IEEE Signal Processing - March 2018 - 57

as the zero-crossing times of x (t) - d (t). It
the first term measures the error in the
To alleviate the
can be easily verified that, to have at least
nonuniform sampled points, t i s, and the
problems encountered
one zero-crossing at each period of the
second term enforces smoothness on the
in large alphabet signal
dither function, d (t) should cover the whole
reconstructed signal.
compression, we propose
For theoretical results, generally a
dynamic range of the input signal. Further,
an approach based on
sine function is used as the dither funcunder some conditions on the dither signal,
tion along with an ideal reconstruction
the SNR of this signal acquisition scheme is
oversampling but a
proportional to 1/x 2 [35]. Moreover, if the
representation with fewer algorithm. However, our experiments with
seismic data reveals that a sawtooth dithdither has period mx, the raw bit rate to repnumber of bits.
er function gives more accurate results
resent zero-crossings, t k s, would be approxsince it offers a uniform time quantization
imately equal to m log 2 (mx). Hence, the total
over the entire dynamic range of the analog signal. Moreover,
number of bits required to represent an analog signal within
the reconstruction based on variational objective function is
some distortion bound is approximately in the same order as
robust to perturbations and achieves good signal recovery.
the traditional sampling methods, i.e., the SNR behaves as
To further reduce the dynamic range of data, an RLS adapexp (cR) for some positive constant c in both cases. Beside
tive filter is applied to the sequence of zero-crossings (t n s
resolving the aforementioned large alphabet issue, the simplicity and efficiency of this sampling technique make it a good
in Figure 5). Then the residual is compressed using a simple
candidate to deploy on low-end devices such as sensors. The
entropy coding algorithm such as adaptive arithmetic coding
downside of this sampling scheme is its complex reconstruction
or a low-complexity CTW. We denote the aforementioned
algorithm [35]-[37]. However, this is not an issue in seismic
scheme as oversampling-RLS (OV-RLS). The decoder first
acquisition since the reconstruction is carried out at the data
reconstructs the residual time series signal, dt n s, and genercenter (or server), which generally has more processing power.
ates the locations of zero-crossings, t n s. Knowing the dither
The signal reconstruction problem from nonuniform samfunction at the receiver, the locations of zero crossings are then
ples has been investigated extensively, and near-optimal perused to create a sequence of nonuniform samples of the seisformance can be achieved using methods such as Lagrange
mic trace, which are used to reconstruct the original signal.
polynomial interpolation. However, many of these theoreticalFor the reconstruction of the signal from the zero-crossly sound methods, such as [35] and [36], either easily become
ing samples, we used the spline expansion basis as z ($),
unstable in the presence of perturbations or perform poorly
which in our experiments exhibits superior and robust
for signals with a short time duration. Recently, a variational
performance compared to other methods. As shown in Figobjective function has been proposed based on a smoothed
ure 6(a) and (b), for USGS and Utah data sets, respectively,
constraint optimization [37]. For a smooth compact generating
while for the high SNR regimes the oversampling method
function z ($), let VT (z) be the set of all functions of the form
does not improve over the RLS, it achieves higher compression gains than both RLS and LPC for lower SNRs. As
g (t) = R k c k z (t/T - k) . The reconstructed signal is found by
2
explained previously, the average sampling rate should be
xt = argmin g ! V T (z) R i d (t i) - g (t i) + n # g (r) (t) dt, where

1
x (t )
−1
d (t )

s (t ) = x (t ) - d (t )

δtn-2 δtn-1 δtn δtn+1

1-bit A /D
fs >> fNyquist

sign(s (nτ))

Run-Length
Coding

{δtn}

Σ

{rn}

Transmit
CTW
Encoder

H (z -1)

z -1

Figure 5. A block diagram of the compression scheme based on 1-bit quantization and RLS adaptive filtering.
IEEE Signal Processing Magazine

|

March 2018

|

57
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
