DYNAMIC PROGRAMMING BASED OPTIMUM NON-UNIFORM SAMPLES FOR SPEECH
RECONSTRUCTION AND CODING
Prasanta Kumar Ghosh
⋆
and T.V. Sreenivas
†
Department of Electrical Communication Engineering
Indian Institute of Science, Bangalore-560 012, INDIA
E-mail:
⋆
prasanta@ece.iisc.ernet.in
†
tvsree@ece.iisc.ernet.in
ABSTRACT
Non-uniform sampling of a signal is formulated as an opti-
mization problem which minimizes the reconstruction signal
error. Dynamic programming (DP) has been used to solve this
problem efficiently for a finite duration signal. Further, the
optimum samples are quantized to realize a speech coder. The
quantizer and the DP based optimum search for non-uniform
samples (DP-NUS) can be combined in a closed-loop manner,
which provides distinct advantage over the open-loop formu-
lation. The DP-NUS formulation provides a useful control
over the trade-off between bitrate and performance (recon-
struction error). It is shown that 5-10 dB SNR improvement
is possible using DP-NUS compared to extrema sampling ap-
proach. In addition, the close-loop DP-NUS gives a 4-5 dB
improvement in reconstruction error.
1. INTRODUCTION
Signal reconstruction from nonuniform samples (NUS) is a
widely studied problem. There have been many approaches
to signal reconstruction from non-uniform samples, namely,
from zero-crossings [1], from level crossings [2], signal re-
construction from periodically non-uniform samples [3], or
through iterative methods [4]. However, very few attempts
have been made [5] to analyze the quantization properties
of NUS and use them for signal compression. [6, 7] put
some light on the aspect on nonuniform sampling for cod-
ing of speech waveform. In [8] we have presented quanti-
zation properties of extrema samples (ES) for speech signal
reconstruction and speech coder based on this is proposed.
However, the resulting coder is a variable rate coder since the
number of ES (being nonuniform in nature) varies with time.
Also, the bitrate of the ES based coder is not easily scalable
in the sense that ES have been chosen as a fixed NUS and no
measure is incorporated to increase or decrease the number of
NUS.
The aim of this paper is to find optimum NUS to recon-
struct a finite duration signal which minimizes certain cost
function. The optimum NUS are obtained efficiently by dy-
namic programming (DP). It is found that for a particular
choice of interpolating function, the extrema are not necessar-
ily optimum NUS to minimize the reconstruction signal error.
The dynamic programming provides the advantage of choos-
ing the number of NUS to achieve a specific performance. We
also incorporate the quantization of NUS location and ampli-
tude while minimizing the cost function using DP approach
and we find that it improves reconstructed signal quality over
that obtained using DP and quantization separately.
2. ANALYSIS BY SYNTHESIS APPROACH FOR
OPTIMUM NONUNIFORM SAMPLES
Let x[n], 0 ≤ n ≤ N - 1 be the signal segment for resam-
pling and reconstruction. Let {η
i
}
M
i=1
be the NUS locations
of x[n]. The reconstructed signal ˆ x[n] is in general a function
of {η
i
}
M
i=1
and {x[η
i
]}
M
i=1
, i.e.,
ˆ x[n]= F ({η
k
}, {x[η
k
]}),k = i, i +1, ..., i +Δ
i.e., a subset of (Δ + 1) NUS are used to reconstruct x[n].
Specifically, for Δ=1, we can write:
ˆ x[n]= x[η
i
]+(x[η
i+1
] - x[η
i
])F
j
n - η
i
η
i+1
- η
i
,
η
i
≤ n<η
i+1
(1)
where F
j
(X ) is the local interpolation function. We consider
three local interpolation functions as in [8]:
Linear interpolation : F
1
(X )= X.
Polynomial interpolation : F
2
(X )= X
2
(3 - 2X ).
Sinusoidal interpolation : F
3
(X )=
sin
πX
2
2
.
The error in reconstruction is denoted by:
e[n]= x[n] - ˆ x[n]. (2)
We seek to minimize the energy of e[n] to obtain optimum
NUS. Thus we can pose the optimization problem to deter-
mine the optimum NUS as follows:
η
opt
i
= arg min
{ηi }
1
N
N−1
n=0
{x[n] - ˆ x[n]}
2
1 ≤ i ≤ M (3)
I 1221 142440469X/06/$20.00 ©2006 IEEE ICASSP 2006