Equalization of sub-DAC frequency response misalignments in time-interleaved high-speed digital to analog converters

Abstract

A method and system for calibrating a time-interleaved digital to analog converter (DAC) provides equalization of frequency response misalignments in sub-DACs forming the DAC. In a calibration mode, test signals are applied to an DAC and output amplitudes and phases of are measured. From the measured values, complex values of the gains of the respective sub-DACs. h.sub.m(F) are determined and a specified target frequency response T(F) for a tandem connection equalizer-DAC is determined. For each of a plurality of test frequencies, complex values of equalizer gains Eq.sub.m are determined from Eq.sub.m(F)=T(F)/h.sub.m(F), to form equalizing frequency responses. Sets of equalizing coefficients C.sub.m(p) pursuant to discrete Fourier transforms on Eq.sub.m(F). In an operation mode, a digital input signal is transformed input into an equalized digital signal E(n) through use of the sets of equalizing coefficients C.sub.m(p).

Claims

1. A method of equalization of a time-interleaved digital to analog converter (DAC) having a demultiplexer, M sub-DACs, where M is an integer, a combiner and a filter coupled in series between a digital DAC input for receiving a digital input DAC signal-to-be-processed, and an analog DAC output, wherein the DAC input is coupled to the demultiplexer which is adapted in response to the applied input DAC signal, to provide demultiplexed digital signals to respective inputs of the sub-DACs, and wherein in response to a succession of mutually time-shifted sampling signals, outputs of the M sub-DACs are coupled to inputs of the combiner, wherein the combiner is adapted to combine the signal inputs of the combiner to establish an analog signal and pass the resultant combined signal through the filter to the analog DAC output, comprising the steps of: A) in a calibration mode performed step by step, i) for each step: a) choosing a test frequency F is from a list of test frequencies; b) generating a test signal at the chosen test frequency, and applying the test signal to the DAC input of the time-interleaved DAC; c) measuring amplitudes Amp.sub.k and phases Phs.sub.k of spurious components in an output signal at the DAC output, wherein k is an integer, 0k<M, associated with the respective spurious components; d) calculating M complex values of sub-DAC gains h.sub.m(F) of the respective sub-DACs for each m, wherein m is an integer, 0m<M, associated with the respective M sub-DACs, according to equations: $h_m\left(F\right)=1/M.Math.\underset{k=0}{\overset{M-1}{.Math.}}{\mathrm{Amp}}_k.Math.\exp \left(j.Math.\left(2.Math.k.Math.m/M+{\mathrm{Phs}}_k\right)\right);$ e) specifying a target frequency response T(F) of a tandem connection equalizerDAC; f) calculating M complex values of equalizing gains Eq.sub.m(F) for each m, according to equations:
Eq.sub.m(F)=T(F)/h.sub.m(F); ii) accumulating equalizing gains Eq.sub.m(F) for different test frequencies F of the respective steps, forming required equalizing frequency responses; iii) calculating M sets of equalizing coefficients C.sub.m(p) at the end of the calibration stage by performing discrete Fourier transforms on the required equalizing frequency responses Eq.sub.m(F), wherein each C.sub.m(p) set includes L coefficient elements, and is associated with a respective one of the M sub-DACs, and wherein p is an integer, 0p<L, associated with a respective one of the L coefficient elements in a set; and B) in an operation mode: i) transforming an input digital signal s.sub.in(n) applied to the DAC input into an equalized digital signal E(n), wherein a sample with the number n of the equalized digital signal is produced by a convolution of a set of equalizing coefficients C.sub.m(p), m=n mod M, with a corresponding part of the input digital signal: $E\left(n\right)=\underset{i=0}{\overset{L-1}{.Math.}}{C}_m\left(i\right).Math.s_{in}\left(n-i\right);$ ii) applying the equalized digital signal to the input of the DAC, whereby the digital signal is converted into the analog signal at the analog DAC output.

2. The method of equalization of claim 1, wherein the measuring sub-step A.i.c of the calibration mode is performed with a measuring device, wherein the measuring device is a spectrum analyzer adapted for producing an amplitude spectrum of an output signal at the analog DAC output, wherein amplitudes Amp.sub.k of spurious components at the DAC output are measured directly by the spectrum analyzer and phases Phs.sub.k of the spurious components at the DAC output are determined by a procedure whereby auxiliary signals are applied to the DAC input, amplitudes of a spur produced at the DAC output by each auxiliary signal are measured, and the measurement results are used for calculation of Phs.sub.k.

3. The method of equalization of claim 2, wherein the phases Phs.sub.k of the spurious components at the DAC output are determined for each k, 0k<M, using a plurality of auxiliary test signals of the form:
TS(m,.sub.i)=exp(j2Fm).Math.[1Amp.sub.k.Math.exp(j(2.Math.m.Math.k/M+.sub.i)], where m is sub-DAC number, Amp.sub.k is amplitude of spurious component with number k and .sub.i is trial phase, wherein phase of the spurious component is determined as phase minimizing spectrum magnitude.

4. The method of equalization of claim 3, wherein the phase Phs.sub.k of the spurious components are calculated by using plurality of squared spectrum magnitudes of signal TS(m, .sub.i) by means of quadrature detection.

5. The method of equalization of claim 3, wherein after spur phase Phs.sub.k is determined, amplitude of the spurious component Amp.sub.k is adjusted to minimize total spur magnitude at given frequency.

6. The method of equalization of claim 3, wherein the phases Phs.sub.k of the spurious components at the DAC output being determined by a procedure comprising: A) a test with a first auxiliary signal, whereas: a) the first auxiliary signal is generated as a harmonic signal with the test frequency F and an envelope 1/M.Math.exp(j.Math.2.Math.k.Math.m/M), m=n mod m.Math.n being the number of a sample in the first auxiliary signal; b) the first auxiliary signal is applied to the DAC input; c) the modulus A.sub.1 of the spurious component with the number k in the signal at the DAC output is measured by the spectrum analyzer; B) a test with a second auxiliary signal, whereas: a) the second auxiliary signal is generated as a harmonic signal with the test frequency F and an envelope 1+Amp.sub.k/|A.sub.1|.Math.R; b) the second auxiliary signal is applied to the DAC input; c) the modulus A.sub.2 of the spurious component with the number k in the signal at the DAC output is measured by the spectrum analyzer; C) a test with a third auxiliary signal, whereas: a) the third auxiliary signal is generated as a harmonic signal with the test frequency F and an envelope 1+Amp.sub.k/|R|.Math.exp(j.Math./2).Math.R; b) the third auxiliary signal is applied to the DAC input; c) the modulus A.sub.3 of the spurious component with the number k in the signal at the DAC output is measured by the spectrum analyzer; D) the phase Phs.sub.k is calculated with the use of equations:
Phs.sub.k=2.Math.a cos(A2/2/Amp.sub.k) if A.sub.3A2, and
Phs.sub.k=2.Math.a cos(A2/2/Amp.sub.k) otherwise.

7. The method of equalization of claim 1, wherein the DAC incorporates a low pass filter to perform a step of suppressing spurs with numbers from k.sub.1 to k.sub.2, and thereby eliminating the use of corresponding information for determination of the sub-DACs gains h.sub.m(F) according the equation:
Eq.sub.m(F)=T(F)/h.sub.m(F). whereby a determination of the sub-DACs gains h.sub.m(F) is performed as a set of consecutive iterations, each iteration step comprising: a) a generating a test signal of the chosen test frequency F; b) applying the test signal to the DAC input; c) measuring amplitudes Amp.sub.k and phases Phs.sub.k with the numbers k of the visible spurious components in the signal at the DAC output, said numbers satisfying the inequalities 0k<k.sub.1 or k.sub.1<k<M; d) calculating the sub-DACs gains h.sub.m(F) by the use of the next equations: $h_m\left(F\right)=1/M.Math.\underset{k=0}{\overset{k_1-1}{.Math.}}{\mathrm{Amp}}_k.Math.\exp \left(j.Math.\left(2.Math.k.Math.m/M+{\mathrm{Phs}}_k\right)\right)+1/M.Math.\underset{k=k_2}{\overset{M-1}{.Math.}}{\mathrm{Amp}}_k.Math.\exp \left(j.Math.\left(2.Math.k.Math.m/M+{\mathrm{Phs}}_k\right)\right);$ e) specifying a target frequency response T(F) of the tandem connection equalizer-DAC; f) calculating M complex values of equalizing gains Eq.sub.m(F) using the equation:
Eq.sub.m(F)=T(F)/h.sub.m(F); g) accumulating equalizing gains Eq.sub.m(F) for different test frequencies F, thereby forming equalizing frequency responses; h) calculating M sets of equalizing coefficients C.sub.m(p) at the end of the calibration stage by discrete Fourier transforms of required equalizing frequency responses Eq.sub.m(F), wherein m is a set number and p, 0p<L, is a number of a coefficient in a set.

8. The method of equalization of claim 6, including the further step of repeating the iterative calculation of equalizing coefficients C.sub.m(p) while at least one spur remains above a previously set threshold.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows a block diagram of a conventional time-interleaved DAC.

(2) FIG. 1A shows a block diagram of an exemplary system including a time-interleaved DAC and a complementary network processor, exemplifying the disclosure.

(3) FIG. 2 shows a flow chart for the disclosure, that illustrates an exemplary method of equalizing frequency response misalignments in a time-interleaved DAC.

(4) FIG. 3 shows a flow chart for the disclosure, that illustrates another exemplary method of equalizing frequency response misalignments in a time-interleaved DAC with a low pass filter.

(5) FIGS. 4A and 4B illustrate an exemplary measurement of a spur phase.

(6) FIG. 5 (parts a and b) and FIG. 6 (parts a, b and c), show for the disclosure, examples of results of operation simulation for a time-interleaved DAC with equalization.

(7) FIGS. 7A, 7B, and 7C, and FIG. 8 (parts a and b) show for the disclosure, results of experimental tests of a time-interleaved DAC with equalization.

DETAILED DESCRIPTION

(8) An exemplary system 9 of the disclosure is shown in FIG. 1A. That system 9 includes a time-interleaved DAC 10 which, as illustrated, has the form of the DAC shown in FIG. 1. DAC 10 includes a digital DAC input 12 for receiving a digital signal-to-be-processed, and an analog DAC output 14 for presenting a processed analog output signal. DAC 10 includes a demultiplexer 16 having an input coupled to the digital DAC input 12 and includes M outputs connected to inputs of respective ones of M sub-DACs #1-#M. Outputs of the sub-DACs #1-#M are respectively coupled to corresponding inputs of a combiner 18. An output of combiner 18 is coupled to an input of a filter 19. An output of filter 19 provides the processed analog output signal to analog DAC output 14. System 9 further includes a processor network PN coupled to the DAC 10, for processing various signals of DAC 10 as described below.

(9) As generally described above, in operation, an input digital signal applied to the digital DAC input 12 is applied in turn to the demultiplexer 16. Demultiplexer 16 demultiplexes the applied input digital signal and provides the resultant signals to the inputs of the respective M sub-DACs #1-#M. Outputs of the M sub-DACs #1-#M are applied to combiner 18 to generate an analog output signal which is passed through filter 19 to the DAC analog output 14.

(10) In detail, the time-interleaved DAC 10 of the disclosure comprises M sub-DACs, wherein the m-th sub-DAC (where m and M are integers, and 0m<M) is characterized by an amplitude frequency response A.sub.m(F) and phase frequency response .sub.m(F). The values A.sub.m(F) and .sub.m(F) of these responses, at a fixed frequency F, describe the complex gain h.sub.m(F) of the sub-DAC:
h.sub.m(F)=A.sub.m(F).Math.exp(j.Math..sub.m(F)).

(11) When a signal s.sub.in(n)=exp(j2Fn) is applied to the DAC input 12, an n-th sample is processed by the sub-DAC with the number m, where m=n mod M. Here, the sample is multiplied by the sub-DAC complex gain h.sub.m(F) and is converted into an output analog sample s.sub.out(n)=A.sub.m(F).Math.exp(j2Fn+.sub.m(F)).

(12) The output signal s.sub.out(n) may be considered as a product of the input signal Sin(n) and a periodic signal P(n):
s.sub.out(n)=s.sub.in(n).Math.P(n),
where P(n)=h.sub.m(F) for m=n mod M. A Fourier transform provides for that signal, for example, provided by a processor network PN in FIG. 1A, complex amplitudes H.sub.k of the decomposition of the signal P(n) into sum of frequency components:

(13) $\begin{array}{cc}{H}_k=\underset{m=0}{\overset{M-1}{.Math.}}P\left(n\right).Math.\exp \left(-j2.Math.k.Math.m/M\right)=\underset{m=0}{\overset{M-1}{.Math.}}{h}_m\left(F\right).Math.\exp \left(-j2.Math.k.Math.m/M\right).& \left(1\right)\end{array}$ An inverse Fourier transform, for example, provided by processor network PN, reconstructs the signal P(n):

(14) $\begin{array}{cc}P\left(n\right)=h_m\left(F\right)=1/M.Math.\underset{K=0}{\overset{M-1}{.Math.}}{H}_k.Math.\exp \left(j2.Math.F_k.Math.n/F_s\right),& \left(2\right)\end{array}$ where F.sub.k=Fs/M.Math.k is the frequency of the component with the number k and F.sub.s is the DAC sampling frequency.

(15) In a form, as provided by processor network PN, a substitution of P(n) is made into the equation for s.sub.out(n), providing an expression for s.sub.out(n) in the form of a sum of exponential waveforms

(16) $s_{\mathrm{out}}\left(n\right)=1/M.Math.\underset{K=0}{\overset{M-1}{.Math.}}{H}_k.Math.\exp \left(j2.Math.\left(F+F_k.Math.n/M.Math.F_s\right)\right).$

(17) In this sum, the addend with the index k=0 corresponds to an undistorted part of the output signal. Other addends present spurious frequency components or spurs.

(18) A spur with the number k (1k<M) has the frequency F+k.Math.F.sub.s/M and equals
Spur(k)=1/M.Math.H.sub.k.Math.exp(j2.Math.(F+k*n/M.Math.F.sub.s)).

(19) According to the equation (1) above, a complex amplitude H.sub.k of a spur depends on the number k of the spur, on the number M of sub-DACs in the composite DAC 10 and on the complex gains h.sub.m(F) of the m-th sub-DAC. The spur frequencies are shifted in relation to the signal frequency F by a displacement which is a multiple of the sampling frequency Fs divided by the numbers M of the sub-DACs.

(20) According to the present disclosure, a time interleaved DAC with equalization presents a tandem connection of an equalizer and the DAC. The equalizer may be a hardware device or a software program running on a computer. The operation of the equalizer depends on the number n of the processed sample, so that it has a frequency response Eq.sub.m(F), where m=n Mod M. The tandem connection of an equalizer and the DAC has a frequency response Eq.sub.m(F).Math.h.sub.m(F).

(21) An initial cause of an appearance of spurious frequency components in a time interleaved DAC is the misalignment of sub-DACs frequency responses h.sub.m(F). To prevent an appearance of the spurious frequency components, an equalization operation corrects an existing misalignment and makes the frequency responses Eq.sub.m(F).Math.h.sub.m(F) of the tandem connection equalizer-DAC to be the same for all m.

(22) In a form, a time-interleaved DAC with equalization of the disclosure can work in one of two stages: in a calibration stage, or in an operation stage. In a calibration stage, a set of M equalizing frequency responses Eq.sub.m(F), 0m<M, is found. In an operation stage, a spectrum of an input digital signal is changed with the use of the equalizing frequency responses Eq.sub.m(F). The transformed signal is applied to the DAC input.

(23) To make possible the equalization in a time interleaved DAC, preliminary measurements at the calibration stage, provide sufficient information for calculation of the equalizing frequency responses Eq.sub.m(F). Such information may be obtained from sub-DACs frequency responses h.sub.m(F). However, such direct measurement of sub-DACs frequency responses presents severe problems and is often simply impossible. The present disclosure effects circumstantial acquisition of the needed information by determination of the amplitudes and phases of spurious frequency components. The following calculations produce sub-DACs frequency responses h.sub.m(F) and, at last, the equalizing frequency responses Eq.sub.m(F).

(24) The set of operations performed in a time interleaved DAC with equalization of the disclosure in a calibration stage is illustrated by a flow chart, shown in FIG. 2.

(25) As may be seen in the flow chart of FIG. 2, a calibration of the time interleaved DAC 10 is performed step by step, using one test frequency after another (from a list of test frequencies which are prepared in advance). The calibration begins with a first frequency of the list and ends with a last frequency.

(26) At a calibration step, a test signal s(n)=cos(2.Math..Math.F.Math.n) is generated and applied to the DAC input 12. The corresponding analog output signal is analyzed, with the amplitude Amp.sub.k and phase Phs.sub.k being determined for each spur with the number k, 1k<M. The amplitude Amp.sub.0 and the phase Phs.sub.0 of the undistorted part of the signal are also measured.

(27) The knowledge of the amplitudes Amp.sub.k and phases Phs.sub.k are used to calculate the complex gain h.sub.m for each sub-DAC with the number m by the use of the equation (2).

(28) To make a transition from sub-DACs complex gain h.sub.m(F) to equalizing frequency responses Eq.sub.m(F), a target frequency response T(F) of the tandem connection equalizer-DAC is specified. To be a target response T(F), a response that equals 1 identically may be chosen, or one of the sub-DACs complex gains h.sub.m(F), or their average, or any function of F which is suitable in DAC intended applications. A set of required frequency responses Eq.sub.m(F) is then calculated by using the equation:
Eq.sub.m(F)=T(F)/h.sub.m(F).(3)

(29) Discrete Fourier transforms of required frequency responses Eq.sub.m(F) for different m, produce M sets of equalizing coefficients C.sub.m(p), where m corresponds to a set number and p, 0p<L, where p corresponds to a number of a coefficient in a set. An amount L of coefficients in a set is determined by a required accuracy of the equalization and a required level of spurs suppression. The so-determined sets of equalizing coefficients C.sub.m(p) are used in the operation mode of the interleaved DAC with equalization of the disclosure.

(30) A DAC usually includes at its output, a low pass filter (LPF) for smoothing over the output analog signal and removing the corresponding distortions. Such a filter suppresses the spurs which fall within its stopband. Often the construction of a DAC does not permit a user to have access to the signal at the LPF input. In such an event, the amplitudes and the phases of the spurs in the LPF stopband cannot be measured. Incomplete information causes appearance of errors when calculating the equalization parameters with the equation (2), and the equalization brings about only a partial suppression of spurs.

(31) To achieve a more significant level of spur suppression, the calibration is typically repeated, each time starting from the very beginning of the procedure. At the end of a suppression procedure, spurs amplitudes are measured anew and are used as a foundation for a decision of calibration continuationand generally calibration is repeated as long as at least one spur is in excess of a prearranged threshold.

(32) A flow chart, which illustrates the sequence of operations for the case when some spurs are suppressed by LPF and are invisible, is shown in FIG. 3. This flow chart differs from the flow chart in FIG. 2 in several ways. First, a calculation of the sub-DACs complex gains h.sub.m(F) is performed with the use of the equation:

(33) $h_m\left(F\right)=1/M.Math.\left(\stackrel{k_1}{\underset{k=0}{.Math.}}{\mathrm{Amp}}_k.Math.\exp \left(j.Math.\left(2.Math.k.Math.m/M+{\mathrm{Phs}}_k\right)\right)+\underset{k=k_2}{\overset{M-1}{.Math.}}{\mathrm{Amp}}_k.Math.\exp \left(j.Math.\left(2.Math.k.Math.m/M+{\mathrm{Phs}}_k\right)\right)\right),$
where spurs with the numbers k in the interval k.sub.1<k<k.sub.2 are supposed to be invisible. Second, after the calibration is repeated for all test frequencies F (from the first to the last), the remaining level of spurs is estimated, and a decision is made to continue the calibration or to stop it.

(34) A reduction of the initial magnitude of spurs at the beginning of spurs suppression increases the extent of spurs suppression at its end. Therefore, the iterative procedure of spur suppression described above ends rapidlyusually after 2-4 iterations.

(35) Often enough, economic considerations dictate the use of inexpensive measuring devices for DAC calibration. The measurement of output signal parameters then should be restricted by the use the most simple and inexpensive measuring devices, such as the simplest spectrum analyzers which are able to measure only signal amplitude spectrum.

(36) According to the present disclosure, in a form, under the described circumstances, the spurs phases Phs.sub.k are determined using an approach: some auxiliary signals are applied to the DAC input, the amplitude of the spur produced at the DAC output by each auxiliary signal is measured, and measurement results are used for final calculation of Phs.sub.k.

(37) By way of example, the following method of spur phase detection is highly tolerant to measurements errors caused by noise and DAC non-linearity. This method is based on spur signal subtraction. The m-th sub-DAC transfer function corresponding to k-th spurious component Spur(k)=Amp.sub.k.Math.exp(j.Math.Phs.sub.k) is represented as:
h.sub.mk=Amp.sub.k.Math.exp(j(2.Math.m.Math.k/M+Phs.sub.k)),
Using the processor network PN, for example, to find the unknown phase Phs.sub.k the following test signals TS() are formed, depending on a trial phase :
TS(m,)=exp(j2Fm).Math.[1Amp.sub.k.Math.exp(j(2.Math.m.Math.k/M+)],
where m=n mod M. The first term of the test signal generates spur signal Spur(k)=Amp.sub.k.Math.exp(j.Math.Phs.sub.k) due sub-DACs frequency response. The second term at the spur frequency results in complex vector

(38) $\mathrm{RT}\left(\right)={\mathrm{Amp}}_k.Math.\exp \left(j.Math.\right).Math.\underset{m=0}{\overset{M-1}{.Math.}}{h}_m\left(F\right)/M$
The average of sub-DAC frequency responses does not depend on spur number k and has the same value for all spurious components. As a result, the output of test signal at a particular spur frequency
equals to the sum of two vectors: Spur(k) and RT(). A magnitude of spectral component measured by spectrum analyzer is maximized when phases of Spur(k) and RT() coincide and minimized when they are opposite. In an ideal case, an average sub-DAC frequency response is unity and when the trial phase is opposite to spur phase, the k-th spur magnitude is zero. To make detection more robust to noise and distortion, spur power (square of spectral magnitude), is measured using a spectrum analyzer. In this case, a dependence of the k-th spurious spectral component on test phase is modulated as cosine of the phase difference between the spur and the test signal. A total spectrum power at the spur frequency in this case equals Power(k, )=Spur(k).sup.2+RT().sup.2+2.Math.Spur(k).Math.R ().Math.cos(). Thus, making several measurements with pre-determined values of test phase .sub.1, . . . .sub.N, in the range of 0 to 2*, a spur phase is found by a standard quadrature detection method, well known in the prior art.

(39) The method of spur phase detection described above is not sensitive to amplitude errors since minimum phase location does not change. A phase of a reference vector may deviate from zero so that all spur phases are determined with constant phase offset. This frequency-independent phase offset results in a constant phase shift of DAC frequency responses and does not affect accuracy of spur correction. Usually all sub-DAC frequency responses are normalized relative to one (e.g. the first) sub-DAC which eliminates the phase offset. In a form, in a high SNR environment, for example, only three trial phase values may be required for a spur phase calculation. However, in a low SNR environment, use of more trial phase measurements will provide increased accuracy.

(40) In a form, an additional amplitude adjustment step may be used to correct non-linearity of a DAC response. Once a spur phase is found, additional measurements may be performed, aimed to force residual spur magnitude to zero (or a device noise floor level). This is done by generating plurality of test signals with a variable compensating spur amplitude and minimizing spectral magnitude at a spur frequency. Minimization can be done using standard methods well known in the prior art, such as a steepest descent or golden section search.

(41) Another method of spur phase detection that requires only three test signals and relies on ratio of spur amplitudes, may be preferable in some circumstances. In this method, a first auxiliary signal is built as a harmonic signal with a test frequency F and an envelope P.sub.in(n)=exp(j2.Math.m.Math.k/M), where m=n Mod M. After passing through the DAC, each sample of the signal is multiplied by a complex gain h.sub.m(F) of the sub-DAC with the number m so that an output envelope equals P.sub.out(n)=h.sub.m(F).Math.exp(j2.Math.m.Math.k/M). According to equation (1) above, the complex amplitude H.sub.k of the spur with the number k in this case equals:

(42) $\begin{array}{c}{H}_k=\underset{m=0}{\overset{M-1}{.Math.}}{P}_{\mathrm{out}}\left(n\right).Math.\exp \left(-j2.Math.k.Math.m/M\right)=\underset{m=0}{\overset{M-1}{.Math.}}{h}_m\left(F\right).Math.\exp \left(j2.Math.m.Math.k/M\right).Math.\\ \exp \left(-j2.Math.k.Math.m/M\right)=\\ =1/M.Math.\underset{m=0}{\overset{M-1}{.Math.}}{h}_m\left(F\right).\end{array}$
The last expression does not contain the spur number k and is the same for all spurs. In the following calculations, the vector

(43) $R=1/M.Math.\underset{m=0}{\overset{M-1}{.Math.}}{h}_m\left(F\right)$
is used as a reference vector. A measurement by an available spectrum analyzer at the output of the DAC, produces the amplitude |P| of this reference vector.

(44) The second auxiliary signal is built as a harmonic signal with the test frequency F and the envelope P.sub.in(n)=1+(Amp.sub.k/|R|).Math.R. When the signal with such an envelope is applied to the DAC input, a spur with the number k appears at the DAC output among other spurs. The total complex amplitude A.sub.T of this spur is a sum of two addends: A.sub.T=A.sub.1+A.sub.2. The first addend A.sub.1 is caused by a unit part of the envelope so that its complex amplitude equals the complex amplitude H.sub.k with a modulus Amp.sub.k. A second addend A.sub.2 is caused by a second part of the envelope. Its complex amplitude is directed along the reference vector R and has a modulus |Amp.sub.k.Math.|R|/R|=Amp.sub.k. Thus, |A.sub.1|=|A.sub.2|. The composition of these two addends is illustrated in FIG. 4a. In this diagram, a sum A.sub.TOf the two addends is shown as a diagonal of a parallelogram with the sides A.sub.1 and A.sub.2. Since |A.sub.1|=|A.sub.2|, the parallelogram is a rhomb with perpendicular diagonals. The phase Phs.sub.k of the initial spur equals the angle between vector A.sub.1 and a reference vector R, where the angle is equal to a doubled angle between the diagonal of a parallelogram and its side. From the right-angled triangle PQS, the last angle equals a cos(|A.sub.T|/(2*Amp.sub.k)). After measuring the total amplitude |A.sub.T|, a modulus of the phase Phs.sub.k is calculated using the equation
|Phs.sub.k|=2.Math.a cos(|A.sub.T|/(2*Amp.sub.k)).

(45) The vector H.sub.k may be located in one of two position relative to the vector R (see FIG. 4b). The angles between vectors H.sub.k and R in these two positions differ in sign. However, the preceding procedure makes it possible to find in either case only the modulus of the angle. To remove the ambiguity a third auxiliary signal is applied to the DAC input, for example, by processor network PN. This signal is built as a harmonic signal with a test frequency F and an envelope P.sub.in(n)=1+R.Math.exp (j.Math./2)/|R|. A spur produced by such a signal, is a sum of a vector with the complex amplitude H.sub.k and a vector caused by a second part of the envelope which is directed perpendicular to the reference vector R. When the vector H.sub.k is located counter-clockwise with respect to the vector R, the modulus of the sum A.sub.T is larger than the amplitude Amp.sub.k of the vector A.sub.1. When the vector H.sub.k is located clockwise with respect to the vector R, the modulus of sum A.sub.T is less than the amplitude Amp.sub.k of the vector A.sub.1. These considerations lead to the unambiguous rule of the spur phase determination:
Phs.sub.k=2.Math.a cos(|A.sub.T|/(2*Amp.sub.k)), if |A.sub.T|>Amp.sub.k, and
Phs.sub.k=2.Math.a cos(|A.sub.T|/(2*Amp.sub.k)), else.

(46) Once the spurs phases Phs.sub.k have been determined (in addition to the direct measurement of spurs amplitudes Amp.sub.k), the complex gains h.sub.m(F) of the sub-DACs are calculated by the use of the equation (2), and equalizing complex gains Eq.sub.m(F) are found with the use of the equation (3). In this manner, the measurement of spurs phases Phs.sub.k and a calibration of a time interleaved DAC with the use of a spectrum analyzer capable of measuring the amplitude spectrum only, is made possible.

(47) As an illustration, FIGS. 5-8 show results of a software simulation and hardware tests of a time interleaved DAC with equalization according to the present disclosure. FIG. 5 shows an input spectrum (in part a) and an output spectrum (in part b) of a simulated time interleaved DAC with equalization (32 sub-DACs with 64 Gs/s sampling rate). Sub-DACs frequency responses were measured using all 32 spurs, resulting in ideal suppression of spurious components (below 100 dB level). FIG. 6 shows a result of a two-pass interleaved DAC simulation, when an output spectrum bandwidth is limited by 13 GHz. The input signal spectrum shown in FIG. 6 (in part a) results in a one-pass equalization on (as shown in part b). Most spurs are suppressed below 65 dB level, and after a second iteration of equalization (as show in part c), residual spurs are reduced below 90 dB.

(48) FIGS. 7A, 7B, 7C show comparisons of experimental signal sine wave spectrums generated by a Keysight AWG 8195 signal generator before and after sub-DACs equalization. After sub-DACs responses were measured, an equalizer with 159 taps was used to pre-distort AWG signal prior to loading it to the AWG 8195 signal generator. Sine wave signals at 7.7, 8.3 and 9.3 GHz spectrum were measured without (top graphs in FIGS. 7A, 7B and 7C), and with (bottom graphs in FIGS. 7A, 7B and 7C), DAC equalization. The spectrum contains multiple low-level spurs caused by the interleaved DAC structure (32 sub-DACs) and non-linear distortions. The suppressed sub-DAC spurs are highlighted by gray boxes. Strong reflection (up to 38 dB spur magnitude) occurs from frequency response misalignments between four groups of sub-DACs, generating a biggest spur at 16 GHzcenter frequency. After equalization, this spur is suppressed below 70 dB level. Other sub-DAC spurs are smaller and occur at 2 GHz steps. Most of these minor spurs are also suppressed below 70 dB level.

(49) The sub-DAC equalization impact on wideband modulated signals was evaluated by generating 1 GHz wide QAM16 signal using the AWG8195 signal generator at 64 Gs/s sampling rate. Error vector magnitude (EVM) measurement was performed using Keysight 89600 VSA software integrated with Guzik Technical Enterprises ADP700 series 32 GS/s digitizer. When modulated QAM16 signal is centered at 8 GHz and a channel frequency response is compensated by an adaptive equalizer, the best EVM equals 1.83%. Sub-DACs equalization reduces EVM to 1.45% (0.38% improvement). This EVM gain is caused by suppression of big reflection occurring in the vicinity of 8 GHz region due to reflection from 16 GHz frequency shown in FIGS. 7A, 7B and 7C. Signal quality improvement can also be seen on eye-diagrams, for example, as shown at FIG. 8 (in part a for EVM=1.83% and in part b for EVM=1.45%). In part a and part b of FIG. 8, the equalized signal (bottom eye-diagram) have better eye opening than the non-equalized signal (top eye-diagram).

(50) Although the foregoing description of the embodiment of the present technology contains some details for purposes of clarity of understanding, the technology is not limited to the detail provided. There are many alternative ways of implementing the technology. The disclosed embodiment is illustrative and not restrictive.