📘 signals and systems

1. **Problem statement:** Given the Fourier transform $$F(\omega) = 9 e^{-5 j \omega^2 + j \frac{\omega}{2}}$$, find the original function $$f(t)$$ in the form $$f(t) = A H(t - t_0

1. The problem states that if $F(\omega)$ is the Fourier transform of $f(t)$, we need to find the Fourier transform $G(\omega)$ of the function $g(t) = -3 e^{6jt} f(t)$. 2. Recall

1. **Problem Statement:** We need to sketch the graph of the function $$y(t) = [x(t) + x(-t)]u(t)$$ where $x(t)$ is given and $u(t)$ is the unit step function.

1. **Problem Statement:** Explain the Z-transform definition and key properties including differentiation, initial and final value theorems, and convolution property.

1. **Problem statement:** Given the continuous LTI system described by the differential equation $$\frac{d^2y(t)}{dt^2} + 6 \frac{dy(t)}{dt} + 8 y(t) = x(t),$$

1. **Problem statement:** Find the output $y[n]$ of a system with impulse response $h[n] = \left(\frac{1}{6}\right)^n u[n]$ when the input is $x[n] = (n+1)\left(\frac{1}{4}\right)^

1. **Problem statement:** Find the output $y[n]$ of a system with impulse response $$h[n] = \left(\frac{1}{6}\right)^n u[n]$$ for the input $$x[n] = A \cdot \sin(\omega n)$$ using

1. **Problem Statement:** Find the average power of the discrete-time signal $x[n] = \cos\left(\frac{\pi n}{3}\right)$.\n\n2. **Formula for Average Power:** The average power $P$ o

1. The problem asks for the average power of the signal $x[n] = \cos\left(\frac{\pi n}{3}\right)$. 2. The average power $P$ of a discrete-time signal $x[n]$ is defined as

1. **Problem Statement:** Determine the Z-transform of the finite duration signals given by: (e) $x_5(n) = \delta(n)$

1. **Problem Statement:** Determine if each system is time-invariant or time-variant. 2. **Key Concept:** A system is time-invariant if a time shift in the input causes the same ti

1. **Find the z-transform of \(\delta(n)\).** The sequence \(\delta(n)\) is the unit impulse, which is 1 at \(n=0\) and 0 elsewhere.

1. **Problem Statement:** We are given the exponential signal $$x(t) = 3e^t$$ and asked to draw its analog and digital forms.

1. The problem is to express the piecewise function $$g(t) = \begin{cases} 3, & 0<t<2 \\ 2t-1, & 2\le t <5 \\ 9, & t\ge 5 \end{cases}$$

1. **State the problem:** Express the piecewise function $$f(t)=\begin{cases}t-2,&0<t<4\\6-t,&t\ge4\end{cases}$$

1. **State the problem:** Write the function \(f(t) = \begin{cases}1 - t, & 0 < t < 1 \\ 0, & t \geq 1 \end{cases}\) in terms of the unit step function \(u(t)\). 2. **Recall the un

1. **Problem Statement:** We are given a strictly bandlimited signal $x_a(t)$ with Fourier transform $X_a(j\Omega)$ such that $X_a(j\Omega) = 0$ for $|\Omega| > \Omega_0$. We want

1. Problema: Determinar si la función graficada representa la suma $x(t) = \text{Tri}(t + 1) + \text{Tri}(t) + \text{Tri}(t - 1)$, donde $\text{Tri}(t)$ es la función triangular es