4.1. A Fixed Fractional Delay Filter
The proposed method adopts the fractional delay filters to estimate the gain, time skew, and bandwidth mismatch in each channel. The fractional delay filters are used in each sub-ADC. Therefore, the input signal is limited to the first Nyquist bandwidth of the sub-ADC. To make the proposed algorithm work for the high Nyquist bandwidth of the sub-ADC in the TI-ADCs system, the fractional delay filters should be effective in high sub-ADC Nyquist bandwidth.
When the bandpass (BP) signals are in the
NBP-th Nyquist bandwidth, the frequency can be expressed as follows:
Here NBP ≥ 1, fL, and fH are the low and high cutoff frequencies of the signal, respectively. Additionally, fs is the sampling frequency of sub-ADC.
The BP signal is constructed by the sum of two complex signals of
x+(
t) and
x−(
t). Therefore, the frequency response of the complex signals can be respectively represented as X
+(
f) and X
−(
f). According to the band-pass sampling theory, the original spectrum of the band-pass signal can be folded back to the baseband in the under-sampled TI-ADC converter whose sampling frequency conforms to the Nyquist criterion. In the odd-order Nyquist band, the baseband spectrum of the under-sampling signal has the same shape as the original spectrum. Additionally, the baseband spectrum of even order NB input signal is reversed. In the odd-order Nyquist band, X(
f) can be represented as follows:
Here, Y
+(
f) and Y
−(
f) is the output of X
+(
f) and X
−(
f), respectively. Furthermore,
nn can be obtained as follows:
where the input signal is the
NBP-th Nyquist bandwidth and
is the result of rounding down
NBP/2. Therefore, the inverse fast Fourier transform (IFFT) of X(
f) can be expressed as
According to the Euler formula,
x(
t) can be rewritten as follows:
In the time domain,
y+(
t) and
y−(
t) can be obtained as follows:
According to Equation (27), the reference of the
i-th channel can be represented as follows:
Therefore, the DFT of Equation (28) can be expressed as follows:
where,
y0[
n +
i/
Mfs] can be represented as follows:
And
yh0[
n +
i/
Mfs] can be represented as follows:
In the even Nyquist bands, the DFT of
xi(
t +
i/
Mfs) can be expressed as follows:
Figure 5 shows the reference generation of the TI-ADC system with BP signal input. The IDFT of Hilbert filter frequency response can be expressed as follows:
Additionally, the causal filters’ coefficients are obtained by windowing Equation (33).
4.2. Simulation Result
The signal-to-noise distortion ratio (SNDR) is a metric that describes the signal performance. Additionally, SNDR can be calculated by the following:
where
PSignal is the power of the ideal signal,
PNoise is the power of the noise signal, and
PDistortion is the spurious power of the output signal. In the TI-ADCs system, the presence of mismatches leads to an increase in
PNoise. When and
PNoise are fixed, the smaller the calibration error, the higher the SNDR. Therefore, the SNDR is applied to evaluate the effect of error calibration.
The calibration of offset is simpler than other mismatches. Hence, the offset is set to 0 in the simulation tests. The 12-bit 8 GS/s 8-channel TI-ADCs simulation system is established according to Equation (3) with MATLAB. Additionally, the mismatches of the simulation model are shown in
Table 1. Moreover, the parameters of the bandpass fractional filters are shown in
Table 2. Additionally, the number of taps of
is 73. In the 12-bit 8 GS/s 8-channel TI-ADCs simulation system, the SNDR of the input signal before sampling is 60 dB because of the Gaussian white noise in each channel.
Figure 6 shows the values of SNDR and spurious-free dynamic range (SFDR) for sinusoidal signals with the different frequencies of the first Nyquist bandwidth. The frequency range is limited within 0.05–0.42 GHz. Additionally, the frequency spacing between two single-tone sinusoidal signals is 0.01 GHz. Where
fc is the sampling frequency of sub-ADC. In
Figure 6a, the SNDR is stable above 65 dB in the test frequency range. Additionally, the SFDR is stable above 80 dB in
Figure 6b.
The mismatches shown in
Table 3 are set to verify that the proposed methods are still effective when there are gain, time skew, and bandwidth mismatch at the same time. The frequency range is limited within 0.05–0.42 GHz. Where
fc is the frequency of sub-ADC, and
fs is the frequency of the TI-ADCs system. Additionally,
Figure 6 shows the SNDR and SFDR before and after calibration. It can be seen from
Figure 7 that the proposed calibration method still has a good effect in the cases of gain, time skew, and bandwidth mismatch. The SNDR before calibration does not exceed 20 dB. Additionally, the SNDR is above 65 dB after calibration. Therefore, the value of SNDR is improved by at least 45 dB after calibration. Moreover, the value of SFDR is above 80 dB after calibration.
Figure 8 and
Figure 9 show the calibration performance within the second and third Nyquist bandwidths of the sub-ADC, respectively. Additionally, the frequency ranges are limited within 0.55–0.92 GHz and 1.05–1.42 GHz, respectively. The offset, gain, time skew, and bandwidth mismatch of the TI-ADCs system are shown in
Table 3.
It can be seen from
Figure 7,
Figure 8 and
Figure 9 that the SNDR and SFDR of the TI-ADCs system decrease with the increase in the input signal frequency at the same gain, time skew, and bandwidth mismatch.
Figure 8 and
Figure 9 show that even if the input signal is not located in the first Nyquist bandwidth of the sub-ADC, the proposed methods still have a good calibration effect. Furthermore,
Figure 7,
Figure 8 and
Figure 9 show that the proposed methods are effective for bandpass signals with different Nyquist bandwidths. Additionally, the SNDR is above 65 dB and the SFDR is above 80 dB in the first, second, and third Nyquist frequency bands. Therefore, the frequency of the input signal has relatively little effect on the proposed technique.
Figure 10 shows the frequency spectrums of the multi-tone signal before and after calibration. Where the frequencies of the multi-tone signal are [2.30 GHz, 2.42 GHz, 2.54 GHz, 2.66 GHz, 2.78 GHz].
Figure 10a shows the frequency spectrum of the TI-ADCs system before calibration. Where the offset, gain, time skew, and bandwidth mismatch are shown in
Table 3. The performance of the TI-ADCs system shown in
Figure 10a is poor. Where the SNDR is −2.77 dB and the SFDR is −17.28 dB. Additionally, the performance of the TI-ADCs system is well-improved after calibration. In
Figure 10b, the SNDR is 60.19 dB and the SFDR is 75.53 dB. Therefore, the SNDR is improved by 62.96 dB, and the SFDR is improved by 92.81 dB.
Figure 10 shows that the proposed calibration algorithm is still effective for multi-tone bandpass signals with high sub-ADC Nyquist bandwidth.
There are matrix inversions in the LS algorithm. Additionally, the implementation of matrix inversion in FPGA is complex. Therefore, Equation (19) is rewritten as follows:
Hence, a system of linear equations can be constructed as follows:
According to Cramer’s rule [
29], the solution of
requires about
N!(
N − 1)(
N + 1) times of addition and
N!(
N + 1)
N times of multiplication. In this way, the matrix inversion can be avoided when solving
. Additionally, the solution of
AN×N requires about
N(
Ls − 1)
N times of addition and
NLsN times of multiplication. Similarly, the solution of
bN×1 requires about
N(
Ls − 1) times of addition and
times of multiplication. Therefore, the estimation in the
i-th channel of gain, time skew, and bandwidth mismatch requires about
times of addition and
times of multiplication. Where
L is the number of bandpass fractional delay filter taps.
The comparison results of the proposed technique with the prior state-of-the-art are summarized in
Table 4. The bandwidth mismatch can be calibrated with the proposed technique. In the reference [
13], the calibration of time skew is based on the correlation between channels. However, Equation (36) of reference [
13] is established under the necessary condition that the time skew of the adjacent channels is much smaller than the sampling period of the sub-ADC. To summarize, the calibration techniques of reference [
13] must meet the necessary condition that the time skew between adjacent channels or compared with the reference channel is much smaller than the sampling period of the sub-ADC. Similarly, there are restrictions on the size of the time skew in the references [
30,
31]. References [
16,
27] and this work approach enables a wide range of time skew mismatch compensation. Therefore, references [
16,
27], and this work have a good effect on the improvement of SNDR and SFDR. Work [
13,
16,
27], and this work needs 1045.5 K, 277.5 K, 720 K, and 128.1 K multiplication modules to estimate time skew, respectively. Additionally, the multiplication modules occupy most of the computing resources in the techniques of time skew estimation or correction. Therefore, it can be considered that the proposed estimation technique in this work has the lowest power consumption. Work [
13], work [
16], work [
27], work [
30], work [
31], and this work needs 31, 33, 12, 25, 21, and 5 multiplication modules to correct time skew respectively. Therefore, the power consumption of the proposed technique of correction time skew is far less than techniques [
13,
16,
27,
30,
31]. ENOB represents the final performance of the TI-ADC system after calibration. Compared with work [
16,
27], and [
30], the proposed technique has the best performance in ENOB. The quantization error is ignored in the work [
31]. Therefore, the ENOB after calibration is not compared between work [
31] and others.
In Equation (19), quantization noise and Gaussian white noise are not considered. Therefore, the calibration of time skew is affected by quantization noise and Gaussian white noise.
Figure 11 shows the performances of calibration time skew in a 2-channel 6.4 GS/s TI-ADCS under different white noise levels.
As shown in
Figure 11, there is a positive correlation basically between the SNDR of TI-ADCS and the SNR of sub-ADC when the SNR of sub-ADC is in the range of 0~100 dB. Therefore, the proposed technique can effectively calibrate the gain, time skew, and bandwidth mismatch at the SNR of 0~100 dB. In
Figure 12, there is a positive correlation between the SNDR of TI-ADCS and the ENOB of sub-ADC when the ENOB of sub-ADC is in the range of 1 bit~16 bit. It can be seen from
Figure 11 and
Figure 12 that the proposed technique is effective when the SNR of sub-ADC is in the range of 0~100 dB or the ENOB of sub-ADC is in the range of 1~16 bits.
The minimum value of
is obtained by the least square method. Hence, the cumulative error
is affected by quantization error and white Gaussian noise. In
Figure 13 and
Figure 14, the cumulative errors of the proposed technique are respectively shown under different quantization errors and Gaussian white noise of sub-ADC. It can be seen from
Figure 13 and
Figure 14 that the cumulative error is stable and approaches 0 when the SNR is greater than 30 dB and the ENOB is greater than 5 bits. It can be concluded from
Figure 11,
Figure 12,
Figure 13 and
Figure 14 that the proposed technique will be affected by the SNR and ENOB. However, the proposed technique has a good effect in the 30~100 dB of SNR or the 5~16 bit of ENOB.