Inverse Laplace Transform

Inverse Laplace Transform using Partial Fraction Method $latex \displaystyle F(s) = \frac {N(s)}{D(s)} &s=2 &bg=ffffff$ $latex \displaystyle N(s) \rightarrow &s=2 &bg=ffffff$ Numerator Polynomial $latex \displaystyle D(s) \rightarrow &s=2 &bg=ffffff$ Denominator Polynomial $latex \displaystyle \Rightarrow &s=2 &bg=ffffff$ If degree of numerator polynomial $latex \displaystyle N(s) &s=2 &bg=ffffff$ is higher than the degree of $latex \displaystyle D(s) &s=2 &bg=ffffff$, than we should divide … Continue reading Inverse Laplace Transform

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Laplace Transform Solved Problems – 2

Example 11 Laplace Transform of Triangular Pulse Solution Differentiate the $latex f(t) &s=2 &bg=ffffff$ Again differentiate the $latex \displaystyle \frac {df(t)}{dt} &s=2 &bg=ffffff$ $latex \displaystyle \frac {d^2f(t)}{dt^2} = A\delta(t) - 2A \delta(t-1) + A \delta (t-2)  &s=2 &bg=ffffff$ Now take the Laplace transform on both the side $latex \displaystyle s^2 F(s) = A - 2A e^{-s} + A … Continue reading Laplace Transform Solved Problems – 2

Laplace Transform Solved Problems – 1

Example 1 $latex \displaystyle f(t) = \delta(t) &s=2 &bg=ffffff$ $latex \displaystyle F(s) = \int_{-\infty}^{\infty} \delta(t) e^{-st} dt &s=3 &bg=ffffff$ We know the property of $latex \displaystyle \delta(t) &s=2 &bg=ffffff$ $latex \displaystyle x(t) \delta(t) = x(0) \delta(t)  &s=2 &bg=ffffff$ $latex \displaystyle F(s) = \int_{-\infty}^{\infty} \delta(t) dt  &s=3 &bg=ffffff$ Area of the $latex \displaystyle \delta(t)  &s=2 &bg=ffffff$ $latex \displaystyle \int_{-\infty}^{\infty} … Continue reading Laplace Transform Solved Problems – 1

Unilateral Laplace Transfom

$latex \displaystyle \Rightarrow  &s=2 &bg=ffffff$ Unilateral Laplace transform is also called as One-sided laplace transform. $latex \displaystyle \Rightarrow  &s=2 &bg=ffffff$ Unilateral Laplace transform is used for the analysis of Causal system. $latex \displaystyle f(t) \rightleftharpoons F(s)  &s=2 &bg=ffffff$ $latex \displaystyle F(s) = \int_{0^{-}}^{\infty} f(t) e^{-st} dt  &s=3 &bg=ffffff$ Initial Value Theorem $latex \displaystyle f(0^{+}) = \lim_{t \to 0^{+}} f(t) = … Continue reading Unilateral Laplace Transfom

Properties of Laplace Transform

Linearity $latex \displaystyle f_1(t) \rightleftharpoons F_1(s)  &s=2 &bg=ffffff$ $latex \displaystyle f_2(t) \rightleftharpoons F_2(s)  &s=2 &bg=ffffff$ $latex \displaystyle \Rightarrow a_1 f_1(t) + a_2 f_2(t) \rightleftharpoons a_1 F_1(s) + a_2 F_2(s)  &s=2 &bg=ffffff$ Proof $latex \displaystyle F_1(s) = \int_{-\infty}^{\infty} f_1(t) e^{-st} dt  &s=2 &bg=ffffff$ $latex \displaystyle F_2(s) = \int_{-\infty}^{\infty} f_2(t) e^{-st} dt  &s=2 &bg=ffffff$ $latex \displaystyle F(s) = \int_{-\infty}^{\infty} a_1 … Continue reading Properties of Laplace Transform

Basics of Laplace Transform

$latex \displaystyle \Rightarrow  &s=2 &bg=ffffff$ Laplace transform is used for the analysis of Stable, Unstable as well as Marginally stable systems. $latex \displaystyle \Rightarrow  &s=2 &bg=ffffff$ Laplace transform is the generalized representation of Fourier Transform.  $latex \displaystyle \Rightarrow  &s=2 &bg=ffffff$ Laplace transform converts any Linear differential equation into an algebric equation thats make analysis to be easy. $latex \displaystyle \Rightarrow  &s=2 &bg=ffffff$ … Continue reading Basics of Laplace Transform

Discrete Time Convolution

Discrete-Time signal as Sum of Weighted impulses We know the property of impulse function $latex \displaystyle x(n) \delta(n) = x(0) \delta(n) &s=2 &bg=ffffff$ $latex \displaystyle x(n) \delta(n-k) = x(k) \delta(n-k) &s=2 &bg=ffffff$ Now apply summation on both side of above equation $latex \displaystyle \sum_{k=-\infty}^{\infty} x(n) \delta(n-k) = \sum_{k=-\infty}^{\infty} x(k) \delta(n-k) &s=3 &bg=ffffff$ $latex \displaystyle x(n) … Continue reading Discrete Time Convolution

Properties of Z transform

Linearity $latex \displaystyle x_1(n) \rightleftharpoons X_1(z) &s=2 &bg=ffffff$ ROC : $latex \displaystyle a_1 < |z| < b_1 &s=2 &bg=ffffff$ $latex \displaystyle x_2(n) \rightleftharpoons X_2(z) &s=2 &bg=ffffff$ ROC : $latex \displaystyle a_2 <|z| < b_2 &s=2 &bg=ffffff$ According to property of Linearity $latex \displaystyle a_1 x_1(n) + a_2 x_2(n) \rightleftharpoons a_1 X_1(z) + a_2 X_2(z) &s=2 … Continue reading Properties of Z transform

Rank of a Matrix

Rank   $latex \rightarrow &s=2 &bg=ffffff$  No. of linearly independent rows or columns in a matrix $latex A &s=2 &bg=ffffff$ is called the Rank of $latex A &s=2 &bg=ffffff$ The rank is commonly represented by Rank (A) = rk (A) = $latex \displaystyle \rho (A) &s=2 &bg=ffffff$ $latex \Rightarrow &s=2 &bg=ffffff$  A fundamental result in linear algebra is that the … Continue reading Rank of a Matrix

Memoryless LTI System

Consider an LTI system $latex \displaystyle x(t) \rightarrow &s=2 &bg=ffffff$  input to the system $latex \displaystyle h(t) \rightarrow &s=2 &bg=ffffff$  impulse response of the system $latex \displaystyle y(t) \rightarrow &s=2 &bg=ffffff$  output to the system $latex \displaystyle y(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau  &s=3 &bg=ffffff$ $latex \Rightarrow &s=2 &bg=ffffff$ For an LTI system to be memory-less its … Continue reading Memoryless LTI System