UDC 621.396.621.52
`Indexing Terms: Radio receivers, direct conversion
`
`Digital matching of the I and Q signal
`paths of a direct conversion radio
`
`M. DICKINSON, BA-
`
`Based on a paper presented at the I ERE Conference on Digital
`Processing of Signals
`in Communications
`held at
`Loughborough in April 1985
`
`' Royal Signals and Radar Establishment, St Andrews Road, Great
`Malvern, Worcs. WR14 3 PS
`
`1 Introduction
`A direct conversion or 'zero i.f.' radio1 requires two
`orthogonal
`baseband
`signal
`paths
`(In-phase and
`Quadrature) in order to distinguish between signals
`displaced equally in frequency above and below the local
`oscillator. In a digital radio the I and Q signal paths are
`digitized separately, requiring different analogue circuits in
`each path—notably the anti-alias filters. The basic zero i.f.
`digital radio architecture is shown in Fig. 1.
`The received real signal at the input of the mixers is of
`the form
`
`(1)
`X(t) = A(t)(cos (cor0+j sin (ojrt))
`After a perfect quadrature mixing process the important
`mixer products (i.e. those at baseband) in the I and Q
`signal paths would be
`(2)
`1(0 = B(t) cos (cot)
`(3)
`Q(0 = B(t) sin {cot)
`where co = cor — coo, coo is the local oscillator frequency, and
`the constant phase angle is ignored.
`Because of unavoidable component differences, the
`frequency response of the two paths will be different,
`
`SUMMARY
`A direct conversion digital radio may have a limited dynamic
`range performance under some circumstances, due to
`deviations from true quadrature between the I and Q signal
`paths. These errors are analysed and a method of correcting
`them, using an adaptively designed digital filter, is proposed.
`
`causing a departure from the true 90 degree phase shift
`them. This can be modelled by introducing
`between
`frequency dependent phase and gain errors in one of the
`signal paths. So, modifying equations (2) and (3) gives
`
`1(0 = B(t) cos (cot)
`Q(0 = (l+3)B(t) sin (co
`
`(4)
`
`3 = the magnitude mismatch at frequency co
`cf> = the phase mismatch at frequency co
`This assumes, without loss of generality, that the errors
`are
`in
`the Q
`signal.
`Putting,
`for
`simplicity,
`XR(t) = B(t) cos (cot)
`and X^t) = B(t) sin (cot) and
`expanding (4),
`
`where
`
`a = (1 + 3) cos 0 —
`(6)
`b =
`(7)
`If the two signal paths are subsequently processed as if
`they are in perfect phase quadrature, the perceived signal,
`Y(t), is
`
`Fig. 1. Direct conversion radio architecture (receive mode)
`
`Mixers
`
`A n t i - A l i as F i l t e rs
`(fco «%s 10KHz)
`
`making the substitutions
`
`a = c~d
`
`(8)
`
`(9)
`
`and re-arranging
`
`Y(t) = XR(t)+iXl(t)-d(XR(t)+iXx(t))-c(XR(t)-]Xl(t))
`= X(t)-dX(t)-cX*(t)
`(10)
`So the perceived signal has two parts; (1— d)X(t), the
`wanted input signal modified due to the mismatch by a
`factor (1— d), and cX*(t), a proportion of the input signal
`at the opposite frequency, — co. This is the image term, the
`magnitude of which is of concern.
`
`\
`
`20-22 Bits
`
`Digital
`Processor
`
`(Channel
`filtering etc)
`
`V t
`
`A/D
`
`\
`fs
`
`A/D
`
`20-22 Bits'
`
`\
`fs = 5 0 -» 100 kHz
`
`Journal of the Institution of Electronic and Radio Engineers, Vol. 56, No. 2, pp. 75-78, February 1986
`
`©1986 IERE
`
`TCL EXHIBIT 1046
`Page 1 of 4
`
`

`
`From (6), (7), (8) and (9)
`
`c = |[(1 + <5)cos 0 — 1 — j(l+<5) sin 0]
`
`so the magnitude of the image, relative to the input signal,
`is
`
`= i{(52 + 4(l+(5)sin 20/2}*
`(11)
`image
`the
`Equation
`(11) gives
`the magnitude of
`response, relative to the input signal, at a certain frequency
`offset from the local oscillator. In a well-designed system S
`and 0 will be small, allowing (11) to be simplified in order
`to easily estimate the image magnitude.
`For magnitude errors only, i.e. 0 = 0.
`
`-5
`
`arid for phase errors only, where 3 = 0
`
`(12)
`
`(13)
`
`and phase mismatch is a slowly varying function, which
`could be adequately defined by a set of about 10 points.
`The large delay inherent in high-order anti-alias filters
`makes stepping a single
`tone
`through
`the various
`frequencies a prohibitively slow technique. Instead a comb
`of frequencies is generated digitally, and injected into both
`signal paths simultaneously, using one D/A converter. The
`anti-alias filters themselves act as the post-filters, so the
`only errors introduced, except those being measured, are
`due to the D/A converter. As these are the same for both
`signal paths, they can be ignored.
`The resulting output of the filters is digitized and FFT
`analysed,
`to provide a set of magnitude and phase
`measurements for each path. The difference between the
`two sets of data defines the mismatch, irrespective of the
`absolute phase and magnitude of the signal.
`If the comb frequencies are chosen to be spaced by 1/N
`times the sampling frequency, a short FFT (length N) can
`be used in the analysis, to provide accurate results without
`the need for windowing.
`For a digital system the maximum possible signal level
`must be kept within the D/A number range, so the amount
`of energy available at each frequency
`in
`the comb
`decreases as the number of frequencies increases. This
`problem
`is especially bad for a sequence containing
`frequencies that are highly correlated. The effect is that, as
`the number of calibration points increases, the signal-to-
`noise ratio becomes smaller so the calibration becomes
`more dense but less accurate. There is, therefore, an
`optimum number of frequencies
`to obtain
`the most
`accurate results, which for the experimental set up was
`around 8, as illustrated in Fig. 2.
`
`4.
`
`6.
`
`8.
`
`10.
`
`12.
`
`F R E Q U E N CY
`
`( k H z)
`
`Fig. 2. The fit of the calibration points for different numbers of
`frequencies in the comb.
`— Actual response. O 8 points in comb.
`D 16 points in comb. + 32 points in comb.
`
`The length of the calibration sequence has to allow for
`the settling of the anti-alias filters to the abrupt change in
`data. For the filters and sampling rates used experi-
`mentally, the settling took around 1 ms. The valid data
`sequence only has to be as long as the FFT used in its
`analysis, i.e. 64 or 128 points. Thus the total calibration
`period will be about 2 ms. This does not include the
`processing time needed to design the next correction filter
`coefficients, but this will be done in parallel with other
`radio functions and wiil not affect radio use.
`
`The requirement for the degree of balance between the
`signal paths can be put into two categories. For systems
`not employing any form of frequency translation in the
`subsequent processing, the images are in-band and only
`degrade signal fidelity. Thus, images of up to — 40 dB
`below signal can be tolerated. On the other hand, if
`frequency
`translation
`is used, e.g. digital
`tuning and
`simultaneous decoding of more
`than one band,
`the
`situation
`is different.
`Images
`from
`large off-centre
`unwanted signals may swamp a smaller wanted signal on
`the opposite side (in
`the r.f. spectrum) of the
`local
`oscillator. Ideally, for this mode of operation, the images
`should be suppressed by at least as much as any of the
`to
`i.e. of order — 80 dB
`other spurious
`responses,
`—100 dB. To obtain this degree of matching, the errors
`introduced by the analogue part of the signal paths must
`be corrected digitally.
`The problem is one of designing an all-pass filter which,
`when placed in one signal path, will correct the inbalance
`between the two. The algorithm must be one which can be
`implemented with the minimum of computation, so that
`the filter can be adapted to a slowly changing mismatch by
`repeating the design process. The filter only has to correct
`the response within the system passband, as defined by the
`anti-alias filters and the digital channel filters. The filter
`response for the rest of the band is unimportant, as long as
`it does not significantly affect
`the overall stopband
`attenuation.
`This paper describes a method of designing such a filter,
`based on a calibration process. The calibration, using a
`D/A converter as the signal source, provides information
`about
`the magnitude and phase mismatch at various
`frequencies across the band. The mismatch at all other
`frequencies is estimated by forming polynomials, for both
`magnitude and phase, to interpolate between the discreet
`frequencies at which a calibration measurement is done.
`These polynomials, and ones describing the out-of-band
`response, are used to provide the samples for the design of
`a finite impulse response (FIR) frequency-sampling filter. 2
`The magnitude and phase response of the filter is designed
`such that, when it is placed in one of the signal paths, it
`corrects the mismatch.
`
`2 Calibration
`While the calibration is being performed the radio will not
`be operational, therefore the time taken must be kept to a
`minimum. Initial work, using circuit simulation software,
`showed that the frequency dependence of the magnitude
`
`3 Filter Design
`Having calibrated the mismatch, a filter has to be designed to
`the arbitrary magnitude and phase response required for
`correction. The first approach was to use the calibration
`points directly as the samples, but this has two drawbacks.
`
`76
`
`J. I ERE, Vol. 56, No. 2, February 1986
`
`TCL EXHIBIT 1046
`Page 2 of 4
`
`

`
`The interpolation between the calibration points is poor,
`and it is difficult to define the out-of-band samples in a
`sufficiently controlled manner. Optimization techniques,2
`which would move the unimportant out-of-band samples
`in such a way as to decrease the in-band ripple, cannot be
`used because there is nothing to optimize to, other than
`the actual calibration points themselves. The use of an
`optimization technique is also unattractive because of the
`excessive processing required.
`
`reaching high values, and is only constrained in position
`and gradient at two points. At the changeover frequency
`(/c) from the first to the second polynominal the gradient
`is made continuous, and at 0-5 the gradient is flat. The
`position at 0-5 is fixed to the 'no change' value of 1 for the
`magnitude mismatch polynominal, and 0 for the phase
`mismatch polynomial.
`The fit of these two polynomials to a typical mismatch
`(phase only shown) is plotted in Fig. 3.
`
`3.2 Frequency-Sampling Filter Design
`With the polynomials describing the wanted response
`established, a filter can be designed whose magnitude
`and phase response approximates
`these
`two curves
`simultaneously.
`The frequency-sampling design method is simply that of
`taking an inverse DFT of samples of the desired frequency
`response. As no optimization is to be done, the frequency
`samples are just the phase and magnitude polynomials
`evaluated at N evenly spaced frequency points, where N is
`the length of the DFT (or FFT if AT is a power of 2) used in
`the design. In this case the samples are complex in order to
`define the magnitude and phase simultaneously. The
`desired impulse response, or set of coefficients, must be
`real. The samples are converted to rectangular form, the
`real part is made symmetric and the imaginary part is
`made anti-symmetric to ensure a real output sequence
`from the inverse DFT.
`
`0.
`
`2.
`
`4.
`
`6.
`
`8.
`
`10.
`
`12.
`
`14.
`
`16.
`
`IB.
`
`20.
`
`F R E Q U E N CY
`
`( k H z)
`
`Fig. 4. Magnitude response of the correction filter (64 taps).
`.... Actual response.
`— Correction filter response.
`
`The number of taps in the filter determines the density of
`fit points along the polynomial, so increasing the filter
`length increases the accuracy of the correction. The digital
`radio architecture, on which this work is based, requires
`high-order FIR channel filters, of around 100 taps, to
`define the receiver bandwidth after the anti-alias filters.
`For this reason the correction filter lengths used have been
`64 and 128 taps.
`The frequency response of a typical filter is shown in
`Fig. 4, the actual response is also shown for comparison.
`
`4 Experimental Results
`To assess the performance of the correction filter it was
`necessary to be able to determine experimentally the
`mismatch between typical filters. Figure 5 shows the
`system used to get this information from a variety of filters.
`The array processor is capable of controlling two 16-bit
`parallel inputs and outputs (to the D/A and the A/D
`converters) at transfer rates up to 100 k s"1, using a
`purpose-built interface. This allows the generation and
`
`3.1 Polynomial Interpolation
`To improve the poor DFT interpolation, a more easily
`controlled curve is fitted to the calibration points using
`polynomial interpolation. This allows mismatch points to
`be computed on a much finer grid, to which the frequency-
`sampling filter can then be fitted. To ensure a good fit of
`the filter, certain constraints are imposed on the curve. The
`ripple, or Gibbs phenomenon that the filter displays, is
`minimized by minimizing the order of curvature of the
`polynomial. Discontinuities in particular must be avoided.
`The basis of the frequency-sampling filter design
`technique is the DFT, which implies a periodicity in the
`frequency response. To eliminate discontinuities at the
`band edges, i.e. at the normalized frequencies 0 and 0-5,
`the gradient of the polynomial must be flat at these points.
`The magnitude and phase mismatches are approximated
`by solving Nth order polynomials of the form:
`a2f2 + ... + aNfN
`x(f) =
`with the constraints
`dx
`_ = 0
`
`/ = 0, a n d /= 0-5
`
`at
`
`and
`
`for i = 1 to M
`c. = ao + axft + a2f\ +. . . + aNf?
`M being the number of calibration points and x{ being the
`calibration result at frequency ft.
`To solve this set of simultaneous equations we must
`have N = M + 2. The d.c. term is trivial, (a0 = x(0)), as is
`the constraint of zero gradient at d.c. (ay = 0). An
`(N — 2)x(N — 2) matrix is formed and the remaining
`coefficients are found by matrix methods.
`The frequencies /• are all within the system passband,
`therefore the polynomial is unconstrained in the rest of the
`band—approximately two-thirds of the whole band. Being
`of high order, the polynomial is uncontrolled in this region
`and would unacceptably alter the stop band characteristic,
`as well as make the frequency-sampling filter fit poor.
`To alleviate this problem, a second polynomial is
`formed for the system stopband. It is low order, to stop it
`
`Fig. 3. The polynomial approximation to the actual phase
`response.
`— Actual response
`
`Polynomial fit. x Calibration points.
`
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`
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`
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`PHBSE
`
`FREQUENCY
`
`( k Hz )
`
`M. DICKINSON
`
`77
`
`TCL EXHIBIT 1046
`Page 3 of 4
`
`

`
`Table 1. Peak errors and image responses for various filters before and after correction
`
`Filter type
`
`Peak Error
`
`Magnitude
`
`Before
`
`+ 0016
`
`+ 0016
`
`-00034
`
`+ 00036
`
`+ 0003
`
`-0003
`
`Phase (degrees)
`
`— Peak
`(from
`
`Image Response (dB)
`equation (11))
`
`After
`
`-000014
`
`- 0 0 06
`
`Before
`
`+ 0-28
`
`+ 065
`
`+ 00002
`
`-0-82
`
`- 0 0 01
`
`+ 000025
`
`- 1 -8
`
`- 1 -6
`
`+ 000025
`
`-1-75
`
`After
`
`Before
`
`After
`
`+ 0015
`
`+ 0015
`
`+ 0025
`
`+ 004
`
`+ 0013
`
`- 42
`
`- 42
`
`- 43
`
`- 36
`
`- 37
`
`- 35
`
`- 80
`
`- 67
`
`- 73
`
`- 64
`
`- 78
`
`0 -8 kHz
`
`0 - 10 kHz
`
`0 -8 kHz
`
`0 - 10 kHz
`
`0 -8 kHz
`
`Linear
`Phase
`Filter
`(Passive 8th order)
`Kemo
`Butterworth
`Filter
`(48 dB/oct)
`Kemo
`Linear Phase
`Filter
`(48 dB/oct)
`
`as
`at
`Filters
`under test
`
`Dual
`A/0
`05 Bit)
`
`0/A
`(16 Bit)
`
`0 - 10 kHz
`
`+ 0044
`
`- 68
`
`signals and averaging meant the results were accurate and
`repeatable.
`Figure 6 shows a plot of the image size, calculated from
`equation (11), versus frequency. The top curve shows the
`image response of the filter alone, calculated from the
`actual mismatch data measured in the manner described
`above. The bottom curve is the image response after
`correction, calculated from the error between the actual
`mismatch and the correction filter response. In most cases
`the image response is worst near the passband edge, i.e.
`8 kHz to 10 kHz. At this part of the passband the filter
`magnitude and phase responses are changing most rapidly,
`therefore
`the polynomial fit to the rapidly changing
`mismatch becomes worse.
`The filter used for the results plotted in Fig. 6, and the
`other figures, was passive, linear phase and 8th order. The
`measurements were repeated using a Kemo
`laboratory
`dual variable filter with Butterworth and linear phase filter
`types. The
`results using all
`three
`filter
`types are
`summarized in Table 1.
`
`5 Conclusion
`Table 1 and Fig. 6 show that the image response is
`suppressed by 35 to 40 dB by the correction filter, and the
`final peak image response is — 80 dB, or better, relative to
`the input signal level. These results were obtained from an
`experimental system limited by 15-bit A/D converters. The
`image response, therefore, is only slightly worse than the
`dynamic range of the calibration system.
`With attention paid to designing and building initially
`well-matched anti-alias filters, and with a higher resolution
`D/A—A/D converter pair,
`the
`image response of a
`digital radio corrected in this way should be no worse than
`other spurious responses of the radio.
`
`6 References
`'Digital
`1 Masterton, J., Ramsdale, P. A. and Vance, I. A. W.
`techniques for advanced radio'. Proc. Conf. Mobile Radio Systems
`and Techniques,
`September
`1984, pp. 6-10,
`(IEE Conf.
`Pub
`).
`2 Rabiner, L. R. and Gold, B., 'Theory and Application of Digital
`Signal Processing' (Prentice Hall, New York, 1975).
`
`Manuscript first received by the Institution on 23rd October 1984
`Paper No. 2223/COMM405
`
`Fig. 5. Experimental set-up for acquisition of calibration data.
`
`-40.
`
`-50.
`
`-60.
`
`-70.
`
`-80.
`
`-90.
`
`-100.
`
`-110.
`
`-120.
`
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`6.
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`7.
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`i 1 i
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`I
`
`FREQUENCY
`
`( k H z)
`
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`
`Fig. 6. A comparison of the image performance before and
`after correction.
`Top curve—filter image response.
`Bottom curve—response after correction.
`
`output of any desired sequence to the filter placed in the
`D/A—A/D path. The response to this signal is digitized in
`two channels simultaneously and stored for subsequent
`analysis.
`Using this system as the source of the calibration
`information, a correction filter is designed according to the
`described procedure. The response of this filter is then
`computed and compared with a dense set of points
`defining the actual magnitude and phase errors.
`The actual mismatch data are found by digitally
`generating a single tone and stepping through about 250
`frequencies between 0 and 0-5. The difference between the
`two signal paths indicates the mismatch. The use of large
`
`78
`
`J. I ERE, Vol. 56, No. 2, February 1986
`
`TCL EXHIBIT 1046
`Page 4 of 4