1. Effects of Filtering on the Linearity of Current-steering IF DAC
T. Rahkonen1 , J. Aikio1 , M. Mustaparta2
1
Department of Electrical and Information Engineering and Infotech Oulu
PL 4500, 90014 OULUN YLIOPISTO (Oulu), FINLAND
Tel: +358 8 553 2678 Fax: +358 8 553 2700 e-mail: timo.rahkonen@oulu.fi
2
Nokia Siemens Networks, Oulu, Finland
Abstract—The distortion of a current-steering digital-to-analog The equivalent circuit of the current-steering DAC is shown
converter (DAC) is often dominated by the signal-dependent in Fig. 1b. Here ������ is a digital word (normalized to vary
output impedance. This has normally been analysed by assuming between one and zero), and hence current ������⋅������������ and (1−������)⋅������������
a purely resistive load. When a digital IF signal is generated, the
different harmonic bands are separated and can also be filtered are steered to the outputs. If dynamic switching effects [3]
separately. This paper illustrates that linearity can be improved are neglected, the dominant cause of the non-linearity in the
considerably by shorting the generated 2nd harmonic and low- current-steering DAC is the fact that as the number of parallel
frequency voltage immediately at the DAC’s output, so that these current sources increases, also the output impedance of the
can not mix further to any higher order products. current source decreases. This can be studied by the current
equation
I. I NTRODUCTION
Current-steering DACs have been used for a long time, when ������ ⋅ ������������ = ������������ ⋅ ������ ⋅ ������������ + ������������ ⋅ ������������ (1)
a high-speed, broadband DAC is needed. Their linearity has
been extensively studied (e.g. [1], [2]), and the dominant cause where ������ ⋅ ������������ now indicates that the output conductance
of nonlinearity has been found to be the 2nd-order curvature increases linearily with the input signal.
caused by signal-dependent output impedance of the current Solving the node voltage ������������ , we get the non-linear function
sources. for the output voltages
In RF basestation transmitters (TX) there is a tendency
to avoid direct up-conversion from DC, and modern current- ������������ ⋅ ������
������������������ = (2)
steering DACs are capable of generating a digital intermediate- ������������ + ������ ⋅ ������������
frequency (IF) signal at tens or even hundreds of MHz. Here ������������ ⋅ (1 − ������)
������������������ = . (3)
we are not employing the full Nyquist bandwidth of the DAC ������������ + (1 − ������) ⋅ ������������
any more, and the linearity of the DACs can be reconsidered.
The series expansion of ������������������ at the mid-scale (������ is now a
II. N ON - LINEARITY OF A CURRENT- STEERING DAC bipolar signal that describes the deviation from the center) is
roughly of form
The schematic and equivalent circuit of a typical current-
steering high-speed DAC is shown in Figure 1. In one extreme,
the DAC consists of 2������ parallel equal-sized current sources, ������������������ = ������������ ⋅ ������������ ⋅ (0.5 + ������ − ������ ⋅ ������ 2 + ������2 ⋅ ������ 3 − ...) (4)
which is switched to either one of the outputs, and converted
where ������ = ������������ ⋅ ������������ . This results in a dominant 2nd-order
to a voltage in an external, relatively low-ohmic resistor.
curvature as analysed e.g. in [1].
The above analysis is based on assuming that ������������ is resistive
over the entire band. This is desired if a broadband conversion
is aimed for. However, when considering digital IF generation
it is worth to note that although the calculated input-output
response is non-linear, the conversion from a digital word to
the output current is exactly linear in the above model, and
all the distortion is really caused by voltage ������������ in the output
node. This is evident from the equation of the total DAC output
current in the pin:
������������������������ = ������ ⋅ ������������ − ������������ ⋅ ������ ⋅ ������������ . (5)
The only cause of distortion in this model is hence the
signal-dependent output impedance ������ ⋅ ������������ that behaves as a
2nd-order mixer between the digital signal ������ and the output
Fig. 1. Current-steering DAC, a) schematic, b) equivalent circuit. voltage ������������ . Hence, the spectral properties of the signal at the
978-1-4244-8971-8/10$26.00 c 2010 IEEE

2. TABLE I
output node of the DAC are very important, and actually the T ONES AT DAC OUTPUT (������ ������ = ������������ /4 − Δ) - A N E XAMPLE
main mechanism generating 3rd-order distortion is such that
first, the linear voltage response mixes with the digital signal, Tone Freq dBc
generating 2nd-order distortion, and then the already generated 1 ������������ -1
2 2������������ -41
2nd-order distortion voltage is again mixed with the digital 3 3������������ -60
signal. 4 ������������ − ������������ -10
5 ������������ -45
6 ������������ + ������������ -15
III. S PECTRUM AT THE DAC OUTPUT
Figure 2 shows the typical spectrum in the output pin of TABLE II
a resistively loaded IF TX DAC. The desired signal (marked IM PRODUCTS AT DAC OUTPUT - AN EXAMPLE
with 1 in Fig. 2 and Table I) is centered at ������������ = ������������ /4 (actually
slightly below at ������������ = ������������ /4 − Δ to separate different tones in Tone Freq dBc
the analysis), and its strong sampling image (tone 4) is seen IM2 (1,2) ������������ /4 − Δ -76
IM2 (2,4) ������������ /4 + 3Δ -86
at ������������ − ������������ = 3 ⋅ ������ ������/4. A clock spur (5) is seen at ������������ , the IM3 (4,6) ������������ /4 + 3Δ -84
2nd harmonic of the IF signal is centered at ������������ /2, and finally IM2 (4,5) ������������ /4 − Δ -90
a rectified low-frequency signal centered at 0 Hz. The third
harmonic overlaps with the ������������ − ������������ image. These tones, their
frequencies and relative magnitudes are summarised in Table
shown to land near ������������ /4. Note that one can recognise some
I. It is worth noting, that the spectral images are quite strong,
of the mixing mechanism by deviating the carrier frequency
just 10 and 15 dB below the desired output. When drawing the
from ������������ /4������������ − Δ.
plot, tone amplitude of 0.5 and classical input-output square-
law and cubic non-linearity coefficients of 0.02 and 0.005 were IV. VOLTERRA ANALYSIS OF THE DISTORTION
arbitrarily assumed, and ������������������������ /������ response is used to attenuate From above it is clear, that we can now affect the overall dis-
the images. tortion by filtering the harmonics and spectral images directly
at the DAC’s output. The effect of this is seen directly in the
0 magnitude spectrum, but here we do a more detailed study by
1 plotting the distortion contributions as voltage phasors by us-
4 ing Volterra analysis. Analysis method called VoHB (Volterra
−20 6
on Harmonic Balance) has recently been implemented on top
of Aplac circuit simulator [4].
−40 Volterra analysis is often based on fundamentals and calcu-
Mag dB
2 lates the harmonics. For this purpose the spectral images seem
5
to call three tones alone for a single-tone analysis. Luckily, the
−60
3 images lie near the multiples of the input tones, so that the in-
band signal can be modeled by a 2-tone, and image tones by
−80 additional strong signals at the 3rd and 5th harmonic bands.
Section V shows the traditional 2-tone response and Section
−100
VI adds up the image tones.
0 0.25 0.5 0.75 1 1.25 1.5 One more thing needs to be noticed. As the direct and
freq/fs
inverting output currents have ac currents of +������ ⋅ ������������ and
Fig. 2. Spectrum of an IF DAC output. −������ ⋅ ������������ , the corresponding output voltages are of form
The above spectral components can mix to the carrier ������������������ = +������ ⋅ ������������ ⋅ ������������ ,
frequency immediately at the output of the DAC in many
������������������ = −������ ⋅ ������������ ⋅ ������������ , (6)
ways. Table II shows the mixing mechanisms (numbers are
referring to the tones in Fig. 2 and Table I), output frequency, and the 2nd-order distortion in both output nodes are in the
and relative magnitude of some IM products. The strongest same phase :
distortion here is generated when the 2nd harmonic mixes
with the fundamental digital signal and ends up into the ������������������ = ������ ⋅ ������������ ⋅ ������������������ = ������2 ⋅ ������������ ⋅ ������������ ⋅ ������������
(2)
(7)
fundamental TX band, as indicated by the first row of Table and
II. The IM2 product of the ������������ − ������������ and second harmonic is (2)
������������������ = −������ ⋅ ������������ ⋅ (−������ ⋅ ������������ ⋅ ������������ ). (8)
ca. 10 dB weaker, as the image is 10 dB below the desired
carrier, and the IM2 results of the clock spur and 2nd harmonic Hence, the 2nd-order distortion appears as a common-mode
naturally depends on the level of the clock spur. Finally, also signal, while the desired signal, the image signals, and all odd-
IM3 products can exist: here the IM3 of both clock aliases is order distortion products appear as differential signals. Hence,

3. TABLE III
these signals also propagete very differently in the succeeding IM3 LEVEL WITH DIFFERENT FILTERING OPTIONS
analog filters.
Filter dBc
V. R ECONSTRUCTION FILTER AND 2- TONE VOLTERRA Fully balanced -119
ANALYSIS 2nd harm. shorted -122
DC and 2nd harm. shorted -142
Any DAC needs a steep reconstruction filter to attenuate
the strong ������������ − ������������ image. In an IF DAC this is complicated
further by the fact that now the desired and image bands are
From above it is seen that mixing from the DC band is
close together, and the image is only moderately filtered by
stronger. Hence, even a complete cancelation of the second
the DAC’s ������������������������ /������ sample-and-hold response. The benefit of
harmonic term with the 2nd harmonic trap (r=0.82) reduces
the IF operation is that now the harmonic bands are clearly
the generated IM3 distortion by 30 % (3 dB). This is seen in
separated from each other, and can be shaped independently.
5, where the M12 term is very short, but the up-converted
Figure 3 illustrates a 5th-order low-pass filter provided by
envelope frequency term M10 is as strong as before. The
Analog Devices for a DAC that is to be used to generate a
remaining 70% of the IM3 can be cancelled by nulling the low-
digital IF at 150 MHz [5]. The above distortion reasoning
frequency distortion, too. This is illustrated in Fig. 6 where the
seems to be well known by the manufacturer, because the
overall distortion has reduced from original 90 nV to ca. 1 nV.
passive ladder filter is built up quite cleverly. It passes the
Cancelling differential low-frequency signals is conviently
differential signal with Butterworth response, and the ladder
done by placing an inductor between the DAC outputs. How-
is mostly built up between the balanced signal branches.
ever, this has no effect on the 2nd-order distortion that appears
However, the last capacitor is broken up in differential and
as a common-mode signal. Hence, we need to place two
common mode branches. This does not affect the differential
inductors from nodes ������������������ and ������������������ to ground, or use tranformer
response as long as C3 + C4a*C4b/(C4a+C4b) remains fixed
coupling. This affects also the DC common-mode level of the
(7.5pF), but now C4a and C4b affect also the common-mode
DAC output. If we want to maintain the DC level we can
response. From common mode point of view all components
connect the inductors to a voltage buffer, the output impedance
between the two branches can be considered as open circuits
of which is considerably lower than ������������ /2. Fig. 6 shows that
(the same voltage at both ends forces their current to zero),
the mixing results from the 2nd harmonic is almost completely
so that the filter reduces to two separate series resonant
cancelled, and also the up-conversion from DC has decreased
circuit consisting of L1,L2, and C4. Ratio r in Fig. 3 tunes
due to the low DC impedance seen by the DAC.
the dimensioning: if r=1, the filter is fully balanced, r=0.82
Table III compares the IM3 distortion level in the DAC-filter
(resulting in 6pF and 4pF for C3 and C4, respectively) creates
combination in the following cases: a) a fully diffential filter,
a trap for the common-mode 2nd harmonic.
b) the filter is modified (as in Fig. 3) to cancel the common-
mode 2nd harmonic, and c) also inductors are added from DAC
outputs to ground to short the low-frequency distortion. In the
simulated example ������������ =20mA, ������������ =50 ohm, and ������������ = 10−4
S, and IM3 is seen to improve by 20 dB.
VI. A NALYSIS WITH THE SPECTRAL IMAGES
The above analysis was simplified in that sense that it had
only a 2-tone fundamental response consisting of tones ������������ =
gm3
APLAC 8.50 User: University of Oulu - Microelectronics & Materials Physics
90.00n
Fig. 3. DAC with a filter with tuned differential and common mode responses.
Imag
Figure 4 shows the result of a Volterra analysis, where a
45.00n
2-tone test signal is applied to a circuit with a traditional
fully balanced filter (r=1). The distortion phasor of the lower
IM3 generated by current source gm3 (������ ⋅ ������������ source) is TOT V(K11M10) V(K11M12)
0.00
broken to contributions, and the notations used are such that
Klm describes the shape of the nonlinary, referring to a term
������ ������
������������������ ⋅ ������������ ⋅ ������������ , and Mlm describes the frequency bands that are -45.00n
mixed. Hence, the IM3 seems to be caused by input-output -90.000n -45.000n 0.000 45.000n
crossterm K11 (i.e. the ������ ⋅ ������������ ⋅ ������������ term), and it consists of two
Real
V(K11M10) V(K11M12)
contributions: mixing of the fundamental and 2nd harmonic TOT
band (M12) and twice as strong mixing result of fundamental
and DC band (M10). Fig. 4. IM3 contributions in a fully balanced filter.

4. gm3 gm3
APLAC 8.50 User: University of Oulu - Microelectronics & Materials Physics APLAC 8.50 User: University of Oulu - Microelectronics & Materials Physics
60.00n 7.00n
TOT
Imag
Imag 3.50n V(K11M12)
V(K11M10)
30.00n V(K11M34)
0.00 V(K11M32)
V(K11M54)
TOT
V(K11M10)
V(K11M12) -3.50n
0.00
-7.00n
-160.000n -106.667n -53.333n 0.000
-30.00n Real
-60.000n -30.000n 0.000 30.000n V(K11M10) V(K11M12)
Real V(K11M32) V(K11M34)
V(K11M10) V(K11M12) V(K11M54) TOT
TOT
Fig. 7. IM3 contributions with image frequencies included.
Fig. 5. IM3 contributions with a 2nd harmonic trap.
gm2
APLAC 8.50 User: University of Oulu - Microelectronics & Materials Physics can be used to switch the phase of the 2nd harmonic term and
600.00p
control its magnitude, but the steep frequency response of the
Imag notch makes this cancellation rather narrowband.
0.00
VII. S UMMARY
This paper reviewed the nonlinear model of a high-speed
V(K11M12)
-600.00p
current-steering DAC used for digital IF transmitters, and
V(K11M10)
assuming that the dominant cause of distortion is the signal-
dependent output admittance, showed that IM3 distortion is
TOT
-1.20n generated by further mixing of the 2nd-order voltage distortion
-600.000p 0.000
Real
600.000p 1.200n at the output of the DAC. Hence, it is important to minimize
V(K11M10) V(K11M12)
both low frequency and 2nd harmonic components immedi-
TOT ately in the output nodes of the DAC, and this can be achieved
by two modifications in the reconstruction filter.
Fig. 6. DAC with a 2nd harmonic trap and highpass responses.
VIII. ACKNOWLEDGMENT
This work has been supported by EU project ICESTARS,
������������ − Δ������ and ������ℎ = ������������ + Δ������ and their harmonics, but not and Academy of Finland.
the image tones. If the center frequency ������������ is exactly ������������ /4,
the image tones can be easily added to frequencies 2������������ + ������ℎ , R EFERENCES
2������������ + ������ℎ , 3������������ + 2������ℎ , and 2������������ + 3������ℎ . This is done in Fig. 7 with [1] Wikner, J.: Studies on CMOS Digital-to-Analog Converters Linkping
plain balanced filtering (r=1), and some new contributions are Studies in Science and Technology Dissertation No. 667, 2001
[2] Deveugele, J.; Steyaert, M.; Chapter ”‘RF DAC’s: output impedance and
seen. The strongest one of these is M32, which is the mixing distortion”’ in Analog Circuit Design, Springer Netherlands,isbn = 978-
result between the 2nd harmonic and the lower image. As 1-4020-3885-3
expected, it ca. 10 dB weaker than the fundamental M12 term. [3] Clara, M.; Wiesbauer, A.; Klatzer, W.; Nonlinear distortion in current-
steering D/A-converters due to asymmetrical switching errors Proc. of
Terms M54 and M34 are results of the images mixing with the 2004 International Symposium on Circuits and Systems, 2004. ISCAS
the clock spurious, and these are the first products not affected ’04, vol. 1, pp I-285 - I-288.
by 2nd harmonic and DC band filtering. [4] Aikio, J.; Rahkonen, T.; Detailed distortion analysis technique based
on simulated large-signal voltage and current spectra. IEEE Trans. on
When the IF is hundreds of MHz the capacitive parallel Microwave Theo. and Tech., vol. 53, no. 10, pp. 3057-3066.
admittance of the current sources may dominate over resistive [5] Mustaparta, M.; Reaalisen ja kompleksisen ylossekoituksen vertailu tuki-
������������ . In such case the importance of the 2nd harmonic is higher, asemalahettimess. M.Sc thesis in University of Oulu, Finland, 2009.
as the distortion current generated by a capacitance is directly
proportional to the frequency of the generated distortion:
high-frequency distortion has a higher amplitude than a low-
frequency distortion.
The above analysis does not yet show any intrinsic cubic
non-linearity. If such exists, it causes IM3 directly without any
harmonics involved. Yet, in a lucky case it may be possible to
force the IM3 generated by up or down converted 2nd-order
products to be in opposite phase with the IM3 generated by
the cubic nonlinearity. Off-tuning of the 2nd harmonic notch