A NEW TRANSFORM TECHNIQUE FOR ERROR
CORRECTION AND CONCEALMENT1
F. Marvasti1, Paul Strauch2, P. Ferreira3, Ran Yan2
Dept of Electronics, King’s College London
Strand, London WC2R 2LS, UK
2
Lucent Technology, Swindon, SN5 6PP, UK
3
University of Aveiro, Aveiro, Portugal
1
ABSTRACT
We will discuss a new transform method that is
suitable for a burst of erasures and error concealment
for speech, image and video signals over ATM and
wireless channels. The advantage of this transform is
that it works on the field of real and or complex
numbers as opposed to finite fields.
1.
INTRODUCTION
FFT has been used for error detection and correction
of real or complex numbers [1-3]. Efficient techniques
for decoding FFT codes can be devised under erasure
channels [4]. However, error detection and correction
of real numbers are very sensitive to noise [5]; there
are methods to get around this sensitivity issue [6-7].
In these techniques, redundancy is added by inserting
zeros in the transform domain; the inverse transform
can correct for erasures.
Error Concealment methods have always been
regarded as an engineering art to subjectively conceal
errors for speech and image signals. Simple methods
such as previous frame substitution for these types of
signals are always regarded as ad-hoc techniques for
improving the quality or concealing errors of a
received signal. The rational behind this approach is
attributed
to
the
correlation
of
adjacent
samples/frames and the insensitivity of the ear or the
eye to the concealed errors. Error concealment is
closely related to the source and channel coding. If the
correlation among adjacent frames is not utilised by
the source encoder, then error concealment can help
the quality especially when some frames are lost. On
the other hand, the more powerful a channel encoder,
the less frequent the possibility of a lost or bad frame
and hence the less frequent the need for error
concealment.
In this paper, we would like to relate error
concealment methods to error correction codes. The
consequence of this relationship is the use of Reed
Solomon (RS) like decoding methods for erasure
recovery (error concealment) except that we will use
the Galois Fields of real and/or complex numbers. The
1
Patent protection for this technique has been applied for.
basic idea is as follows. If e percent of the frames need
error concealment, we can pre-distort the signal by e
percent at the encoder globally and then convert the
error concealment problem into an error correction
problem. A simple example can clarify this idea. If e is
equal to 10 percent, we can reduce the bandwidth of
the source signal at the encoder by 10 percent. This
reduction will not have a noticeable distortion on the
signal. In essence the 10 percent distortion is
distributed on all the samples (frames) and is
concealed from subjective evaluation. Due to the
bandwidth reduction by 10 percent, the encoder is
now over-sampled. Either we can remove the
redundancy and then use the classical source and error
correction codes or, alternatively, over-sampling can
be regarded as non-binary redundant symbols similar
to RS codes [4] and [7]. Hence RS decoding can be
used for decoding the lost (erased) frames. Essentially
at the encoder, the source encoder compresses the
over-sampled signal and then the channel encoder
adds parity bits for error protection. At the decoder,
the received bits sequentially go through the channel
decoder, source decoder, and the Error Concealment
(EC) decoder. The EC decoder corrects (conceals) for
the 10 percent bad frames on the average. Below, we
will describe the practical issues related to EC and
then describe the technique in more mathematical
detail.
1. DELAY AND SENSITIVITY ISSUES
There are various methods for reducing the bandwidth
of the source signal: 1- An ideal low pass filter, 2- An
FIR or IIR filter, 3- Zeroing the high frequencies in
the FFT or the proposed transform domain. The first
method creates infinite delay at the decoder. The
second method is equivalent to the convolutional
codes but is not suitable for a bursty frame loss since
the decoder is an iterative method, which fails for
bursty losses. The third approach is the preferred
method and creates a delay of n samples- the block
size of the FFT or the proposed transform; this is
equivalent to the block codes for error correction
codes. The problem with FFT is that it requires very
accurate precision for the samples and become
unstable for large values of n (n>64)[3] and [7].
Below we will describe a new transform technique
that does not have the drawbacks of FFT.
2. THE NEW TRANSFORM METHOD
The new transform is actually derived from FFT and
hence the fast algorithm can still be used. If the
elements of the FFT transform matrix are
ai ,k  exp( j * 2 *  * i * k / n )
for ~ i, k ~ 1,2,...n,
the
modified
transform
is
bi ,k  exp( j * 2 *  * p * i * k / n )
for ~ i, k ~ 1,2,...n,
where p is relatively prime with respect to n2. With
this condition satisfied, we are guaranteed that
bi ,1 ~~ are distinct for all values of i ~ 1,2,..., n
and are the sorted version of the FFT kernels
a i ,1 ; we
can show that the new transform has still conjugate
symmetry. The advantage of this transform is that,
unlike FFT, it is no longer sensitive to the
quantization of samples and additive noise. The
forward transform can be handled by FFT and sorting;
the sorting of coefficients is as follows:
sort i  mod( p * i, n), ~~ i ~ 1,2,..., n .
The
inverse transform of the frequency coefficients is
equivalent to the inverse sorting and the inverse FFT,
respectively. Since the kernel of the new transform is
exponential, RS decoding procedure can be used for
recovering lost (bad) frames [4]. The value of p can be
optimised based on the value of n, and the number of
bad frames  in a block of n; this issue will be
discussed in the next section.
3. THE CHOICE OF P
Besides the fact that p and n have to be relatively
prime, there are two contradictory requirements for the
choice of p, the first one is that we need a p such that
the decoding is stable. For stability, it is necessary that
the error locator polynomial [6] spans the unit circle,
i.e., the following complex points should be spread
over the unit circle.
exp( j * 2 *  * p * m / n )
m ~ n   * s,..., n
where  is the number of bad frames and s is the
number of samples per frame. In other words, we
require p *  * s  n . On the other hand, filtering
in the new transform domain creates some distortion
in the original source signal. In order to reduce this
distortion, p has to be chosen such that low energy
coefficients are in the high frequency region. FFT
(p=1) is an extreme example, where almost all the
high frequency coefficients are zero. However, the
FFT transform is highly unstable. If p is around n/2,
almost half of the high frequency coefficients of FFT
remain in the high frequency region of the new
transform. However, the transform may not be stable
depending on  . Also, we can show that the transform
for p=n/2+e is the conjugate of p=n/2-e for even n.
Therefore, a compromise value for p between
n /  * s  p  n / 2 has to be chosen. For
example, for n=480,   1 , s=160, p=197 can be
found experimentally to be the best compromise.
4. PROPERTIES OF EC CODES
This class of EC codes, similar to the classical error
correction codes, become more powerful by increasing
n and  . Assume a pre-distortion measure D as a
function of the code rate r =1-  * s / n . If we keep
the rate r constant (i.e., constant pre-distortion), we
can decrease the Mean Square Error (MSE) of the
erased frames to zero by increasing the block size n if
the average probability of Bad Frame Indication (BFI)
is less than the EC redundancy rate  * s / n 3. The
price to be paid is  frames delay both at the encoder
and the decoder. It is therefore, desirable to have
shorter frame sizes, e.g., the speech standard G.729
which has 10ms (s=80 samples) frame size is more
suitable for EC than GSM frames of 20ms (s=160
samples).
5. DECODING ALGORITHM
The EC decoding algorithm is similar to RS decoding.
For FFT decoding see [4]. The error locator
polynomial is
s
XXk
ttk exp
If p and n are not integers, new classes of transforms are
generated but are not related to FFT, and by zeroing the high
frequencies we may not yield a real signal. These transforms
can still be used for error correction and not EC.
 ii  q
m
m=1
3
This is equivalent to the statement that the EC rate should
be less than the channel capacity for BFI. For an erasure
channel, the capacity is c= r*(1-Pe), where r is the
transmitted sample rate and Pe is the probability of BFI. The
source information rate is r* ( n
2
j
  * s ) / n . The
channel capacity theorem implies that
  * s) / n 
 * s / n  Pe.
r* ( n
r*(1-Pe), or alternatively,
where
The above transform is actually a sorted version of
FFT. The original FFT looks like Fig 2.
ttk  exp( j * k * q)
and q  2 *  * p /n.
The value of p determines the type of the transform.
For p=1, the error polynomial is for the FFT
transform. The above error locator polynomial is

 h * tt 
equivalent to
t
t 0
t
j
ht
. The
values can
5.895704
e(im ) * (ttm ) r and then summing over m, we
get
Er 
2
3
0
0
be found from the inverse transform (inverse sorting
and inverse FFT). By multiplying the above equation
by
4
Y
k
Y1
1.151008 10
6
1
100
150
200
250
300
k
350
400
450
500
4.8 10
2
trace 1
Fig 2. Normalised FFT coefficients of the same
segment of the speech signal.
  E r t * h t  E r 
t
The quality of the pre-distorted signal for
s = 80 is as shown in Fig 3.
h
 -1 and
r  n / 2   / 2  1...n / 2   / 2  1 .
where t = 1…,
are
the transform (FFT and the sorting
algorithm) coefficients. Since  * s samples of E r
are known at “high frequency” positions, the
remaining values of
Er
  1 and
alk
e(im ) ’s are the lost samples of the Bad Frame and
E r ’s
50
2000
Re ( ye
( xx)
e1 )
1000
l
l
Re ( y)
0
l
1000
corr ( Re ( y)  xx )  0.908
are determined from the
2000
above recursive formula. Once E r ’s are determined
for all values of r, its inverse transform yields the
amplitudes of the lost samples of the Bad Frame. For
computational complexity see [4] and [7].
0
100
200
l
300
400
500
trace 1
trace 2
trace 3
Fig 3. The recovered, the original, and the predistorted signal for   1 and s = 80.
6. SIMULATION RESULTS
We have simulated the above new transform method
on Mathcad and have listened to the quality of predistorted signal. For n=480, and p=197, the transform
of a typical speech signal looks like Fig 1.
The quality of the recovery algorithm for
= 80 is shown in Fig 4.
  1 and s
corr ( ( e )  ( e1) )  1
2000
5.895704
6
1000
Y
k
Y1
1.151008 10
Re e
4
j
Re e1
j
2
0
1000
3
2000
0
0
1
50
100
150
200
250
k
300
350
400
450
500
2
4.8 10
trace 1
Fig 1. Normalised Transform coefficients of a segment
of speech signal.
0
100
200
j
300
400
500
trace 1
trace 2
Fig 4. The lost and the recovered speech samples for
s=80 and   1 .
xx
lk
n
2
1 
For s=160 and other parameters remain as before, the
quality of the pre-distorted signal is shown in Fig 5. alk
2000
Re ( ye
( xx)
e1 )
1000
l
l
0
Re ( y)
l
1000
corr ( Re ( y)  xx )  0.837
2000
0
100
200
l
300
400
500
trace 1
trace 2
trace 3
Fig 5. The recovered, the original, and the predistorted signal for   1 and s = 160.
7. CONCLUSIONS
xx
lk
n
2
1 
A new class of transform technique is derived that is
useful for both error correction and error concealment
of speech and video signals. Since the proposed
transform method can be shown to be a sorted version
of FFT, fast algorithms can be implemented. Also,
because the elements of the new transform matrix are
exponentially related to each other, efficient RS type
of decoding can be used for both error correction and
error concealment. Simulation results for both speech
and video signals show the validity of the proposed
transform method. Simulation results in this paper are
related to speech signals; the results for video signals
will be published later.
8. REFERENCES
The quality of the recovery algorithm for
= 160 is shown in Fig 6.
  1 and s
corr ( ( e )  ( e1) )  1
2000
1000
Re e
j
Re e1
j
0
1000
2000
0
100
200
j
300
400
500
trace 1
trace 2
Fig 6. The lost and the recovered speech samples for
s=160 and   1 .
For n=480 and   1, a pre-distorted speech signal
yields an acceptable quality of speech for s=80
samples. This pre-distorted signal can conceal one
frame in a block of 6 frames.
For present GSM applications where s=160, n needs
to be at least 800 for an acceptable quality. This code
can correct for one GSM frame per a block of 5
frames. In general the quality of pre-distorted signal
can be improved by increasing n; but by increasing n,
we get more delay.
The above transform method has been successfully
tested on video signals for both error correction and
error concealment. The results will be published in a
different paper.
[1] R.E. Blahut, “Transform Techniques for error
control codes,” IBM J of Res. Develop., vol.23,
pp. 299-315, May 1979.
[2] K. Wolf, “Redundancy, the discrete Fourier
transform, and impulse noise cancellation,” IEEE
Trans. Commun., vol. COM-31, no.3, March
1983.
[3] F. Marvasti, “FFT as an alternative to error
correction codes,” IEE Colloquium on DSP
Applications in Communication Systems, March
1993.
[4] F Marvasti, M Hasan and M Echhart, “Speech
recovery with bursty losses”, in Proc of
ISCAS’98, Monterey, CA, May31-June 3, 1998.
[5] Wong, F. Marvasti, and W.C. Chambers,
“Implementation of recovery of speech with
impulsive noise on a DSP chip,” IEE Electronic
Letters, vol.31, no.17, pp.1412-1413, 17 Aug.,
1995.
[6] M Hasan and F Marvasti, “Noise sensitivity
analysis for a novel error recovery technique”, in
Proc of ICT’98, Halkidiki, Greece, 21-25 June
1998.
[7] F. Marvasti, M. Hasan, M. Echhart, S. Talebi,
“Efficient techniques for burst error recovery,”
accepted for publication in the IEEE Trans on
SP.