This document discusses a proposed architecture for a higher Nyquist-range digital-to-analog converter (DAC) that employs sinusoidal interpolation. [1] Conventional DACs operate within the Nyquist range, but the proposed architecture aims to utilize higher Nyquist ranges by approximating an oscillating signal from an RF DAC concept using sinusoidal interpolation in the time domain. [2] The proposed architecture quantizes both the input signal and pulse amplitude modulation waveform and combines them digitally, replacing analog oscillatory circuits with a digital data stream. This reduces analog complexity compared to existing techniques. [3] Simulation results and theoretical analysis are presented to support that the proposed architecture can provide similar performance

1. A Higher Nyquist-Range DAC Employing
Sinusoidal Interpolation
M. Reza Sadeghifar J Jacob Wikner
Department of Electrical Engineering, Department of Electrical Engineering,
Link¨ ping University, SE-581 83 Link¨ ping, Sweden
o o Link¨ ping University, SE-581 83 Link¨ ping, Sweden
o o
E-mail: mreza@isy.liu.se E-mail: Jacob.Wikner@liu.se
Abstract—This work discusses a link between two previously In applications such as multi-standard base stations, the
reported ideas in high-speed digital-to-analog converter (DAC) requirements on transmitter and thus DAC linearity are high
design: linear approximation with analog interpolation tech- so as to meet several telecommunication standards. The same
niques and an RF DAC concept where oscillatory pulses are used
to combine a DAC with an up-conversion mixer. An architecture arguments hold for next-generation radar applications, which
is proposed where we utilize analog interpolation techniques, in fact put higher requirements on linearity than for example,
but using sinusoidal rather than linear interpolation in order GSM. To reach highest possible performance, one has to apply
to allocate more energy to higher Nyquist ranges as is commonly some kind of optimization of the signal chain. Although the
done in RF DACs. The interpolation is done in the time domain, RF DAC performance in [1] is satisfactory for this kind of
such that it approximates the oscillating signal from the RF DAC
concept to modulate the signal up to a higher Nyquist range. applications, still it can be modified so that we get less analog
Then, instead of taking the output from within the Nyquist range, complexity. To this end, we are proposing a technique that can
as in conventional case, the output of the DAC is taken from potentially provide us with the same performance as RF DAC.
higher images. The proposed architecture looks promising for This paper is organized as follows: Section II shows a
future implementations in high-speed DACs as it can be used
in RF DAC or modified versions of digital-to-RF converters
theoretical background to DACs, Section III describes the
(DRFCs). Simulation results and theoretical descriptions are suggested architecture, Section IV shows the simulation results
presented to support the idea. for the proposed architecture and eventually the paper is
concluded in Section V.
Index Terms—Digital–analog conversion, interpolating DACs,
Mixing DAC, RF DAC, Direct digital synthesis.
II. BACKGROUND
I. I NTRODUCTION The ambition with the work presented in this report is to
link the theories between two different reported architectures
High-speed (and high-resolution) data converters are vital such that we can implement high-speed DACs with lower
components in all telecommunication systems. Typically, the requirements on the reconstruction filters.
higher speed we can use in the sampling process, the lower The concept of using an oscillatory pulse amplitude modula-
requirements can be put on strictly analog components, such tion (PAM), is well-known since some time ago, but a revised
as analog filters. Of course, a higher digital complexity is then approach was proposed by Luschas, et al., in 2004 [1].
required in order to reach better performance but it can mostly Due to non-ideal reconstruction the continuous-time analog
be catered for with the scaled process dimensions and becomes output spectrum contains not only the Nyquist range but also
less of a concern today and in the future. images of the signal spectrum repeated at multiples of fs . The
There are plenty of academic publications demonstrating reconstructed signal, xr (t), can be written as
high-speed DACs. However, often, they still operate in the sub-
GHz region and still have somewhat ”modest” linearity. We xr (t) = x(n) · p(t − nT ), (1)
have, for example, a suggested RF DAC in [1], direct digital to ∀n
RF (DIF2RF) [2], etc. Publication results demonstrating DACs
operating in the higher GHz regions typically use more exotic where p(t) is the PAM waveform. Ideally, p(t) should be a
processes such as SiGe [3] or SOI principles [4]. sinc function, i.e., sin(t)/t, such that a brick-wall filtering
The first, and perhaps obvious, approach to convert from a is obtained which band-limits the signal to Nyquist range.
digital to an analog representation at high speed is to increase However, in conventional Nyquist-rate DACs (using a zero-
the sample frequency of the DAC. Typically, you find from order hold technique) p(t) is typically a rectangular-shaped
all reported results (and conveniently from theory too) that pulse that results in a sinc weighted frequency characteristics.
there is a relationship between resolution (i.e. linearity) and In the approach used in [1], an oscillatory pulse is embedded
frequency: with higher frequency or bandwidth, the linearity in the PAM pulse, p(t). The signal spectrum of their proposed
decreases [5]–[7]. pulse waveform in the frequency domain turns out to be very
978-1-4244-8971-8/10$26.00 c 2010 IEEE

2. Fig. 2. Illustration of time-domain behavior for (a) linear interpolation, (b)
Fig. 1. Frequency characteristics of a (a) brickwall, (b) sinusoid, and effect sinusoid mixing, (c) quantized microstepping of sinusoid waveform.
of (c) combined brickwall and sinusoid waveforms.
function is realized such that a linear transition is obtained
interesting. Assume that we have a PAM waveform as between two consecutive samples, rather than applying the
brickwall pulse. For example, the continuous-time waveform
1 + cos(ω0 t) |t| ≤ T /2 between samples x(nT − T ) and x(nT ) can be written as
p(t) = (2)
0 |t| > T /2,
t − (n − 1)T
xr (t) = x(n − 1) + (x(n) − x(n − 1)) · , (4)
where we currently allow the function to also take negative T
time values. Here, ω0 is a certain angular frequency, typically for (n − 1)T < t ≤ nT in this particular case. To obtain this
associated with multiples of the sample frequency and T is linear interpolation, a kind of microstepping in [8] is used.
the time duration of the pulse, which typically is given by At a frequency L times higher than the sample frequency, the
T = 1/fs = 2π/ωs , where fs is the sample frequency and ωs analog output is updated with a slight increase in amplitude
is the sample angular frequency. The corresponding Fourier to eventually reach the appropriate level upon arrival of the
transform of the PAM waveform in (2) can be written as next sample at the original sample frequency. The PAM pulse
ω ω−ω0 ω+ω0
sin 2fs T sin 2fs T sin 2fs includes, in this case, a ramp and its spectrum is expected
P (jω) = T ω + + . (3) to be shaped as sinc2 . The attenuation of out-of-band images
2fs 2 ω−ω0
2fs
2 ω+ω0
2fs at higher frequencies would then be approximately doubled
Figure 1 shows how different pulse waveforms affect the in terms of decibels thus motivating a lower complexity in
output spectrum. In Fig. 1(a), we find the effect of the the analog reconstruction filter [8]. However, it also impacts
traditional, practical brickwall pulse. The spectrum is sinc- the usable frequency range, as the data needs to be restricted
weighted and attenuated images are found over the frequency to a more narrow band, eventually resulting in a fairly low-
domain. The analog reconstruction filter needs to remove those bandwidth DAC implementation.
images and also restore the amplitude response in the signal Figure 2 illustrates three different interpolation waveforms
band. In Fig. 1(b), we illustrate the relationship between time- in the time domain. In (a) the linear interpolation waveform
domain and frequency domain for a sinusoid and in (c) the as proposed in [8] is shown illustrating the microstepping
combined waveform is shown, as also described by (2) and approach. In (b) we find the approach used in [1] with
(3), respectively. The output spectrum of the signal, say X(jω) continuous-time waveforms. Fig. 2(c) illustrates our proposed
will be multiplied by the P (jω) from (3) and hence the higher approach as further described in Section III. The proposed
Nyquist range (at ω0 ) can be utilized instead of the original approach utilizes the microstepping technique as described in
Nyquist band. As can be seen, there is a loss of signal power, [8] and combines it with the generation of a sinusoid approach.
but in terms of in-band signal-to-noise ratio (SNR), there is no Consequently, the hardware needed for analog oscillatory
loss, as the output quantization noise of the DAC is attenuated pulse generation such as voltage-controlled oscillators (VCO),
as well. The component will however be more sensitive to power and area consuming buffers, etc. can be replaced by
circuit noise and other external factors. This technique can circuits that are more digital in nature.
also be thought of as a mixing DAC since the digital base-
band is converted to analog and at the same time shifted to III. P ROPOSED A RCHITECTURE
higher frequencies. In addition to the microstepping technique to generate the
interpolation waveform, we also quantize the amplitude levels
A. Interpolation Techniques as illustrated in Fig. 2(c). In practice, this means that we
A pseudo-analog interpolation technique to design DACs actually control the PAM waveform with yet another DAC,
is proposed in [8]. In this technique, a linear interpolation rather than using the analog VCO as required in [1]. Our

3. Fig. 4. Illustration of the the multiplication process between two signals
containing quantization noise.
˜
where X(nT ) is the digital word and X(nT ) its inverse.
Fig. 3. Illustration of two approaches to perform up-mixing in a current-
steering DAC. (a) shows an analog method and (b) a mixed-signal method.
B. Theoretical Background
As mentioned above, both the input signal, X(n), and the
figure of merit is to reduce the number of closed-loop analog tail current, IT (t), in the proposed technique are quantized. As
components and instead offer a direct, digital data stream can be seen in (7), the output current consists of the product of
that can be weighted and combined in the analog domain. two quantized signals. The main signal is quantized to N bits
The data stream can be generated by a high-speed direct and the oscillating signal to M bits. To understand how this
digital synthesis (DDS) phase accumulator [9]. This phase affects the performance, consider the following expression
accumulator can potentially also be used in a combination Sout = (x + qx ) · (p + qp ) (8)
with non-linearly weights in the current source to achieve
high speed. As argued above, a DAC with digital generating = x · p + x · qp + p · qx + q x · qp ,
circuits replaces the oscillatory waveform and essentially we signal noise
can expect the overall DAC output to behave quite similarly where Sout denotes output signal, x input signal, and p the
to the continuous-time approach. PAM waveform. qx and qp are denoting the quantization
A. Block Diagram noise. To find the signal-to-noise ratio (SNR) with respect to
quantization, we have to look at the signal and noise powers
In Fig. 3 we show the main ideas behind the approach in
with respect to M and N . We can get the signal power by
[1] and our proposed approach. In (a) the tail current source
taking the ”expected value” of the squared signal, hence
is controlled by an analog waveform, typically some kind
of sinusoid centered around a DC point, whereas in (b) we Ps = E (x · p)2 = E x2 · E p2 . (9)
have divided the tail current source into a multiple of sub-
current sources that are controlled by digital data streams. The Assuming sinusoidal signals for both x and p, the maximum
combined current of these sub-sources will generate the total amplitudes are approximately Δx · 2N −1 and Δp · 2M −1 ,
tail current. For the case in Fig. 3(a) using a standard CMOS respectively. The total power can be written as
transistor, the tail current will be approximated by 2 2
Δx · 2N −1 Δp · 2M −1
2 Ps = · , (10)
IT (t) = α · (vac sin(ω0 t) + VDC − VT ) . (5) 2 2
The current contains a DC component, one signal component where Δx and Δp are the corresponding quantization levels.
at ω0 and one at 2ω0 . For our proposed case, as in Fig. 3(b), Similarly, the noise power (see (8)) can be written as
the tail current would instead be given as 2
Pn = E x · qp + p · q x + q x · q p . (11)
M −1
nT nT mT By expanding (11) we have
IT = IT,0 · Wm · y + , (6)
L m=0
L L 2 2 2
Pn = E x · qp + p · qx + q x · qp
where Wm are the weight ratios of the different current sources
2 2
and IT,0 is a unit current source. The M control signals, ym , +2 · E xp · qp qx + E xqx qp + E pqp qx . (12)
are running at L times the sample frequency, such that the tail
current is quantized with respect to both time and amplitude. The input and PAM signals as well as noise are uncorrelated
In both cases described above the output DAC current is and the last term of (12) can further be split into separated
composed by the difference between the two currents at the terms. E{qx } and E{qp } are mean values of the quantization
output of the switches, i.e., ID = IP − IN . Further, the noise qx and qp , respectively, and we assume the quantization
switches are controlled by the data signal and the total output error to take a uniform distribution and the mean values must
current from the DAC - at time point nT - can be written as therefore become zero. Then, the noise power becomes
2 2 2
˜
ID (nT ) = IT (t) · X(nT ) − IT (t) · X(nT ), (7) Pn = E x · qp +E p · qx +E qx · q p . (13)

4. In general, the noise power for uniform, fine quantization is
(a) Technique used by Zhou
Δ/2
1 Δ2 0
E q2 2
= σe = e2 de = , (14)
Magnitude [dB]
−Δ/2 Δ 12
−50
and we can then identify different terms in (13). The first term
can be written as:
2
Δx · 2N −1 Δ2
−100
2 p 0 0.5 1 1.5 2 2.5 3
E x · qp = · , (15)
2 12 (b) Technique used by Luschas
and so on. Eventually the noise power can be written as 0
Magnitude [dB]
Δ2 · 4N Δ2
x p Δ2 · 4M Δ2
p Δ2 Δ2 p
Pn = · + · x+ x· . (16) −50
8 12 8 12 12 12
The SNR is formed by taking the ratio between (10) and (16) −100
and can then be expressed in dB as 0 0.5 1 1.5 2 2.5 3
(c) Proposed approach
SNR ≈ 6·(N +M )+1.76−10·log10 4N +4M +2/3 dB.
0
We see from the formula above, that if M would be a Magnitude [dB]
very large number, the second term would approximate to −50
−10 · log10 4M ≈ −6 · M and the expression approximates
the famous expression SNR ≈ 6 · N + 1.76 dB. [5] −100
0 0.5 1 1.5 2 2.5 3
IV. S IMULATION R ESULTS Frequency [GHz]
Behavioral-level simulations have been performed in MAT-
Fig. 5. MATLAB simulation results for (a) linear interpolation technique, (b)
LAB and the results are presented to support and motivate the sinusoid mixing technique, (c) quantized microstepping of sinusoid waveform
approach. Quantization and truncation noise is also counted technique.
for, in the SFDR simulations. Simulation results comparing
the three different approaches are shown in Fig. 5. A single
tone test signal with a frequency of 50 MHz is applied while as behavioral-level simulation results have been presented to
the sample frequency is fs = 750 MHz. support the idea. This technique is realizable and effective in
The DAC output spectrum using the approach in [8] is terms of cost and performance.
illustrated in Fig. 5(a). The signal must be placed in the lower The additional high-speed digital control of the up-mixing
Nyquist-range since images at higher frequencies is attenuated waveform, can be efficiently combined with non-linearly
by more than 45 dB which also relaxes the requirements on the weighted sources to reach high speed [9].
following analog reconstruction filter. Fig. 5(b) shows the DAC
output spectrum utilizing the approach in [1]. Here, a higher R EFERENCES
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