4.1 Introduction
This chapter presents an overview of a time-interleaved ADC (TI-ADC), including the concept of time-interleaved sampling, and mismatch error sources. As described in Chap-ter 2, the sampling speed of SC pipelined ADCs is usually limited by the opamp’s per-formance. Time-interleaving an array of slow ADCs can increase the effective sampling rate [6]. However, the entire A/D conversion linearity will be degraded if there exist mismatches among these ADCs.
A TI-ADC has two primary properties due to the time-interleaving operation. One is that time-interleaved sampling can increase the sampling rate, but not increase the indi-vidual ADC’s clock rate. This allows for high data rate without increasing each ADC’s processing speed. The other is that using time-interleaving A/D architecture makes the ADC’s output data easy to be parallel processed, since each ADC can provide its own data stream.
47
V0(t) V1(t)
Figure 4.1: A conventional time-interleaved ADC.
4.2 Overview of Time-Interleaved ADCs
A conventional TI-ADC along with its timing diagram is shown in Fig. 4.1, where Tc is the clock period of the individual A/D channel, and its corresponding clock frequency is given by fc = 1/Tc. This TI-ADC consists of M low-speed A/D channels, and hence has an effective sampling rate of fs = M × fc. Each channel comprises a sample-and-hold (S/H) circuit and an ADC. In each channel, the analog input, Vi(t), is sampled by the front-end S/H. The sampled signal, Vm(t), is then quantized by the following ADC to produce the digital output, Dmo[n]. The final digital output, Do[n], of the TI-ADC is obtained by multiplexing Dom[n]’s. In this figure, t means the continuous-time index, and nmeans the discrete-time index.
The operation of the front-end S/H is typically modeled as that the Vi(t) input
mul-4.2. OVERVIEW OF TIME-INTERLEAVED ADCS 49
Figure 4.2: The behavioral model of S/H in a TI-ADC.
tiplies with the channel’s sampling clock, φm(t), and then produces the discrete output, Vm(t). The model is as shown in Fig. 4.2. The φm(t) sampling signal can be expressed using the delta function, δ(t), with the sampling period of Tc = M × Ts, where Ts = 1/fs
is the TI-ADC’s sampling period. The φm(t) is hence written as φm(t)= whose corresponding Fourier transformation (FT) is obtained by
V˜m(f ) = ˜Vi(f ) ∗ ˜Φm(f )= 1 obtained by linearly adding each channel’s output spectrum, ˜Dm(f ), and is given by
D˜o(f )=
M−1
X
m=0
D˜m(f ) (4.4)
Assume these ADCs are ideal, for all m. Then, the ˜Dm(f ) can be replaced by ˜Vm(f ), and (4.4) can be rewritten as
D˜o(f ) =
Fig. 4.3 shows the conceptual spectra of a 4-channel TI-ADC. According to the sam-pling theorem, there are no aliasing as long as the frequency of Vi(t) input is equal or less than half of the f s sampling frequency, and the output spectrum, ˜Do(f ), periodically repeats the input spectrum, ˜Vi(f ), at the multiplicative frequency of fs. All channels’
spectra have the same magnitude but different phases except those at the multiplicative frequency of fs. These spectra are shown as the ˜V0(f ) to ˜V3(f ) in Fig. 4.3. Ideally, as they are combined into the output, the ˜Do(f ) output spectrum can exactly repeat the input spectrum at the frequencies of the integral multiples of fs.
4.3 Nonidealities in Time-Interleaved ADCs
The use of the time-interleaved A/D architecture introduces errors in the sampled signals, which would not appear in a single A/D channel. Sampling time skew, gain and offset mismatches among A/D channels are three major error sources which will distort a TI-ADC’s output [33]. In this section, the errors caused by timing mismatch, gain and offset mismatches are discussed by assuming the TI-ADC’s sampling rate is fs, and M chan-nels are time-interleaved. Moreover, a 4-channel TI-ADC is illustrated to describe the mismatch effects on the final ADC’s output spectrum.
4.3.1 Timing Mismatch
Timing accuracy of the front-end S/H circuit in each channel is required more crucial than its following ADC, since the S/H circuit proceeds the full bandwidth of the input signal whereas the ADC proceeds the slowly held signal. However, this accuracy is degraded due to the imperfections of the clock generator network in the TI-ADC, the actual turn-off time of the switches in the S/H circuit, and the thermal noise of devices. These imperfections can be categorized into systematic errors and random errors. Systematic errors result
4.3. NONIDEALITIES IN TIME-INTERLEAVED ADCS 51
0
f
f f
f phase
k=0 k=1 k=2 k=3 k=4 k=5 k=6
f (f) + (f) + (f) + V 3 (f)
V 2 V 1
V 0 (f) = D o
fs/4 2fs/4 3fs/4 fs 5fs/4 6fs/4 0
1
3 (f) 2 (f) (f) (f)
V V V V
~
~ ~ ~ ~ ~
~
~
~
Figure 4.3: Conceptual spectra of a 4-channel TI-ADC.
∆tm
Vi (t) Ideal
Sample
Actual Sample
φm(t)
φm(t)
^
∆Ve Amplitude
Time
Figure 4.4: Illustration of timing skew effect on the sampled signal.
in frequency dependent distortion in the output spectrum. Random errors resulted from clock jitter and thermal noise can be viewed as the white noise added to the output signal.
The random errors cannot be avoided in any circuits, and they are not further discussed in this section.
Systematic timing errors resulted from the clock skews among the A/D channels can be modeled as a time delay ∆tm in the φm(t) sampling function. This mismatch effect on the sampled signal of the m-th ADC can be illustrated as shown in Fig. 4.4. The ideal φm(t) sampling function shown as the square wave with solid line, and can be rewritten with the skew effect as
φˆm(t)= X+∞
n=−∞
δ(t − [(nM+ m)Ts+ ∆tm]) (4.6)
represented with the dot-lined square wave, where using ˆφrepresents the nonideal case.
When ˆφis used to sample the Vi(t) input, the sampled data will be incurred with the time skew errors, and is shown as the square symbol in Fig. 4.4. The mathematical expression of the sampled data is therefore given by
Vm,∆t(t)= Vi(t) × ˆφm(t)= Vi(t) ×
+∞
X
n=−∞
δ(t − [(nM+ m)Ts+ ∆tm]) (4.7)
Furthermore, we have the output spectrum of the TI-ADC affected by the time skew given
4.3. NONIDEALITIES IN TIME-INTERLEAVED ADCS 53
k=3 k=5 k=6
k=2 k=4
∆ t
∆ t
∆ t
∆t ∆ t
f V 3, (f) V 2, (f) +
V 1, (f) +
D ~ o, (f) = V ~ 0, (f) + ~ ~ ~
∆t (f) V 0,
~
∆t (f) V 3,
~
∆t (f) V 1,
~
∆t (f) V 2,
~
0 fs/4 2fs/4 3fs/4 fs 6fs/4
f
f f 5fs/4
f k=0 k=1
phase
Figure 4.5: Conceptual spectra of a 4-channel TI-ADC with sampling skew mismatch.
as:
Comparison of (4.8) and (4.5) shows that the first one has an extra term of e
−j2πk∆tmMTs
. If the ∆tm time skew is a constant, it causes an effective gain error in the sampled signal of the individual A/D channel. If there exist mismatches in ∆tm’s, they cause distortions at the frequency locations of kt/MTsin the TI-ADC’s output spectrum except the multiples of 1/Ts, where ktis an integer.
The conceptual spectra of a 4-channel TI-ADC interfered with sampling time skew in A/D channels are sketched in Fig. 4.5. If the 2nd and the 3rd A/D channels have sampling time skew, their output spectra incur phase errors. When they are combined into the final ADC’s output, these errors will reflect on the output spectrum. Finally, they result in the gain error of magnitude and the phase shift in the output spectrum, ˜Do,∆t(f ).
4.3.2 Gain Mismatch
The gain effect of the m-th channel in a TI-ADC can be modeled as a multiplication of the input signal Vi(t) with a gain factor of gm, for all m = 0, 1, · · · , M − 1, as shown in Fig. 4.6. A multiplier added in the front of the S/H circuit in each channel is used to model the gain factor. Ideally, g0= g1 = · · · = gM−1 = 1.
Assume the linear gain factor is only considered. The sampled signal of the S/H output is therefore modified as
Vm,g(t)= gm× Vi(t) × φm(t)= gm × Vi(t) × X+∞
n=−∞
δ(t − (nM+ m)Ts) (4.9) for all m= 0, 1, · · · , M − 1. In this model, the S/H of the m-th channel samples gm× Vi(t) rather than Vi(t). Fig. 4.7 shows an illustration of gain error influenced on the sampled signal. Due to the gain error, the sampled data have ∆Ve different from the nominal case as shown in the figure. This will result in insufficient Dmo output codes, and missing codes will appear in the final digital output.
In the frequency domain, the nonideal gain will distort the final output spectrum. As-sume gm is constant, for all m. The mathematical expression of the final output spectrum
4.3. NONIDEALITIES IN TIME-INTERLEAVED ADCS 55
V0,g(t) ADC
0
ADC 1
ADC m
ADC M−1
φ
0φ
1φ
mφ
M−11 S/H S/H
0 m
S/H S/H
M−1 g1
g0 gm gM−1
V (t)i
Do [n] Do0
Do1
Dom
DoM−1 V1,g(t) Vm,g(t) VM−1,g(t)
Multiplexer
Figure 4.6: Model of A/D channel’s gain factor in a TI-ADC.
φm(t) Vi (t)
Vi (t) gm x
∆Ve
Actual Sample
Ideal Sample
Time Amplitude
Figure 4.7: Illustration of gain error effect on the sampled data.
k=1 k=2 k=3 k=4 k=5 k=6
0 fs/4 2fs/4 3fs/4 fs 6fs/4
f
f f 5fs/4
f 0,g(f)
V 2,g(f)
V 3,g(f) V ~
~
~
k=0 phase
f D o,g(f) = V 0,g(f) + V 1,g(f) + V 2,g(f) + V 3,g(f)
~ ~ ~ ~ ~
V 1,g(f)
~
Figure 4.8: Conceptual spectra of a 4-channel TI-ADC with gain effect.
4.3. NONIDEALITIES IN TIME-INTERLEAVED ADCS 57
with nonideal gain is therefore given by D˜o,g(f ) = Compare (4.10) to (4.5), an additional gm factor appears in the spectrum. The spectral magnitude is scaled by gm. If there are mismatches among gm’s, the spectral images of the Dooutput can not be completely eliminated. Hence, spurious distortions will appear at the same frequency locations as the spurs resulted from the sampling time skew. A 4-channel TI-ADC is used as an example to illustrate this influence.
The conceptual spectra of a 4-channel TI-ADC is drawn in Fig. 4.8. If the 2nd and the 3rd channels have gain error, the magnitude of their spectrum, ˜V1,g(f ) and ˜V2,g(f ), are scaled down with their gain factor respectively. This makes the ˜Do,g(f ) output spectrum has spurs at the frequencies of kg× fs/M excluding those of multiples of fs, where kgis an integer. The overall A/D linearity is therefore degraded.
4.3.3 O ffset Mismatch
Offset effects on a TI-ADC are investigated here by modeling it as a constant signal, om, added into the input signal at the front of the S/H circuit. Since other non-ideal effects are not considered, this model is shown in Fig. 4.9. The S/H samples the Vi(t)+ om signal. Its output is therefore given by
Vm,os(t) = [Vi(t)+ om] × φm(t)= [Vi(t)+ om] × X+∞
n=−∞
δ(t − (nM+ m)Ts) (4.11) for all m = 0, 1, · · · , M − 1. Equation 4.11 indicates that the S/H samples not only the input signal Vi(t), but also the offset om. Fig. 4.10 illustrates the offset effects on the sampled data. Due to the offset, the actual sampled data compared to the ideal case have errors with a constant amount of ∆Ve = om. This also causes insufficient Domcodes of the m-th channel output, and hence, results in missing codes in the final digital output.
The TI-ADC’s output spectrum incurred by offset becomes as D˜o,os(f ) =
ADC
Figure 4.9: Model of offset in a TI-ADC.
∆Ve
Figure 4.10: Illustration of offset error effect on the sampled data.
4.3. NONIDEALITIES IN TIME-INTERLEAVED ADCS 59
k=1 k=2 k=3 k=4 k=5 k=6
0 fs/4 2fs/4 3fs/4 fs 6fs/4
f
f f 5fs/4
f
f V 0,os (f)
V 1,os (f)
V 2,os (f)
V 3,os (f)
D o,os(f) = V 0,os (f) + 1,os (f) + V V 2,os (f) + V 3,os (f)
~
~
~
~
~ ~ ~ ~ ~
phase
k=0
Figure 4.11: Conceptual spectra of a 4-channel TI-ADC with offset.
The offset of each channel causes a constant impulse spectrum independent of the input signal exactly at the frequencies of ko/MTs which interferes the signal spectrum at the same frequencies, where ko is an integer. If there exist offset mismatches among the channels, the final TI-ADC’s output spectrum will be disturbed with the spectrum of the offset. This is illustrated with the conceptual spectrum of a 4-channel TI-ADC as shown in Fig. 4.11. These mismatches result in incomplete cancellation of the spurs in the TI-ADC’s output spectrum at these frequencies of ko× fs/4 except the multiples of fs, and can cause dramatic degradation in the overall A/D linearity.
4.4 Summary
Using time-interleaving technique can increase the effective sampling rate, but timing, gain and offset mismatches among the interleaved channels result in dynamic degrada-tion. This chapter overviews the time-interleaved A/D architecture, and discusses these mismatch effects. The mathematical expressions of the operation principles of the time-interleaved A/D conversion are also drawn.
As described in the preceding section, the timing and gain mismatch errors contribute to the image spurs distorted the final output signal spectrum at the same frequency lo-cations. The offset spurs generated by the offset mismatches among A/D channels are independent of the input signal. Therefore, for a given offset mismatch, the offset spurs will always at the same level.
To design a TI-ADC, channel-to-channel matching requirements at high-resolution levels (> 12 bits) are not easily achievable. Hence, many approaches were proposed to reduce these mismatch errors. In particular, the digital background calibration tech-niques are flexible in their implementation. A variety of techtech-niques for reducing time-interleaving mismatch error will be discussed in the following chapter.