Signal Processing - May 2017 - 102

accuracy can be significantly increased
by the proposed calibration method based
on sparsity constrained optimization.

Notations
All of the small letters represent timedomain signal, and all of the capital letters
represent Fourier domain coefficients.
Vectors and matrices are in bold case. The
set of natural numbers is denoted by N.

Preliminary of
frequency-domain SRC
Let us denote an N x -point input sequence as x (n), n = 0, f, N x - 1 that is
obtained by sampling the given analog
signal x (t) at a sampling frequency Fx,
and its DFT as X (k), k = 0, f, N x - 1.
Similarly, we also define the ideal
N y -point output sequence as y (m),
m = 0, f, N y - 1 that can be obtained
by sampling the analog signal x (t) at
a sampling frequency Fy, and its DFT
as Y (k), k = 0, f, N y - 1. In general,
the sequence y (m) is considered as the
ideal SRC output signal that is used as a
reference to evaluate the accuracy of the
output sequence produced by the SRC
method. The factor of the sampling frequency conversion is defined as Fy /Fx,
which can be easily proved to be equal
to N y /N x and I/D, respectively, where I
and D are the factors of up-sampling and
down-sampling, respectively, as shown in
Figure 1. Without loss of generality, it is
assumed that the ratio, I/D, is irreducible.
In practice, the SRC produces an output
sequence, yt (n), i.e., the approximation
of y (m) for m = 0, f, N y - 1. With the
frequency-domain method in [4], its corresponding DFT, Yt (k), k = 0, ..., N y - 1,
is obtained by manipulating the DFT,
X (k), of the input sequence x (n) according to [2] and [3].
■ Decimation (D 2 I ) . In the case of
Fy 1 Fx ,
Yt (k) =
ZI
Ny
] X (k),
0#k1
2
D
]]
Ny
k=
[ C 1,
2
]
] I X (k + N x - N y), N y 1 k 1 N y
2
\D
(1)
■

102

Interpolation (D 1 I ) . In the case of
Fy 2 Fx ,

Yt (k) =
ZI
0 # k 1 Nx
] D X (k),
2
]
Nx # k # N - Nx
C
,
y
[ 2
2
2
]I
N
x
] X (k +N x -N y), N y 1 k 1 Ny .
2
\D
(2)

In both (1) and (2), some values of Yt (k)
are replaced by the values of X (k) scaled
by I/D. For decimation, according to
(1), the values of X (k) for k = N y /2
to N x - N y /2 are ignored in realizing
Yt (k). The value of X (N y /2) is set to be
C 1, and X (k) for k = N x - N y /2 + 1 to
N x - 1 are relocated to become Yt (k)
for k = N y /2 + 1 to N y - 1. For interpolation, (2) shows that a small value
of C 2 is used to be the values of Yt (k)
for k = N x /2 to N y - N x /2, and the
values of scaled X (k) for k = N x /2 + 1
to N x - 1 are used to approximate the
values of Yt (k) for k = N y - N x /2 + 1 to
N y - 1. It should be fully understood
that the formulation of Yt (k) is based
on the spectrum distribution of input
sequence. Therefore, the estimated output spectrum Yt (k) always deviates from
the true spectrum Y (k).
In summary, the operations specified
in (1) and (2) are ignoring or inserting
some frequency components of small
values. The other frequency components
may also have some small differences
from those of the expected ones. All of
these variations contribute to the conversion errors that may be considered as perturbations. Now the question is, given the
input signal sequence x (n) and the estimated Yt (k), how the conversion errors
can be minimized. This article presents a
positive answer to this question by introducing the SR concept into the design of
frequency-domain SRC.

the calibrated DFT Y l(k) can be obtained.
The corresponding calibrated timedomain signal, yl(n), can be obtained
from Y l(k) by an IDFT. In a vector form,
it is expressed as
yl = F

-1

Let us denote the DFT in (1) or (2) as the perturbed DFT and assume that the error
calibrated DFT, Y l(k), is obtained by subtracting perturbation Yu (k) from Yt (k), i.e.,
Y l(k) = Yt (k) - Yu (k),

(3)

where Yt (k) is obtained from (1) or (2). In
particular, the DFT perturbation Yu (k) can
be obtained by our proposed calibration
method to be presented and subsequently
IEEE Signal Processing Magazine

|

May 2017

|

-1

t -Y
u ),
(Y

(4)

where F -1 ! C N y # N y represents the
IDFT matrix. Since the perturbed DFT,
t , is already obtained from (1) or (2),
Y
our task becomes properly estimating
u to obtain improvethe perturbation Y
ment in accuracy.

Loss function
An intuitive strategy to reduce the
conversion errors in the time domain
is to minimize the loss function that
is defined to measure the difference
between the ideal and estimated outputs.
Because the ideal output, i.e., y (n) in our
case, is unavailable, we use the accessible input sequence, x (n), to define the
loss function. In particular, the Euclidean distance is employed to measure the
differences between the samples selected from the original input x (n) and the
error-calibrated output samples yl(m).
Let us denote the indices selected from
original input as n x and the indices
selected from error calibrated output as
n y , where the condition n x /Fx = n y /Fy
should be satisfied. In other words, these
selected samples of yl(n y) should be as
close to the input samples x (n x) as possible. Using the equality n x I = n y D, the
following loss functions for decimation
and interpolation can be obtained.
■ For decimation (I 1 D), the loss
function can be expressed as
Ny - 1

/

n y = 1 , n y /I ! N

Calibrated SRC in frequency domain

Yl = F

■

;x c

2
ny D
m - yl (n y)E . (5)
I

For interpolation (I 2 D), the loss
function can be expressed as
Nx - 1

/

n x = 1, n x /D ! N

;x (n x) - yl c

n x I 2 . (6)
mE
D

Substituting (4) into (5) and (6), the loss
functions for decimation and interpolation can be respectively obtained. For
brevity, a vector form of the loss functions can be expressed as
x - Fu

-1

t -Y
u ) 22 ,
(Y

(7)



Table of Contents for the Digital Edition of Signal Processing - May 2017

Signal Processing - May 2017 - Cover1
Signal Processing - May 2017 - Cover2
Signal Processing - May 2017 - 1
Signal Processing - May 2017 - 2
Signal Processing - May 2017 - 3
Signal Processing - May 2017 - 4
Signal Processing - May 2017 - 5
Signal Processing - May 2017 - 6
Signal Processing - May 2017 - 7
Signal Processing - May 2017 - 8
Signal Processing - May 2017 - 9
Signal Processing - May 2017 - 10
Signal Processing - May 2017 - 11
Signal Processing - May 2017 - 12
Signal Processing - May 2017 - 13
Signal Processing - May 2017 - 14
Signal Processing - May 2017 - 15
Signal Processing - May 2017 - 16
Signal Processing - May 2017 - 17
Signal Processing - May 2017 - 18
Signal Processing - May 2017 - 19
Signal Processing - May 2017 - 20
Signal Processing - May 2017 - 21
Signal Processing - May 2017 - 22
Signal Processing - May 2017 - 23
Signal Processing - May 2017 - 24
Signal Processing - May 2017 - 25
Signal Processing - May 2017 - 26
Signal Processing - May 2017 - 27
Signal Processing - May 2017 - 28
Signal Processing - May 2017 - 29
Signal Processing - May 2017 - 30
Signal Processing - May 2017 - 31
Signal Processing - May 2017 - 32
Signal Processing - May 2017 - 33
Signal Processing - May 2017 - 34
Signal Processing - May 2017 - 35
Signal Processing - May 2017 - 36
Signal Processing - May 2017 - 37
Signal Processing - May 2017 - 38
Signal Processing - May 2017 - 39
Signal Processing - May 2017 - 40
Signal Processing - May 2017 - 41
Signal Processing - May 2017 - 42
Signal Processing - May 2017 - 43
Signal Processing - May 2017 - 44
Signal Processing - May 2017 - 45
Signal Processing - May 2017 - 46
Signal Processing - May 2017 - 47
Signal Processing - May 2017 - 48
Signal Processing - May 2017 - 49
Signal Processing - May 2017 - 50
Signal Processing - May 2017 - 51
Signal Processing - May 2017 - 52
Signal Processing - May 2017 - 53
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Signal Processing - May 2017 - 56
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Signal Processing - May 2017 - 58
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Signal Processing - May 2017 - 60
Signal Processing - May 2017 - 61
Signal Processing - May 2017 - 62
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Signal Processing - May 2017 - 65
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Signal Processing - May 2017 - 69
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Signal Processing - May 2017 - 88
Signal Processing - May 2017 - 89
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Signal Processing - May 2017 - 91
Signal Processing - May 2017 - 92
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Signal Processing - May 2017 - 96
Signal Processing - May 2017 - 97
Signal Processing - May 2017 - 98
Signal Processing - May 2017 - 99
Signal Processing - May 2017 - 100
Signal Processing - May 2017 - 101
Signal Processing - May 2017 - 102
Signal Processing - May 2017 - 103
Signal Processing - May 2017 - 104
Signal Processing - May 2017 - 105
Signal Processing - May 2017 - 106
Signal Processing - May 2017 - 107
Signal Processing - May 2017 - 108
Signal Processing - May 2017 - 109
Signal Processing - May 2017 - 110
Signal Processing - May 2017 - 111
Signal Processing - May 2017 - 112
Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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