2.2 Non-Idealities of Time-Interleaved ADCs
2.2.2 Specific Time-Interleaving Limitations
The specific time-interleaving limitations discussed in this section appear because of mismatches between channels. They can have a mismatch in offset, gain and timing, of which the timing-mismatch is the largest problem in fast TI-ADCs. All these limitations occur in the frequency domain as spurious tones and therefore degrade the SFDR of the TI-ADC. To understand the occurrence of the spurious tones, the discrete-time domain equations (2.3) and (2.4) are represented in the continuous-time domain by equations (2.6) and (2.7), where the quantization is neglected:
γi(t) = x(t) ·
First the combination of the mismatches is shown, subsequently the mismatches are analyzed separately. The mismatch errors are inserted in equation (2.6):
γi(t) = (gix(t − ∆ti) + oi)
where for channel i: oiis the offset, gi is the gain and ∆ti is the timing error [4]. Then the output of the TI-ADC is:
y(t) =
To obtain the output spectrum, the Fourier transforms of ˆxi(t) and si(t) are needed:
Xˆi(jΩ) = giX(jΩ)e−jΩ∆ti+ oi2πδ(Ω) (2.10)
2.2. NON-IDEALITIES OF TIME-INTERLEAVED ADCS 7
where X(jΩ) is the Fourier transform of x(t)
Si(jΩ) = 2π
The Fourier transform of y(t) then becomes:
Y (jΩ) = With this, the general formula for the combination of the offset-mismatch, gain-mismatch and timing-mismatch becomes:
Simplifying equation (2.16), only the tones at kΩs± Ω0 are left because all other tones cancel out:
Y (jΩ) = M TAπ
8 CHAPTER 2. TIME-INTERLEAVED ADCS
where k = m/M for mmodM = 0. For the fundamental interval equation (2.17) becomes:
Y (jΩ) = Aπ
Tsj[(δ (Ω − Ω0) − δ (Ω + Ω0))] (2.18) The canceling of the non-ideal tones also happens if the offset, gain and timing in all channels are equal (oi = o, gi= g and ∆ti= ∆t):
For the fundamental interval equation (2.19) becomes:
Y (jΩ) = gAπ
jTse−jΩ0∆t[(δ (Ω − Ω0) − δ (Ω + Ω0)) + o2πδ (Ω)] (2.20) With an offset unequal to zero a DC tone for the offset of the channels is introduced.
Figure 2.6 shows the single sided spectrum of an ideal TI-ADC for two input frequencies.
0 50 100 150 200 250 300 350 400 450 500
Figure 2.6: Simulated spectrum of an ideal 4-channel TI-ADC with fs=1GHz, (a) single sided spectrum of y[n] with f0≈45.7153MHz 16384 point FFT, (b) single sided spectrum of y[n] with f0≈494.4458MHz 16384 point FFT.
In the next sections the different mismatch errors are discussed separately.
Offset Mismatch
For the discussion about offset-mismatch the offset errors are assumed to be different for each channel, and all other characteristics are the same. Offset is a DC error per sub-ADC which becomes periodic with time-interleaving. Therefore, the offset-mismatch is periodic with period M Tsand independent of the input signal. In the frequency-domain the offset-mismatch appears as tones at frequencies independent of the input frequency and independent of the input amplitude.
This can be shown by eliminating the gain and timing mismatches in equation (2.9) and (2.15):
y(t) =
2.2. NON-IDEALITIES OF TIME-INTERLEAVED ADCS 9
Therefore, the tones are at:
Ωerror= MmΩs, m ∈ Z (2.23)
The power of the offset error is constant and independent of the input signal as well. Figure 2.7 shows the simulation of the TI-ADC as in figure 2.6 but with the offset-mismatch (o=[−0.002 0.0033 −0.0021 −0.004]). [5]
0 50 100 150 200 250 300 350 400 450 500
Figure 2.7: Simulated spectrum of a 4-channel TI-ADC with fs=1GHz and offset-mismatch (o=[−0.002 0.0033 −0.0021 −0.004]),
(a) single sided spectrum of y[n] with f0≈45.7153MHz 16384 point FFT, (b) single sided spectrum of y[n] with f0≈494.4458MHz 16384 point FFT.
The simulation shows that the spurious tones are at a fixed frequency and the SFDR and SNDR are also independent of the input frequency.
Gain Mismatch
For the discussion about gain-mismatch the gain errors are assumed to be different for each channel, and all other characteristics are the same. The errors also occur with a period of M Ts, just as offset mismatch, but the errors are amplitude modulated with the input frequency Ω0. The largest absolute errors occur at the peaks off the input signal. Therefore, in frequency domain, the location of the error is dependent on the input frequency while the power of the error is independent of Ω0 but dependent on the amplitude of the input signal. This can be shown by eliminating the offset and timing mismatches in equation (2.9) and (2.15):
y(t) =
10 CHAPTER 2. TIME-INTERLEAVED ADCS
Therefore, the tones are at:
Ωerror=MmΩs± Ω0, m ∈ Z, mmodM 6= 0 (2.26) Figure 2.8 shows the simulation of the TI-ADC as in figure 2.6 but with the gain-mismatch (g=[0.994 0.9891 1.009 0.996]). The simulation shows that the location of the spurious tones are
0 50 100 150 200 250 300 350 400 450 500
Figure 2.8: Simulated spectrum of a 4-channel TI-ADC with fs=1GHz and gain-mismatch (g = [0.994 0.9891 1.009 0.996]),
(a) single sided spectrum of y[n] with f0≈45.7153MHz 16384 point FFT, (b) single sided spectrum of y[n] with f0≈494.4458MHz 16384 point FFT.
at a frequency dependent on the input frequency but the SFDR and SNDR are independent of the input frequency.
Timing Mismatch
For the discussion about timing-mismatch, the timing error, due to clock-skew, is assumed to be different for each channel, and all other characteristics are the same. The errors again occur with a period of M Ts and are amplitude modulated with the input frequency Ω0 just as the gain-mismatch. The largest errors occur at the largest slew-rate of the sine wave. Therefore, the location of the error in the frequency domain is again dependent on the input frequency, and the power of the error is proportional to Ω0 and dependent to the amplitude of the input signal. This can be shown by eliminating the offset and gain mismatches in equation (2.9) and (2.15):
y(t) =
Therefore, the tones are at:
Ωerror=MmΩs± Ω0, m ∈ Z, mmodM 6= 0 (2.29) Figure 2.9 shows the simulation of the TI-ADC as in figure 2.6 but with the timing-mismatch (∆t=[−0.009 −0.002 −0.008 0.004]ns).
The simulation shows that the locations of the spurious tones are dependent on the input frequency. It also shows that the SFDR and SNDR are also dependent on the input frequency.
2.2. NON-IDEALITIES OF TIME-INTERLEAVED ADCS 11
Figure 2.9: Simulated spectrum of a 4-channel TI-ADC with fs=1GHz and timing-mismatch (∆t=[−0.009
−0.002 −0.008 0.004]ns),
(a) single sided spectrum of y[n] with f0≈45.7153MHz 16384 point FFT, (b) single sided spectrum of y[n] with f0≈494.4458MHz 16384 point FFT.
Because the error is proportional to the input frequency, this error is dominating at higher speeds and therefore important and challenging to correct with high accuracy. The timing-mismatch spurs are located at the same frequencies as the gain-timing-mismatch spurs, but can be distin-guished because the power of the gain-mismatch is independent of the input frequency (dominant at low frequencies) and timing-mismatch is dependent on the input frequency (dominant at high frequencies).
Total Mismatch
Figure 2.10 shows the same graphs as in figure 2.6 but with all mismatch errors included. Offset (o=[−0.002 0.0033 −0.0021 −0.004]), gain (g=[0.994 0.9891 1.009 0.996]), timing (∆t=[−0.009
−0.002 −0.008 0.004]ns).
X Gain and timing mismatches SFDR= 46.6dB SNDR= 41dB
X Gain and timing mismatches SFDR= 36.2dB SNDR= 34.7dB
Figure 2.10: Simulated spectrum of a 4-channel TI-ADC with fs=1GHz and all three mismatches, offset (o=[−0.002 0.0033 −0.0021 −0.004]), gain (g=[0.994 0.9891 1.009 0.996]), timing (∆t=[−0.009 −0.002 −0.008 0.004]ns),
(a) single sided spectrum of y[n] with f0≈45.7153MHz 16384 point FFT, (b) single sided spectrum of y[n] with f0≈494.4458MHz 16384 point FFT.
The simulation shows that the timing-mismatch becomes dominant over the gain-mismatch for higher frequencies, as expected.
12 CHAPTER 2. TIME-INTERLEAVED ADCS