A farrow-structure-based multi-mode transmultiplexer

Abstract

This paper introduces a multi-mode transmultiplexer (TMUX) consisting of Farrow-based variable integer sampling rate conversion (SRC) blocks. The polyphase components of general interpolation/decimation filters are realized by the Farrow structure making it possible to achieve different linear-phase finite-length impulse response (FIR) lowpass filters at the cost of a fixed set of subfilters and adjustable fractional delay values. Simultaneous design of the subfilters, to achieve overall approximately Nyquist (Mth-band) filters, are treated in this paper. By means of an example, it is shown that the subfilters can be designed so that for any desired range of integer SRC ratios, the TMUX can approximate perfect recovery as close as desired.

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A Farrow-Structure-Based Multi-Mode Transmultiplexer
Amir Eghbali, H˚ akan Johansson, and Per L¨ owenborg
Division of Electronics Systems, Department of Electrical Engineering
Link¨ oping University, SE-581 83, SWEDEN
Email: {amire, hakanj, perl}@isy.liu.se
Abstract—This paper introduces a multi-mode transmultiplexer
(TMUX) consisting of Farrow-based variable integer sampling rate
conversion (SRC) blocks. The polyphase components of general inter-
polation/decimation filters are realized by the Farrow structure making
it possible to achieve different linear-phase finite-length impulse response
(FIR) lowpass filters at the cost of a fixed set of subfilters and adjustable
fractional delay values. Simultaneous design of the subfilters, to achieve
overall approximately Nyquist (Mth-band) filters, are treated in this
paper. By means of an example, it is shown that the subfilters can be
designed so that for any desired range of integer SRC ratios, the TMUX
can approximate perfect recovery as close as desired.
Index Terms—Multi-mode communications, transmultiplexers, sam-
pling rate conversion.
I. INTRODUCTION
One of the main aims in communications engineering is to con-
structing flexible radio systems (e.g., software defined radios) so
that services among different telecommunications standards can be
handled [1]. As the number of communications standards (or modes)
increases, the requirements on flexibility and cost-efficiency of these
systems increase as well. Consequently, it is vital to develop new
low-cost multi-mode terminals. Transmultiplexers (TMUXs) allow
different users to share a common channel and hence, constitute
one of the main building blocks in communications systems [2].
The importance of TMUXs gets pronounced by considering the fact
that well known multiple access schemes such as code division
multiple access (CDMA), time division multiple access (TDMA),
and frequency division multiple access (FDMA) are special cases of
a general TMUX theory [3].
Multi-mode communications require multi-mode TMUXs that sup-
port different bandwidths for various telecommunications standards.
As an example, the bit rate of the wireless standards IS-54/136,
GSM, and IS-95 are 48.6, 271, and 1228.8 Kbps, respectively [4].
Furthermore, in each of these standards, respectively, 3, 8, and 798
users share one channel where the channel spacing is 30, 200, and
1250 KHz. In conclusion, to support multi-mode communications,
there is a need for a system which can allow various users with
different bit rates to share a common channel.
TMUXs are composed of a synthesis filter bank (SFB) followed
by an analysis FB (AFB) with both the AFB and SFB being a
parallel connection of a number of branches [2]. Each branch is
realized by digital bandpass interpolators/decimators where in the
case of a uniform TMUX, the bandwidths and center frequencies of
the bandpass interpolators/decimators are fixed. However, multi-mode
TMUXs require interpolators/decimators with variable bandwidths
and center frequencies.
A. Contribution of the Paper
In this paper, we introduce a multi-mode TMUX which consists
of Farrow-based variable integer sampling rate conversion (SRC)
and variable frequency shifters. To be more specific, each integer
SRC block is designed using the Farrow structure [5] resulting in
a fixed set of subfilters. To perform any integer SRC, there is only
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Fig. 1.Farrow structure with fixed subfilters and fractional delay µ.
a need to modify the fractional delay values required by the Farrow
structure and, consequently, it is possible to use one set of subfilters to
perform any integer SRC. The Farrow structure is generally designed
to approximate an allpass transfer function in the frequency range of
interest [6]. In this paper, the Farrow structure realizes the polyphase
components of general interpolation/decimation filters. In addition,
it is designed such that the cascade of interpolation and decimation
filters in the SFB and AFB, approximates a Nyquist (Mth-band)
filter. This method of designing the Farrow structure has not been
treated before. Using the design method in this paper, Nyquist filters
with arbitrarily small approximation errors and different passband
edges can be achieved. Therefore, the TMUX can approximate perfect
recovery (PR) [7] as close as desired via proper design of the
subfilters. Previous design techniques [8], [9] cannot achieve this as
they have no constraints on a band near π which is considered as the
don’t-care band for the Farrow structure.
In comparison with the TMUX in this paper, [8], [9] propose a
multi-mode TMUX where a cascade of the Farrow structure and a
lowpass filter is used in the SFB and AFB. The advantage of the
TMUX in this paper over that of [8] is the elimination of the lowpass
filter, which also results in a different way to design the subfilters
of the Farrow structure, with constraints in the whole frequency
range [−π,π]. However, by utilizing the Farrow structure, both these
approaches eliminate the need to design different sets of subfilters
which would be required for general non-uniform TMUX structures,
e.g., [10].
B. Paper Outline
Section II discusses the Farrow structure and how it can be used
to obtain SRC blocks. In Section III, the structure of the multi-
mode TMUX is introduced and the design and implementation of
its building blocks are considered. Then, the simultaneous design of
the subfilters is discussed and illustrated by an example. Section IV
deals with the functionality and performance of the TMUX which is
followed by concluding remarks in Section V.
II. FARROW STRUCTURE FOR SRC
As shown in Fig. 1, the Farrow structure is composed of
fixed linear-phase finite-length impulse response (FIR) subfilters
Sk(z),k = 0,1,...,L with either a symmetric (for k even) or anti-
symmetric (for k odd) impulse response. Furthermore, the subfilters
can have even or odd orders and in the case of odd order, all the
subfilters are general filters whereas for the even-order case, S0(z)

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Fig. 2.Multi-mode TMUX composed of variable integer SRC and adjustable frequency shifters.
reduces to a pure delay. The transfer function of the Farrow structure
is given by [6]
H(z) =
L
X
k=0
Sk(z)µk,|µ| ≤ 0.5
(1)
where µ is the fractional delay value. Assuming Tin (Tout) to be
the input (output) sampling period and considering even/odd order
subfilters, µ is defined as [11]
Even order
Odd order
:[nin+ µ(nin)]Tin = noutTout,
:[nin+ 0.5 + µ(nin)]Tin = noutTout
(2)
where nin (nout) is the input (output) sample index. If µ is constant
for all input samples, the Farrow structure generates a delayed (with
a delay of µ) version of the input signal. However, if µ changes for
every input sample, as in (2), the Farrow structure can perform SRC.
The subfilters in Fig. 1 can be designed such that the Farrow structure
1) approximates an allpass transfer function having a fractional
delay [6], or 2) can realize the polyphase components of general
interpolation/decimation filters (with the Nyquist filter [12] being a
special case). In the latter case, lowpass filters with different passband
edges can be obtained through a fixed set of subfilters and variable
multipliers [13].
III. PROPOSED MULTI-MODE TMUX
The discussion in Section II reveals that it is possible to use
the Farrow structure to obtain general lowpass filters at the cost of
a fixed set of subfilters and variable multipliers. In other words,
the Farrow structure can realize the polyphase components of a
general lowpass filter which means that a fixed set of subfilters can
be used to implement interpolators/decimators with different integer
SRC ratios. Hence, these general filters can be used to construct
a multi-mode TMUX as shown in Fig. 2. The TMUX consists of
upsampling/downsampling by Rp; lowpass interpolation/decimation
filters, i.e., Gp(z) for interpolation andˆGp(z) for decimation; and
adjustable frequency shifters, i.e., frequency shifts by ωp and ˆ ωp.
Assuming the sampling period at branch p of the TMUX to be Tp,
we have
T0
R0
R1
where Ty is the sampling period of y(n).
In the SFB, the TMUX generates the required bandwidths through
upsampling by Rp followed by a lowpass filter Gp(z). To place the
users in appropriate positions in the frequency spectrum, variable
frequency shifters are utilized. Finally, all users are summed to form
y(n) for transmission1. In the AFB, to recover a specific user signal,
the received signal ˆ y(n) is first frequency shifted such that the
=T1
= ... = Ty
(3)
1Like e.g., OFDM-based TMUXs, the output of the TMUX is complex.

−40
−20
0
20
ωT [rad]
(b)
|X(ejωT)| [dB]
(a)
0
0.25π
0.5π
0.75ππ
1.25π
1.5π
1.75π
2π

−40
−20
0
20
ωT [rad]
(c)
|X(ejωT)| [dB]
0
0.25π
0.5π
0.75ππ
1.25π
1.5π
1.75π
2π

−40
−20
0
20
ωT [rad]
|X(ejωT)| [dB]
0
0.25π
0.5π
0.75ππ
1.25π
1.5π
1.75π
2π
Fig. 3. TMUX outputs. (a) Integer interpolator. (b) and (c) Frequency shifters.
Rp= 5,ωp= 0.24π, ˆ ωp= 0.24π.
desired user signal can be processed in the baseband. Then, a lowpass
filterˆGp(z) followed by downsampling by Rp is used to obtain the
desired signal. Figure 3 illustrates the principle of the structure by
plotting the frequency spectrum at the output of the interpolator and
the frequency shifters with a Gaussian input. It is noted that the
procedure to compute ωp ensures that the user signals do not overlap
and hence, the TMUX is slightly redundant2. However, redundancy
is needed anyhow to achieve 1) a multi-mode TMUX with a fixed
set of subfilters and without the need to redesign them for each
new configuration of standards, and 2) high quality transmission in
communications systems [2].
A. Implementation of Gp(z) andˆGp(z)
General linear-phase FIR interpolation and decimation filters can
be realized using the Farrow structure [13]. To do so, each polyphase
branch is realized by a Farrow structure having a distinct fractional
delay value and, thus, integer SRC blocks can be implemented using
a fixed set of subfilters and variable multipliers. In other words, if
the SRC ratio is to be changed, there is only a need to change the
set of multipliers as they correspond to the set of fractional delays
required for SRC. Assuming that Gp(z) is used for SRC by Rp, its
polyphase representation can be written as [2]
Gp(z) =
Rp−1
X
m=0
z−mGp,m(zRp),
(4)
2Specifically, the values of ωp are chosen such that the cutoff frequencies
of the filters, in different branches, do not overlap. Details can be found in
[8] with a difference that the present paper does not assume any guard bands.

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Fig. 4.
utilizing the symmetry of µp,m.
Realization of polyphase components Gp,m(z) and Gp,Rp−m(z)
where Gp,m(z) denote the polyphase components of Gp(z). In order
to make Gp(z) a general interpolation/decimation filter of order
N, it should approximate z−N/2in the passband and zero in the
stopband. Consequently, in the passband, each term z−mGp,m(zRp)
should have a delay of z−N/2which means that Gp,m(z) should
approximate an allpass transfer function with a fractional delay of
(N
To conclude, a general interpolation/decimation filter of order N
can be designed by choosing its zeroth polyphase component, i.e.,
Gp,0(z) to be a Type I linear-phase FIR filter of order N0 =
utilizing the Farrow structure to realize the polyphase components
Gp,m(z),m = 1,2,...,Rp− 1 so that they have an odd order3of
N1 =
2− m)/Rp [13].
N
Rpand
N
Rp− 1 as
Gp,m(z) =
L
X
k=0
Sk(z)µk
p,m, µp,m =−m
Rp
+1
2.
(5)
By choosing the values of µp,m as in (5), they possess antisymmetry
according to µp,m = −µp,Rp−m. Considering the antisymmetry of
µp,m, and as shown in Fig. 4, the polyphase components Gp,m(z)
and Gp,Rp−m(z) can be written as
Gp,m(z)=Φp,m(z) + Ψp,m(z),
Φp,m(z) − Ψp,m(z),
where Φp,m(z) and Ψp,m(z) are shown in Fig. 5 and defined as
Gp,Rp−m(z)=
(6)
Φp,m(z)=
?L
X
?L+1
X
2?
k=0
Gp,2k(z)µ2k
p,m,
Ψp,m(z)=
2
?
k=1
Gp,2k−1(z)µ2k−1
p,m .
(7)
Hence, interpolation by Rp can be performed as shown in Fig. 6
which consists of a fixed set of subfilters, viz. the zeroth polyphase
component Gp,0(z) and the Farrow subfilters Sk(z); multipliers
due to the fractional delays µp,m; and the output commutator [2].
The structure for the decimator can be derived by transposing the
interpolator structure of Fig. 6.
B. Design of Gp(z) andˆGp(z)
In this section, we will discuss the design of the interpola-
tion/decimation filters Gp(z) andˆGp(z) used in the TMUX of Fig.
2. Assuming one branch of the TMUX between users xp(np) and
ˆ xp(np), to approximate PR as close as desired, the filter Gp(z)ˆGp(z)
should approximate an Rpth-band filter as close as desired. As the
TMUX is redundant, the level of cross talk is determined by the
stopband attenuation of the interpolation/decimation filters. In each
branch of the TMUX, the filter Gp(z)ˆGp(z) is sandwiched between
upsamplers and downsamplers by Rp. This means that the overall
transfer function, for each branch, is equal to the 0th polyphase
3With proper modifications, even-order filters can also be designed [13].
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(a) Realization of Ψp,m(z). Q = 2?L+1
2
? − 1.
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(b) Realization of Φp,m(z). P = 2?L
2?.
Fig. 5. Realizations of Ψp,m(z) and Φp,m(z) according to (7).
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Fig. 6.Interpolator with fixed subfilters, multipliers, and commutator.
component of Gp(z)ˆGp(z). Consequently, the filters Gp(z) and
ˆGp(z) should be designed such that
• They have sufficiently small ripples in their stop bands to control
the cross talk.
• The 0th polyphase component of Gp(z)ˆGp(z) approximates an
allpass transfer function in the whole frequency band.
In other words, we should meet4
|[Gp(ejωT)ˆGp(ejωT)e
jNωT
2
]0th− 1| ≤ δ1,
|Gp(ejωT)| ≤ δ2,
|ˆGp(ejωT)| ≤ δ3,
ωT
∈ [0,π],
∈ [ωsT,π],
∈ [ωsT,π](8)
ωT
ωT
where the passband and stopband edges are given by
ωcT =π(1 − ρ)
Rp
, ωsT =π(1 + ρ)
Rp
.
(9)
In addition, δ2 and δ3 are the stopband ripples (to control cross
talk) with ρ being the roll-off factor of the Rpth-band filter.
Furthermore, δ1 is the deviation of the 0th polyphase component
[Gp(ejωT)ˆGp(ejωT)]0thfrom an allpass transfer function, and there-
fore it controls the distortion.
To use the fixed set of subfilters (as mentioned in the previous
subsection) in the TMUX of Fig. 2, it is necessary that the subfilters
are designed such that (8) is satisfied over the range of Rp values of
interest. Assuming Gp(z) =ˆGp(z), due to the fact that there is only
one fixed set of subfilters, the 0th polyphase component of the filter
Gp(z)ˆGp(z) can be written as
Fp(ejωT)=[Gp(ejωT)ˆGp(ejωT)e
Rp−1
X
jNωT
2
]0th=
n=0
[Gp(e
j(ωT−2nπ
Rp))e
j(ωT−2nπ
Rp)N
2]2. (10)
4For convenience in design, the term e
and through (10), the center tap belongs to the 0th polyphase component.
jNωT
2
constructs a non-causal filter

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π
0.998
0.999
1
1.001
1.002
ωT [rad]
Fp(ωT)
0
0.2π
0.4π
0.6π
0.8π

π
−80
−60
−40
−20
0
ωT [rad]
Gp(ejωTl) [dB]


0
0.2π
0.4π
0.6π
0.8π
Rp=4
Rp=5
Rp=6
Rp=7
Rp=15
Fig. 7. Approximate Rp-th band filters and their 0th polyphase components.

−40
−20
0
ωT [rad]
(b)
|X(ejωT)| [dB]
(a)
0
0.25π
0.5π
0.75ππ
1.25π
1.5π
1.75π
2π
X0
X1
X2
X3
X4
2468 10 1214
x 10
−3
−70
−60
−50
−40
δ1=δ2=δ3
EVM [dB]
Fig. 8.Functionality and performance of the TMUX in Fig. 2.
In order to use a fixed set of subfilters to perform any desired integer
SRC, a simultaneous optimization to minimize δi,i = 1,2,3 in (8)
over Rp values of interest needs to be performed. Figure 7 shows the
characteristics of the approximately Rpth-band filters and their cor-
responding 0th polyphase components resulting from a simultaneous
optimization for Rp = {7,5,15,6,4} and δ1 = δ2 = δ3 = 0.0024.
In this design example, the values for ρ, L, N1, and N0 were 0.2, 5,
17, and 18 respectively, and furthermore, the design problem has been
formulated in the minimax sense. However, other design methods
such as least-squares to minimize the energy can alternatively be
used, mutatis mutandis.
IV. TMUX FUNCTIONALITY AND PERFORMANCE
Toverifythefunctionality
consisting
oftheproposed
different
{7,5,15,6,4} resulting in
TMUX,
user
a
multi-mode
{X0,X1,X2,X3,X4} with Rp
ωp = {0.1714π,0.5829π,0.9029π,1.1829π,1.6829π} is assumed5.
Figure 8(a) shows the spectrum of the SFB outputs at different
branches of the TMUX in Fig. 2.
To illustrate the performance of the proposed TMUX with respect
to the values of δi,i = 1,2,3 in (8), the error vector magnitude
(EVM), a metric of transmitter signal quality, is used [8]. EVM
provides a statistical estimate of the error vector normalized by the
setupoffivesignals
=
5For illustration purposes, the values of Rp are chosen such that 99% of
the frequency range [0,2π] is occupied by the spectrum of the users.
magnitude of the ideal signal and is defined as
EV Mrms =
sPNs−1
k=0
PNs−1
|s(k) − sref(k)|2
|sref(k)|2
k=0
,
(11)
where s(k) and sref(k) represent the length-Ns measured and ideal
complex sequences, respectively. Using the filters depicted in Fig. 7
and the values of Rpmentioned above, the mean value for EV Mrms
and EV MdB in a 16-QAM signal are 0.0015 and −56.4384,
respectively. The trend of EVM for different filter designs is shown
in Fig. 8(b) and it can be seen that the error in approximating PR
can be made as small as possible6by reducing δi,i = 1,2,3.
V. CONCLUSION
In this paper, a multi-mode TMUX consisting of Farrow-based
variable integer SRC and variable frequency shifters was introduced.
Using the Farrow structure to realize the polyphase components of
general interpolation/decimation filters, it is possible to perform any
integer SRC by the use of a fixed set of subfilters. The Farrow struc-
ture is designed such that the cascade of interpolation and decimation
filters approximates a Nyquist filter. By means of examples, the
functionality and performance of the proposed TMUX is illustrated.
It is possible to extend the idea such that both rational and integer
SRC ratios can be handled through one set of subfilters resulting in
a TMUX that supports arbitrary SRC ratios. This will be treated in
another paper.
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6The least-squares formulation results in smaller EVM values than the
minimax approach. However, it is the application that defines the approach to
be used. Further discussion on this is out of the scope of this paper.