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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" into 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 digital interpolation filter which generates the extra
data points. The high oversampling rate moves the image frequencies higher, thereby allowing a
less complex lower cost filter with a wider transition 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 represents the ultimate extension 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 requirements 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.
SAMPLED
SIGNAL
t
RECONSTRUCTED
SIGNAL
t
1
–3.92dB
A
1
fc
sin
A=
IMAGES
IMAGES
πf
fc
πf
fc
IMAGES
f
0
0.5fc
fc
1.5fc
2fc
2.5fc
3fc
Figure 1: Unfiltered DAC Output Showing Images and sin (x)/x Roll Off
Rev.A, 10/08, WK
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Note that the reconstructed signal amplitude is down 3.92 dB at the Nyquist frequency, fc/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 result of the sampling function and are also attenuated by the
sin(x)/x function.
OVESAMPLING INTERPOLATING DACS
The basic concept of an oversampling/interpolating DAC is shown in Figure 2. The N-bit words
of input data are received at a rate of fc. The digital interpolation filter is clocked at an
oversampling frequency of Kfc, and inserts the extra data points. 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 sampling 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 oversampling/interpolating DAC concept was by Ritchie,
Candy, and Ninke in 1974 (Reference 1) and followed by a 1981 patent (filing date) by
Mussman and Korte (Reference 2).
(FROM PLL CLOCK MULTIPLIER)
Kfc
N-BITS
@ fc
DIGITAL
INTERPOLATION
FILTER
N-BITS
@ K fc
DAC
Anti-imaging Filter Response
(A):NYQUIST
ANALOG
OUTPUT
ANTIIMAGING
FILTER
(B): OVERSAMPLING
WITH INTERPOLATION
f
f
fc
fc
Kfc
2
2
Figure 2: Oversampling Interpolating DAC
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Kfc
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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 transition band of the filter starts at
10 MHz and ends at 20 MHz. Assume that the 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 attenuation. Filters become even more complex as the transition
band becomes narrower.
A
ANALOG LPF
fCLOCK = 30MSPS
dB
fo
IMAGE
IMAGE
10
20
30
40
IMAGE
50
IMAGE
60
70
80
FREQUENCY (MHz)
B
fCLOCK = 60MSPS
dB
fo
ANALOG
LPF
IMAGE
IMAGE
10
20
30
40
50
60
70
80
Figure 3: Analog Filter Requirements for fo = 10 MHz:
(A) fc = 30 MSPS, and (B) fc = 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 interpolation filter which computes the additional data points. The
response of the digital filter relative to the 2× oversampling frequency is shown in Figure 3B.
The analog antialiasing filter transition zone is now 10 to 50 MHz (the first image occurs at 2fc –
fo = 60 – 10 = 50 MHz). This transition zone is a little greater than 2 octaves, implying that a 5or 6-pole filter is sufficient.
selectable 2×, 4×, or 8× oversampling interpolating 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 maximum. For an output frequency of 50 MHz,
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an input update rate of 160 MHz, and an oversampling ratio of 2×, the image frequency occurs at
320 MHz – 50 MHz = 270 MHz. The transition band for the analog filter is therefore 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.
N
LATCH
fc
N
PLL
DIGITAL
N
INTERPOLATION
FILTER
K•fc
N
LATCH
DAC
LPF
TYPICAL APPLICATION: fc = 160MSPS
fo = 50MHz
fo
K=2
Image Frequency = 320 – 50 = 270MHz
Figure 4: Oversampling 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 likely to generate noise within the system.
SIGMA-DELTA DACS
Sigma-delta DACs operate very similarly to sigma-delta ADCs, however in a sigma-delta DAC,
the noise shaping function is accomplished with a digital modulator rather than an analog one.
A Σ-Δ DAC, unlike the Σ-Δ ADC, is mostly digital (see Figure 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 algorithm and outputs data at a high rate), a Σ-Δ modulator
(which effectively acts as a low pass filter to the signal but as a high pass filter to the
quantization noise, and converts the resulting data to 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 LPF. Because of the high oversampling frequency, the complexity
of the LPF is much less than the case of traditional Nyquist operation.
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(A) SINGLE BIT
N-BITS @ fs
N-BITS @ K fs
DIGITAL
INTERPOLATION
FILTER
1- BIT @ KfS
DIGITAL
ΣΔ
MODULATOR
ANALOG SIGNAL:
2 LEVELS
1-BIT
DAC
ANALOG
OUTPUT
FILTER
ANALOG
OUTPUT
(B) MULTIBIT
N-BITS @ fs
N-BITS @ K fs
DIGITAL
INTERPOLATION
FILTER
M- BITS @ KfS
DIGITAL
MULTIBIT
ΣΔ
ANALOG SIGNAL:
2M LEVELS
M-BIT
DAC
MODULATOR
ANALOG
OUTPUT
FILTER
ANALOG
OUTPUT
Figure 5: Sigma-Delta DACs
It is possible to use more than one bit in the Σ-Δ DAC, and this leads to the multibit architecture
shown in Figure 5B. The concept is similar to that of interpolating DACs previously discussed,
with the addition of the digital 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 have 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 respect to all audio specifications.
The AD1955 multibit sigma-delta audio DAC is shown 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.
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Figure 6: AD1955 Multibit Sigma-Delta Audio DAC
SUMMARY
Oversampling used in conjunction with digital filtering is a powerful tool in modern sampled
data systems. We have seen how the same fundamental theory is applicable to both ADCs and
reconstruction DACs. A primary advantage is the relaxation of the requirements on the
antialiasing/anti-imaging filter. Another advantage is the increase in SNR which occurs because
of the process gain.
The Σ-Δ ADC and DAC architecture is the ultimate extension of the oversampling concept and is
the architecture of choice for most voiceband and audio signal processing data converter
applications.
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REFERENCES
1.
G. R. Ritchie, J. C. Candy, and W. H. Ninke, "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 Interpolative 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-directed 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.