Question

1. Let h(t) be the triangular pulse shown in Figure 1.3(a) and let x(t) be the unit impulse train shown in Figure 1(b) and expressed as \(x(t) = \delta_\tau(t) = \sum_{n=-\infty}^{\infty} \delta(t - nT)\) Determine using graphical method and sketch y(t) = h(t)*x(t) (Convolution) for the following values of T: i) ii) T = 3 T = 1.5 2. Find the CTFT of the following signals a. \(x(t) = e^{-2(t-1)}u(t-1)\) b. \(x(t) = e^{-2|t|}\) 3. Find the inverse Fourier transform of \(X_1(j\omega) = u(\omega) - u(\omega - 2)\)? 4. Given that x(t) has the Fourier transform X(j\omega), express the Fourier transforms of the signals listed below in terms of X(j\omega). Use properties of Fourier transform. a. \(x_1(t) = x(1 - t) + x(-1 - t)\) b. \(x_2(t) = x(3t - 6)\) 5. Determine whether each of the following statements is true or false. Justify your answers. a. An odd and imaginary signal always has an odd and imaginary Fourier transform. b. The convolution of an odd Fourier transform with an even Fourier transform is always odd.

          1. Let h(t) be the triangular pulse shown in Figure 1.3(a) and let x(t) be the unit impulse
train shown in Figure 1(b) and expressed as
\(x(t) = \delta_\tau(t) = \sum_{n=-\infty}^{\infty} \delta(t - nT)\)
Determine using graphical method and sketch y(t) = h(t)*x(t) (Convolution) for the
following values of T:
i)
 ii)
T = 3
T = 1.5
2. Find the CTFT of the following signals
a. \(x(t) = e^{-2(t-1)}u(t-1)\)
b. \(x(t) = e^{-2|t|}\)
3. Find the inverse Fourier transform of \(X_1(j\omega) = u(\omega) - u(\omega - 2)\)?
4. Given that x(t) has the Fourier transform X(j\omega), express the Fourier transforms of the
signals listed below in terms of X(j\omega). Use properties of Fourier transform.
a. \(x_1(t) = x(1 - t) + x(-1 - t)\)
b. \(x_2(t) = x(3t - 6)\)
5. Determine whether each of the following statements is true or false. Justify your
answers.
a. An odd and imaginary signal always has an odd and imaginary Fourier transform.
b. The convolution of an odd Fourier transform with an even Fourier transform is
always odd.

1. Let h(t) be the triangular pulse shown in Figure 1.3(a) and let x(t) be the unit impulse
train shown in Figure 1(b) and expressed as
x(t) = δτ(t) = ∑n=-∞^∞δ(t - nT)
Determine using graphical method and sketch y(t) = h(t)*x(t) (Convolution) for the
following values of T:
i)
ii)
T = 3
T = 1.5
2. Find the CTFT of the following signals
a. x(t) = e^-2(t-1)u(t-1)
b. x(t) = e^-2|t|
3. Find the inverse Fourier transform of X1(jω) = u(ω) - u(ω - 2)?
4. Given that x(t) has the Fourier transform X(jω), express the Fourier transforms of the
signals listed below in terms of X(jω). Use properties of Fourier transform.
a. x1(t) = x(1 - t) + x(-1 - t)
b. x2(t) = x(3t - 6)
5. Determine whether each of the following statements is true or false. Justify your
answers.
a. An odd and imaginary signal always has an odd and imaginary Fourier transform.
b. The convolution of an odd Fourier transform with an even Fourier transform is
always odd.

Added by Robert V.

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