Abstract
The main objective in this thesis is to design optimal samplers, downsamplers and
interpolators (holds) which are required in signal processing. The sampleddata
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 sampleddata 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 sampleddata
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 sampleddata 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 statespace. 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 sampleddata
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 sampleddata
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.
Noncausal 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
statespace 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 sampleddata system
theory has the potential to answer many more interesting optimization problems
arising in signal processing.
Original language  Undefined 

Awarding Institution 

Supervisors/Advisors 

Award date  29 Nov 2012 
Place of Publication  Enschede 
Publisher  
Print ISBNs  9789036534734 
DOIs  
Publication status  Published  29 Nov 2012 
Keywords
 Samples
 EWI22628
 Signals
 IR82465
 METIS289816
 System(s)
Cite this
Shekhawat, H. (2012). Optimal Sampling and Interpolation. Enschede: University of Twente. https://doi.org/10.3990/1.9789036534734