Reconstruction of a Speech Signal Using Projections on Convex Surfaces Scott Philips
 ABSTRACT In many situations error received in discrete time systems are too large to be recovered using simple filtering techniques. When it is necessary that the information is recovered, alternate means must be used. POCS including the Papoulis-Gerchberg algorithm will provide a useful alternative to solving these problems. This paper will discuss the removal of localized errors within a bandlimited speech signal using this approach. 1. Introduction Projection onto a Convex Set (POCS) is a useful tool for optimizing a signal given a set of constraints. The Papoulis-Gerchberg algorithm was first developed in the 1970’s and was applied in an attempt at super resolution. The algorithm attempts to restore a region between two points of a signal by imposing two constraints. The signal must be bandlimited and must contain known points outside the region of interest. Signals with these constraints represent a convex set. By alternating between each constraint the region between the two points can be restored. 2. Problem Statement When a bandlimited signal is received with severe localized errors, most methods of filtering are unable to restore the missing data. These errors can be caused by a channel cutting in and out, power surges, or short time interference or fading. When accurate reconstruction is desired another approach is needed. 3. Approach Reconstructing a speech signal with multiple localized errors will be done using the Papoulis-Gerchberg algorithm. The first constraint imposed is that of bandlimited signal. In the set of all possible signals of length N, SN, there exists a subset comprised of all bandlimited signals. This subset constitutes a convex set C1={s(t)|S(u)=0;u>B}, which is a plane within the set SN. The second constraint is that of identical tails. This is the set of all signals that have an unknown shape in a region between two points, but have a given function outside this region. This subset also represents a convex set C2={s(t)|s(t)=b(t);tK} which is a plane within the set SN. In this paper this set will be generalized to a signal with multiple unknown regions or regions with localized errors. Assuming that the localized errors occur at a rate much less than the sampling frequency, set C2 can still be applied. Lemma 1 asserts that recursively projecting onto each of these sets will result in a signal with both qualities. According to Papoulis-Gerchberg this with be the original signal with error. 4. Results Error was introduced to a 22.05 kHz bandlimited speech signal by zeroing a series of four points numerous times throughout the signal. The signal is now no longer bandlimited. Figure 2 shows a fragment of the original speech signal, s(t), and the signal corrupted with errors, r(t). Defining the added error as e(t) =s(t)-r(t), the energy of this error is found to be 13.8 joules. See signal 1. Figure 1 By merely using a 50 point low pass filter, very little of the error is removed. This is shown in figure 2. The energy of the error was only reduced to 12.6 joules. This is due to the fact that at the points of error, the signal is sampled well below the Nyquist rate. There is just not enough information to reconstruct the signal. See signal 2. Figure 3  This can be overcome by using the Papoulis-Gerchberg algorithm. First, the signal is projected onto the convex set C1 by low pass filtering to 22.05 kHz. This will slightly reduce the error in the corrupted regions of the signal, but will also introduce some error into the uncorrupted regions. Next, the original signal is imposed on the sections of the signal that were known to be uncorrupt. This projects back onto convex set C2. Repeating these two steps n times results in a signal rn(t). As n gets large rn(t) should converge to the original signal s(t). Figure 3 show rn(t) for increasing values of n. From a visual inspection, the signal looks much cleaner. The energy of the error is now down to only 2.36 joules.   See signal 3. Figure 4   6. Analysis The error was greatly reduced and a ‘cleaner’ sounding signal was heard. The graphs demonstrate the change in the signal as the iterations increase. This interpolation ability works well, but also comes with restrictions on how large the reconstruction region can be. The more error introduced in the system, the more iterations are needed to reconstruct the signal causing a great increased in processing time. This is caused by the greater amount of information lost. Processing time is not the only bound on the amount of error that the method can handle. There also is a bound associated with numerical precision. The larger the region is that is filled with error, the greater the numerical precision is needed. The precision of your software will put a hard bound on how much information can be recovered. Given a signal that is bandlimited to one half the sampling frequency, the performance of the system will deteriorate quickly if the error region is much larger than 5 samples. 7. Conclusion Given a particular set of constraints to reconstruct the signal POCS proved to viable solution. Signal error can be gotten rid of while retrieving lost information. If the conditions are met POCS is a much better alternative to filtering.

 Signal 1: Original Signal Signal 2: Signal with Error Signal 2: Filtered Signal Signal 3: POCS Reconstruction