📘 signals and systems

1. 問題陳述： 我們要計算序列 $y[n] = A^n \{u[n-2] - u[n-13]\} * 2\{u[n+2] - u[n-12]\}$ 的卷積。

1. **State the problem:** We need to find the inverse Z-transform $e(k)$ of the function $$E(z) = \frac{z}{z^{2} + 3z + 2}.$$\n\n2. **Factor the denominator:** The denominator is a

1. **Problem Statement:** Find the z-transform of the given functions: (i) $$F(s) = \frac{1}{(s+a)^2}$$

1. **Problem statement:** Find the z-transform of the given Laplace domain functions: (i) $$F(s) = \frac{1}{(s+a)^2}$$

1. **Problem Statement:** Find the Laplace Transform $\mathcal{L}\{f(t)\}$ of the piecewise function:

1. **Problem Statement:** We need to find the Laplace Transform of the CPU load function:

1. **Problem statement:** Given the sequence $$x[n] = (6 - n)[u[n] - u[n - 6]]$$ where $$u[n]$$ is the unit step function, we want to find and sketch the sequences: 1) $$y_1[n] = x

1. **Problem Statement:** We need to find and draw the function $$z(t) = 10 \cdot x_1(t) \times 5 \cdot x_2(t) = 50 \cdot x_1(t) \cdot x_2(t)$$ where $$x_1(t)$$ is a rectangular pu

1. مسئله: ضریب‌های سری فوریه سیگنال‌های داده شده را با دوره تناوب $T=4$ محاسبه کنید. 2. فرمول‌های سری فوریه برای سیگنال تناوبی با دوره $T$ به صورت زیر است:

1. مسئله: تابع تناوبی $x(t)$ به صورت قطعه‌ای تعریف شده است: $$x(t) = \begin{cases} 1 & 0 < t < 2 \\ -1 & 2 < t < 4 \end{cases}$$

1. **Problem Statement:** Given several signals and transforms, we need to analyze and solve for Laplace transforms and regions of convergence (ROC) for the exponential signals, an

1. مسئله: رسم سیگنال‌های گسسته و پیوسته با اعمال تبدیلات مختلف روی محور زمان. 2. بخش a) سیگنال $x[-n + 2]$ با $n$ از $-2$ تا $2$ و مقادیر $x[n]$ داده شده است.

1. مسئله: بررسی کنیم آیا دنباله $Y[n] = x[4n+1]$ خطی است یا خیر. 2. تعریف خطی بودن: یک سیستم یا دنباله خطی است اگر دو خاصیت ابرپوزیشن و هم‌خطی را داشته باشد.

1. **Problem Statement:** Find the complex exponential Fourier series representation of the signal $$m(\beta) = \frac{\sin 6\beta}{2} + \frac{\cos 2\beta}{3}$$. 2. **Recall the com

1. **State the problem:** Find the Complex Exponential Fourier Series representation of the signal $$s(t) = 4 \sin\left(2t + \frac{\pi}{6}\right)$$. 2. **Recall the formula:** The

1. **Problem Statement:** Obtain the trigonometric Fourier series for the periodic triangular wave shown in Figure 2.5 with period $T=4$, peaks of height 1 at $t=-2$ and $t=2$, and

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 $$\frac{d^2y(t)}{dt^2} + 6 \frac{dy(t)}{dt} + 8 y(t) = x(t),$$