ADI Technical Article: Oversampling Interpolation DAC

Oversampling and digital filtering can help to reduce the requirements on the antialiasing filter preceding the ADC. Reconstruction DACs can use the principles of oversampling and interpolation in a similar manner. For example, digital audio CD players often use oversampling, where the basic data update rate from the CD is 44.1 kSPS. Early CD players used traditional binary DACs and inserted “0”s into the parallel data, increasing the effective update rate to 4, 8, or 16 times the basic throughput rate. The 4×, 8×, or 16× data stream passes through a digital interpolation filter to produce additional data points. High oversampling rates shift the image frequencies higher, allowing the use of simpler, lower-cost filters with wider transition bands. In addition, the SNR within the signal bandwidth is improved due to processing gain. The Σ-Δ DAC architecture uses much higher oversampling rates to extend this principle to the extreme, making it popular in modern CD players.

The same oversampling and interpolation principles can also be used in high-speed DACs in the communications field to relax the requirements on the output filter and improve the SNR through processing gain.

Reconstructing the DAC's output spectrum

The output of the reconstruction DAC can be represented as a series of rectangular pulses whose width is equal to the inverse of the clock rate, as shown in Figure 1.

Figure 1: Filterless DAC output showing image and sin(x)/x roll-off

Note that at the Nyquist frequency fc/2, the reconstructed signal amplitude is reduced by 3.92 dB. If desired, an inverse sin(x)/x filter can be used to compensate for this effect. Images of the fundamental signal appear as a result of the sampling function and are also attenuated by the sin(x)/x function.

Oversampling Interpolation DAC

The basic principle of an oversampling/interpolating DAC is shown in Figure 2. An N-bit input data word is received at rate fc. A digital interpolation filter operates at a clock rate equal to the oversampling frequency, Kfc, and inserts additional data points. The effect on the output spectrum is shown in Figure 2. At the Nyquist sampling frequency (A), the requirements on the analog anti-imaging filter can be quite high. By oversampling and interpolating, the requirements on this filter can be greatly relaxed, as shown in (B). In addition, the signal-to-noise ratio is improved because the quantization noise is spread over a wider area than the original signal bandwidth. The SNR improves by 3 dB when the original sampling rate is doubled (K = 2) and by 6 dB when K = 4. Early CD players took advantage of this and were generally able to make the algorithms in the digital filter accurate to better than N bits. Today, most DACs in CD players are of the Σ-Δ type.

The earliest literature on the principle of oversampling/interpolating DACs is the 1974 paper by Ritchie, Candy, and Ninke (Reference 1) and the patent filed by Mussman and Korte in 1981 (filing date) (Reference 2).

Figure 2: Oversampling interpolation DAC

The following example uses some real numbers to illustrate the principle of oversampling. Assume that a conventional DAC is driven with an input word rate of 30 MSPS (see Figure 3A) and the DAC output frequency is 10 MHz. The image frequency component at 30 – 10 = 20 MHz must be attenuated by an analog antialiasing filter with a transition band starting at 10 MHz and ending at 20 MHz. Assuming that the image frequency must be attenuated by 60 dB, the filter must go from a passband corner frequency of 10 MHz to a stopband attenuation of 60 dB in the transition band from 10 MHz to 20 MHz (one octave). Each pole of the filter provides approximately 6 dB/octave of attenuation. Therefore, at least 10 poles are required to provide the required attenuation. The narrower the transition band, the more complex the filter.

Figure 3: Analog filter requirements at fo = 10 MHz: (A) fc = 30 MSPS, (B) fc = 60 MSPS

Suppose we increase the DAC update rate to 60 MSPS and insert “0”s between each raw data sample. Now, the parallel data stream is 60 MSPS, but we must determine the value of the zero-valued data point, which is accomplished by processing the 60 MSPS data stream with the added zeros through a digital interpolation filter, which calculates the additional data points. The digital filter response for a 2× oversampling frequency is shown in Figure 3B. The analog antialiasing filter transition region is now 10 MHz to 50 MHz (the first image occurs at 2fc – fo = 60 – 10 = 50 MHz). This transition region is slightly larger than 2 octaves, indicating that a 5 or 6 pole filter is sufficient.

The AD9773/AD9775/AD9777 (12-/14-/16-bit) family of transmit DACs (TxDAC®) are 2×, 4×, or 8× selectable oversampling interpolating dual-channel DACs, and a simplified block diagram is shown in Figure 4. These devices are capable of processing 12/14/16-bit input word rates up to 160 MSPS, and a maximum output word rate of 400 MSPS. Assuming an output frequency of 50 MHz, an input update rate of 160 MHz, and an oversampling ratio of 2, the image frequency occurs at 320 MHz – 50 MHz = 270 MHz, so the transition band of the analog filter is 50 MHz to 270 MHz. Without 2× oversampling, the image frequency occurs at 160 MHz – 50 MHz = 110 MHz, and the filter transition band is 50 MHz to 110 MHz.

Figure 4: Simplified block diagram of an oversampling interpolating TxDAC®

It should also be noted that the oversampling interpolating DAC supports lower input clock frequencies and input data rates, so it is much less likely to generate noise within the system.

Sigma-Delta DAC

The operating principle of a Σ-Δ DAC is very similar to that of a Σ-Δ ADC, but in a Σ-Δ DAC, the noise shaping function is implemented using a digital modulator instead of an analog modulator.

Unlike Σ-Δ ADCs, Σ-Δ DACs are mostly digital (see Figure 5A). They consist of an “interpolation filter” (a digital circuit that accepts data at a low rate, inserts zeros at a high rate, then applies a digital filter algorithm and outputs data at a high rate), a Σ-Δ modulator (which acts as a low-pass filter for the signal and a high-pass filter for the quantization noise, converting the resulting data into a high-speed bit stream), and a 1-bit DAC whose output switches between equal positive and negative reference voltages. The output is filtered in an external analog low-pass filter (LPF). Because of the high oversampling frequency, the complexity of this LPF is much lower than at the traditional Nyquist sampling frequency.

Figure 5: Sigma-Delta DAC

Σ-Δ DACs can use multiple bits. This is the “multi-bit” architecture shown in Figure 5B. The principle is similar to the interpolation DAC discussed previously, but with the addition of a Σ-Δ digital modulator.

In the past, multi-bit DACs were difficult to design due to the accuracy requirements of the n-bit internal DAC (it only has n bits, but must have linearity of the final number of bits N). However, the AD195x family of audio DACs solves this problem by utilizing a proprietary “data scrambling” technique (called “data-directed scrambling”) to provide excellent performance across all audio specifications.

Figure 6 shows the AD1955 multibit Σ-Δ audio DAC. The AD1955 also uses data-directed scrambling, supports various DVD audio formats, and has a very flexible serial port. The typical THD + N is 110 dB.

Figure 6: AD1955 multi-bit Σ-Δ audio DAC

Summarize

Oversampling combined with digital filtering is a powerful tool in modern sampled data systems. We have seen that the same basic principles apply to both the ADC and the reconstruction DAC. The main advantage is that the requirements for the antialiasing/antiimaging filters are relaxed, and another advantage is that the SNR is improved due to processing gain.

The Σ-Δ ADC and DAC architecture is the ultimate extension of the oversampling principle and is the architecture of choice for most voice-band and audio signal processing data converter applications.

References

1. GR Ritchie, JC Candy, and WH Ninke, “Interpolative Digital-to-Analog Converters,” IEEE Transactions on Communications, Vol. COM-22, November 1974, pp. 1797-1806. (One of the earliest papers on oversampling interpolating DACs).

2. HG Musmann and WW Korte, “Generalized Interpolative Method for Digital/Analog Conversion of PCM Signals,” US Patent 4,467,316, filed June 3, 1981, issued August 21, 1984. (Description of interpolating DACs).

3. Robert W. Adams and Tom W. Kwan, “Data-directed Scrambler for Multi-bit Noise-shaping D/A Converters,” U.S. Patent 5,404,142, filed August 5, 1993, issued April 4, 1995. (describing a segmented audio DAC that employs a “data scrambling” technique).

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. See also The Data Conversion Handbook, Elsevier/Newnes, 2005, ISBN 0-7506-7841-0, Chapter 3.