IEEE Solid-State Circuits Magazine - Spring 2016 - 10

A C ircu it for All Season

Behzad Razavi

The Delta-Sigma Modulator

Delta-Sigma modulators (DSMs) are a
class of oversampling analog-to-digital converters (ADCs) that perform
"quantization noise shaping," thus
achieving a high signal-to-noise ratio
(SNR). An efficient solution for resolutions above approximately 12 b, DSMs
are extensively used in analog and RF
applications. In this article, we study
the fundamentals of this vast field.

The Delta Modulator
It is helpful to first study the predecessor of DSMs, namely, the delta
modulator. Shown in Figure 1(a), the
latter consists of a 1-b quantizer (e.g.,
a single comparator) and an integrator, both placed in a negative-feedback
loop. The high loop gain ensures that
VF . Vin and hence the digital output
is a representation of the analog input.

Vin

1−Bit
Quantizer

Dout

-
VF
(a)
CK
Vin

Dout
VF
R1

c1
(b)

Figure 1: (a) A delta modulator and (b) its
simple implementation.
Digital Object Identifier 10.1109/MSSC.2016.2543061
Date of publication: 21 June 2016

S P R I N G 2 0 16

tor, a multibit quantizer (an ADC),
Figure 1(b) depicts a simple implemenand a multibit digital-to-analog contation where the integrator is approxiverter (DAC).
mated by a low-pass filter.
In the 1970s, the potential of DSMs
The principal difficulty with the
was further explored.
delta modulator is
Candy proposed the
that the output digital
use of the structure
representation in fact
Delta-Sigma
for robust analog-tocontains only the demodulators are a
digital conversion in
rivative of the input, as
class of oversampling
1974 [3] and, along
can be seen by noting
analog-to-digital
converters
with Ching and Althat VF = # D out dt in
that perform
Figure 1(a). This difexander, in 1976
"quantization
ferentiation alters the
demonstrated a resonoise shaping,"
signal spectrum, atlution of 13 b with a
thus achieving a
tenuates the low-fre1-b quantizer in the
high signal-toquency content of the
loop [4]. These two
noise ratio.
signal, and amplifies
papers pointed out
high-frequency noise.
that the overall resolution increases as
Brief History
the quantizer is clocked faster and its
To avoid the differentiation effect
output circulates around the loop more
in Figure 1(a), Inose et al. [1] clevfrequently. An important observation
erly moved, in 1962, the integramade by Candy was that the overall outtor from the feedback path to the
put noise is the "first difference" of the
forward path, introducing the " 3 R
quantizer's additive noise, exhibiting a
modulator" shown in Figure 2. Here,
spectrum of the form sin 2 (~TCK /2),
where TCK is the quantizer clock period
the high-loop gain forces the run[3]. That is, the noise is suppressed at low
ning average of D out to follow Vin. Of
course, D out also contains the quanfrequencies. Candy also recognized that
tization noise created by the quanthe performance negligibly degrades
tizer, but with certain interesting
with the imperfections of the analog
and useful alterations.
components within the loop.
The 3 R modulator structure actuThe first integrated DSM was evially predates the work by Inose et al.
dently reported by van de Plassche
In a patent filed in 1961 [2], Brahm
in 1977 [5]. Using a continuous-time
discloses the system shown in Figure 3,
(CT) integrator, the ADC achieved
where the loop contains an integraa resolution of about 17 b in bipolar technology.
In 1978, Tewksbury and Hallock
described higher-order DSMs, present+
Vin
Dout
ing the architecture shown in Figure 4
-
(but attributing it to G.R. Ritchie) [6].
They also showed that the quantization noise spectrum is attenuated
Figure 2: A DSM.

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