Efficient Structure for Single-Bit Digital Comb Filters and Resonators

Abstract

A design technique for a single-bit digital comb ifiter is presented. The proposed filter response and performance are assessed in terms of signal-to-quantization-noise ratio (SQNR) and stability. It is found that the comb filter possesses a distinct frequency response in broadband signal applications. The same technique is utilized to design and simulate a single-bit N-period digital resonator. Feedback loop ifiters can be used to tune the frequency response of the >DZ modulators. The proposed design technique is efficient in hardware implementation.

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Efficient Structure for Single-Bit Digital Comb Filters
and Resonators
Amin Z. Sadik
School of Engineering Systems
Queensland University of Technology
Currently a Visiting Researcher at SECE
RMIT, Melbourne, Australia
E-mail: amin.sadik@rmit.edu.au
Zahir M. Hussain, SMIEEE
School of Electrical & Computer Engineering
RMIT, Melbourne, Australia
Email: zmhussain@ieee.org
Peter O’Shea
School of Engineering Systems
Queensland University of Technology
Brisbane, Australia
pj.oshea@qut.edu.au
Abstract—A design technique for a single-bit digital comb filter
is presented. The proposed filter response and performance are
assessed in terms of signal-to-quantization-noise ratio (SQNR) and
stability. It is found that the comb filter possesses a distinct
frequency response in broadband signal applications. The same
technique is utilized to design and simulate a single-bit N-period
digital resonator. Feedback loop filters can be used to tune the
frequency response of the Σ∆ modulators. The proposed design
technique is efficient in hardware implementation.
I. INTRODUCTION
Single-bit processing has been attracting interest due to the
promise of efficient and simple implementation [1]. Sigma-
delta modulators (Σ∆M’s) are the main single-bit modulators or
analog-to-digital converters (ADC’s). However, a drawback of the
Σ∆M system is that high resolution can only be obtained for low
to medium bandwidths [3]. This is so because the oversampling
ratio (OSR) should be high for better resolution (might be several
orders of magnitude higher than the Nyquist rate). Hence, It is
difficult to handle broadband applications, such as broadband
power-line communication (BPC), using an ordinary Σ∆M.
The signal-to-quantization-noise (SQNR) can be improved
by either increasing the order of the Σ∆M or increasing the
OSR. Therefore, one approach to alleviate the low bandwidth
bottleneck is to use a higher order Σ∆M with a reduced OSR.
However, this will introduce the problem of unpredictable insta-
bility that is seemingly inherent in such high-order Σ∆ systems
[2].
When designing a Σ∆M, the noise-transfer function (NTF)
should be given significant consideration. In general, it is de-
signed as a high-pass function so that the quantization noise can
be moved to higher frequency bands. In general, NTF can be
classified into two classes:
1) Pure differentiation of order M: In this case the noise-
shaping function (noise transfer function) can be expressed
as follows:
N(z) = (1 − z−1)M.
(1)
As the order M increases, more noise power will move to
high frequency bands, hence, noise in the low frequency
bands will reduce and, consequently, SNQR in the base-
band is increased. Moreover, SQNR can be improved by
increasing OSR. Due to the characteristics of this type of
x ( n )
_
+
H1(z)
A(z)
_
+
H2(z)
B(z)
sgn [⋅]
y(n)
Legend: Multi−Bit
Single−Bit
Fig. 1.
Block diagram of a second-order Σ∆M.
NTF, its usage is usually limited in mid and low bandwidth
applications such as audio applications.
2) non-monotonic transfer function: Pure differentiation re-
sponse can be modified by introducing poles into NTF as
follows:
N(z) =(z − 1)M
D(z)
.
(2)
In this case the NTF is an Mth-order polynomial with a
leading coefficient of 1. It is simply a high-pass function,
where the coefficients of the Σ∆M can be designed using
analog filter techniques.
II. THEORY AND DESIGN
The design of single-bit Σ∆M is a non-trivial task. Many
works in the literature have reported methods to estimate the
performance of Σ∆M analytically [4]. However, these methods
only approximate the actual behavior of Σ∆M.
To characterize the modulator it is common to look at the
STF and NTF. The STF describes how the modulator alters the
original input signal spectrum. Ideally, STF is unity. In a similar
manner the NTF describes how the modulator shapes noise away
from the center frequency, fc. For a low-pass modulator, fc= 0
Hz (DC), and for a band-pass modulator fcis often equal to fs/4
for simpler design.
The NTF is the main design task which determines the amount
of baseband noise shaping performed by the modulator.

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A. Background
Fig.(1) shows a second-order Σ∆M which contains two Mth-
order FIR filters (can be assumed of different order as well)in its
feedback loop to tune its response. Based on the linear model of
Σ∆M, the z-transfer function of the above system can be found
as follows:
Y (z) =
X(z) +
Q(z)
H1(z)H2(z)
D(z)]
(3)
where D(z) is given by:
D(z) = A(z) + B(z)
1
H1(z)+
1
H1(z)H2(z)
(4)
with A(z) and B(z) being the transfer functions of the FIR
filters (whose coefficients are {ai|i = 0,···,M} and {bi|i =
0,···,M} as follows:
A(z)=
M
?
M
?
i=0
aiz−i
(5)
B(z)=
i=0
biz−i.
(6)
From above the signal and noise transfer functions can be
expressed respectively as follows:
1
D(z)
S(z)=
(7)
N(z)=
1
H1(z)H2(z)D(z).
(8)
Now, depending on the corresponding topology required for
the standard second-order Σ∆M, the transfer functions H1(z)
and H2(z) can take any of the following forms:
H1(z)=
H2(z) =
z−1
1 − z−1
1
1 − z−1
H2(z) =
(9)
H1(z)=
H2(z) =
(10)
H1(z)=
1
1 − z−1;
z−1
1 − z−1.
(11)
These topologies were found to have identical characteristics
regarding noise shaping [5]. The only difference among them is
the delay factor (z−1) and the scaling gain. For instance, if we
adopt the first form, D(z) will be given as follows [6]:
D(z) = 1+(ao−2)z−1+(1−bo+b1+ao)z−2+G(z) (12)
where G(z) is given by:
G(z) =
M−1
?
i=1
bi+1z−i−2+
M
?
i=1
(ai− bi)z−i−2.
(13)
For ao= 1 and bo= 2, the structure will be reduced to the
standard Σ∆ topology, i.e., D(z) = 1, which implies two poles
at z = 0. For coefficient values other than ao= 1 and bo= 2,
D(z) will be a second-order polynomial in z−1, providing M
equations with 2M unknown coefficients. These coefficients can
be found using different approaches [7]. However, D(z) will not
increase the order of noise shaping in the transfer function of
the Σ∆M, but it may improve the stability of the system if it is
well-designed.
B. The Proposed Structure
To improve system performance, D(z) should be designed as
an FIR low-pass filter to reduce the height of voltage steps at
the output of the integrators [6], i.e., the input signal to D(z)
should not be attenuated at low frequency (when z → 1). If we
assume D(z) as an (M + 2)nd-order FIR filter with coefficients
{di|i = 0,···,M + 2} as follows:
M+2
?
Then, (in a semi-digital implementation, where the coefficients
are implemented by analog means) a comb filter can be produced
if the coefficients are equal, i.e., d0 = d1 = ··· = dM+2 =
1
M+2. This means that we impose the following condition on the
coefficients of D(z):
M+2
?
In this paper, our main intention is to design a single-bit digital
comb filter for the purpose of efficient hardware implementation
.
Now, the next step is to select proper functions to represent
H1(z) and H2(z). As the z-transfer function of the comb filter
is basically composed of equally spaced zeros around the unit
circle circumference, then refereing to eqn.3 we should chose
H1(z) and H2(z) such that they match the required frequency
response of the NTF as follows:
D(z) =
i=0
diz−i,
(14)
i=0
di= 1.
(15)
H1(z) = H2(z) =
1
(1 − z−M).
(16)
We choose H1(z) and H2(z) to have the same transfer function
for convenience but not necessarily. Then, the output of the Σ∆M
will be as follows:
Y (z) =X(z) + Q(z)(1 − z−M)2
D(z)
(17)
where D(z) is now given by:
D(z) = A(z) + B(z)(1 − z−M) + (1 − z−M)2
with A(z) and B(z) are as defined earlier. For simplicity, we
propose A(z) = B(z), but of course this is not the case
always, it depends on the application to be achieved. Therefore,
if
arithmetical manipulations we can find A(z) as follows:
(18)
?M+2
i=0di = 1 as we proposed earlier, then after a few
A(z) = B(z) = z−M.
(19)
Thisimplies
?M
that inthis case both
?M
i=0ai
=
1 and
by a pure M-delay line.
i=0bi = 1 and these functions are represented
Now, we re-design D(z) such that it introduces some poles
into the noise transfer function to comply with the non-monotonic
noise transfer function type mentioned above (item-2. This can

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x ( n )
_
+
z−M
_
+
z−M
sgn [⋅]
y(n)
z−M
Legend: Multi−Bit
Single−Bit
Fig. 2.
Block diagram of the proposed single-bit digital comb filter.
be carried out simply by putting A(z) = B(z) = 1 as is the case
with a standard Σ∆M, where D(z) will be given by:
D(z) = 3 − z−M+ z−2M.
(20)
In this case D(z) will introduce 2M poles into the NTF. These
poles are distributed uniformly as conjugate pairs around a circle
inside the unit circle on the z-plane. The radius r of this circle is
r = 0.577. Therefore, it is expected that D(z) will contribute to
shaping the noise as well as to changing the STF in such away
that converts the frequency response into M-period resonator, as
will be seen next.
The designed single-bit digital comb filter is shown in Fig.(2).
While the structure of the designed M-period resonator is shown
in Fig.(3)
Fig.(4) shows the signal and noise frequency transfer functions
S(ej2πv) and N(ej2πv) obtained from equation (20) with M =
10 as compared to those obtained from equation (18) with M =
10 which can be seen at Fig.(5). From these two figures we
can expect the role that the NTF can play in tuning the Σ∆
system response. This may suggest introducing ternary filters in
the feedback loop of the Σ∆ modulator.
C. Stability of the Proposed Structure
The stability of the system is decided by the poles in its transfer
function. The single-bit comb filter does not contain prominent
poles since D(z) = 1 implies two trivial poles at the center.
Moreover, all zeros lies on the unit circle. On the other hand, the
M-period resonator possesses 2M poles and all these poles are
located inside the unit circle, in addition to the same zeros as in
the NTF of the comb filter.
g ( k )
_
+
z−1
_
+
z−1
sgn [⋅]
s(k)
Legend: Multi−Bit
Single−Bit
Fig. 3.
Block diagram of the designed M-period resonator.
−1 −0.8 −0.6−0.4
Normalized Frequency (×π rad/sample)
−0.200.20.40.60.8
−140
−120
−100
−80
−60
−40
−20
0
20
Magnitude (dB)
Magnitude Response in dB
NTF
STF
STF
NTF
Fig. 4.
for the designed single-bit comb filter
Noise transfer function, N(v) and signal transfer function, S(v)
00.10.2 0.3
Normalized Frequency (×π rad/sample)
0.40.50.60.70.8 0.9
−160
−140
−120
−100
−80
−60
−40
−20
0
20
Magnitude (dB)
Magnitude Response in dB
NTF
STF
STF
NTF
Fig. 5.
for the designed single-bit M-period resonator
Noise transfer function, N(v) and signal transfer function, S(v)
The stability of the modulator is assessed by looking at the
quantizer input xq(n) Knee plot [10]. These were used to find
which input values would result in the divergence of the quantizer
input towards infinity. From which we may expect that this Σ∆M
comb filter is to be stable as long as the input signal amplitude
is limited by |xq| < 2. Fig.(9) reveals this situation.
III. SIMULATION AND DISCUSSION
To verify the above analytic results, the proposed single-
bit comb filter and multi-period resonator are simulated using
MATLAB. The frequency response of the comb filter for M = 10
and OSR R = 64 is shown in Fig.(6).
Fig.(7) depicts the frequency response of the M-period single-
bit resonator for M = 10 and OSR R = 64.
The signal-to-quantization-noise ratio (SQNR) is an essential
performance measure for Σ∆M. The in-band SQNR is given in
[8] as follows:
?0.5
?1/(2 R)
where X(ej2πv) is the Fortier transform of the (oversampled)
input signal x(i). The SQNR can be estimated empirically. To
do this, first the input signal spectrum must be removed from
the output and replaced by interpolating the end points , and
second the actual noise transfer function should be found to
evaluate the SQNR as given by the expression above using
SQNRin−band=
2
0
|X(ej2πv)|2dv
−1/(2 R)
|N(ej2πv)|2dv
(21)

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00.10.20.30.40.5 0.60.7 0.80.9
−40
−30
−20
−10
0
10
20
30
40
50
60
Normalized Frequency (×π rad/sample)
Magnitude (dB)
Magnitude Response in dB
Fig. 6.
filter with order M = 10 and OSR R = 64.
Frequency Response of the proposed single-bit digital comb
00.511.522.53
x 104
−100
−50
0
50
100
Filter Frequency Response
Normalized Frequency
Magnitude (dB)
Fig. 7.
resonator filter order M = 10 and OSR R = 64.
Frequency Response of the proposed single-bit M-period
Hannining-windowed FFT’s. Sinusoidal inputs are used in this
test. The input signal spectrum is chosen such that its spectral
energy lies within a single FFT bin. Fig.(8) shows the simulated
SNR as a function of OSR for sinusoidal inputs with different
frequencies. As expected [9], doubling the sampling frequency
reduces the noise power by about 9 dB, of which 3 dB is due to
the reduction in power spectral density of the quantization noise,
with additional 6 dB due to the action of the NTF.
It is also evident that, for the same value of OSR, there is an
improvement of about 10 dB when the frequency of the input
signal fois increased by an order of magnitude. This shows that
this Σ∆M topology lends itself well to broadband frequency
applications, such as Broad-Band Power-line Communication
0 50 100150200
OSR
250 300350400
30
40
50
60
70
80
90
SNR, dB
SNR vrs. OSR
fo=200 Hz
fo=2 kHz
fo=20 kHz
Fig. 8.
signal amplitude of 0.5.
SQNR against OSR for different input signal bandwidths with
−15−10−50510 1520
20
30
40
50
60
70
80
90
100
Input Amplitude, dB
SQNR, dB
SQNR vrs. Input Amplitude
OSR=128
OSR=64
OSR=32
Fig. 9.
amplitude for different values of OSR.
SQNR of the proposed single-bit comb filter versus the input
(BPL). This also suggests that the proposed single-bit Σ∆ comb
filter can be utilized with relatively low OSR if the input
frequency is high. In [11], we noticed that a similar structure
has been proposed for UWB-OFDM applications.
Fig.(9) shows the SQNR as a function to the amplitude of the
input signal for different OSRs. It can be seen clearly that the
SQNR collapses at an absolute input levels less than 0.3dB.
IV. CONCLUSIONS
In this paper we propose a design technique for single-bit
systems using a feedback path filter to tune the response of the
ΣΛ modulator. This may suggest utilizing ternary filters in the
feedback loop in future work. A single-bit digital comb filter is
designed and its performance is evaluated in terms of signal-to-
quantization noise ratio (SQNR), the dynamic range (input signal
level), and stability. Moreover, we showed that the same design
technique can be used for other single-bit systems, where we
used it to design a multi-period resonator. It was shown that the
proposed filters lend themselves very well to broadband input
signals and can be utilized in emerging technologies such as the
Broad-Band Power-line Communication (BPL).
ACKNOWLEDGMENT
This work is supported by the Australian Research Council
under the ARC Discovery Grant DP0557429.
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