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Ubiquitous Computing and Communication Journal 1
DIGITAL COMPENSATION OF IN-BAND IMAGE SIGNALS CAUSED
BY M-PERIODIC NONUNIFORM ZERO-ORDER HOLD SIGNALS
Christian Vogel1, Christoph Krall
2
1
ETH Zurich, SwitzerlandSignal and Information Processing Laboratory (ISI)
[email protected] & Schwarz GmbH & Co. KG
Radiocommunications Systems Division
[email protected]
ABSTRACT
In this paper we introduce the design of a digital time-varying filter to compensate
the spurious in-band spectra caused by M-periodic nonuniform zero-order hold
signals. For this purpose, a continuous-time framework to describe and analyze
such signals is developed. Based on the error analysis, an equivalent discrete-timeframework is derived and used for the design of a discrete-time time-varying
precompensation filter. We exemplify the design of the precompensation filter for
the two-periodic case through simulations and measurements. Furthermore, we
discuss Farrow filters to reduce the design complexity.
Keywords: periodic, nonuniform, zero-order hold, compensation, reconstruction.
1 INTRODUCTION
Emerging communication systems require
high-speed and high-resolution data converters [1].
Since the integration density of digital circuitsgrows much faster than the one of analog circuits,
we have an increasing gap between analog and
digital circuits in terms of speed, area, and power
consumption. In particular, data converters - as the
central interface between analog and digital circuits
- are affected by this technological gap [2]. A new
design paradigm exploits the technology gap by
using digital signal processing to overcome the
analog impairments of data converters, e.g. [3].
In this paper we discuss digital-to-analog
converters (DACs) using a zero-order hold (ZOH)
for the signal reconstruction. We investigate the
compensation of in-band images caused by ZOHswith periodic nonuniform hold signals, i.e., a
nonuniform ZOH, as illustrated in Fig. 1. The
individual sample instants deviate by rnT from the
ideal time instants nT , where T is the nominal
sampling period of the DAC and r n
is the relative
time offset for time n with period M , i.e.,
rn= r
n+ M . Beside the typical sin x ( ) / x shaped
output spectrum, such nonuniform ZOH signals
introduce additional spurious images that
significantly reduce the DAC performance [4]. We
can find such spurious images in DACs driven by
clock signals with deterministic jitter [5] and intime-interleaved DACs with timing mismatches [6-
8]. Although the effects of
y(t)
x(0T )
x(1T )x(2T )
x(4T )
0T + r0T 1T + r1T
2T + r2T 3T + r0T
4T + r1T t
x(3T )
Figure 1: Output of a three-periodic nonuniform
ZOH (M=3) in the time domain.
nonuniform ZOHs have been extensively analyzed
[4,5,9], their digital compensation has not yet been
addressed in the literature.
We propose a digital time-varying filter to
compensate for the spurious image spectra caused
by periodic nonuniform ZOH signals. Therefore wedevelop a discrete-time model of a nonuniform
ZOH in Sec. 3 and use this model in Sec. 4 to
derive the ideal frequency response of a time-
varying compensation filter. In Sec. 5 we present
filter design examples including a Farrow filter
design and finally draw our conclusions in Sec. 6.
We want to note that preliminary results of the
presented work have been presented at two
conferences [10,11].
2 PRELIMINARIES
Sampling a continuous-time signal x c (t ) atuniform time instances t = nT produces the
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Ubiquitous Computing and Communication Journal 2
sequence x[n]= xc(nT ), where T denotes the
sampling period. The discrete-time Fourier
transform of x [n] is
X e j ( ) =
1
T X c j
T p
2
T
p=
. (1)
Throughout this paper, we assume that the signal
x c(t ) is bandlimited, i.e., the continuous-time
Fourier transform of x c(t ) satisfies
X c j ( ) = 0, for
T . (2)
With these assumptions, the signal x c(t ) can be
recovered from the sequence x [n] by using a
sequence-to-impulse train converter and an ideal
continuous-time low-pass filter with gain T and
cut-off frequency /T .
y(t)
x(4T )
x(3T )
x(1T )
x(0T )
0T 1T 2T 3T 4T t
x(2T )
Figure 2: Ideal uniform ZOH.
In practice, however, we can neither generateideal Dirac-delta impulses nor can we implement an
ideal continuous-time low-pass filter. Instead a
ZOH and a low-order analog low-pass filter are
used. Employing an ideal uniform ZOH for the
reconstruction process results in the reconstructed
signal y(t ) , which is illustrated in Fig. 2 and is
given by
y(t ) = x0 (t ) h0 (t )T (3)
where
x0 (t ) = xc(t ) t nT ( )
n=
(4)
is the sampled continuous-time signal and
h0(t ) =
1
T u t ( ) u t T ( )( ) (5)
is the 1/T -weighted impulse response of an ideal
ZOH with u(t ) being the unit step function. The
reason to explicitly introduce the T -factor in
Eq. (3) can be seen in the frequency domain.
Applying the continuous-time Fourier transform(CTFT) to Eq. (3) results in
Y ( j ) =1
T X c j p
2
T
H 0 j ( )
p=
T
= X c j p2
T
H 0 j ( )
p=
(6)
with
H 0 ( j ) =
sin T
2
T
2
e j
T
2 . (7)
We realize that the factor T of the ZOH
compensates the 1/T -factor introduced by the
sampling process. Furthermore, the output signal
consists of the 2 /T -periodic extended
continuous-time signal X c ( j ) weighted by the
sinc( x ) function given by Eq. (7).
3 SYSTEM MODELS
After clarifying the behavior of a uniform ZOH,
we introduce a ZOH with M -periodic nonuniform
hold signals and derive a discrete-time model.
3.1 Continuous-time model
Figure 1 illustrates the output of a ZOH with
M -periodic nonuniform hold signals in the time
domain. In contrast to the uniform case, each
sampling instant is time-shifted by r n, where r
nis
periodic with M , i.e., rn= r
n+ M . These time offsets
of the hold signal can be modeled by the M -
periodic impulse response hn= h
n+ M starting at r
n
and ending at (1+ rn+1
)T [4,5], i.e.,
hn(t ) =
1
T u t r
n( ) u t (1+ rn+1
)T ( )( ) . (8)
Therefore, the output signal y t ( ) in Fig. 1 can be
modeled as the sum of M weighted impulse trains
of period MT
xn(t ) = x
c(t ) t mMT nT ( )
m=
(9)
convolved with M impulse responses hn(t )T , i.e.,
y(t ) = xn (t ) hn (t )T n=0
M 1
. (10)
The CTFT of Eq. (10) results in
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Ubiquitous Computing and Communication Journal 3
Y ( j ) = X c j k 2
MT p
2
T
k =0
M 1
p=
(
H k j ( )
(11)
with
(
H k j ( ) =1
M H n j ( )
n=0
M 1
e jkn
2
M (12)
and
H n ( j ) =
sin 1+ rn+1 rn( )T
2
T
2
e j 1+rn+1+rn( )
T
2 . (13)
0 0.5 1 1.5 2 2.5 3−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
E n e r g y D e n s i t y S p e c t r u m ( d
B c )
Normalized frequency (Ω/ Ωs)
Desired
Image 1Image 2
Image 3
Figure 3: Output of a DAC with a 4-periodic
nonuniform ZOH (M=4) in the frequency
domain.
Figure 3 illustrates the spectral components
given by Eq. (11)-(13) for M = 4 . The desired
output is the 2 -extended continuous-frequency
spectrum weighted by
(
H 0( j ), which is the average
frequency response over all frequency responses
H n ( j ) for n = 0,..., M 1. This spectrum is
similar to the output spectrum in the uniform case;
however, we additionally have M 1 spectra,
which are modulated and filtered images of the
spectrum of a uniform ZOH. As these images
appear within the fundamental band, i.e., < /T ,
they cannot be removed by an analog low-pass
filter and significantly degrade the output signal
quality. A detailed analysis of the impact on the
performance can be found in [4,5].
3.2 Discrete-time model
In order to derive discrete-time compensation
filters for the in-band images, we represent the
continuous-time output signal given by Eq. (11) for
< /T in discrete-time. Therefore, the output in
Eq. (11) has to be ideally bandlimited, which can be
realized by the filter
H id j ( ) =1, <
T
0,
T
. (14)
Using Eq. (14) and Eq. (11), we can express the
ideally bandlimited signal as
Y b( j ) =1
T Y ( j )TH id ( j ) . (15)
Because we assume that the underlying continuous-
time signal x c(t ) is bandlimited, which is
implicitly satisfied if x [n] is a synthetic signal, we
can relate [12, Chap. 4]
X e j ( ) =
1
T X c j
T
(16)
and
(
H k e j ( ) =
(
H k j
T
(17)
for < . Consequently, we can express Eq. (11)-
(13) in discrete-time as
Y (e j
) = X e j
k
2
M
(
H k e j
( )k =
(18)
with
(
H k e j ( ) =
1
M H n e
j ( )n=0
M 1
e jkn
2
M , (19)
and
H n (e j
) =
sin 1+ rn+1 rn( )1
2
1
2
e
j 1+rn+1+rn( )1
2 . (20)
Using Eq. (18) and Eq. (1) with Y (e j
) it can be
verified that
Y b( j ) =Y (e j T
)TH id ( j ) (21)
results in Eq. (15) and the continuous-time and the
discrete-time frequency output are related by
Y (e
j
)=
1
T Y j
T
, <
. (22)
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Ubiquitous Computing and Communication Journal 4
4 DERIVATION OF THE COMPENSATION
FILTER
In this section we propose a time-varying filter
to compensate the in-band images. We derive the
design equations and analytically solve them for the
two-periodic case.
4.1 Design equationsTo compensate for the spurious images, we
propose an M -periodic time-varying filter gn[ l ] ,
which relates the input x [n] to the output s[n] as
s[n]= gn[ l ] x[n l ]l =
. (23)
The precompensation filter and the discrete-time
model of the nonuniform ZOH are illustrated in
Fig. 4 for the two-periodic case.
discrete-time model ZOH
H 1(e jω)
(−1)n
Gn(e jω)
(−1)n
H 0(e jω)
G1(e jω)
G0(e jω)s[n]x[n] y[n]
Figure 4: Precompensation filter in front of the
discrete-time model of a nonuniform ZOH
(M=2).
The time-varying filter gn[ l ] should modify the
signal x [n] in such a way that the signal s[n]
driving the nonuniform ZOH does not generate
image signals within the fundamental band.
Since gn[ l ] is periodic with M , we can
represent it as the inverse discrete-time Fourier
series (DTFS) [13, Chap. 4]
gn[ l ]=(
gk [l ]k =0
M 1
e jkn
2
M (24)
with the related DTFS
(
gk [l ]=1
M gn[ l ]
n=0
M 1
e jkn
2
M . (25)
After substituting Eq. (24) in Eq. (23), we can write
s[n] =(
gk
k =0
M 1
[l ]e jkn
2
M x[n l ]l =
=
(
gk [l ] x[n l ]l =
e
jkn2
M
k =0
M 1
.
(26)
The discrete-time Fourier transform (DTFT) of
Eq. (26) gives
S e j ( ) =
(
Gk
k =0
M 1
e j k
2
M
X e
j k 2
M
(27)
where
(
Gk e j ( ) =
1
M Gn
n=0
M 1
e j ( )e
jkn2
M (28)
is the DTFT of Eq. (25). Because of the time-
varying filter gn[ l ], the input signal into the ZOH
is s[n] instead of x [n] and Eq. (18) becomes
Y (e j
) = S e j k
2
M
(
H k e j ( )
k =0
M 1
. (29)
Substituting the output of the filter given by
Eq. (27) in Eq. (29) results in
Y e j ( ) =
(
H k 1
e j ( )
(
Gk e j k
1+k ( )
2
M
k =0
M 1
k 1=0
M 1
X e j k
1+k ( )
2
M
.
(30)
With l = k 1+ k we can simplify Eq. (30) to
Y e j ( ) =
(
H l k e j ( )
(
Gk e j l
2
M
l =k
M 1+k
k =0
M 1
X e j l
2
M
=
(
H l k e j ( )
(
Gk e j l
2
M
l =0
M 1
k =0
M 1
X e j l
2
M
(31)
where we have exploited the periodicity of (
H l k
(e j
) .
Thus, the overall transfer function can be
expressed as
Y e j ( ) =
(
F l e j l
2
M
X e
j l 2
M
l =0
M 1
(32)
with
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Ubiquitous Computing and Communication Journal 5
(
F l e j ( ) =
(
H l k k =0
M 1
e j +l
2
M
(
Gk e j ( ) (33)
as it is shown in Fig. 5 for the two-periodic case.
F 0(e jω)
F 1(e jω)
(−1)n
y[n]
x[n]
Figure 5: Overall transfer function of the
cascaded system for M=2.
From the theory of multi-rate systems we know
that a system is defined as perfect reconstruction
system if the output is a scaled and delayed version
of the input [14]. In our compensation problem wewant to maintain unity gain of the overall system
and therefore require that
(
F l e j l
2
M
=
e j
, for l = 0
0, for l =1,2,K, M 1
(34)
where is the delay of the system. If Eq. (34) is
fulfilled all image spectra are cancelled and the
sin( x ) / x distortions in the fundamental band are
equalized. The time-varying filter gn[ l ] can be
efficiently implemented as an M -channel
maximally decimated multi-rate filter bank [14].
4.2 Two-periodic nonuniform ZOHWe will exemplify the filter design procedure
for the two-periodic case, i.e., M = 2. By
expressing Eq. (33), and Eq. (34) in matrix notation
we obtain
e j
0
=
(
H 0e j ( )
(
H 1e j ( )
(
H 1e j + ( )( )
(
H 0e j + ( )( )
(
G0e j ( )
(
G1e j ( )
(35)
Solving the matrix equation leads to
(
G0
e j ( )
(
G1
e j ( )
=
(
H 0
e j + ( )( )
(
P e j ( )
(
H 1
e j + ( )( )
(
P e j ( )
e j
(36)
with
(
P e j ( ) =
(
H 1
e j ( )
(
H 1
e j + ( )( )
(
H 0
e j ( )
(
H 0
e j + ( )( ).
(37)
Applying the inverse DTFS as defined in Eq. (24)
to Eq. (36) and substituting the relation Eq. (19) for (
H k (e j
) we obtain the transfer functions of the
time-varying filter gn[ l ] that are
G0
e j ( )
G1
e j ( )
=
H 1 e
j + ( )
( )P e j ( )
H 0
e j + ( )( )
P e j ( )
e j
(38)
with
P e j ( ) =
1
2 H
1e
j ( ) H 0
e j + ( )( )( H
0e
j ( ) H 1
e j + ( )( )).
(39)
Analytically solving the matrix equations for larger orders of M is infeasible. Instead the matrix
has to be numerically solved for each frequency
used in the filter design procedure.
5 FILTER DESIGN EXAMPLES
In the following we will discuss the design of a
two-periodic time-varying FIR filter, the
compensation of a two-channel time-interleaved
DAC, and the design of a Farrow filter.
5.1 FIR filter design
For the design example we have assumed atwo-periodic ZOH signal with time offsets of r
0= 0
and r 1= 0.0391. To approximate the ideal
frequency responses given by Eq. (38) we use
causal finite-impulse response (FIR) filters with
transfer functions
Gn
ae j ( ) = gn
a[l ]e
j l
l =0
L
(40)
and approximate the ideal frequency responses in
the minimax (Chebychev) sense, i.e.,
minGne j ( )Gn
ae j ( )
D (41)
where D
is the definition domain of and
frequencies above D
belong to the “don't care”
band. We have designed filters of order L =17 ,
with = 8.5, and D= (0.7 ,K,0.7 ) by
solving the approximation problem with the Matlab
optimization toolbox CVX [15].
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Ubiquitous Computing and Communication Journal 6
0 0.1 0.2 0.3 0.4 0.5−140
−120
−100
−80
−60
−40
−20
0
E 0
( e
j ω ) ( d B )
Normalized frequency (Ω/ Ωs)
Figure 6: Deviation from the ideal overall
frequency response.
0 0.1 0.2 0.3 0.4 0.5−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
E 1
( e j ω ) ( d B )
Normalized frequency (Ω/ Ωs)
Figure 7: Attenuation of the spurious images.
For the specified problem we have obtained a
maximum approximation error of 61.1 dB in the
filter design of G0
a(e
j
) and 60.4 dB for G1
a(e
j
).
In Fig. 6 we see the impact of the approximation
error on
E 0e j ( ) =
(
F 0e j ( ) e j (42)
i.e., the difference between the average frequency
response and an ideal delay, and in Fig. 7 on the
residual images
E 1e j ( ) =
(
F 1e j ( ) . (43)
Within the definition band D
, the approximation
error of an ideal delay is less than -98 dB and the
spectral images are attenuated by at least -60 dB.
To further test the filter design, a discrete-time
multitone signal with frequencies [0.0581, 0.1162,
0.1743, 0.2324, 0.2905, 0.3486] 1/T has been used
as an input to the nonuniform ZOH. Without any
compensation we have obtained the energy density
spectrum shown in Fig. 8. We see the sin( x ) / x
0 0.5 1 1.5−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
E n e r g y D e n s i t y S p e c t r u m ( d
B c )
Normalized frequency (Ω/ Ωs)
Figure 8: Output of a nonuniform ZOH with
period M=2.
0 0.5 1 1.5−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
E n e r g y D e n s i t y S p e c t r u m ( d
B c )
Normalized frequency (Ω/ Ωs)
Figure 9: Precompensated output of anonuniform ZOH with period M=2.
output spectrum and the additional image spectra
due to the nonuniform ZOH signal. The largest
unwanted spur in the fundamental band, i.e., the
band between 0 and 0.5, is about -29 dBc. The
energy density spectrum with compensation is
shown in Fig. 9. Within the fundamental band the
unwanted spectral images are considerably reduced.
The largest spur is about -61 dBc, which is an
improvement of 32 dB compared to the
uncompensated case. A higher attenuation of the
images is achievable by increasing the filter order.When comparing the out-of-band energy of
Fig. 8 and Fig. 9 we recognize that the out-of-band
energy in Fig. 9 is slightly increased. This minor
amplification of the out-of-band energy, however,
should not change the design requirements of the
analog reconstruction filter significantly.
5.2 Time-interleaved DAC
We have implemented a two-channel time-
interleaved digital-to-analog converter that
produces two-periodic nonuniform ZOH signals [8].
Therefore, we have used a XILINX Virtex-4 FPGA
evaluation platform [16] to generate the digitalsignal and two high-speed DACs from Analog
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Ubiquitous Computing and Communication Journal 7
Devices (AD9734) [17] to convert the digital
signals into the analog domain.
An 800 MHz differential clock generator has
generated the 180 degrees phase-shifted clock
signals for the two DACs, where the differential
outputs have been used as single ended clock
signals. Thus, the overall sampling rate of the two-channel time-interleaved DAC is 1600 MS/s. The
two analog output signals have been combined with
a power combiner and measured with an Agilent
54855A 6 GHz scope with a sampling frequency of
20 GS/s.
For the presented measurements we have
generated a single sinusoid with a frequency of
300 MHz and have identified the time offsets as
r 0= 0 and r
1= 0.12 . Along with the identified time
offsets a filter, which compensates the digital signal
as depicted in Fig. 4, has been designed according
to Eq. (41). The digitally compensated and
converted signal has been measured with the high-speed sampling scope. The results of the
measurements are shown in Fig. 10. The mirror
images at 500 MHz have been reduced by about
20 dB. Due to uncertainties in the time offset
estimation, a better reduction has not been possible.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.42.4−
80
−60
−40
−20
0
Frequency (GHz) E n e
r g y D e n s i t y S p e c t r u m ( d
B c )
uncompensated
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−80
−60
−40
−20
0
E n e r g y D e n s i t y S p e c t r u m ( d
B c )
Frequency (GHz)
compensated
Figure 10: Measurements for the
uncompensated and the precompensated
sinusoid.
5.3 Farrow filter designAs shown in the last section, we can design a
time-varying filter consisting of two impulse
responses to compensate two-periodic nonuniform
ZOH signals with given deviations. In some
applications it is necessary to redesign the
compensation filter from time to time, since the
time offsets r n
are subject to temperature changes,
aging effects, and voltage supply variations.
Therefore, the previously introduced filter design
process might become too complex for certain
applications. For the two-periodic case, the solution
to this problem is the adaptation of a Farrow
filter [18], which reduces the redesign complexity
significantly.
By defining r n+1 r
n= (1)
n
and r n+1+ r
n= ,
we can rearrange Eq. (20) as
H n (e
j ) = A
n (e j
, )e j 1+ ( )
1
2 (44)
with
An (e j
, ) =
sin 1+ 1( )n ( )
1
2
1
2
. (45)
With these definitions, the frequency responses
An(e
j , ) depend on one additional spectral
parameter as required for Farrow-based filter
designs. Furthermore, all frequency responses
H n (e
j
) share a common factor exp(
j 1+
( ) / 2) .Hence, we can conclude from Eq. (18) and Eq. (20)
that the factor exp( j 1+ ( ) / 2) shifts the entire
input signal by (1+ ) / 2 in time, but does not
produce spurious images. Since we are only
interested in compensating the spurious images, we
can modify our perfect reconstruction condition
given by Eq. (34) for the two-periodic case as
(
F l e j l
2
M
= e
j + 1+ ( )1
2
, for l = 0
0, for l =1
(46)
By using Eq. (44) and Eq. (46) and following the
derivations of the last section we obtain the
compensation filters as
G0
e j , ( )
G1
e j , ( )
=
A1
e j + ( )
, ( )P e j ( )
A0
e j + ( )
, ( )P e
j ( )
e j . (47)
The ideal time-varying compensation filter depends
on the frequency and on an additional spectral parameter . Hence, these filters can be designed
and implemented as Farrow f ilters.
A Farrow filter has an impulse response of
gn
a[ l , ]= cn , p[ l ]
p
p=0
P1
(48)
with
Gn
ae j
, ( ) = gn
al , [ ]
l =0
L
e j l . (49)
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Ubiquitous Computing and Communication Journal 8
In general, we have to design two sets of
Farrow coefficients c0, p[ l ] and c
1, p[ l ] for our two-
periodic compensation filter. However, we can
further simplify the filter design by exploiting the
even symmetry regarding , i.e.,
A0 e j , ( ) = A1 e j , ( ) (50)
which can be verified with Eq. (45). In consequence,
we can show that
P e j
, ( ) = P e j
, ( ) (51)
and can conclude that the transfer functions of the
two-periodic time-varying filter are even functions
regarding the spectral parameter
G0 e
j
,
( )=
G1 e
j
,
( ) . (52)
This property can be exploited for the filter design
by choosing a definition domain for D
that is
symmetric around zero, i.e., D= ( max
,K, max).
For such a choice, the filter design will lead to two
transfer functions related by
G0
ae j
, ( ) =G1
ae j
, ( ) (53)
and can be described in general as
Gn
a e j , ( ) =G0
a e j ,1( )n
( ) . (54)
Accordingly, we only have to design one set of
filter coefficients c0, p[l ] and can efficiently
implement the compensation filter as shown in
Fig. 11.
s[n]
c0,1[l]c0,2[l] c0,0[l]
(−1)nλ(−1)nλ
Gan(e
jω)x[n]
Figure 11: Proposed modified Farrow filter for
P=3.
Furthermore, G0 (e j
, ) consists of a real zero-
phase response A0 (e
j + ( ), ) /P(e
j ) and a linear-
phase response exp( j ) , which can be chosen
arbitrarily. By choosing a delay such that the
compensation filter will exhibit symmetrical or
anti-symmetrical impulse responses, the number of
coefficient multipliers can be basically halved [11,
Chap. 6].
For the design example we have chosen an FIR
Farrow filter of order P = 3 with subfilters of order
L=
17 and a delay of =
8.5, i.e., subfilters with9 multipliers if we exploit the linear-phase
property, with D= (0.7 ,K,0.7 ). The range of
possible time offsets has been D= (0.05,K,0.05) .
The approximation problem was solved in the
Chebyshev sense ( L
norm), i.e.,
minG0e j , ( )G0
a e j , ( )
D , D (55)
where D
is the definition domain of the spectral
parameter , by again using Matlab and the
optimization toolbox CVX. Design procedures can
be found for example in [19].
0 0.1 0.2 0.3 0.4 0.5−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
E 0
( e j ω ) ( d B )
Normalized frequency (Ω/ Ωs)
λ=±0.05
λ=±0.0391
λ=±0.0278
λ=±0.0167
λ=±0.0056
Figure 12: Deviation from the ideal overall
frequency response for different .
0 0.1 0.2 0.3 0.4 0.5−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
E 1
( e j ω ) ( d B )
Normalized frequency (Ω/ Ωs)
λ=±0.05
λ=±0.0391
λ=±0.0278
λ=±0.0167
λ=±0.0056
Figure 13: Attenuation of the spurious images
for different .
For the specified problem we have obtained a
maximum approximation error of 55.3 dB in thefilter design of G
0
a(e
j , ) . In Fig. 12 we see the
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Ubiquitous Computing and Communication Journal 9
impact of the approximation error due to the filter
design on realizing an ideal delay, i.e.,
E 0e j ( ) =
(
F 0e j ( )e
j + 1+ ( )1
2
(56)
and in Fig. 13 on the residual images as in Eq. (43)
for various values of . Within the definition band,
the approximation error of an ideal delay is less
than -70 dB and the spectral images are attenuated
by at least -63 dB. These approximation errors are
comparable to the values for the fixed FIR filter;
however, for the given specification, the flexible
Farrow-based solution needs at least 3 times more
multipliers in its implementation. Hence, we can
avoid the online filter design and gain flexibility,
but have to pay for it with higher implementation
costs.
Finally, we have simulated two-periodic ZOHsignals with time offsets r
0= 0 and r
1= 0.0391
driven by a digitally generated multi-tone signal as
in Sec. 5.1. The energy density spectra of the
uncompensated and the compensated output of the
ZOH are shown in Fig. 8 and Fig. 14, respectively.
In the compensated spectrum, the spurs are reduced
to -63 dBc.
Figure 14: Output of a two-periodic ZOH
precompensated by a Farrow filter.
6 CONCLUDING REMARKS
We have introduced the design of a discrete-time
time-varying filter to compensate the spurious in-
band image spectra caused by M-periodic
nonuniform ZOH signals. A general filter design
procedure is derived and the filter design for the
two-channel case is demonstrated by two examples.
Furthermore, for the two-channel case, we have
extended the design procedure to Farrow filters,
which allows for compensating a wide range of
possible time offsets by the simple adjustment of
some multipliers.
ACKNOWLEDGEMENT
Christian Vogel was supported by the Austrian
Science Fund FWF's Erwin Schrödinger Fellowship
J2709-N20.
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Ubiquitous Computing and Communication Journal 10
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