a) Suppose x(t)=5sinc(200πt). Using properties of Fourier transform, write down the Fourier transform and sketch the magnitude spectrum, ∣X(ω)∣, of: i) x1​(t)=−4x(t−4), ii) x2​(t)=ej400πtx(t), iii) x3​(t)=cos(400πt)x(t) b) Consider a system with input, x(t), output, y(t), and unit impulse response, h(t)=e−2hu(t). If it is excited by a rectangular pulse, x(t)=u(t+2)−u(t−2), find an expression for Y((ω).

a)The Fourier transforms and magnitude spectra are:

i) X1(ω) = -4X(ω)ej4ω, |X1(ω)| = 4|X(ω)|

ii) X2(ω) = X(ω - 400π), |X2(ω)| = |X(ω - 400π)|

iii) X3(ω) = (1/2)[X(ω - 400π) + X(ω + 400π)], |X3(ω)| = (1/2)|X(ω - 400π)| + (1/2)|X(ω + 400π)|

b) The expression for Y(ω) is given by Y(ω) = [tex]e^(^-^2^j^ω^)^/^j^ω[/tex] * [[tex]e^(^4^j^ω^)[/tex] - [tex]e^(^-^4^j^ω^)[/tex]].

a) The Fourier transform and magnitude spectrum of a signal x(t) can be manipulated using properties of the Fourier transform. In the given question, we are asked to find the Fourier transforms and magnitude spectra of three different signals derived from the original signal x(t) = 5sinc(200πt).

i) For the first case, x1(t) = -4x(t - 4), we observe a time shift of 4 units to the right. The Fourier transform of x1(t) is given by X1(ω) = -4X(ω)ej4ω, where X(ω) is the Fourier transform of x(t). The magnitude spectrum, |X1(ω)|, is obtained by taking the absolute value of X1(ω), which simplifies to 4|X(ω)|.

ii) In the second case, x2(t) = ej400πtx(t), we introduce a modulation term in the time domain. The Fourier transform of x2(t) is given by X2(ω) = X(ω - 400π), which represents a frequency shift of 400π. The magnitude spectrum, |X2(ω)|, is equal to the magnitude of X(ω - 400π).

iii) For the third case, x3(t) = cos(400πt)x(t), we multiply the original signal x(t) by a cosine function. The Fourier transform of x3(t) is given by X3(ω) = (1/2)[X(ω - 400π) + X(ω + 400π)]. The magnitude spectrum, |X3(ω)|, is the sum of the magnitudes of X(ω - 400π) and X(ω + 400π), divided by 2.

b) In order to find the expression for Y(ω), we need to determine the Fourier Transform of the system's impulse response, h(t), and the Fourier Transform of the input signal, x(t). The given impulse response is h(t) = [tex]e^(^-^2^t^)^u^(^t^)[/tex], where u(t) is the unit step function. The Fourier Transform of h(t) is H(ω) = 1 / (jω + 2), where j is the imaginary unit and ω represents the angular frequency.

The rectangular pulse input, x(t), is defined as x(t) = u(t + 2) - u(t - 2), where u(t) is the unit step function. To find the Fourier Transform of x(t), we can utilize the time-shifting property and the Fourier Transform of the unit step function. Applying the time-shifting property, we get x(t) = u(t + 2) - u(t - 2) = u(t) - u(t - 4). The Fourier Transform of x(t) is X(ω) = 1 / jω * (1 - [tex]e^(^-^4^j^ω^)[/tex]).

To obtain the expression for Y(ω), we multiply the Fourier Transform of the input signal, X(ω), by the Fourier Transform of the impulse response, H(ω). Multiplying X(ω) and H(ω), we get Y(ω) = X(ω) * H(ω) = 1 / (jω * (jω + 2)) * (1 - [tex]e^(^-^4^j^ω^)[/tex]). Simplifying this expression yields Y(ω) = [tex]e^(^-^2^j^ω^)^/^j^ω[/tex] * [[tex]e^(4^j^ω)[/tex] - [tex]e^(4^j^ω)[/tex]].

Learn more about magnitude

brainly.com/question/28714281

#SPJ11

Consider the 90Sr source and its decay chain from problem #6. You want to build a shield for this source and know that it and its daughter produce some high energy beta particles and moderate energy gamma rays. a. Use the NIST Estar database to find the CSDA range [in cm) and radiation yield for the primary beta particles in this problem assuming a copper and a lead shield. b. Based on your results in part a, explain which material is better for shielding these beta particles.