6. Summary and future work
vious approach. Post-processing is included in the optimization problem. For this, we change the signal model. We used the high-band signal modeling in the proposed ap- proach. Also, the proposed ABE approach enhances the narrowband signal consisting of frequency components up to 4 kHz approximately. The proposed approach uses H∞ op- timization for designing the synthesis filter corresponding to the high-band signal model.
The obtained synthesis filter has high-band spectral envelope information. Besides, the gain adjustment technique is used to set the energy level of the estimated high-band sig- nal, and the DFT concatenation is used to avoid the unwanted information leaked by the non-ideal filters (synthesis filter and low pass filter) in the wideband signal estimation.
The DNN model is used for predicting the synthesis filter and gain factor.
• Further, we again changed the signal modeling, which leads to better results. We used the mapped high-band signal modeling to get a better solution by the H∞ sampled-data system theory. The mapped high-band signal has the high-band information mapped to the narrowband region using modulation. Additionally, we modified the set-ups as per the ITU-T protocols for a better comparison with peers. Apart from that, we use the gain adjustment and spectral floor suppression techniques for controlling the energy of the estimated high-band signal. Separate DNN models are used for estimating the synthesis filter and gain factor. Also, the computation process of the gain factor reduces the performance loss due to obtaining errors in the estimated synthesis filter.
6.2 Future directions
does not provide an optimal selection of coefficients. Therefore, as a future direction, optimal FIR representation of the synthesis IIR filter can be used for all the proposed ABE approaches.
• Signal models are obtained using Prony’s method in the proposed ABE approaches. Other signal modeling schemes (such as recursive methods [95,96], recursive weighted linear least-squares (WLLS) procedure , Newton-like algorithm , and quasi-Newton al- gorithm ) can be used to see the effect of different signal modeling schemes on the performances.
• A deep study could be done for different phonemes. We can experiment to decide the optimal signal model for each phoneme. It means the optimal number of poles and zeros in the signal model could be decided for each phoneme empirically. It may result that length of the FIR synthesis filter can be different for different phonemes. It needs to design a separate statistical model for each phoneme.
• The band-limited narrowband signal encoded at 12.2 kbps has frequency components between 300-3400 Hz approximately. Low-frequency components in the range of 0-300 Hz can be recovered to improve speech quality. In addition, losses obtained due to encoding of the narrowband signal can be reduced.
• We assumed the ideal AMR block in our work. It may produce error in obtaining the synthesis filter. Results might be better if the exact model of AMR is used.
• An analysis of the performance of the proposed bandwidth approach can be done by using different narrowband signals encoded at different bit rates, such as 4.75 kbps, 5.15 kbps, 5.90 kbps, 6.70 kbps, 7.40 kbps, 7.95 kbps, and 10.20 kbps.
• We could extend the work for the noisy signals.
• The H∞ sampled-data system theory could be used in other speech applications, such as speaker identification, speaker verification, and speech classification etc.
6. Summary and future work
• The H∞ sampled-data system theory could be explored for other signals, such as image, audio, ECG, etc.
Sampled-data system theory
A.1 History of sampled-data system theory . . . 108 A.2 Abbreviations . . . 109 A.3 A general closed-loop system . . . 109 A.4 Lifting and inverse lifting . . . 110 TH-2564_156102023
A. Sampled-data system theory
In this appendix, we provide a brief introduction to the sampled-data system theory.
A.1 History of sampled-data system theory
It is a well-established result that digital signal transmission has numerous benefits over analog transmission. Therefore, the analog/continuous signal is discretized using the sampling process. The resulting discrete signal is transmitted but reconstructed back to the continuous domain at the receiver. Here the main aim is to recover an analog signal from its samples with minimal error. This is called a (continuous/analog) signal reconstruction problem. The signal reconstruction problem is the fundamental problem in digital signal processing. The sampling theorem ( ) states that we can recover the original continuous signal from the sampled-data if it is band-limited. In practice, signals are not band-limited. Hence, a popular solution to achieve band-limited signal by using an anti-aliasing filter introduces another distortion due to the Gibbs phenomenon. Furthermore, the impulse response of the ideal anti-aliasing filter is difficult to implement as it is non-causal and does not decay very fast.
To find an optimal answer to the signal reconstruction problem in general, researchers started looking at these problems as mathematical optimization problems (see Sun et al. , and Unser ). Mathematically, this means the design of an analog to discrete converter (sampler) and a discrete to analog convertor (hold) given the error criterion. The major chal- lenge is in treating discrete and analog signals (or multi-rate) in a common framework. The Sampled data system theory provided such a framework. In 1995, Chen and Francis  applied the sampled-data system theory to the signal reconstruction problem entirely in the discrete domain (this problem is at the heart of this thesis). The continuous signal reconstruction problem was studied first in 1996 by Khargonekar and Yamamoto . Instead of aiming at exact reconstruction as in the Shannon case, minimizing the error without throwing away any frequencies is the main criterion in the signal reconstruction using sampled-data system theory.
The optimization is done using theH2-norm or H∞criterion. The sampled data system theory is applied to several signal processing applications using different error criteria with or without causality constraints after the Khargonekar and Yamamoto paper  in 1996. For exam-