Provided by the author(s) and University College Dublin Library in accordance with publisher
policies. Please cite the published version when available.
Title Polyphase FIR Networks Based on Frequency Sampling for Multirate DSP Applications
Authors(s) Cruz-Roldán, Fernando; Osés del Campo, José David; Godino-Llorente, Juan Ignacio;
Boquete-Vázquez, Luciano; Bleakley, Chris J.
Publication date 2010-04
Publication information Circuits, Systems, and Signal Processing, 29 (2): 169-181
Publisher Springer
Item record/more information http://hdl.handle.net/10197/7111
Publisher's statement The final publication is available at www.springerlink.com.
Publisher's version (DOI) 10.1007/s00034-009-9140-5
Downloaded 2022-08-09T22:17:22Z
The UCD community has made this article openly available. Please share how this access
benefits you. Your story matters! (@ucd_oa)
© Some rights reserved. For more information, please see the item record link above.
Polyphase FIR networks based on frequenc y
sampling for multirate DSP applications
Fernando Cruz–Rold
´
an, Jos
´
e David Os
´
es del Campo,
Juan Ignacio Godino–Llorente, Luciano Boquete–V
´
azquez, and C. J. Bleakley,
Abstract
It is well known that the frequency sampling appr oach to the design of Finite Impulse Response
digital filters allows recursive implementations which are computationally efficient when most of the
frequency samples are integers, powers of 2 or nulls. The design and implementation of decimation
(or interpolation) filters using this approach is studied herein. Firstly, a procedure is described which
optimizes the tradeoff between the stopband energy and the deviation of the passband from the ideal filter.
The search space is limited to a small number of samples (in the transition band), imposing the condition
that the resulting filter have a large number of zeroes in the stopband. Secondly, three different structures
to implement the decimation (or interpolation) filter are proposed. The implementation complexity, i.e.
the number of multiplications and additions per input sample, are derived fo r each structure. The results
shown that, without taking into account finite word-length effects, the most efficient implementation
depends on the filter length to decimation (or interpolation) ratio.
This work was partially supported by the Spanish Ministry of Education and Science under grant PR2007-0218, and by
Comunidad Aut´onoma de Madrid and Universidad de Alcal´a through projects CCG07-UAH/TIC-1740, CCG08-UAH/TIC-3941,
and CCG08-UAH/TIC-4054.
F. Cruz-Rold´an is with the Department of Teor´ıa de la Se˜nal y Comunicaciones, Escuela Polit´ecnica Superior de la Universidad
de Alcal´a, 28871 Alcal´a de Henares (Madrid), SPAIN (phone: + 34 91 885 66 93; fax : + 34 91 885 66 99; e-mail:
fernando.cruz@uah.es).
J. D. Os´es del Campo and J. I. Godino–Llorente are with the Department of Ingenier´ıa de Circuitos y Sistemas, Escuela
Universitaria de Ingenier´ıa T´ecnica de Telecomunicaci´on, Universidad Polit´ecnica de Madrid, 28031 Madrid, SPAIN (phone: +
34 91 336 7831; fax: + 34 91 336 78 29; e-mail: igodino, doses@ics.upm.es).
L. Boquete-V´azquez is with the Department of Electr´onica, Escuela Polit ´ecnica Superior de la Universidad de Alcal´a, 28871
Alcal´a de Henares (Madrid), SPAIN (phone: + 34 91 885 65 72; fax: + 34 91 885 65 91; e-mail: luciano.boquete@uah.es).
C. J. Bleakley is with the School of Computer Science and Informatics, University College Dublin, Belfield, Dublin 4,
IRELAND (phone: +353 1 716 2915; fax: +353 1 269 7262; e-mail: chris.bleakley@ucd.ie).
CIRCUITS, SYSTEMS, AND SIGNAL PROCESSING 1
Index Terms
Decimation and interpolation filtering, multirate signal processing, filtering theory, frequency sam-
pling technique.
CIRCUITS, SYSTEMS, AND SIGNAL PROCESSING 2
Polyphase FIR networks based on frequenc y
sampling for multirate DSP applications
I. INTRODUCTION
Decimation and interpolation filters (Fig. 1) are used in a wide range of applications, such
as data compression, speech enhancement, multi-carrier data transmission, wireless transceivers,
digital receivers, software radio receivers, or image size con version [1]–[9]. In Fig. 1, P (z) is
the system function, which is usually a linear-phase low-pass filter, with a cutoff frequency of
ω
s
= π/M to prevent downsampler-induced aliasing or to remove images after the upsampl i ng
process.
One problem these structures have is related to the M value of the decimation or interpolation
factor. In some practical systems, such as in decimation schemes for Σ∆ A/D converters or in
Software Defined Radio (SDR) receivers, interpolation/decimation ratios can have large values.
In these cases, most decimation filter architectures are designed in multistage subfilters, each
one obtaining a smaller decimation factor. At the end of the decimation chain, a very selective
low-pass Finite Impulse Response (FIR) filter is generally obtained. Besides providing a hi gh
value for M, it is also well known that decimation/interpolation filters operating at high rates
must be computationally very efficient. This means that high order filters with good selectivity
and di s crimination are required, and ef ficient structures to implement them as well.
To address these issues, this paper deals with the design and implementation of these filters
using the frequency sampling approach [10], [11]. In the proposed technique, the coefficients are
obtained by means of a procedure which onl y consists of optimizing the values of the samples in
the transition band. In this sense, FIR filters with a high length (number of coefficients) can be
designed, by opt i mizing only a small number of samples. In addition, it is well known that the
frequency sampling approach is a method that allows recursive implementations of FIR filters.
In general, these implementations greatly reduce the number of arithmetic operations in digital
filters, especially when most samples are either integers (or powers of 2) or null. Therefore, these
structures are beneficial when a low number of frequency samples are nonzero and the length of
the prototype filter is large. In this case, efficient implementations from a computational point
CIRCUITS, SYSTEMS, AND SIGNAL PROCESSING 3
P z( )
M
M
P z( )
x n[ ]
x n[ ]
y n[ ]
r n[ ]
(a)
(b)
Fig. 1. (a) Decimation and (b) interpolation filters.
of view can be obtained. We address this possibi l i t y herein, studying recursive implementations
for the polyphase components of the derived filters. For the sake of brevity, we mainly focus
our attention on decimation filters, but the results could be extended to interpolation filters.
The rest of this letter is organized as follows. In Section II, we describe a procedure to obtain
linear-phase decimation or interpolation filters. The proposed technique is based on a frequency
sampling approach which can provide linear-phase filters which also have a very large number of
transmission zeros. Section III deals with the implementation structures. First, the relationship
between the frequency samples and the M polyphase components is obtained. Certain cases
of interest are considered, and a new expression for the pol yphase components that allows the
system to be implemented with real coefficients is developed. The implementation complexity is
studied based on the number of multiplications per input sample (MPIS) and additions per input
sample (APIS). The results are compared to conventional polyphase implementation. In Section
IV, several examples are included to illustrate the benefits of the proposed technique and to
compare the computational cost of different structures. Finally, we summarize our conclusions.
II. FREQUENCY SAMPLING TECHNIQUE
The frequency sampling approach is a well-known di screte design technique for obtaining FIR
filters [10], [11]. Basically, this technique consists of finding the impulse response coefficients
p [n] from a desired frequency response specified at a set of equally spaced frequencies; i.e.,
given the frequency samples P [k], the impulse response coefficients are obtained as
p [n] =
1
N
N−1
!
k=0
P [k] · e
j
2π
N
k·n
, n = 0, 1 · · · , N − 1. (1)
Expressing the above equation in a matrix form, we get
p
T
=
1
N
W
−1
N
P, (2)
