Generalization of the frequency sampling method

Abstract

The classical “frequency sampling method” (1975) based
on the direct utilization of the Lagrange interpolation technique is
extended in a natural way to a rather efficient Hermite interpolation
scheme. This structure is a new extension related to the resonator-based
digital filter family introduced by M. Padmanabhan et al. (1996). The
development resulted in a system having good properties if measurement
of signals consisting of sinusoidal components is to be performed

Full text

IEEE
Instrumentation and Measurement
Technology Conference
Brussels,
Belgium,
June
4--6,
1996
lization
of
the
eney
Sampling
GBbor
Pkceli, Gyula
Simon
Department of Measiirement and Instrument Engineering,
Technical IJniversity
of'
Budapest,
H-152:l
Budapest, Hungary
Phone:
+
36
1
463'2057
Fax:
-t-
36
1
4634112
E-Mail: peceli@mmt.bme.hu
Abstract-
The classical "frequency sampling methlod"
[l]
based on the direct utiliaation
of
the Lagrange intc?rpola-
tion technique
is
extended
in a natural way to a rather
efficient Hermite interpolation scheme. This structure is a
new extension related to the recently introduced resonator-
based digital filter family
[7].
The development resulted in
a system having good properties if measurement of signals
consisting
of
sinusoidal components
is
to be performed.
I. INTRODUCTION
The "frequency sampling
(FS)
method is a classical
approach to design finite impulse response (FIR) digital
filters (see e.g.
[l])
in the frequency domain via the direct
utilization of the Lagrange interpolation technique (see
e.g.
[2]).
The FS method is strongly rlelated to the re-
cursive evaluation of the Discrete Fouriei- Transformation
(DFT) if the frequency-domain sampling
is
uniform. Some
early attempts to realize the recursive DlFT suffered from
serious implementational problems due to the
fact
that
digital resonators operating
at
their stability limit were
involved. Later this problem has been solved with the
introduction of
a
common structure for recursive trans-
formations
[3]
which proved to be useful in implementing
different digital filters and filter-banks including infinite
impulse response (TIR) and adaptive filters, as well
([4]-
[7]).
This structure consists of parallel first- or second-
order digital resonators within
a
common feedback loop.
The infinite
loop
gain
at
the resonator frequencies itssures
good sensitivity properties and the conditions of hit cy-
cle immunity and
IOW
roundoff noise can also be fiiilfilled
[4].
The transfer value from the filter input to one of the
resonator outputs can be characterized by
1
at the corre-
sponding resonator pole frequencies and by
0
at
the pole
frequency of all the other parallel resonator sectionj. The
concept of "frequency sampling" means the prescription
of transfer function values at these distinct, not necessar-
ily uniformly distributed resonator pole positions. Typi-
cal frequency domain filter designs start with a tolerance
scheme given as
a
desired magnitude, phase, or possibly
'This
work
was
supported
by
the
Hungarian
Fund
for
Ssientific
Research
(OTKA)
under contract
4-100
group delay requirement. With the
FS
method the situ-
ation is ,somewhat different because it solves the approx-
imation problem by interpolation and therefore the spec-
ifications are not automatically met. On the other hand
at the resonance frequencies the prescribed values are im-
plemented without any systematic error. This feature of
the FS method is very attractive in solving measurement
problems where the overall accuracy of the system is of key
importance.
A
typical example can be the measurement of
composi1,e sinusoidal waveforms (see e.g.
[8])
where there
is
a
real chance to prescribe the necessary transfer values
for all the frequency components to be measured.
Unfortunately, however, possibly with the exception of
the synchronized multi-sine measurements, where the arti-
ficially glenerated sine waves are
to
be measured, the prac-
tical signals usually have considerable frequency content
also between the "sampled" frequency positions. There-
fore it isn't indifferent what is the behavior of the fre-
quency characteristics in these ranges. The idea to gener-
alize the FS methods comes
at
this point. On one hand
it is desirable to maintain all the positive features of the
original technique but on the other some additional ap-
proximation power
is
needed to improve the overall per-
formance. In this paper the case of resonators with higher
(>
1)
multiplicity
is
investigated. This means that we
have
a
structure consisting of cascades of identical res-
onators in parallel within
a
common feedback loop. The
output taps of the multiple resonators may fix not only
the complex transfer value but according to the actual
resonator multiplicity its first, second, etc. derivatives
at
the corresponding frequency. It
is
easy to show that this
technique
is
in complete correspondence with the Hermite
interpolation method, however, similarly to the Lagrange
interpolation case (see e.g.
[4]
or
[7])
the very same struc-
ture
is
capable to combine the zero-set coming from the
Hermite interpolation with an arbitrary pole-set and thus
implement any rational transfer function.
The
first part
of
the paper overviews all those special-
ties which come from measurement problems and strongly
influence the design of digital filters and filter-banks. Zero
339

group delay and consequently constant phase in
a
certain
frequency range is
a
typical requirement e.g. in measur-
ing bridges. The second part is devoted to the analysis of
the structure consisting of parallel resonators and to the
case where higher-order resonators are applied. In this
section the design equations are also provided. Finally in
the third part the applicability of this Hermite interpola-
tion structure to adaptive Fourier analysis (see e.g.
[SI)
is
investigated.
11.
PARALLEL
MEASUREMENT
OF
SINUSOIDAL SIGNAL
COMPONENTS
There are many practical problems where the measure-
ment of signals containing sinusoidal components is re-
quired. Typical examples are the measurements related
to rotating machinery or simply to the line frequency. In
both cases the harmonically related sinusoidal components
are to be separated and the unavoidable measurement
noise is to be suppressed. The recursive DFT structure
suggested in
[3]
can be
a
good candidate to perform mea-
surements on such signals if the frequency of the compo-
nents is known in advance or can be determined by
an
adaptation mechanism
[6].
The block diagram of such
a
transformer is given in Fig. 1. The structure consist of
-1
P
-
-
I
11
I
I
..-
*
I
Fig.
1.
Feedback-based
DFT
filter-bank
complex coefficient first-order resonators having
a
single
complex pole on the unit circle. All these resonators op-
erate in parallel within
a
common feedback loop. At the
frequencies corresponding to the resonator pole frequen-
cies this loop has infinite loop gain therefore the transfer
value equals to
1
independently of the other parameters
of the system. The transfer function of the mth channel
of this filter-bank has form the of
9,,&
x
-
1
where
m=Q,l,
...,
N-1.
(2)
The resonator poles can be located arbitrarily along the
unit circle, only the case of multiple poles
is
to
be avoided.
For a DFT filter-bank the poles are the Nth roots of unity,
therefore its parameters have simpler form
zm
ej%m
3
Sm=-
m=0,1,
...,
N-1.
(3)
N'
.Zm
=
The value of the transfer function
(1)
is zero if
z
=
z,,
n
=
Q,1,
...,
N-1,exceptthecaseofn
=
mwhenit equalsl.
If
the input sequence of this filter-bank consists of sinusoids
of frequency corresponding to
zn
(n
=
Q,1,
...,
N
-
1),
then it will separate the mth component and measure it
without any systematic error. However, if the frequency of
the mth component differs from the pole frequency of the
mth resonator, then a systematic magnitude and phase
error will appear. For the DFT case the magnitude error
can be expressed by
1
EZ(2ffTT;
I
'
(4)
where
Af
and
T
stand for the frequency error and for
the sampling time, respectively. Equation
(4)
is easy to
derive since for the DFT case
(1)
can be expressed in
a
much simpler form of
The phase of
(5)
can also be directly derived.
It
turns out
that the phase is practically linear except the unavoidable
phase jumps of
T
at the resonator pole positions of in-
dex
n
(n
#
m).
The presence of this linear phase shift
may cause serious measurement problems also for small
Af
values. This problem is well-known also in the high
precision measuring bridges, where due to the frequency-
dependent parasitic components the balanced state of the
bridge will differ from that of the nominal frequency case
To reduce these errors due to the mismatch of the input
and
pole frequencies the transfer functions
of
(1)
should
be modified.
A
better performance can be achieved if
the magnitude and the phase approximate in
a
wider fre-
quency range one and zero (or
at
least
a
constant value),
respectively. A possible solution to this approximation
is
the classical windowing technique (see e.g.
[l])
widely
used in spectrum analysis. In the filter-bank of Fig.
1
win-
dowing can be performed
at
the parallel output as
a
sim-
ple linear combination of the neighboring channels. The
[91.
340

Fig.
2.
Feedlback-based filter-bank
with
resonator
multiplicity
of
2
conventional Hanning window (see e.g.
[l])
requires the
linear combination of the
(m
-
l)th, mtli and
(m
-t
1)th
channels. Windowing, however, reduces the frequency res-
olution, i.e.
to provide the same selectivity the order
of
the system on
Fig.
1
should be increased. Unfortuinately
the increased order will destroy the improvement due to
windowing. The next Section will introduce
an
alternative
solution which at the prize
of
higher system order ciin im-
prove the magnitude and phase responses of the chmnels
without affecting the frequency resolution.
111. CHARACTERIZATION
OF
THE
RESONATOR-B
WED
FILTER-BANKS
The single-input multiple-output (SIMO) filter-bank of
Fig.
1
can serve as a basic building block in very many
practical applications.
Using the parameters
of
(3)
the
system operates as a DFT filter-bank which time recur-
sively evaluates the DFT of the last
N
input samples
and
simultaneously provides the DFT components, as well.
The application of the frequency sampling
(FS)
method is
simply the calculation of the linear combination
of
these
DFT components. The weights applied are sampled val-
ues of the transfer function to be implemiented. The sam-
pling must be performed at the resonator pole frequencies
where (1) equals one. The order of the attainable FIR
filters is
N
-
1. The system
of
Fig.
1
provides
ii
one-
step delay because the channel outputs are the oiitputs
of the delay elements of the resonators, tlherefore a "lead-
ing"
z-'
is also present in the transfer functions
of
the
filters. This delay can be avoided if the output is calcu-
lated as the linear combination of the inputs of the delay
elements. The parallel nature
of
channel outputs einables
the simultaneous application of several sets of weights, i.e.
the implementation
of
several filters having the same in-
put
*
The first step toward the generalization
of
the
olriginal
FS
method is the application of arbitrarily located res-
onators (see
(2)).
If the input signal consists of cli2mpo-
nents with frequencies corresponding to these poles then
similar overall behavior can be expected
as
in the DFT
case. The error caused by the frequency mismatch has
similar nature
as
described above. The only disadvan-
tage
is
the size
of
the dynamic range required during the
transient phase (first
N
samples) of the operation if the
lo-
cation of the resonator poles
is
highly asymmetrical. The
second step of the generalization
is
the introduction of
common poles into the transfer functions of
(1).
If
p,,
n
=
0,1,
...,
N
-
1,
denotes the poles to be implemented
then equation
(2)
must be replaced by
m=0,1,
...,
N-1.
(6)
In
[4]
a method is described to relate the poles
of
the over-
all system and those
of
the resonators. The application of
this relation simplifies the final structure and at the same
time helps
to
avoid zero-input limit cycles.
As
a
further
step of tlhe generalization the case of multiple resonator
poles should be mentioned. The corresponding structure
with resonator multiplicity of
2
(in every channel) is given
in Fig.
2.
The overall system order
is
2N.
The structure
is in comiplete correspondence with the Hermite interpo-
lation. Its parameters can be derived from the transfer
function of the form of
where
m=0,1,
...,
N-1,
(8)
34
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03
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05
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Fig.
3.
Magnitude responses with pole multiplicities
1,
2,
and
3
This derivation is based on the fact that
(7)
should equal
to one, and its derivative to zero
at
2
=
zm.
If the res-
onator poles equal to the Nth roots of unity then
(8)
will
have
a
much simpler form of
The development of the higher multiplicity cases is
straightforward, the only difference is that the higher or-
der derivatives are also to be considered. For
FIR
fil-
ter implementation this generalization of the FS meth-
ods means that not only the transfer function values
at
t
=
zm,
m
=
0,1,
...,
N
-
1
are taken into account but its
first, second, etc. derivatives, as well. The filter output
is composed as the linear combination of the output of
each resonator, i.e. each channel contributes to the filter
output with two complex weights.
If
filter poles are also
needed, i.e. the transfer functions of
(7)
should implement
common poles, then the
gm
and
qm
(m
=
0,1,
...,
N
-
1)
values can be calculated similarly as in the case of the
Lagrange structure.
The filter-banks with multiple resonator poles offer
a
wide variety of possible systems since the multiplicity of
these poles can be different from channel to channel. The
system of Fig.
2
is only a first attempt, to improve the
performance of sine wave measurements. If the resonator
pole frequencies coincide with that of the signal com-
ponents, then the direct utilization of outputs
X,(n),
m
=
0,1,
...,
N
-
1
will provide better noise immunity.
For the DFT case this fact is illustrated
in
Fig.
3
where
the magnitude responses with resonator pole multiplici-
ties of
1,
2,
and
3
are provided. If there
is
a
difference
Af
in the frequencies, then the sum of the two resonator
outputs of the very same channel will give better perfor-
mance, since this channel transfer function will equal to
1
at the resonator pole position and simultaneously its
derivative is forced to be zero. This latter means that
at
this point the derivative of the magnitude and the phase
is
zero and therefore the systematic error in the case of fre-
quency difference is less than with the Lagrange structure.
The actual magnitude and phase error can be calculated
using standard methods from equation
(7).
Iv. ADAPTIVE
FOURIER
ANALYSIS
WITH
MULTIPLE
RESONATORS
The structures investigated above can be efficiently uti-
lized if the center frequencies of the measuring channels
are “synchronized” to
a
certain extent to that of the sig-
nal components to be measured. If a reference signal is
available to directly control the resonator pole positions
then this synchronization can be solved. Typical examples
are the measurements of harmonically related sinusoids
where the fundamental frequency is time-varying but can
be measured with certain accuracy. Since this accuracy is
usually very limited due to the dynamics of the frequency
changes, the better performance near to the center fre-
quency locations can be of real importance.
If there is no available reference signal then the con-
cept
of
the Adaptive Fourier Analyzer
(AFA)
described
in
[6]
can be used and generalized. This analyzer is
a
very efficient adaptive filter-bank, which tries to lock to
the fundamental frequency of the signal and suppress all
the harmonically related component except one as it
is
342

the case of the time-recursive DFT. The system a,dapts
not only its parameters but also its struclme. This latter
is
performed simply by accommodating as many channels
into the common loop as many harmonically related fre-
quency positions can be located to the interval from zero
to the sampling frequency.
Obviously for lower funda-
mental frequencies the system order
N(n)
will increase
and for higher ones decrease. The complete adaptation
algorithm can be directly utilized also for the case
cif
this
Hermite interpolation structure. The only problem to be
solved is the selection of the optimal channel characteris-
tics for the fundamental frequency
t,o
provide fast conver-
gence. Some early simulations show that the performance
of the system in Fig.
2
seems to be very similar to that of
the Lagrange structure if both solutions apply the same
number of channels. This latter means that the consider-
able increase of the system order affects the convergence
speed only to
a
small extent. The adaptation procedure
is based on the output signal Xl(n). The performance
with combined output, i.e. with the combination
of
the
two resonator outputs of the fundamental channel is under
investigation.
V.
CONCLUSIONS
In this paper the possible generalizations of thli? fre-
quency sampling
(FS)
method were investigated.
It
was
shown that if signals consisting of sinusoidal components
are to be measured the structure based on the classical
Hermite interpolation technique provides better per for-
mance. The prize to be paid for this improvement
is
the increase
of
the filter-bank order, however, especially
for the case of uniform resonator pole distribution this fact
can be tolerated. The suggested system can also be used
to implement arbitrary
FIR
and IIR filters similarly
to
the
structures described
[4]
and
[7],
but additionally, with the
generalization of the frequency sampling method, higher
approximation power can be concentrated to certain fre-
quency locations. The Adaptive Fourier Analyzer algo-
rithm
[6]
can be used to adapt the fundamental frequency
of this new system, as well.
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343