1-2 linear systems theory

1-2 linear systems theory - BENG101 Foundations of...

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BENG101 Foundations of Biomedical Imaging Fall 2009 Lecture 2: Linear Systems Theory
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Why do we need to have an understanding of signals, linear systems theory, and transforms? Because in medical imaging the data that is collected (i.e. measured) are mathematically represented as signals (i.e. functions), and their manipulation into images and other functional outputs requires the use of linear systems theory via the use of mathematical transforms.
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So what is a transform? It is a mathematical operation that takes a function (i.e. signal) in one space (i.e. represented as a set of specific variables) and transforms it into an equivalent signal in another space.
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One example is the Fourier transform, which can be used to represent a time domain signal as a summation of component sinusoidal functions of varying frequencies with differing amplitudes (i.e. the same original function/signal in the frequency domain). (To be introduced in some detail below.)
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Another example is the Radon transform, which takes x-ray projections measured through an object (a sinogram) and transforms them into interpretable anatomical image reconstructions. i.e. this is the mathematical basis of computed tomography (CT) which we will study in detail later in the course.
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Characterization of Signals • Signal: description of how 1 parameter relates to another – Dependent parameter (y-axis) – Independent parameter (x-axis) • Signals arise when one parameter (dependent) is a function of another parameter (independent) • When time is the independent variable signals are in the time domain • When frequency is the independent variable signals are in the frequency domain • Types: – Continuous: continuous range of values – Discrete or Digitized: only discrete values
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Example of Digitized Signal 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 1.0 2.0 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 1.0 2.0 3.0 continuous signal discrete signal
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Signal Mean (μ) • Average value of a signal • In electronics the mean is called the DC value μ= 1 N x i 0 N 1
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Measure of Signal Variability • AC refers to how a signal fluctuates around the mean value • Standard deviation ( σ ): measures how far the signal fluctuates from the mean • Variance: σ 2 = ( x i − μ ) 2 0 N 1 N 1
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Relation Between σ and Peak to Peak Value (Vpp) for Common Signals
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1-2 linear systems theory - BENG101 Foundations of...

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