MT-017 tutorial oversampling interpolating dacs by walt kester introduction oversampling and digital filtering eases the requirements on the antialiasing filter which precedes an adc. the concept of oversampling and interpolation can be used in a similar manner with a reconstruction dac. for instance, oversampling is common in digital audio cd players, where the basic update rate of the data from the cd is 44.1 ksps. early cd players used traditional binary dacs and inserted "zeros" in to the parallel data, thereby increasing the effective update rate to 4-times, 8-times, or 16-times the fundamental throughput rate. the 4, 8, or 16 data stream is passed through a digita l interpolation filter wh ich generates the extra data points. the high oversampling rate moves th e image frequencies higher, thereby allowing a less complex lower cost filter with a wider transiti on band. in addition, there is an increase in the snr within the signal bandwidth because of the process gain. the sigma-delta dac architecture uses a much higher oversampling rate and represen ts the ultimate extensi on of this concept and has become popular in modern cd players. the same concept of oversampling and interpolation is also utilized in high speed dacs used in communications applications, relaxing the require ments on the output filter as well as increasing the snr due to process gain. output spectrum of a reconstruction dac the output of a reconstruction dac can be represented as a series of rectangular pulses whose width is equal to the reciprocal of the clock rate as shown in figure 1. 0.5f c f c 1.5f c 2f c 2.5f c 3f c a = sin f f c f f c 1 f a t t ?3.92db sampled signal reconstructed signal 0 1 f c images images images figure 1: unfiltered dac output showing images and sin (x)/x roll off rev.a, 10/08, wk page 1 of 7
MT-017 note that the reconstructed signal amplitude is down 3.92 db at the nyquist frequency, f c /2. an inverse sin(x)/x filter can be used to compensate for this effect if required. the images of the fundamental signal occur as a re sult of the sampling function and are also attenuated by the sin(x)/x function. ovesampling interpolating dacs the basic concept of an oversam pling/interpolating dac is show n in figure 2. the n-bit words of input data are rece ived at a rate of f c . the digital interpolation filter is clocked at an oversampling frequency of kf c , and inserts the extra data point s. the effects on the output frequency spectrum are shown in figure 2. in the nyquist case (a), the requirements on the analog anti-imaging filter can be quite severe. by oversampling and interpolating, the requirements on the filter are greatly relaxed as shown in (b). also, since the quantization noise is spread over a wider region with respect to the original signal bandwidth, an improvement in the signal-to-noise ratio is also achieved. by doubling the original samp ling rate (k = 2), an improvement of 3 db is obtained, and by making k = 4, an improvement of 6 db is obtained. early cd players took advantage of this, and generally carried the arithmetic in the digital filter to more than n-bits. today, most dacs in cd players are sigma-delta types. one of the earliest publications on the oversamp ling/interpolating dac c oncept was by ritchie, candy, and ninke in 1974 (reference 1) and followed by a 1981 patent (filing date) by mussman and korte (reference 2). digital interpolation filter dac anti- imaging filter n-bits @ f c n-bits @ k f c kf c analog output f c f c 2 kf c kf c 2 f f (a):nyquist (b): oversampling with interpolation anti-imaging filter response (from pll clock multiplier) figure 2: oversampli ng interpolating dac page 2 of 7
MT-017 the following example illustrates the concept of oversampling using some actual numbers. assume a traditional dac is driven at an input word rate of 30 msps (see figure 3a). assume the dac output frequency is 10 mhz. the image frequency component at 30 ? 10 = 20 mhz must be attenuated by the analog antialiasing filter, and the tr ansition band of the filter starts at 10 mhz and ends at 20 mhz. assume that th e image frequency must be attenuated by 60 db. the filter must therefore go from a passband corner frequency of 10 mhz to 60 db of stopband attenuation over the transition band between 10 and 20 mhz (one octave). a filter gives approximately 6-db attenuation per octave for each pole. therefore, a minimum of 10 poles is required to provide the desired at tenuation. filters become even more complex as the transition band becomes narrower. f clock = 30msps db image 10 20 30 40 50 60 70 80 f o analog lpf 10 20 30 40 50 60 70 80 image analog lpf frequency (mhz) image image image image f o f clock = 60msps db a b figure 3: analog filter requirements for f o = 10 mhz: (a) f c = 30 msps, and (b) f c = 60 msps assume that we increase the dac update rate to 60 msps and insert a "zero" between each original data sample. the parallel data stream is now 60 msps, but we must now determine the value of the zero-value data points. this is done by passing the 60-msps data stream with the added zeros through a digital inte rpolation filter which computes the additional data points. the response of the digital filter re lative to the 2 oversampling frequency is shown in figure 3b. the analog antialiasing filter transition zone is no w 10 to 50 mhz (the first image occurs at 2f c ? f o = 60 ? 10 = 50 mhz). this transiti on zone is a little greater than 2 octaves, implying that a 5- or 6-pole filter is sufficient. the ad9773 / ad9775 / ad9777 (12-/14-/16-bit) series of transmit dacs (txdac ? ) are selectable 2, 4, or 8 oversampling interpol ating dual dacs, and a simplified block diagram is shown in figure 4. these devices are designed to handle 12-/14-/16-bit input word rates up to 160 msps. the output word rate is 400 msps maxi mum. for an output frequency of 50 mhz, page 3 of 7
MT-017 an input update rate of 160 mhz, and an oversamp ling ratio of 2, the image frequency occurs at 320 mhz ? 50 mhz = 270 mhz. the transition band fo r the analog filter is therefore 50 mhz to 270 mhz. without 2 oversampling, the image fr equency occurs at 160 mhz ? 50 mhz = 110 mhz, and the filter transition band is 50 mhz to 110 mhz. f o k?f c f c latch latch dac lpf digital interpolation filter pll n n n n typical application: f c = 160msps f o = 50mhz k = 2 image frequency = 320 ? 50 = 270mhz figure 4: oversampli ng interpolating txdac ? simplified block diagram notice also that an oversampling interpolating dac allows both a lower frequency input clock and input data rate, which are much less lik ely to generate noise within the system. sigma-delta dacs sigma-delta dacs operate very similarly to si gma-delta adcs, however in a sigma-delta dac, the noise shaping function is accomplished with a digital modulator rather than an analog one. a - dac, unlike the - adc, is mostly digital (see fi gure 5a). it consists of an "interpolation filter" (a digital circuit which accepts data at a low rate, inserts zeros at a high rate, and then applies a digital filter algor ithm and outputs data at a high rate), a - modulator (which effectively acts as a low pass filter to the signal but as a high pass filter to the quantization noise, and converts th e resulting data to a high speed bit stream), and a 1-bit dac whose output switches between eq ual positive and negative reference voltages. the output is filtered in an external analog lpf. because of the high oversampling frequency, the complexity of the lpf is much less than the cas e of traditional nyquist operation. page 4 of 7
MT-017 n-bits @ f s n-bits @ k f s analog signal: 2 levels analog output digital interpolation filter digital ? modulator 1-bit dac analog output filter 1- bit @ kf s n-bits @ f s n-bits @ k f s analog signal: 2 m levels analog output digital interpolation filter digital ? modulator m-bit dac analog output filter m- bits @ kf s (a) single bit (b) multibit multibit figure 5: sigma-delta dacs it is possible to use more than one bit in the - dac, and this leads to the multibit architecture shown in figure 5b. the c oncept is similar to that of interp olating dacs previously discussed, with the addition of the digi tal sigma-delta modulator. in the past, multibit dacs have been difficult to design because of the accuracy requirement on the n-bit internal dac (this dac , although only n-bits, must ha ve the linearity of the final number of bits, n). the ad195x-series of audio dacs, however use a proprietary data scrambling technique ( called data directed scrambling ) which overcomes this problem and produces excellent performance with re spect to all audio specifications. the ad1955 multibit sigma-delta audio dac is show n in figure 6. the ad1955 also uses data directed scrambling, supports a multitude of dvd audio formats and has an extremely flexible serial port. thd + n is typically 110 db. page 5 of 7
MT-017 figure 6: ad1955 multibit sigma-delta audio dac summary oversampling used in conjunction with digital f iltering is a powerful tool in modern sampled data systems. we have seen how the same fund amental theory is applicable to both adcs and reconstruction dacs. a primar y advantage is the relaxati on of the requirements on the antialiasing/anti-imaging filter. another advantage is the increase in snr which occurs because of the process gain. the - adc and dac architecture is the ultimate extension of th e oversampling concept and is the architecture of choice for most voiceband and audio signal processing data converter applications. page 6 of 7
page 7 of 7 MT-017 references 1. g. r. ritchie, j. c. candy, and w. h. nink e, "interpolative digital- to-analog converters," ieee transactions on communications , vol. com-22, november 1974, pp. 1797-1806. (one of the earliest papers written on oversampling interpolating dacs). 2. h. g. musmann and w. w. korte, "generalized inte rpolative method for digital/analog conversion of pcm signals," u.s. patent 4,467,316 , filed june 3, 1981, issued august 21, 1984. (a description of interpolating dacs). 3. robert w. adams and tom w. kwan, "data-dir ected scrambler for multi-bit noise-shaping d/a converters," u.s. patent 5,404,142 , filed august 5, 1993, issued april 4, 1995. (describes a segmented audio dac with data scrambling). 4. y. matsuya, et. al., "a 16-bit oversampling a/d conversion technology using triple-integration noise shaping," ieee journal of solid-state circuits , vol. sc-22, no. 6, december 1987, pp. 921-929. 5. y. matsuya, et. al., "a 17-bit oversampling d/ a conversion technology using multistage noise shaping," ieee journal of solid-state circuits , vol. 24, no. 4, august 1989, pp. 969-975. 6. walt kester, analog-digital conversion , analog devices, 2004, isbn 0-916550-27-3, chapter 3. also available as the data conversion handbook , elsevier/newnes, 2005, isbn 0-7506-7841-0, chapter 3. copyright 2009, analog devices, inc. all rights reserved. analog devices assumes no responsibility for customer product design or the use or application of customers? products or for any infringements of patents or rights of others which may result from analog devices assistance. all trad emarks and logos are property of their respective holders. information furnished by analog devices applications and development tools engineers is believed to be accurate and reliable, however no responsibility is assumed by analog devices regarding technical accuracy and topicality of the content provided in analog devices tutorials.
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