9. Let f(æ)= X[a,b) (x) be the characteristic function of the interval [a, b] c [-7, 7], that is, Į 1 if r € (a, b), 0 otherwise. X[a,b} (x) = %3D (a Show that the Fourier series of ƒ is given by b - a f(r) ~ 27 e-ina – e-inb eina 2nin The sum extends over all positive and negative integers excluding 0.

Q: Find the finite cosine transform of 1 – where 0<x< T.

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Q: Difference Approximations of the Partial Derivatives

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Q: 5. Residue theorem Use the residue theorem to compute (a) ro∞ t2+1 dt.

A: Given integral is∫0∞t2+1t4+1dtThe integral is computed using Residue theorem as shown below.

Q: ra+t for – a<t < 0 a² 4) Find the Fourier transform of f(t) = < where 'a' is any odd number a-t for…

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Q: The Fourier transform of a function } (t) iš defihed by F(w) = f(t)e-jwtdt %3D a) Using the above…

A: The Fourier transform of a function f(t) is defined by F(ω)=∫-∞∞f(t) e-jωt dt 1) We have to find the…

Q: The Fourier transform of a continuous- time signal x(t) defined as x(t) = e at for t20 18 for t<0…

A: Given information : xt = e-at  ;  t≥00      ;  t&lt;0

Q: Show that the Fourier transform of f(t) defined by: -3, -a < t < 0 3, f (t) = 0 <t<a Elsewhere where…

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Q: power spectral densities of the input and output processes in the system are connected by the…

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Q: A homogeneous linear ODE with constant coefficientshas a characteristic equation having only the…

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Q: Let X (0,1). Derive the characteristic function of Y-X².

A: Y = X^2, where X is a random variable with a standard normal distribution (mean 0 and standard…

Q: A1 The function f(x) is defined by f(x) = { { 1 1- ½½|x| for |x| ≤ 2 for |x| >2. Calculate the form…

A: Here is the detailed explanation for the final answer: 1. Calculating f'(x) :  - For  x≤2 ,…

Q: [6] Find the Fourier integral representation of f(x) = { 02 0 1.

A: Fourier integral representation

Q: 6. (a) Use the Schröder-Bernstein Theorem to prove that [a, b]| = |(a,b)| for any a, b e R. (b) Use…

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Q: 2B. Find the analytic function f(z) = u+iv for which v= e(xcos y+ ysin y)

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Q: If (p) is the Fourier Sine transform of f(x) for p>0, then prove tha Af(x}} = - (- p) for p<0.

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Q: ) Compute the Fourier transform of the function. (a) f(x) = h(x + 1) – 2h(x) + h(x - 1), where 1, if…

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Q: The z-transform of a unit, u(n) function 1/(1-z1) b. c. 1/(1+z1) d. Σαz1 а.

A: We find Z transform of unit step function.

Q: ra+t for - a<t< 0 a2 4) Find the Fourier transform of f(t) = where 'a' is any odd number a-t for 0<t…

A: Given that ft=1a+ta2-a&lt;t&lt;01a-ta20&lt;t&lt;a Then the fourier transform is,…

Q: Given 30<t ₹2 10 elsewhere find Fw), the Fourier transform of ƒ(t). Compute the values of F(0), F(1)…

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Q: (a+t for -a<t<0 a2 4) Find the Fourier transform of f(t) = %3D a-t for 0<t <a where 'a' is any odd…

A: The Fourier transform of ft=a+ta2  for   -a&lt;t&lt;0a-t a2 for   0&lt;t&lt;a0         otherwise…

Q: 13. Let the function f (x) be 2n-periodic and suppose it is represented by the Fourier series: f (x)…

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Q: The function f(t) has a Fourier transform e P[f](w) = **ƒ(6 F - from the below list ○ F[f](w) = e =…

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Q: A1 The function f(x) is defined by f(x) = { 0 1-13|x| for |x| ≤ 2 for x > 2. Calculate the form of…

A: Step 1: Step 2:  Step 3:  from 2 we have  F[f]k = (1-cos2k)/k^2  Step 4:

Q: (27) Consider the piecewise function f : [-L, L] –→ R given by Ja + L, if – L <x < 0 f(x) : x*, if 0…

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Q: Solve for the inverse Laplace Transform of 8 F(s) 53 - 3s2 – 4s 8 f(t) = – 2t + 2 a. + te4t Ob. 8…

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Q: A: Find the Fourier transform f(t) = {sin nt, 0, for 0 <t<1 otherwise. B: Find the Fourier transform…

A: The formula for the Fourier Trasnform is

Q: Q1) Laplace transform of function f(t) = t² where t 2 0 is ....... a) b)금 d) 115 11 1

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Q: 2. Prove the following properties of Discrete-time Fourier transform (DTFT). DTFT DTFT DTFT 1 if…

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Q: 6. Find the Taylor polynomial T,(x) for the function f(x)= cosx at a="/ it to estimate cos(29780)

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Q: Condsider the ODE with co = 9, c₁ = 4, ... Assuming a particular solution of the form where the…

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Q: By integration find the Fourier transform of Ax)=2r+1 _if þx|<1_and fx)=0 otherwise.

A: According to our company policy, I am supposed to answer only one question at a time. For the rest…

Q: for |x|< a 0 for x<a Q4. Determine the Fourier transform of f(x) =

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Q: 4. Let (1, X (jw) = 10, 0 < x ≤1 otherwise Find x(t) using the inverse continuous-time Fourier…

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Q: f(n) - (0.5)" и(п) + (1.5)" и(-п-1)- 28(п)

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Q: 11: The Laplace transform of the function f(x) = xe4* is a) 1/(s – 4)2 b) 1/(s + 4)? c) 1/(s – 2)4…

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Q: 20. Let f be a continuous function with a period 27, and ao +(an cos nx + bn sin nx) 2 its Fourier…

A: Integral Representation of The constant term  in the fourier series represents the average value of…

Q: 18. f(x)=x²

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Q: Find Fourier transform of the Coulomb function -ar e V (r) = 4n r %3D where r = x2 +y2 +z2 . Hint…

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Q: Find the finite cosine transform of f (x) if ƒ (x) = -x+ in (0, t).

A: