24
Biomedical Signal and Image Processing
Direct calculation of convolution operations involves many complicated and timeconsuming mathematical computations. However, it is comforting to know that the
FT creates an alternative method of calculating the convolution that is much less
computationally intensive. The property described in Table 2.2 says that, in order
to calculate g 1 (t) * g 2 (t) (convolution of g 1 (t) and g 2 (t)), one must identify the involved
signals and calculate the FT of the respected signals (i.e., find G 1 ( f ) and G 2 ( f )), then
multiply the two (i.e., find G 1 ( f ) . G 2 ( f )), and then find the inverse Fourier transform
(IFT) of the result. This would compute g 1 (t) * g 2 (t) without the long and tedious integration or math involved in the true definition of convolution. Understanding the concept of
traversing from time domain to frequency domain and vice versa can save a significant amount of calculation time. The property of convolution is extremely important
for analyzing linear systems, as discussed in the following. The proof for this property
is left as an exercise and will be discussed further in Problems section.
2.2.1.3 Linear Systems Analysis
Before discussing the main usage of the FT in linear systems, we briefly discuss the
concept of linear systems that play an important role in signal and image processing.
Consider a system where an “input” stimulation of the system causes an “output”
response from the system. For example, consider a cart on an open area such a parking lot. If you push the cart with a certain input power, the cart will travel an output
distance. Linear systems are the systems in which the output linearly depends on the
input, i.e., the amplitude of the output is linearly proportional to the amplitude of the
input. Using the example mentioned earlier, if one pushes a cart with twice the original
force or power, the cart will travel twice the original distance. In a more mathematical context, if a push with power p(t) causes the cart to travel for q(t) meters, then a
push for α . p(t) will cause the cart to travel for α . q(t) (where α is a constant).
Linear systems have another property that deals with the response of a system to
two or more inputs that are applied to the system simultaneously. This property (which
is also referred to as “superposition”) states that the response of a linear system to
simultaneous inputs is the summation of the responses of the system to every individual input. Continuing our previous example, assume that if one person pushes the
cart with power p 1 (t), the cart will move for q 1 (t), and if another person pushes the cart
with p 2 (t), the cart will move for q 2 (t). Now, if both these people push the cart at the
same time, i.e., with power p 1 (t) + p 2 (t), the cart will travel for q 1 (t) + q 2 (t) meters, i.e., a
distance that is the summation of the distances the cart travels for each push.
In order to better understand the concept of linear systems, let us focus on the
contrast between linear and nonlinear systems. Nonlinear systems (as the name
suggests) are systems in which the relation between input and output is either not proportional or not superpositional. For example, in our previous example of pushing carts
in a parking lot, let us assume that there are some obstacles (such as cars and other
carts) parked in the lot. In that case, a small push can make a cart to travel for a certain distance, but a proportionally larger push may cause the cart to collide with other
objects and therefore fail to produce a proportionally large distance. This is a typical
example of a nonlinear system in which the complexity of the system prevents simple
and well-behaved characteristics such as proportionality and superposition. Nonlinear
systems are therefore much more difficult to model and analyze.
Biomedical Signal and Image Processing
Direct calculation of convolution operations involves many complicated and timeconsuming mathematical computations. However, it is comforting to know that the
FT creates an alternative method of calculating the convolution that is much less
computationally intensive. The property described in Table 2.2 says that, in order
to calculate g 1 (t) * g 2 (t) (convolution of g 1 (t) and g 2 (t)), one must identify the involved
signals and calculate the FT of the respected signals (i.e., find G 1 ( f ) and G 2 ( f )), then
multiply the two (i.e., find G 1 ( f ) . G 2 ( f )), and then find the inverse Fourier transform
(IFT) of the result. This would compute g 1 (t) * g 2 (t) without the long and tedious integration or math involved in the true definition of convolution. Understanding the concept of
traversing from time domain to frequency domain and vice versa can save a significant amount of calculation time. The property of convolution is extremely important
for analyzing linear systems, as discussed in the following. The proof for this property
is left as an exercise and will be discussed further in Problems section.
2.2.1.3 Linear Systems Analysis
Before discussing the main usage of the FT in linear systems, we briefly discuss the
concept of linear systems that play an important role in signal and image processing.
Consider a system where an “input” stimulation of the system causes an “output”
response from the system. For example, consider a cart on an open area such a parking lot. If you push the cart with a certain input power, the cart will travel an output
distance. Linear systems are the systems in which the output linearly depends on the
input, i.e., the amplitude of the output is linearly proportional to the amplitude of the
input. Using the example mentioned earlier, if one pushes a cart with twice the original
force or power, the cart will travel twice the original distance. In a more mathematical context, if a push with power p(t) causes the cart to travel for q(t) meters, then a
push for α . p(t) will cause the cart to travel for α . q(t) (where α is a constant).
Linear systems have another property that deals with the response of a system to
two or more inputs that are applied to the system simultaneously. This property (which
is also referred to as “superposition”) states that the response of a linear system to
simultaneous inputs is the summation of the responses of the system to every individual input. Continuing our previous example, assume that if one person pushes the
cart with power p 1 (t), the cart will move for q 1 (t), and if another person pushes the cart
with p 2 (t), the cart will move for q 2 (t). Now, if both these people push the cart at the
same time, i.e., with power p 1 (t) + p 2 (t), the cart will travel for q 1 (t) + q 2 (t) meters, i.e., a
distance that is the summation of the distances the cart travels for each push.
In order to better understand the concept of linear systems, let us focus on the
contrast between linear and nonlinear systems. Nonlinear systems (as the name
suggests) are systems in which the relation between input and output is either not proportional or not superpositional. For example, in our previous example of pushing carts
in a parking lot, let us assume that there are some obstacles (such as cars and other
carts) parked in the lot. In that case, a small push can make a cart to travel for a certain distance, but a proportionally larger push may cause the cart to collide with other
objects and therefore fail to produce a proportionally large distance. This is a typical
example of a nonlinear system in which the complexity of the system prevents simple
and well-behaved characteristics such as proportionality and superposition. Nonlinear
systems are therefore much more difficult to model and analyze.