USRE45227E1 - Error estimation and correction in a two-channel time-interleaved analog-to-digital converter - Google Patents
Error estimation and correction in a two-channel time-interleaved analog-to-digital converter Download PDFInfo
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- USRE45227E1 USRE45227E1 US13/683,118 US201213683118A USRE45227E US RE45227 E1 USRE45227 E1 US RE45227E1 US 201213683118 A US201213683118 A US 201213683118A US RE45227 E USRE45227 E US RE45227E
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M1/00—Analogue/digital conversion; Digital/analogue conversion
- H03M1/06—Continuously compensating for, or preventing, undesired influence of physical parameters
- H03M1/0617—Continuously compensating for, or preventing, undesired influence of physical parameters characterised by the use of methods or means not specific to a particular type of detrimental influence
- H03M1/0624—Continuously compensating for, or preventing, undesired influence of physical parameters characterised by the use of methods or means not specific to a particular type of detrimental influence by synchronisation
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/27—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes using interleaving techniques
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M1/00—Analogue/digital conversion; Digital/analogue conversion
- H03M1/06—Continuously compensating for, or preventing, undesired influence of physical parameters
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M1/00—Analogue/digital conversion; Digital/analogue conversion
- H03M1/10—Calibration or testing
- H03M1/1009—Calibration
- H03M1/1028—Calibration at two points of the transfer characteristic, i.e. by adjusting two reference values, e.g. offset and gain error
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M1/00—Analogue/digital conversion; Digital/analogue conversion
- H03M1/12—Analogue/digital converters
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M1/00—Analogue/digital conversion; Digital/analogue conversion
- H03M1/12—Analogue/digital converters
- H03M1/1205—Multiplexed conversion systems
- H03M1/121—Interleaved, i.e. using multiple converters or converter parts for one channel
- H03M1/1215—Interleaved, i.e. using multiple converters or converter parts for one channel using time-division multiplexing
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B1/00—Details of transmission systems, not covered by a single one of groups H04B3/00 - H04B13/00; Details of transmission systems not characterised by the medium used for transmission
- H04B1/06—Receivers
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Definitions
- Time-interleaved Analog-to-Digital Converters have received considerable attention in the recent past in applications that require very high sample rates, i.e., sample rates that cannot be provided by a single ADC.
- a fast ADC is obtained by combining slower ADCs operating in parallel.
- the slower ADCs should each have the same offset, the same gain, and the same uniform sample instants.
- this requirement is difficult to achieve.
- the resulting errors degrade the performance of the TIADC system significantly, thus making the estimation and correction of these errors imperative to improve performance.
- the present invention is a two-channel TIADC wherein offset, gain, and phase errors are estimated and corrected.
- offset estimation and correction an error expression has been developed wherein it is shown that the average offset value of the two ADCs produces a tone at DC while the difference in the offset between the two ADCs produces a tone at the Nyquist frequency.
- the algorithm is first used to minimize the tone at Nyquist which depends upon the difference in the offset between the two ADCs. This is achieved by making the offset on one of the ADCs equal to that of the other.
- the tone at DC can be eliminated in a straightforward manner using well-known DC-offset correction techniques.
- a certain address to the LUT is calculated based on the offset error and the value corresponding to that address in the LUT is used to drive a digital-to-analog converter (DAC) and/or other appropriate analog circuits in the two-channel TIADC to effect the correction.
- DAC digital-to-analog converter
- the address of the LUT can be used to drive the DAC and/or analog circuits. Similar mixed-domain operations for gain and phase errors are carried out.
- FIG. 1 is a block diagram of a two-channel time-interleaved analog-to-digital converter (TIADC).
- TIADC time-interleaved analog-to-digital converter
- FIG. 2 is a block diagram of an offset correction unit in a two-channel TIADC.
- FIG. 3 is a block diagram of a gain correction unit in a two-channel TIADC.
- FIG. 7 is a plot of the spectrum of the signal of FIG. 6 with offset error correction.
- FIG. 8 is a plot of variation of e offset with OLUT 2 address.
- FIG. 9 is a plot of variation of OLUT 2 k address with iteration k.
- FIG. 13 is a plot of variation of address of GLUT 2 k with iteration k.
- FIG. 14 is a plot of the spectrum of a signal with phase error.
- FIG. 17 is a plot of variation of address of PLUT 2 k with iteration k.
- FIG. 19 is a plot of variation of e offset with OLUT 2 address for the wideband signal of FIG. 18 .
- FIG. 21 is a plot of variation of e phase with PLUT 2 address for the wideband signal of FIG. 18 .
- TIADC time-interleaved analog-to-digital converter
- FIG. 1 is a block diagram that shows an example two-channel TIADC 10 .
- a typical 2-channel TIADC 10 may have a bit width of 12 bits and operate at sample frequency of 400 Msps. Alternative embodiments may operate at faster or slower sample rates and with larger or smaller bit widths.
- Two analog-to-digital converters (ADCs) 20 and 21 operate on an analog input signal 12 , represented as x(t), to provide a digital output signal 14 , represented as y(n).
- the ADCs 20 and 21 sample and hold the input signal 12 at alternating sample time intervals 2T, where T is the reciprocal of a sample rate, f samp , provided by a clock signal 45 .
- the ADCs 20 and 21 are charge-domain pipeline ADCs that sample and digitize the input signal 12 on odd rising edges 40 and even rising edges 41 , respectively, of the clock signal 45 .
- a phase shifter may be arranged between the clock and the ADCs 20 and 21 to operate the ADCs 20 and 21 in an alternating fashion.
- a multiplexer 30 interleaves the outputs of the two ADCs 20 and 21 , which are at half the sample rate, to produce an output 14 at the sample rate.
- a digital signal processor (DSP) 60 monitors and corrects offset, gain, and phase errors in the outputs of the ADCs 20 and 21 .
- Taps 100 and 101 feed the outputs of ADCs 20 and 21 , respectively, into the DSP 60 , which computes the error and corresponding correction using a bank of look-up tables (LUTs) 30 - 35 or a bank of digital-to-analog converters (DACs; not shown).
- LUTs bank of look-up tables
- DACs digital-to-analog converters
- the ADCs 20 and 21 have corresponding offset LUTs (OLUTs) 30 and 31 , gain LUTs (GLUTs) 32 and 33 , and phase LUTs (PLUTs) 34 and 35 .
- the DSP 60 processes any errors according to adaptive algorithms, examples of which are described below.
- FIG. 2 is a block diagram of an offset correction unit 210 , including an error measurement block 201 , within the DSP 60 .
- the error measurement block 201 includes a subtractor 230 that takes the difference of the output signals on taps 100 and 101 from the ADCs 20 and 21 .
- An adder 240 coupled to the output of the subtractor 230 forms a feedback loop with a delay register 250 ; the adder sums the output of the subtractor 230 with a delayed version of the output of the subtractor 230 from the delay register 250 , which resets to zero every N samples.
- the delay register 250 transmits a resulting offset error signal, e offset , to an adaptive processor 220 .
- the adaptive processor 220 may be implemented using a DSP, field-programmable gate array, application-specific integrated circuit, programmed general-purpose data processor, or any other suitable implementation. In some embodiments, the adaptive processor 220 operates according to the algorithm described below wherein it selects the address of OLUT 2 31 in a manner that minimizes the offset error signal. The values corresponding to the selected addresses of the OLUTs 30 and 31 are used to correct the offset between the ADCs 20 and 21 in a corresponding manner.
- the adaptive processor 220 determines the sign of the offset error signal using a signum block 282 , which returns a ⁇ 1, 0, or 1 depending on whether the offset error signal is negative, zero, or positive, respectively.
- the output from the signum block 282 is multiplied with an offset step size ⁇ offset k to control the value that adds to a bias N offset /2 to the produce the address for OLUT 2 31 , as shown in FIG. 2 .
- the product may result in a forwards step, backwards step, or no change.
- the resulting product enters a feedback loop implemented with an adder 286 and a delay register 288 .
- a rounding block 290 rounds the output of the feedback loop to form an address step, which may be biased by a bias value N offset /2 using an adder 292 .
- N offset may be biased by a bias value N offset /2 using an adder 292 .
- the bias value might be 128, which sets the offset error to the midpoint in the range of OLUT 2 31 .
- the biased address is then fed to OLUT 2 31 and an overflow/underflow block 294 , which monitors the resultant address and, if necessary, resets the address of OLUT 1 30 to keep the address of OLUT 2 31 within an acceptable range.
- the bias value and range of address locations depend on the particulars of the implementation. Certain implementations may operate at zero bias, eliminating the adder 292 .
- FIG. 3 is a block diagram of a gain correction unit 310 , including an error measurement block 301 , within the DSP 60 .
- Multipliers 360 and 361 square the signals from taps 100 and 101 , respectively.
- the error measurement block 301 includes a subtractor 330 that takes the difference of the squared signals and forwards the difference to a feedback loop that includes an adder 340 and a delay register 350 , which resets to zero every N samples.
- the feedback loop transmits a gain error signal, e gain , to an adaptive processor 220 , which, in some embodiments, operates according to the algorithm described below.
- the adaptive processor 220 selects addresses of the GLUTs 32 and 33 in a manner that minimizes the gain error signal.
- the DSP 60 uses the output from the GLUTs 32 and 33 to control the gain of the ADCs 20 and 21 in a corresponding manner.
- the adaptive processor 220 determines the sign of the gain error signal using a signum block 282 , the output of which is multiplied with a gain step size ⁇ gain k to control the value that adds to a bias N gain /2 to produce the address for GLUT 2 33 , as shown in FIG. 3 .
- the resulting product enters a feedback loop implemented with an adder 286 and a delay register 288 .
- a rounding block 290 rounds the output of the feedback loop to form an address step, which may be biased by a bias value N gain /2 using an adder 292 .
- the biased address is then fed to GLUT 2 33 and overflow/underflow block 294 , which adjusts GLUT 1 32 as necessary.
- the bias value and range of address locations depend on the particulars of the implementation.
- FIG. 4 is a block diagram of a phase correction unit 410 , including an error measurement block 401 , within the DSP 60 .
- the error measurement block 401 includes a subtractor 430 that feeds the difference of signals from the taps 100 and 101 into both input ports of a multiplier 460 , which returns the square of the difference.
- a second subtractor 431 takes the difference of the signal from tap 101 and a version of the signal from tap 100 delayed by a delay register 450 . The output of the subtractor 431 is squared with a second multiplier 461 .
- a third subtractor 432 takes the difference of the outputs from the multipliers 460 and 461 ; the difference from the subtractor 432 enters a feedback loop including an adder 440 and a delay register 451 as in FIGS. 2 and 3 .
- the feedback loop provides a phase error signal, e phase , that drives an adaptive processor 220 .
- the adaptive processor 220 operates according to the algorithm described below. As in the offset and gain correction units 210 and 310 shown in FIGS. 2 and 3 , respectively, the adaptive processor 220 selects addresses in PLUT 2 35 in a manner that minimize the phase error signal. The values corresponding to the selected addresses are used to control the phase error between the ADCs 20 and 21 in a corresponding manner.
- An overflow/underflow block 294 monitors the address setting and adjusts PLUT 1 34 as necessary to keep PLUT 2 35 within a given range.
- the adaptive processor 220 determines the sign of the phase error signal using the signum block 282 , the output of which is multiplied with a phase step size ⁇ phase k to control the value that adds to a bias N phase /2 to the produce the address for PLUT 2 35 , as shown in FIG. 4 .
- the resulting product enters a feedback loop implemented with an adder 286 and a delay register 288 .
- a rounding block 290 rounds the output of the feedback loop to form an address step, which may be biased by the bias value N phase /2 using an adder 292 .
- the biased address is then fed to PLUTs 32 and 33 .
- the bias value and range of address locations depend on the particulars of the implementation.
- Embodiments of the disclosed TIADC may use a single adaptive processor 220 to control all three sets of LUTs 30 - 35 .
- the offset, gain, and phase error correction are applied to the ADCs 20 and 21 in a sequential fashion (e.g., first correct the offset error, then the gain error, then the phase error, and repeat). Because all three correction units share a common adaptive processor 220 , the resulting TIADC is smaller, lighter, more efficient, and simpler to manufacture than a TIADC with separate offset, gain, and phase correction units.
- each adaptive processor 220 may be implemented with dedicated hardware or programmable processors.
- FIG. 5 shows connections of outputs from gain, offset, and phase correction units to a first pipeline stage 501 of a charge-domain pipelined ADC 500 that can be used in the two-channel TIADC 10 of FIG. 1 .
- First and second pipeline stages 501 and 502 incorporate charge-redistribution, charge-comparison, and charge-redistribution-driver circuits in a single-ended pipeline to provide two bits of analog-to-digital conversion. Adding additional stages to the pipeline provides additional bits of analog-to-digital conversion, where successive stages operate on charge packets that propagating through the pipeline in like fashion.
- Charge packets are transferred into and out of the first pipeline stage 501 with a charge-transfer circuit 505 on alternating half-cycles of a clock signal CLK, causing the voltage at node 520 to change according to the size of the transferred charge packet.
- a comparator 530 compares the resulting voltage at node 520 to a reference voltage V RC .
- a latch 522 latches the output from the comparator 530 once per clock cycle to produce a digital output V B1 .
- a charge-redistribution driver 513 receives V B1 and outputs a charge-redistribution voltage signal V QR which to stage 502 .
- the transition in V QR causes a corresponding change in the voltage at one node of a comparator in stage 502 , meaning that the comparison result of stage 501 governs charge-redistribution in subsequent stage 502 .
- Outputs from the offset, gain, and phase correction units 210 , 310 , and 410 can be used to control various components of the pipeline stages.
- an offset control signal 531 from an OLUT such as OLUTs 30 and 31 in FIGS. 1 and 2
- a gain control signal 533 from a GLUT such as GLUTs 32 and 33 in FIGS. 1 and 3
- the clock phases V C1 and V CC1 can be retarded or advanced with a variable phase delay 510 controlled by a signal 511 from a PLUT, such as PLUTs 34 and 35 .
- the variable phase delay 510 can be implemented with a delay-locked loop, dispersive delay line, or any other suitable delay, buffer, or memory element.
- the aim is to minimize the magnitude of the latter tone, i.e., the tone at the Nyquist frequency.
- the average offset value once the difference in offset values is eliminated or minimized (e.g., by linearly shifting the relative offset between the ADCs 20 and 21 using the OLUTs 30 and 31 ).
- FIG. 6 shows the simulated spectrum of a 50 MHz tone with an offset error when the sampling frequency of the two-channel TIADC is 400 MHz.
- there are two tones in the resulting spectrum that arise due to the offset error.
- the tone at DC corresponds to the average offset value between the two ADCs 20 and 21 while the tone at the Nyquist frequency corresponds to the difference in offset values between the two ADCs 20 and 21 .
- FIG. 7 shows the simulated spectrum of the same signal after offset error correction as explained below. It can be seen that the tone at the Nyquist frequency due to the difference in offset values has been minimized The amount of suppression of the tone at the Nyquist frequency depends upon the residual difference of offset between the two ADCs after correction. It can be seen that the tone at the Nyquist frequency is suppressed by more than 50 dB. Between FIGS. 6 and 7 , it can be seen that the DC components are different. This is due to the fact that the offset of ADC 2 has been made approximately equal to that of ADC 1 .
- the calculation of e offset corresponds to taking the difference of the mean or, alternatively, the mean of the difference of the outputs from the two ADCs 20 and 21 over N samples. The larger the value of N, the more accurate the estimate.
- One-way communication, as in cable modems, does not require such fast convergence and can be accomplished with more samples (i.e., larger values of N).
- the OLUTs 30 and 31 each of size N offset , include entries that can be used to directly or indirectly control the offset in each of the ADCs 20 , 21 . Since we are dealing with the estimation and correction as a mixed-domain process, there is no loss of generality if the OLUTs 30 , 31 act as interfaces between the analog and digital domains.
- the addresses of the OLUTs 30 and 31 are evaluated using an adaptive algorithm, such as the one described below, in the digital domain while the outputs of the OLUTs 30 and 31 directly or indirectly provide the corresponding offset correction in the analog domain. To illustrate, let the maximum difference in offsets between the two ADCs be ⁇ X 0 least significant bits (LSBs).
- the maximum tolerable difference in offsets will be about 60 LSBs, or about 3% of the total bit width.
- the entries of the OLUTs 30 and 31 are designed to cover this range using a linear, logarithmic, or any other distribution depending upon the analog circuitry. For a linear distribution, an entry in the OLUTs 30 and 31 differs from the next entry by 2X o /N offset LSBs. In a preferred embodiment, the entries in the OLUTs 30 and 31 are distributed linearly near the zero-error point and logarithmically near the edges of the distribution.
- the address of the OLUT 1 30 of ADC 1 20 is such that the output from the OLUT 1 30 is zero.
- OLUT 1 30 is associated with ADC 1 20 and OLUT 2 31 is associated with ADC 2 21 .
- the DSP 60 sets the address of OLUT 1 30 to N offset /2.
- the address of OLUT 1 is 128 which means that the output from OLUT 1 is zero assuming a linear distribution of values in the OLUTs.
- e offset is a linear variation between the two extremities of the offset errors. It can be seen that the error function passes through zero for a specific address of OLUT 2 31 . In this case the optimum address for OLUT 2 31 is 192.
- the DSP 60 uses an adaptive algorithm to seek the address of OLUT 2 31 that minimizes the absolute value of e offset .
- the adaptive algorithm is based on the sign of e offset and hence is extremely hardware efficient.
- the DSP 60 sets the address of OLUT 1 30 to N offset /2.
- OLUT 2 k denote the address of OLUT 2 31 at the kth iteration.
- ⁇ 1 k denote a variable at the kth iteration and let ⁇ offset k denote a step size for the adaptive algorithm at the kth iteration.
- ⁇ offset k ⁇ [ ⁇ offsetmin , ⁇ offsetmax ] (7) where ⁇ offsetmin and ⁇ offsetmax are the minimum and maximum values, respectively, of ⁇ offset k .
- the adaptive algorithm for correcting the offset error can now be written as
- OLUT 2 k indicates the optimal address of OLUT 2 31 that produces the minimum absolute value of e offset .
- FIG. 9 shows the convergence of OLUT 2 k with iteration k. As can be seen from FIG. 9 , in about 7 iterations, the OLUT 2 k converges to an address around 192, which is the optimum address corresponding to the zero crossing value of e offset shown in FIG. 8 .
- ⁇ 1 2 is obtained.
- the address of OLUT 2 31 is set to 192.
- the sign of e offset should be zero (Recall that the zero crossing of e offset for this case occurs when the address of OLUT 2 is 192).
- G s G 1 + G 2 2
- Equation 12 shows that the gain mismatch produces an image tone reflected around ⁇ s /2 and that the amplitude of the image tone is proportional to the difference in gain values between the two ADCs 20 and 21 . It can also be seen that the input signal is scaled by the average value of the gains of the two ADCs 20 and 21 . This need not be a concern since an Automatic Gain Control (AGC) loop is usually employed to correct for such errors.
- AGC Automatic Gain Control
- FIG. 10 shows the simulated spectrum of a 50 MHz tone with a gain error when the sample frequency of the two-channel TIADC 10 is 400 MHz.
- the amount of suppression depends upon the difference in the gain values of the two ADCs 20 and 21 . Alternatively, it depends upon the difference between the ratio of the gain values of the two ADCs 20 and 21 and unity.
- FIG. 11 shows the simulated spectrum of the same signal after applying the gain correction described below. It can be seen that gain correction reduces the image tone by more than 25 dB.
- a gain error function can be formulated as
- the DSP 60 uses GLUTs 32 and 33 to directly or indirectly control the gain of the output of each ADC 20 and 21 , respectively.
- the distribution in each GLUT 32 and 33 can be linear, logarithmic or any other distribution. If the maximum variation of the ratio of the gains of the two ADCs is (1 ⁇ X g ), then the GLUTs are designed to cover the entire range of 2X g . In a preferred embodiment, X g ⁇ 2%, meaning that the tolerable range of gains is 0.98-1.02 times the nominal gain value.
- address of GLUT 1 32 is such that the output from it is unity.
- N gain 256, although N gain can take any suitable value.
- the error variation is not linear, it is fairly well behaved in terms of linearity.
- the region of zero crossing of e gain corresponds to the optimal address of GLUT 2 33 .
- the minimum of the absolute value of e gain corresponds to GLUT 2 address of 162.
- the DSP 60 sets the address of GLUT 1 32 to N gain /2.
- GLUT 2 k denote the location of GLUT 2 33 at the kth iteration.
- ⁇ 2 k denote a variable at the kth iteration, and let ⁇ gain k denote a step size for the adaptive algorithm at the kth iteration.
- ⁇ gain k ⁇ [ ⁇ gainmin ⁇ gainmax ] (15) where ⁇ gainmin and ⁇ gainmax are the minimum and maximum values, respectively, of ⁇ gain k .
- the adaptive algorithm for correcting the gain error can be written as
- ⁇ 2 k provides the update of the address of GLUT 2 33 based on the sign of e gain .
- GLUT 2 k indicates the optimal address of GLUT 2 33 that produces the minimum absolute value of e gain .
- FIG. 14 shows the simulated spectrum of a signal with a tone at 50 MHz. Again, a sample frequency of 400 MHz is assumed. As can be seen from FIG. 14 , there is an image tone at 150 MHz arising from the phase error.
- FIG. 15 shows the simulated spectrum of the same signal after phase correction; the image tone has been suppressed by more than 25 dB. The amount of suppression depends upon how closely ⁇ t approximates zero.
- the DSP 60 use PLUTs 34 and 35 , each of size N phase , to directly or indirectly control the phase of the clock signal 45 to each of the ADCs 20 , 21 .
- the addresses of the PLUTs 34 , 35 are evaluated using an adaptive algorithm in the digital domain while the outputs of the PLUTs 34 , 35 directly or indirectly provide the corresponding delay in the clock signal 45 used to control the ADCs 20 , 21 .
- the maximum phase delay is about 0.3% of the period, or a time delay of ⁇ 5.75 ps for a sample frequency of 500 MHz (i.e., a sample period of 2 ns).
- the units for ⁇ X p may be in seconds, radians, or fractions of the sample frequency and the entries of the PLUTs 34 , 35 can follow linear, logarithmic or any other distribution depending upon the analog circuitry effecting the correction.
- the DSP 60 sets the address of PLUT 1 34 such that the output from PLUT 1 34 is zero. In other words, there is no correction performed on ADC 1 20 .
- One embodiment employs a linear distribution of values in the PLUTs 34 , 35 , where the size of each PLUT 34 , 35 be N phase .
- the region of zero crossing of e phase corresponds to the optimal address of PLUT 2 .
- the minimum absolute value of e phase corresponds to PLUT 2 address of 157.
- the DSP 60 sets the address of PLUT 1 34 to N phase /2.
- PLUT 2 k denote the address of PLUT 2 35 at the kth iteration.
- ⁇ 3 k denote a variable at the kth iteration, and let ⁇ phase k denote a step size for the adaptive algorithm at the kth iteration.
- ⁇ phase k ⁇ [ ⁇ phasemin , ⁇ phasemax ] (23) where ⁇ phasemin and ⁇ phasemax are the minimum and maximum values, respectively, of ⁇ phase k .
- the adaptive algorithm for correcting the phase error can be written as
- ⁇ 3 k provides the update of the address of PLUT 2 35 based on the sign of e phase .
- Eqn. (22) is valid for all odd Nyquist zones.
- the sign of the phase error is the negative of the phase error given in Eqn. (22). In other words, for even Nyquist zones, the phase error becomes
- FIG. 18 shows the spectrum of a wideband signal to a two-channel TIADC 10 with a sample rate of 400 MHz.
- FIG. 19 shows the variation of e offset with the address of OLUT 2 31 . It is evident from FIG. 19 that e offset is a linear variation between the two extremities of the offset errors, even when the input signal is wideband. It can be seen that the error function passes through zero at a certain address of OLUT 2 31 . In view of this, the same algorithm with a binary search for the optimal address of OLUT 2 31 can be used for the case when the input signal is wideband.
- FIG. 20 shows the variation of e gain with the address of GLUT 2 33 .
- the gain error function is no longer a smooth straight line.
- the gain error shows nonlinear behavior.
- e gain has a certain trend across the entire range of GLUT 2 addresses.
- the detrending operation is preferably a first-order, or linear, least-squares fit to the data, as shown in FIG. 20 .
- FIG. 20 This straight line, extracted from the detrending operation, is shown in FIG. 20 .
- the adaptive algorithm presented earlier for the gain error estimation and correction can be applied to the case when the input is wideband provided that ⁇ gainmin , ⁇ gainmax , and k 2 are chosen appropriately.
- ⁇ gainmax and ⁇ gainmin have to be small so that a linear trend in FIG. 20 can be captured. It must be mentioned that due to small values of ⁇ gain k ⁇ [ ⁇ gainmax , ⁇ gainmin ] in the adaptive algorithm, the convergence time becomes longer.
- FIG. 21 shows the variation of e phase with the address of PLUT 2 .
- the phase error function just like the gain error function, is no longer a smooth function, but shows nonlinear behavior similar to that of the gain error variation.
- e phase has a trend similar to e gain when the input to the two-channel TIADC 10 is wideband.
- FIG. 21 there is a linear part in the nonlinear variation.
- a linear part can be extracted. Such a straight line, obtained through detrending operation, is shown in FIG. 21 .
- the adaptive algorithm for offset correction can be made to converge within the first 7-8 iterations using a binary search.
- the errors for gain and phase are nonlinear and hence in order to expedite the convergence, we propose a two-step algorithm wherein the neighborhood of the optimal address of GLUT 2 33 or PLUT 2 35 is obtained in the first step.
- the second step is the adaptive algorithm mentioned earlier where a small value for ⁇ gain k or ⁇ phase k is used.
- a straight line that represents a least-squares fit for the e gain or e phase variation provides a zero crossing which is the optimal address of GLUT 2 33 or PLUT 2 35 , respectively.
- N lut denote either N gain or N phase
- e err denote either e gain or e phase
- X(k) denote kth address of GLUT 2 33 or PLUT 2 35 .
- the single-step algorithm In applications where the single-step algorithm is preferred (i.e., applications where slow convergence times are acceptable), we can obtain the initial estimate with a calibration signal that comprises of a single tone input to the two-channel TIADC 10 .
- the algorithm mentioned in this document can be used to obtain convergence. After the convergence with a single tone is obtained, the actual input can be introduced. With an appropriately chosen ⁇ gainmin and ⁇ gainmax for gain or ⁇ phasemin and ⁇ phasemax for phase, the adaptive algorithm can be restarted.
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Abstract
Description
where V1 and V2 are the offset values of
denote the average and difference in the offsets between two
y(n)=cos(ωonT+φ)+Vs+cos(ωsnT/2)Vd (4)
where ωs=2π/T is the sampling frequency and (−1)n=cos (ωsnT/2). It is clear from Eqn. (4) that the average offset between the two
From
μoffset k∈[∥offsetmin, μoffsetmax] (7)
where μoffsetmin and μoffsetmax are the minimum and maximum values, respectively, of μoffset k. The adaptive algorithm for correcting the offset error can now be written as
where μ1 1=0, μoffset 1=μoffsetmax and k1 is any arbitrary positive number. As can be seen from Equation 8, μ1 k provides the update of the address of
where G1 and G2 are the gains provided by
Again, using
Eqn. (10) can be re-written as
It can be noted from the above equation that egain can be made approximately equal to zero provided a variable kgain can be found such that
μgain k∈[μgainminμgainmax] (15)
where μgainmin and μgainmax are the minimum and maximum values, respectively, of μgain k. The adaptive algorithm for correcting the gain error can be written as
where μ2 1=0, μgain 1=μgainmax and k2 is any arbitrary positive number. As can be seen from the above adaptive algorithm, μ2 k provides the update of the address of
Here it is assumed that
It can be seen that cos[(−1)n ω0Δt/2]=cos[ωoΔt/2] Since the sine function is an odd function, with (−1)n=cos(nπ), we get sin[(−1)n ωoΔt/2]=cos(nπ)sin[ωoΔt/2]. Using sin(a)cos (nπ)=sin(a−nπ) and nπ=ωsnT/2, the above equation can be written as
Assuming that Δt is small compared to 1/ωo, cos(ωoΔt/2)≈1 and sin(ωoΔt/2)≈ωoΔt/2. Consequently,
We can now see from the above equation that the phase error produces an image tone with an amplitude proportional to the phase error Δt. It is interesting to note that the image tone is π/2 out of phase with the tone produced due to gain error.
An alternative expression for the phrase error given by
also provides information about the phase error between the two
μphase k∈[μphasemin,μphasemax] (23)
where μphasemin and μphasemax are the minimum and maximum values, respectively, of μphase k. The adaptive algorithm for correcting the phase error can be written as
where μ3 1=0, μphase 1=μphasemax and k3 is any arbitrary positive number. As can be seen from the above adaptive algorithm, μ3 k provides the update of the address of
Offset, Gain, and Phase Error Correction for Wideband Signals
where a and b are constants and eerr(k) represents the error value for an address location X(k). It must be recalled that eerr(k) is obtained using Eqn. (13) or Eqn. (22). It can be noted from Equation 26 that Y(k)=a+bX(k) provides a straight line fit to the variation of eerr(k), provided the constants a and b are known. Equating the derivative of R, with respect to the constants a and b, to zero, we get
By solving the above two equations we get
The neighborhood of the optimal point can be obtained by equating y(k)=0. Therefore
where Int(x) represents the integer part of x. There is no need to calculate Σk=1 N
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USRE45343E1 (en) | 2015-01-20 |
US20110063149A1 (en) | 2011-03-17 |
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TWI511465B (en) | 2015-12-01 |
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US7839313B2 (en) | 2010-11-23 |
US7839323B2 (en) | 2010-11-23 |
US20100164763A1 (en) | 2010-07-01 |
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