TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 11, Novembe
r
2014, pp. 78
2
4
~ 783
1
DOI: 10.115
9
1
/telkomni
ka.
v
12i11.60
70
7824
Re
cei
v
ed Ap
ril 7, 2014; Re
vised July
1
8
, 2014; Accept
ed Augu
st 5, 2014
Direct Radio Frequency Sampling System on Software-
defined Radio
Luo Jun
y
i*,
Lei Lin, Che
n
Er
y
a
ng, Zhao Yongxin
Schoo
l of elect
r
onics a
nd inf
o
rmation e
ngi
ne
erin
g,
Chen
gd
u Univ
ersit
y
, C
hen
gd
u, 610
10
6, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: luoj
un
yi
20
09
@qq.com
A
b
st
r
a
ct
A traditio
nal
as
sumptio
n
u
nde
rlying
most dat
a conv
erters is
that the si
gna
l
shou
ld
be sa
mp
le
d a
t
a rate exc
e
e
d
i
ng tw
ice the
hi
ghest freq
ue
nc
y. In this pa
per
, w
e
emp
l
oy a
meth
od for
low
-rate sa
mp
lin
g
of
mu
lti- b
a
n
d
s
i
g
nals
via
a
pply
i
ng
peri
o
d
i
c n
o
nun
ifor
m
sa
mp
ling
i
n
s
h
ift-inv
a
ria
n
t spac
es
gen
erate
d
by
m
ke
rn
el
s wi
th
p
e
r
i
o
d
T. So
, th
e
sam
p
li
ng
a
n
d
re
co
n
s
tru
c
ti
on
o
f
si
g
n
a
l
s we
re
tra
n
s
fo
rm
ed
i
n
to
m
a
tri
x
and
vector op
eratio
ns, the gen
era
l
i
z
e
d
invers
e c
an be
use
to fi
nd the a
n
sw
er and a
n
inter
p
olator is
used
to
insur
e
th
at co
mp
lete
reco
nst
r
uction
w
ill
be
ach
i
eve
d
. Fin
a
lly, w
e
v
a
l
i
da
te the
metho
d
in
MATLAB; t
h
e
concl
u
sio
n
of simulati
on sh
ow
s the frame-w
o
rk presente
d
h
e
re is feasi
b
le.
Ke
y
w
ords
:
gen
eral
i
z
e
d
in
verse rec
onstr
uction, p
e
rio
d
i
c
non
unifor
m
sampli
ng, sh
ift-invari
ant sp
a
c
es,
mu
lti- ba
nd sig
nals, inter
pol
at
or
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
One
g
oal i
n
d
e
sig
n
ing
a
so
ftware
define
d
ra
dio
(SDR) re
ceiver is to
move the
an
alog-to
-
digital
conve
r
ter (A
DC) a
s
clo
s
e
as po
ssible
to th
e a
n
tenna
[1]. T
o
a
c
hieve
thi
s
g
oal,
one
can
manag
e to u
s
e a wi
deb
an
d high-sp
eed
ADC to co
n
v
ert the RF signal
s to digital sign
als. Wi
th
the develop
m
ent of wirele
ss technol
ogy, this
en
able
s
the modul
atio
n of
narro
w-b
and
signal
s by
high ca
rri
er freque
nci
e
s. T
o
demod
ulate
the desir
ed sign
als, the required samp
ling rate for the
ADC
co
uld of
ten be too
hig
h
to be
attain
ed if the
Nyq
u
ist samplin
g
theorem is to
be
satisfied
[2].
The u
n
iform
band
pa
ss
sa
mpling m
e
tho
d
ha
s be
en p
r
opo
se
d to fig
u
re
out the p
r
oblem [3
-5], and
this is a prom
ising
way for multi-ba
nd ra
dio
com
m
uni
cation. The u
n
iform ba
ndp
ass sam
p
ling
is
the intentiona
l aliasin
g
of the
informatio
n band
width
of the sign
al
[
6
, 7]. The sa
mpling fre
que
ncy
requi
rem
ent is no long
er b
a
se
d on the frequ
en
cy of
the RF carrier, but rather o
n
the informa
t
ion
band
width
of the sig
nal.
Thus, th
e re
sulting
pr
o
c
e
ssi
ng rate
can be
si
g
n
ificantly redu
ced.
However, the uniform
sam
p
ling still
suffers f
r
om m
any constraint
s
such probl
em
of timing jitter in
A/D conversi
on p
r
o
c
e
s
s [8]. For
non
u
n
iform
sam
p
l
e
s, the
r
e
are
both ite
r
ative metho
d
s a
nd
noniterative
method
s to
recreate
the
signal
s; the
s
e
metho
d
s pre
s
up
po
se
exa
c
t kno
w
led
g
e
of
the sam
p
le l
o
catio
n
s. Thi
s
is n
o
t alwa
ys the ca
se,
and the
r
e m
a
y occur
situ
ations
whe
r
e
the
locatio
n
data
is unavail
able
or partially a
v
ailable [9].
A signal
cla
s
s that plays a
n
importa
nt role in
sampli
ng theo
ry is signal
s in shift-invari
ant
(SI) s
p
ac
es
[10]. A s
a
mple in s
h
ift-invariant s
p
aces
wa
s pro
p
o
s
e
d
to overco
m
e
these p
r
obl
ems.
The recon
s
truction
of sa
mpled
sig
nal
s is a
c
hiev
e
d
by formin
g li
near combin
ations
of a
set o
f
recon
s
tru
c
tio
n
functio
n
th
at spa
n
a
subspa
ce; su
ch fun
c
tion
s can
be exp
r
esse
d a
s
li
near
combi
nation
s
of shifts of
a set
of ge
n
e
rato
rs with
perio
d T. T
h
i
s
mo
del e
n
compa
s
ses m
any
sign
als
used
in co
mmuni
cation an
d si
g
nal p
r
o
c
essin
g
. Any sign
al
x
(
t
) in a SI
space ge
nerated
by
m
functions shifted wit
h
pe
riod
T
can be
pe
rfect
l
y recovere
d
from
m
sam
p
ling sequ
en
ces,
obtaine
d by filtering
x
(
t
) wit
h
a ban
k of
m
filters and un
iformly sampli
ng their outp
u
t
s at times
nT
.
This p
ape
r is orga
nized a
s
follo
wed. S
e
ction II sets
up the
sampli
ng mod
e
l. In Section
III, we use
generali
zed i
n
verse to
recover sampl
ed
signals. I
n
sectio
n IV, we analyze the
recon
s
tru
c
tio
n
error. Finall
y
, section V shows sim
u
lati
on re
sults.
2. Proposed
Scheme
The archite
c
t
u
re of pa
rallel
samplin
g system is sh
own
in Figure 1.
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TELKOM
NIKA
ISSN:
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Dire
ct Ra
dio
Freq
uen
cy Sam
p
ling Syst
em
on Software-define
d
Ra
dio (Lu
o
Ju
nyi)
7825
Figure 1. The
Mode of the Periodi
c Non
uniform Sam
p
ling
The non
unifo
rm sam
p
ling
process co
nverts a con
t
inuou
s anal
ogue si
gnal
x
(
t
)
∈
L
2
-
spa
c
e i
n
to its discrete represe
n
tation, th
e arch
ite
c
ture
of peri
odi
c n
onunifo
rm
sa
mpling
syste
m
is sh
own in Figure 1.
Let
a
i
(
t
) a
s
on
e of
s
nonu
niform sample
seque
nces,
()
()
(
0
1
)
n
i
tT
n
i
at
i
s
(
1
)
Whe
r
e, T is the sam
p
ling
perio
d,
is se
q
uen
ce sepa
ra
tion.
One of
s
sampled func
tions
,
()
(
)
(
)
n
n
i
y
xn
T
i
t
n
T
i
t
(2)
Whe
r
e,
01
is
.
And the co
rre
spo
ndin
g
sp
e
c
tra i
s
given by:
2/
1
()
(
2
)
/
jn
i
T
i
n
T
YX
n
T
e
(
3
)
In orde
r to reco
nstruct
x
(
t
) from thes
e s
a
mples
y
[
n
](
y
[
n
]=
[
y
0
[
n
],
y
1
[
n
],…,
y
s-
1
[
n
]]), it is
assume
d tha
t
x
(
t
) lies in
a subspace
V
(
φ
)of
L
2
. In this pape
r, we defin
e that the
V
(
φ
) are
gene
rated by
m
space fun
c
tions
φ
(
t
).
1
2
0
()
{
[
]
(
)
:
[
]
}
p
m
pp
pn
z
Vt
n
T
r
n
L
rn
We can repre
s
ent any
x
(
t
)
∈
V
(
φ
) as
follow:
1
0
()
[
]
(
)
m
pp
pn
z
tn
T
xt
rn
(
4
)
The only re
striction o
n
the c
hoi
ce of the function trai
n{
φ
p
(
t
)}
is for g
u
a
rante
e
ing a
uniqu
e
stable
re
pre
s
entation of a
n
y
signal i
n
V
(
φ
)
b
y
s
e
qu
en
c
e
{
r
p
[
n
]},
so
the ge
nerato
r
s
φ
(
t
) mu
st
form
a Rie
sz b
a
si
s
of
L
2
. In othe
r wo
rd
s, there
exist two con
s
tants
0
and
,
such t
hat
:
2
1
22
22
0
2
[]
[]
(
)
[]
rr
m
pp
pn
z
nn
t
n
T
n
r
(
5
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 78
24 – 783
1
7826
Whe
r
e,
1
0
2
2
2
[]
[]
r
m
p
pn
z
nn
r
,
2
is
L
2
norm.
Propo
sition
: if and only if
()
I
IW
, the gen
erato
r
φ
p
(
t
-
nT
) f
o
rm
a Rie
sz b
a
si
s
.
Whe
r
e,
I
is the identity matrix:
1,
1
1
,
,1
,
()
m
mm
m
W
ww
ww
,
*
()
()
1
2/
2/
ab
ab
nz
wn
T
n
T
T
Her
e
,
ψ
(
ω
) is the Fourie
r transfo
rm of
φ
(
t
)
Proof:
(5)
can
be rewritten
as follo
w:
2
22
*
22
[]
[
]
(
)
[]
rr
r
n
z
nn
t
n
T
n
dt
(
6
)
From the the
o
ry of Parseval:
22
**
1
[]
(
)
(
)
(
)
2
R
r
nn
zz
n
t
nT
dt
dt
(
7
)
Whe
r
e,
()
R
is the discrete
-time
Fourie
r tran
sform of
[]
r
n
, and
()
R
is 2
π
-peri
odi
c.
Then (7)
can
be re
written a
s
:
2
*
2
**
0
2
*
0
1
()
(
2
/
)
2
1
(
)
(2
/
)
(2
/
)
(
)
2
()
()
()
2
R
RR
RR
n
n
z
z
nT
d
t
nT
nT
d
T
Wd
(8)
We can have
(9) from Parseval:
2
2
*
2
0
1
[
]
()
()
2
RR
r
nd
(
9
)
It is
eas
y
to know
()
W
is
a p
o
sitive self-adjoi
nt whi
c
h
ha
s
real
non
neg
a
t
ive eigenval
ues.
Let
is the minim
a
l eigenvalu
e
s
and
is the maximal eige
nvalue
s. We
can h
a
ve:
**
*
(
)
(
)
()
()
(
)
()
(
)
R
R
RR
RR
BI
AI
W
The co
ncl
u
si
on ca
n be obt
ained that
φ
p
(
t
-
nT
) form a Rie
sz b
a
si
s if and only if
()
I
IW
.
The a
bove
-
m
entione
d sub
s
pa
ce
V
(
φ
) i
s
a
single
spa
c
e, the
mo
re
intere
sting
aspect
we
are con
s
ide
r
i
ng is that
x
(
t
) lies in a uni
on
of subspa
ce
s
()
p
V
(0
≤
p
≤
m
-1).
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Dire
ct Ra
dio
Freq
uen
cy Sam
p
ling Syst
em
on Software-define
d
Ra
dio (Lu
o
Ju
nyi)
7827
()
(
)
p
V
xt
In Fourie
r do
main, (4)
can
be rep
r
e
s
e
n
ted as follo
w:
1
0
()
()
()
m
pp
p
XR
(
1
0
)
Whe
r
e,
R
P
(
ω
) is the
discrete-time F
o
u
r
ier tran
sform
of
r
p
[
n
],
ψ
p
(
ω
) i
s
the
Fo
urie
r tra
n
sfo
r
m of
φ
p
(
t
).
We can obtai
n the DTFT o
f
the i-th chan
nel sam
p
le
s
[]
i
y
n
by (3) an
d (5
):
1
2/
0
1
2/
0
()
(
2
/
)
(
2
/
)
()
(
)
1
1
2/
=
m
j
ni
T
pp
np
m
jn
i
T
pp
pn
i
Y
R
nT
nT
e
T
Rn
T
e
T
(11)
Whe
r
e, the fa
ct that the
()
p
R
is
2
-pe
r
iodi
c.
An appropri
a
te matrix rep
r
ese
n
t of (11) i
s
given by:\
()
(
)
()
YH
R
(12)
Whe
r
e,
01
1
(
)
(
(
),
(
)
,
(
))
'
s
YY
YY
01
1
()
(
(
)
,
()
,
(
)
)
'
m
RR
R
R
0,
0
0
,
1
0
,
1
1,
0
1
,
1
1
,
1
()
..
.
m
ss
s
m
H
hh
h
hh
h
2/
,
1
()
(
2
/
)
jn
i
T
ip
p
n
T
hn
T
e
Our aim i
s
to obtain value
s
of
R
(
ω
). The
method of re
constructio
n
is to solve Equation (10
)
.
3. Recon
s
tr
u
c
tion Mode
The a
pproa
ch in thi
s
pa
p
e
r i
s
to recovery sample
d
sign
als i
n
two step
s. Fi
rst
,
we u
s
e
the ge
neralized inve
rse
()
H
to find
r
i
[
n
] (0
≤
i
≤
m
-1
); second, a
n
inte
rpolator is e
m
ployed t
o
achi
eve the
compl
e
te
re
constructio
n
o
f
sampl
ed
si
gnal. Th
e fu
ndame
n
tal
stage
s for the
recovery of sampled
sign
a
l
s are
sho
w
n i
n
Figure 2.
Figure 2. The
Block of Reconstructio
n
Bank
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 78
24 – 783
1
7828
We defin
e the function a
s
follow:
1
0
2
ˆ
[]
[]
s
ii
in
Z
J
yn
yn
(
1
3
)
Whe
r
e,
ˆ
[]
i
yn
is
coefficient th
a
t
is
obtain
e
d
via
sa
mpli
n
g
the
re
co
nstructed
conti
nuou
s tim
e
sign
al.
Again by Parseval
we hav
e:
1
2
0
ˆ
(
)
(
)
ˆˆ
(
(
)
(
)
)
(
(
)
(
)
)
s
ii
i
H
JY
Y
d
YY
YY
d
(14
)
Whe
r
e,
()
i
Y
and
ˆ
()
i
Y
is th
e
DTF
T
of
[]
i
yn
and
ˆ
[]
i
yn
r
e
spec
tively. (
)
H
de
note
s
the
Hermitian co
njugate.
'
01
1
()
(
(
)
,
(
)
,
,
()
)
s
YY
Y
Y
'
01
1
ˆˆ
ˆ
ˆ
()
(
(
)
,
()
,
,
()
)
s
YY
Y
Y
.
We have
：
11
,
00
2/
ˆ
()
()
()
(
2
/
)
ms
ij
p
j
kj
jn
i
T
p
nZ
YY
H
nT
e
(
1
5
)
Whe
r
e,
,
()
pj
H
is
the
pj
th elemen
t of matrix
()
H
.
A matrix represe
n
t of (8) is given by:
ˆ
(
)
()
()
()
YQ
H
Y
(
1
6
)
Whe
r
e,
2/
,
(2
/
)
()
jn
i
T
p
n
ip
nT
e
Q
Subs
titute (9) into (7), we have:
((
)
(
)
(
)(
)
)
(
(
)
(
)
(
)
(
))
H
YQ
H
Y
YQ
H
Y
d
J
(
1
7
)
Whe
n
the value of the equ
ation (17
)
is
mini
mum, the
generalized i
n
verse ca
n b
e
attained by:
()
()
/
(
()
()
)
HH
HQ
Q
Q
(
1
8
)
As
s
o
on as
the
r
[
n
] is obtai
ned, we
can
have the re
co
vered
x
(
t
) through a
n
interpolator.
T
N
is d
e
fined
as the
oversa
mpling p
e
ri
od
ic that satisfy
/
N
TT
M
, we can
re
write (3)
as
follow:
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7829
1
0
[]
[
]
(
)
NN
m
pp
pc
z
x
nMT
c
nT
cT
r
(19)
Up
sampli
ng t
he
seq
uen
ce
(
x
[
nT
N
]:
n
∈
Z
) by fa
ctor of
M
, the dth
sub-seq
uen
ce
is
given
by:
1
0
[]
(
)
[]
m
NN
p
p
N
N
pc
z
d
d
cn
M
T
T
c
T
xn
M
T
T
r
(20
)
The DT
FT of (14
)
is:
1
,
0
()
()
()
m
pp
d
p
d
XR
(21)
Finally, we ca
n have the re
con
s
tru
c
ted
signal
s in Fou
r
ier dom
ain:
1
0
1
1
,
0
0
1
1
,
0
0
()
()
()
(
)
()
(
)
M
jd
d
d
M
m
jd
pp
d
p
d
M
m
jd
pp
d
p
d
MM
MM
R
R
xe
x
M
e
e
(
2
2
)
4. Error analy
s
is
We will defin
e an angle b
e
twee
n two clo
s
ed sub
s
p
a
ce
s A and B of a Hilbert space
V
[11]:
,1
co
s(
,
)
i
n
f
B
fA
f
A
BP
f
(23)
,1
si
n
(
,
)
s
u
p
B
fA
f
A
BP
f
(24)
Whe
n
the re
con
s
tru
c
ted
sign
al
()
x
tV
, we can
co
ncl
ude
the sam
p
lin
g error
((
)
)
ex
t
as
follow:
22
2
2
2
(
(
)
)
()
()
(
)
(
)
(
)
(
)
V
V
V
ex
t
x
t
x
t
Pxt
x
t
P
x
t
Px
t
(25
)
From (25), we c
an have:
()
(
(
)
)
VV
Px
t
P
e
x
t
(26)
Whe
n
()
x
tV
W
(
W
is the sam
p
ling
spa
c
e
)
, we
can have the
Equation (27):
(
(
)
)
()
()
(
(
)
)
VW
ex
t
x
t
x
t
E
x
t
(27
)
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ISSN: 23
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KA
Vol. 12, No. 11, Novem
ber 20
14: 78
24 – 783
1
7830
Whe
r
e,
((
)
)
VW
E
xt
is the oblique p
r
oje
c
tion onto
V
a
l
ong
W
.
Furthe
r, the i
n
fimum a
nd t
he Sup
r
em
u
m
of
samplin
g erro
r
can
b
e
given
by Equation
(28
)
.
22
22
2
(
(
))
si
n(
,
)
(
(
)
)
(
(
))
c
o
s(
,
)
VV
Px
t
V
W
e
x
t
Px
t
V
W
(28
)
4. Simulation
In the section, we
will val
i
date the
reconstruction al
gorit
hm i
n
M
A
TLAB. We
desi
gn a
sampli
ng sy
stem that the
sampli
ng cha
nnel
s are
s
=2. The co
rre
spo
ndin
g
no
nuniform sa
mple
seq
uen
ce
s in
Figure 1 a
r
e
0
()
(
)
at
t
n
T
and
1
()
(
)
at
t
n
T
, we defi
ne
/3
T
that
is the se
que
n
c
e sepa
ration
betwee
n
two
interleaved
u
n
iform sampl
e
seq
uen
ce
s.
The gene
rat
e
function
s
0
()
t
and
1
()
t
are given as follow:
0
2
2
3
()
(
)
si
n
c
j
t
T
te
t
T
(29)
1
2
2
3
()
(
)
sin
c
jt
T
te
t
T
(30)
Whe
r
e,
T
is the sam
p
ling
perio
d.
We supp
ose that the input multi-ba
nd si
gnal:
88
7
(
)
s
i
n(
10
)
s
i
n
(
1
0
)
3
x
tt
t
(
3
1
)
(a)
(b)
(c
)
(d)
Figure 3. The
Sampling System Simulati
on
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
x 1
0
-7
-2
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
2
t
i
m
e
s(
s)
A
m
p
l
i
t
ud
e(
v
)
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
x 1
0
-7
-2
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
2
ti
m
e
s
(
s
)
A
m
p
l
i
t
ud
e(
v
)
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
x 1
0
-7
-2
-1.
5
-1
-0.
5
0
0.
5
1
1.
5
2
ti
m
e
s(
s
)
A
m
pl
i
t
ude
(
v
)
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
x 1
0
-7
-2
-1.
5
-1
-0.
5
0
0.
5
1
1.
5
2
ti
m
e
s(
s
)
A
m
pl
i
t
u
de(
v
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Dire
ct Ra
dio
Freq
uen
cy Sam
p
ling Syst
em
on Software-define
d
Ra
dio (Lu
o
Ju
nyi)
7831
5. Conclusio
n
In this
pape
r,
we
u
s
e a
ge
neral
fram
ework to treat
sampling
of m
u
lti-ban
d
sign
al. Our
intere
st is th
a
t
focused
on
how to
re
co
n
s
tru
c
t si
gnal
compl
e
tely. The ap
pro
a
ch
we
cho
s
e
n
a
r
e
that proj
ect t
he si
gnal
ov
er b
a
si
s fun
c
tions a
nd
th
e
n
sa
mple th
e
basi
s
co
efficients. Th
e lat
t
e
r
focu
se
s on
u
s
ing
gen
erali
z
ed i
n
verse t
o
obtain
r
i
[
n
] (
0
≤
i
≤
m
-1
). We
sho
w
e
d
that by usi
n
g
a
interpol
ator t
o
gain the co
mplete multi-band
signal
x
(
t
) from
r
[
n
]. Finally, the simulation p
r
ov
ed
the method we prop
osed is feasible.
Ackn
o
w
l
e
dg
ements
Suppo
rted by
Younge
r Sci
ence Fund of
Che
ngd
u Uni
v
ersity (20
1
3
X
JZ22
)
Referen
ces
[1]
CH T
s
eng, SC
Cho
u
.
Dir
ect
dow
nconv
ersi
o
n
of
mu
ltipl
e
R
F
signa
ls us
in
g ba
nd
pass s
a
mp
lin
g
. Proc.
ICC. 2003; 3: 2
003
–2
007.
[2]
DM Akos, M S
t
ockmaster, JBY T
s
ui, J Caschera.
D
i
rect b
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a
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ling
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p
l
e
distinct RF
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IEEE Trans.
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, 1999; 47(
7): 983–
98
8.
[3]
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NL Scott, DR
White.
T
he theor
y
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EE T
r
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nal Pr
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199
1; 39(9): 19
73–
19
83.
[4]
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ong, T
S
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w
n
-c
o
n
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g multipl
e
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nd
p
a
ss sign
als usi
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pa
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ng.
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s
ui, J Caschera.
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i
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l
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ulso
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onu
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