Optimal Sampling and Interpolation

The main objective in this thesis is to design optimal samplers, downsamplers and interpolators (holds) which are required in signal processing. The sampled-data system theory is used to fulfill this objective in a generic setup. Signal processing, which includes signal transmission, storage and analysis, plays a significant role in a human society. Since ages mankind is trying to find better and better means of signal transmission. Sending signal over a long distance was a challenge in ancient time. However with the advent of electrical/optical signals and electromagnetic waves, signal transmission is a matter of seconds now even over long distances. In the later half of the twentieth century, the digital revolution changed the scenario of signal processing. Digital signal processing provides better quality, low cost, ease of implementation and reconfigurability. In digital signal transmission the original analog signal is sampled using a sampler before transmission. At the receiver side, the received signal is processed digitally to remove unwanted signals like noise and then it is interpolated using a hold to obtain a reconstructed signal. This reconstructed signal must look like our original analog signal. Most of the systems like mobile, TV etc. nowadays use digital signal transmission techniques. Using digital techniques it is also easy to store and analyze signals. Digital signal processing needs sampling of the original signal. Therefore, a fundamental question is: can we reconstruct the original signal from its samples using a hold? This is known as the signal reconstruction problem. The most famous answer of the signal reconstruction problem is given by Shannon’s sampling theorem for the bandlimited analog signals. Bandlimitedness rarely happens in practice therefore Shannon’s sampling theorem is not enough unless we apply some filters to make our analog signal bandlimited. Hence, researchers started looking at the signal reconstruction problem as a mathematical optimization problem from system theoretical viewpoint i.e. how to design samplers and holds such that the reconstructed signal resembles the original signal (measured in some norm sense). Here it is assumed that the spectrum of the signals are known. Sampleddata system theory is such an approach i.e. it is used to solve the signal reconstruction problem where the assumption of bandlimitedness is not required. It also enables us to obtain the solution with greater generality. A distinctive feature of the sampled-data system theory is that it optimizes the analog performance. This approach is much closer to reality as most of the signals we use are analog in nature and utilized in the analog domain. Another distinctive feature of sampled-data system theory is the use of signal model to describe the spectrum of the original analog signals. The choice of signal model depends upon several factors like ease of implementation, accuracy required in signal reconstruction and the information available about signals. One advantage of using sampled-data system theory in the design process is that we can calculate the reconstruction error without any practical implementation. Calculation of the reconstruction error boils down to calculation of the frequency truncated norm if the signal models are linear continuous time invariant (LCTI). In Chapter 3, we obtained closed form expressions to calculate the frequency truncated norms if the LCTI signal model is given by state-space. These methods are easy to implement in Matlab. The use of these closed form expressions to calculate the frequency truncated norms is not restricted to sampled-data system theory but also to other areas of system theory like model reduction. Downsampling of the sampled signal is required in several signal processing applications like audio, image etc. This complicates our signal reconstruction problem because there is a downsampler in between the sampler and hold, and we have have to work with multiple sampling rates. In Chapter 4 we provide a general formulation and solution of optimal downsampling in the sampled-data setup for all linear continuous time invariant signal models. Here we allow noncausal solutions. The effect of noise on the downsampling is also discussed in this chapter. Non-causal solutions that have access to the infinite future, provide a theoretical limit to our solutions. However, they are rarely used in practice because of their unrealizability. Most of the time we design sampler and hold with causality or relaxed causality constraint. This is because it is practically impossible to have access to all future inputs at a given point of time. The constraint of causality/ relaxed causality makes our problem a bit more difficult, but also more interesting. In Chapter 5, we provide a frequency domain abstract and implementable state-space solution to the optimal sampler design problem with a relaxed causality constraint. In this thesis, we used sampled data system theory to answer downsampler and (relaxed causal) sampler design problems. However, the sampled-data system theory has the potential to answer many more interesting optimization problems arising in signal processing.