The sign function, sgn(t), is defined as: sgn(t) = {-1; t 0 (a) Find the Fourier transform of sgn (t). (b) Given that the unit step function u(t) is, u(t) = 1/2[sgn(t) + 1], calculate the Fourier transform U(j omega) of u(t) and show that the result is equal to that given on the Formula Sheet in the table “Fourier Transform Pairs”. An LTI is defined by the following differential equation relating output y(t) to input x(t), y'(t) + 1000y(t) = 1000 x(t). and the input is x(t) = 4 Product (200 t). (a) Find X(j omega). (b) Take the Fourier transform of the differential equation and solve for Y(j omega). (c) Find the system transfer function H(j omega) for this system.

41 1 - The sign function, sgn(t), is defined as: sgn(t) = {-1; t 0 (a) Find the Fourier transform of sgn (t). (b) Given that the unit step function u(t) is, u(t) = 1/2[sgn(t) + 1], calculate the Fourier transform U(j omega) of u(t) and show that the result is equal to that given on the Formula Sheet in the table "Fourier Transform Pairs". An LTI is defined by the following differential equation relating output y(t) to input x(t), y'(t) + 1000y(t) = 1000 x(t). and the input is x(t) = 4 Product (200 t). (a) Find X(j omega). (b) Take the Fourier transform of the differential equation and solve for Y(j omega). (c) Find the system transfer function H(j omega) for this system.

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images - The sign function, sgn(t), is defined as: sgn(t) = {-1; t 0 (a) Find the Fourier transform of sgn (t). (b) Given that the unit step function u(t) is, u(t) = 1/2[sgn(t) + 1], calculate the Fourier transform U(j omega) of u(t) and show that the result is equal to that given on the Formula Sheet in the table "Fourier Transform Pairs". An LTI is defined by the following differential equation relating output y(t) to input x(t), y'(t) + 1000y(t) = 1000 x(t). and the input is x(t) = 4 Product (200 t). (a) Find X(j omega). (b) Take the Fourier transform of the differential equation and solve for Y(j omega). (c) Find the system transfer function H(j omega) for this system.

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