Signal Processing - May 2017 - 103

where x represents the vector form of
the selected input sequence x, and Fu -1
represents the row sampled IDFT matrix.
In particular, the row-sampled IDFT
matrix Fu -1 is different for decimation
and interpolation, where the rules for
performing row sampling can be easily
obtained in (5) and (6). For brevity, the
full expressions are omitted here. Based
on (5) and (6), the number of summation
terms in (7) should be equal to
Ny
min c; N x E, ; Em = min c; N x E, ; 1 IN x Em
D
I
D I D
N
= min ; x E,
(8)
D

where 6x@ represents the nearest integer
less than or equal to x. However, in (7),
the number of parameters N y = IN x D
is larger than the number of summation
terms N x D given that the up-sampling
factor I is larger than one, which is often
satisfied in arbitrary fractional-factor
SRC. In other words, this problem is illposed because the number of parameters
is larger than the number of observations.
The SR technique can desirably solve the
ill-posed problem by proper exploitation
of sparsity. In our problem, however, the
SR technique cannot be directly applied
u is not
since sparsity assumption of Y
satisfied. Therefore, careful design and
proper modification should be carried out
to utilize the SR technique.

Parameter regularization
The key ingredient in the SR technique
is the proper utilization of sparsity, where
an appropriate sparsity domain should
be identified. In the frequency-domain
SRC, the following observations and formulations can be made.
1) As discussed in [2], manipulating
X (k) in the frequency-domain SRC
induces errors. These errors are mainly
located at both ends of the timedomain sequence yt (n). More specifically, most of the errors locate only
in both ends of the converted signal.
An important observation is that the
number of error locations is often
much smaller than the length of the
sequence, which can be considered
to be sparse or compressible. To
u,
obtain a reasonable solution of Y
the sparsity regularization is used

and the following constraint can be
naturally obtained:
Fu

-1

u
Y

# f1 .

(9)

As a matter of fact, the sparsity level
depends on the input length and the
fraction I/D. It would be difficult to
express it with a closed-form expression. Moreover, the choice of e 1 not
only depends on the sparsity level
but also on the average amplitude
of the errors. Similar to all other SR
problems, the value of e 1 should be
properly chosen. However, in our
experiment, it is found that the performance of the optimization is
not very sensitive to the value of e 1
for various input lengths and values
of I/D. As we commented previously, this sparsity pattern is unique for
frequency-domain SRC technique,
which is rather different from the
sparsity pattern exploited in SR-based
calibration problems [5].
2) In many applications, such as audio
and speech signal processing systems,
the signal is often real valued and
its DFT is conjugate symmetric.
Similarly, we can conveniently
assume that the perturbation parameu is also conjugate symmetric.
ter Y
This symmetrical relationship can be
explicitly exploited as
*
Yu (k) = 6Yu (N y - k)@ ,
k = 1, ..., N y /2 - 1.

(10)

The use of this property effectively
decreases the number of parameters
to be optimized. For complex input
sequences, this conjugate symmetric
constraint is not valid and should be
removed from the optimization.
Of course, more constraints can be
imposed on the perturbation parameter
u to limit the solution space for different
Y
application purposes. However, these two
above-described constraints are effective
enough to obtain desirable performance
improvement compared with the uncalibrated frequency-domain SRC.

Optimization formulation
By combining the loss function and constraint terms, the optimization problem
can be formulated as
IEEE Signal Processing Magazine

May 2017

minimize loss function in (7)
u
Y

subject to constraints in (9) and in (10) .
(11)

The objective of the optimization problem (11) is to search for the perturbau in a feasible region
tion parameter Y
constructed by the constraints, which can
minimize the loss function. In our method, the sparsity is properly exploited to
obtain a robust estimate of the parameter.
One additional advantage is that the
optimization scheme is able to calibrate
the phase perturbations of the manipulated DFT as well, which will be shown
in detail in the section "Experimental
Results." It is noted that the phase information in the DFT is extremely important
in some signal processing applications, such as multimedia watermarking
[6]. Therefore, the frequency-domain
SRC-based methods can be applied to
reduce performance degradation due to
phase perturbation.
In summary, our proposed frequencydomain-based SRC method can be carried out by the following steps:
■ Step 1: Compute the N x -point DFT
of input sequence.
■ Step 2: Manipulate the DFT according to (1) for decimation or (2) for
interpolation.
■ Step 3: Estimate perturbation parameters by solving (11) and calculate
the calibrated DFT by (3).
■ Step 4: Compute the N y-point IDFT
of the calibrated DFT obtained in the
last step.
Compared with the uncalibrated
approach in [2], the proposed scheme
can substantially reduce the conversion
errors but requires additional computational costs due to the optimization
in (11). For the optimization method,
it is inevitable that the computational
complexity becomes high if the input
sequence is long. For real-time implementation, it is possible to segment the
long input sequence into many short
segments, where the optimization can
be performed for each segment consecutively. Moreover, the optimization can
be more conveniently and efficiently
carried out by resorting to other computationally efficient methods, such as
fast iterative soft-thresholding [7] and
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Table of Contents for the Digital Edition of Signal Processing - May 2017