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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 very978-1-4244-8971-8/10$26.00 c 2010 IEEE
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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 obtainedinteresting. 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 Ttime values. Here, ω0 is a certain angular frequency, typically for (n − 1)T < t ≤ nT in this particular case. To obtain thisassociated 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, theT = 1/fs = 2π/ωs , where fs is the sample frequency and ωs analog output is updated with a slight increase in amplitudeis the sample angular frequency. The corresponding Fourier to eventually reach the appropriate level upon arrival of thetransform 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 doubledFigure 1 shows how different pulse waveforms affect the in terms of decibels thus motivating a lower complexity inoutput spectrum. In Fig. 1(a), we ﬁnd the effect of the the analog reconstruction ﬁlter [8]. However, it also impactstraditional, practical brickwall pulse. The spectrum is sinc- the usable frequency range, as the data needs to be restrictedweighted 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 waveformsband. In Fig. 1(b), we illustrate the relationship between time- in the time domain. In (a) the linear interpolation waveformdomain and frequency domain for a sinusoid and in (c) the as proposed in [8] is shown illustrating the microsteppingcombined 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 proposedwill be multiplied by the P (jω) from (3) and hence the higher approach as further described in Section III. The proposedNyquist range (at ω0 ) can be utilized instead of the original approach utilizes the microstepping technique as described inNyquist 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 oscillatoryloss, 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 bycircuit 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 RCHITECTUREhigher frequencies. In addition to the microstepping technique to generate the interpolation waveform, we also quantize the amplitude levelsA. 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
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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 theﬁgure of merit is to reduce the number of closed-loop analog tail current, IT (t), in the proposed technique are quantized. Ascomponents and instead offer a direct, digital data stream can be seen in (7), the output current consists of the product ofthat can be weighted and combined in the analog domain. two quantized signals. The main signal is quantized to N bitsThe data stream can be generated by a high-speed direct and the oscillating signal to M bits. To understand how thisdigital synthesis (DDS) phase accumulator [9]. This phase affects the performance, consider the following expressionaccumulator can potentially also be used in a combination Sout = (x + qx ) · (p + qp ) (8)with non-linearly weights in the current source to achievehigh 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 noisecan expect the overall DAC output to behave quite similarly where Sout denotes output signal, x input signal, and p theto the continuous-time approach. PAM waveform. qx and qp are denoting the quantizationA. 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, henceis controlled by an analog waveform, typically some kindof 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 maximumcombined 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 astransistor, 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 2The 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 asthe 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 · qpwhere Wm are the weight ratios of the different current sources 2 2and 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 tailcurrent 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 separatedcomposed by the difference between the two currents at the terms. E{qx } and E{qp } are mean values of the quantizationoutput of the switches, i.e., ID = IP − IN . Further, the noise qx and qp , respectively, and we assume the quantizationswitches are controlled by the data signal and the total output error to take a uniform distribution and the mean values mustcurrent 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)
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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 −50and we can then identify different terms in (13). The ﬁrst termcan 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 Luschasand 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 12The SNR is formed by taking the ratio between (10) and (16) −100and 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. 0We 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 approximatesthe 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 waveformapproach. Quantization and truncation noise is also counted technique.for, in the SFDR simulations. Simulation results comparingthe three different approaches are shown in Fig. 5. A singletone test signal with a frequency of 50 MHz is applied while as behavioral-level simulation results have been presented tothe 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-mixingNyquist-range since images at higher frequencies is attenuated waveform, can be efﬁciently combined with non-linearlyby 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 DACoutput spectrum utilizing the approach in [1]. Here, a higher R EFERENCESenergy-lobe around 2fs is found. A tone with only 6 dB power [1] S. Luschas, R. Schreier, and H.-S. Lee, “Radio Frequency Digital-to- Analog Converter,” IEEE J. Solid-State Circuits, vol. 39, no. 9, pp. 1462less than the main tone (in the Nyquist band) is located at – 1467, Sept. 2004.1.45 GHz. This tone can be used as the signal carrier if ﬁltered [2] S. M. Taleie, T. Copani, B. Bakkaloglu, and S. Kiaei, “A linear ΣΔproperly (sketched in graph). An SFDR of approximately Digital-IF to RF DAC transimtter with embedded mixer,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 5, May 2008.60 dB can be obtained in this technique. Fig. 5(c) shows the [3] S. Halder, H. Gustat, and C. Scheytt, “A 20GS/s 8-bit current steeringsimulation result from proposed technique, i.e., using discrete- DAC in 0.25µm SiGe BiCMOS Technology,” in Proc. European Mi-time oscillating waveform at two times the sample frequency. crowave Integrated Circuits Conf., EuMIC, 2008, pp. 147–150. [4] J. H¨ gglund, “Pulse and noise shaping D/A converter (PANDA) Block aAs shown, the output tone, similar to Fig. 5(b), is located at implementation in 65nm SOI CMOS,” Master’s thesis, Link¨ ping Uni- o1.45 GHz with close interferer that needs to be ﬁltered out. versity, Sweden, 2008.However, the simulation result shows that this technique is [5] J. J. Wikner, “Studies on CMOS Digital-to-Analog Converters,” Ph.D. dissertation, Link¨ ping University, Sweden, Apr. 2000, Dissertation No: opromising since it demonstrates the principles and trade offs 667.between SFDR, analog circuitry complexity and digital design. [6] S. Luschas and H.-S. Lee, “Output impedance requirements for DACs,”The simulated SFDR is approximately 58 dB. The tone at in Proc. IEEE Intern’l Conf. Electronics, Circuits and Systems (ICECS)., Sep 1999, pp. 1193–6.1.5 GHz is a DC component up-converted to 2fs . [7] A. V. den Bosch, M. Borremans, M. Steyaert, and W. Sansen, “A 10-bit 1-GS/s Nyquist current-steering CMOS D/A Converter,” IEEE J. Solid- V. C ONCLUSIONS State Circuits, vol. 36, no. 3, Mar. 2001. We have demonstrated and investigated the basics ideas [8] Y. Zhou and J. Yuan, “An 8-bit 100-Mhz CMOS Linear Interpolation DAC,” IEEE J. Solid-State Circuits, vol. 38, no. 10, Oct. 2003.of a modiﬁed, mixed-signal approach to address the higher [9] A. Majid and A. W. Malik, “Design and implementation of a direct digitalNyquist ranges at the output of high-speed DACs suitable for frequency synthesizer using sum of weighted bit products,” Master’sRF DAC implementations. A theoretical background as well thesis, Link¨ ping University, Sweden, 2009. o
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