6 Questions
What is the Laplace transform of the function f(t) = 4e^(t)sin(wt)?
4s/(s² + w²)
Define Fourier cosine transform and give an example of its application.
The Fourier cosine transform of a function f(x) is defined as Fc(ω) = ∫[0 to ∞] f(x)cos(ωx)dx. Example: The Fourier cosine transform is used in image processing to filter out high-frequency noise.
What is the Z-transform of the sequence u₁ = 7, n≥0?
U(z) = 7z / (z - 1), |z| > 1
What is a harmonic function, and give an example?
A harmonic function is a function that satisfies Laplace's equation. Example: The electric potential around a charged sphere is a harmonic function.
What is a singularity in a complex function, and give an example?
A singularity is a point where a complex function is not defined or is infinite. Example: The function f(z) = 1/z has a singularity at z = 0.
What is an inner product space, and give an example?
An inner product space is a vector space with an inner product that satisfies certain properties. Example: The set of all complex-valued functions on the unit interval [0, 1] with the inner product 〈f, g〉 = ∫[0 to 1] f(x)g(x)dx is an inner product space.
Test your knowledge of signals and systems with this quiz, covering topics such as Laplace transforms, Fourier transforms, Z-transforms, and harmonic functions.
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