and similarly the device 11 in the x2 path is to compensate for the delay in the multiplier 7 in the x signal path.

Minor variations to the arrangement described may be made which will achieve the same effective result. For example instead of multiplying xl by kcosB to obtain y1, we could divide y2 by kcosB i.e. multiply by 1 /kcosB. This would require a total delay compensation in the y1 path equivalent to the delay in two multipliers and an adder. Alternatively x2 could be multiplied by 1 /k initially to equalise the inputs to the quadrature correction circuit. However this still leaves the y2 output with a factor cosB which requires matching in the real channel, so there is no particular advantage in equalising the input powers first.

Claims (11)

1. A Hilbert transform correction system for modifying two input signals which should be in phase quadrature thereby to produce two corresponding output signals which are in phase quadrature, comprising weighting signal producing means fed with the two input signals so as to produce two weighting signals, two multipliers each fed with one of the two input signals and each fed also with one of the weighting signals so as to produce two product signals and an adder fed with the other of the two input signals and one of the product signals, thereby to produce a sum signal which corresponds to one of the two output signals, the other of the two output signals comprising the other of the two product signals.

2. A Hilbert transform correction system as claimed in claim 1 where in the adder is fed with the other of the two input signals via a first delay device and the other of the two product signals is fed via a second delay device to provide the other of the two output signals, whereby signal delays contributed by the system are compensated for.

3. A Hilbert transform correction system as claimed in claim 2 wherein the first delay device is arranged to contribute a delay corresponding to the delay which is contributed by the multiplier which feeds the second delay device and the second delay device is arranged to contribute a delay which corresponds to the delay which is contributed by the adder.

4. A Hilbert transform correction system has claimed in any preceding claim wherein the weighting signal producing means comprises a first squarer and averager to which the said one input signal is fed to provide a first averaged squared signal, a second squarer and averager to which the said other input signal is fed to provide a second averaged squared signal, a multiplier and averager responsive to the two input signals for producing an averaged product signal, a first divider operative to divide the averaged product signal by the first averaged squared signal to produce a first quotient signal which constitutes one of the two weighting signals, a second didider operative to divide the second averaged square signal by the first averaged squared signal to produce a second quotient signal, a further squarer operative to squared the first quotient signal to produce a quotient squared signal and a data processor responsive to the quotient squared signal and to the second quotient signal for producing the other of the two weighting signals.

5. A Hilbert transform correction system as claimed in claim 4 wherein the data processor is operative to take the square root of the difference between the second quotient signal and the squared quotient signal thereby to produce the said other of the two weighting signals.

6. A Hilbert transform correction system as claimed in claim 4 wherein the data processor is arranged to be operative for halving, one plus the difference between the second quotient signal and the squared quotient signal thereby to produce the said other of the two weighting signals.

7. A Hilbert transform correction system as claimed in any preceding wherein the system is a digital system and the two input signals are accordingly digital signals which are processed digitally throughout the system.

8. A Hilbert transform correction system as claimed in any preceding claim wherein the two input signals are base band signals produced by a pair of base band mixers which are nominally phase quadrature related, the base band mixers being fed with quadrature related local oscillator signals.

9. A Hilbert transform correction system substantially as herein before described with reference to the accompanying drawings.

10. A base band phase quadrature demodulator system inclunding a Hilbert transform correction system is claimed in any proceeding claim.

11. Radio communications apparatus including a phase quadrature demodulator system as claimed in claim 9.