2. Two-dimensional sequences and systems
Two-dimensional sequences and systems are
mathematical models used to represent and
analyze signals or data that vary in two
dimensions, such as images, video, and
distributed measurements.
3. Two-dimensional sequences
A two-dimensional sequence is a function of two discrete
variables, usually denoted as x(n1, n2), where n1 and n2
are integers. Each point (n1, n2) in the two-dimensional
plane is associated with a value x(n1, n2).
For example, a grayscale image
can be represented as a two-
dimensional sequence, where
each pixel location corresponds
to a value in the sequence.
Figure: Grayscale image.
4. Two-dimensional systems
A two-dimensional system is an operator that
maps an input sequence x(n1, n2) to an output
sequence y(n1, n2). A two-dimensional system
can be represented by a two-dimensional impulse
response function h(n1, n2), which describes the
output of the system when the input is a unit
impulse at position (n1, n2).
5. Two-dimensional sequences and systems can be
analyzed using various mathematical tools such as
linear algebra, differential equations, and
probability theory. They can be convolved using a
two-dimensional convolution operation,
transformed using two-dimensional Fourier
transforms, and processed using various signal
processing techniques such as filtering, edge
detection, and feature extraction.
6. Properties of Two-dimensional sequences
and systems
Two-dimensional sequences can be viewed as a
collection of one-dimensional sequences, where
each row or column of the two-dimensional
sequence is a one-dimensional sequence.
Two-dimensional systems are operators that
map an input sequence x(n1,n2) to an output
sequence y(n1,n2). They can be linear or
nonlinear, time-invariant or time-varying.
7. Properties of Two-dimensional sequences
and systems
Two-dimensional systems can be represented by a
two-dimensional impulse response function
h(n1,n2), which describes the output of the system
when the input is a unit impulse at position (n1,n2).
Two-dimensional sequences and systems can be
convolved using a two-dimensional convolution
operation. The output sequence is given by the sum
of the products of the input sequence and the
flipped and shifted impulse response function.
8. Properties of Two-dimensional sequences
and systems
It can be transformed using two-dimensional Fourier
transforms, which decompose the signal into its
frequency components. The two-dimensional
Fourier transform of a sequence x(n1,n2) is given by
X(k1,k2), where k1 and k2 are the frequency
variables.
Two-dimensional sequences and systems can be
analyzed using various mathematical tools such as
linear algebra, differential equations, and probability
theory.
