Q3. (a) Consider the continuous function f(t)=sin(27nt). i) What is the period of f(t)? ii) What is the frequency of f(t)? The Fourier transform, F(u), of f(t) is purely imaginary, and because the transform of the sampled data consists of periodic copies of F(u), the transform of the sampled data, F(u), will also be purely imaginary. Draw a diagram of Fourier transform of the function and answer the following questions based on your diagram (assume that sampling starts at t = 0). iii) What would the sampled function and its Fourier transform look like in general if f(t) is sampled at a rate higher than the Nyquist rate? iv) What would the sample function look like in general if f(t) is sampled at a rate lower than the Nyquist rate? v) What would the sample function look like if f(t) is sampled at the Nyquist rate with samples taken at t=0, AT, 2AT,K? (b) Consider a linear, position-invariant image degradation system with impulse response h(x-α.y-ß)=¯[(x-a)²+(y-a)²] Suppose that the input to the system is an image consisting of a line of infinitesimal width located at x = a, and modeled by f(x,y)= 6(x-a), where 8 is an impulse. Assuming no noise, what is the output image g(x, y)?
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Please answer both subparts a and b.
![Q3. (а)
Consider the continuous function f (t) = sin (2rnt).
i) What is the period of f(t)?
ii) What is the frequency of f(t)?
The Fourier transform, F(u), of f(t) is purely imaginary, and because the transform of the
sampled data consists of periodic copies of F(u), the transform of the sampled data, F(µ),
will also be purely imaginary. Draw a diagram of Fourier transform of the funetion and
answer the following questions based on your diagram (assume that sampling starts at t = 0).
iii) What would the sampled function and its Fourier transform look like in general if f(t) is
sampled at a rate higher than the Nyquist rate?
iv) What would the sample function look like in general if f(t) is sampled at a rate lower than
the Nyquist rate?
v) What would the sample function look like if f(t) is sampled at the Nyquist rate with
samples taken at t=0,AT,2AT,K ?
(b)
Consider a linear, position-invariant image degradation system with impulse response
h(x- a,y-B) = el*-a)*+(y-8}*]
Suppose that the input to the system is an image consisting of a line of infinitesimal width
located at x = a, and modeled by f(x,y) = 8(x - a), where ô is an impulse. Assuming no
noise, what is the output image g(x, y)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3586d1c1-9dee-4677-8b90-689aee854188%2Fd3377ba8-da2d-4cb0-a383-ba0d6b81fa05%2Fmh20hf_processed.png&w=3840&q=75)
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