Transformation of signals defined piecewise
Introduction Energy and Power Basic Operations Practice Problems Transformation of signals defined piecewise Even and Odd Signals Commonly encountered signals

Table of Contents

Example

Given signal $x(t)$ defined as follows-

(1)
\begin{align} x(t)= \begin{cases} 2t & \text{if}\ t>0 \\ -t & \text{if}\ t<0 \\ 0 & \text{if}\ t=0 \end{cases} \end{align}

Find $x(-t)$ ,$\frac{1}{2}\big[x(t)+x(-t)\big]$ and $x(1-2t)$.

Solution:

Given $x(t)$
Then,

(2)
\begin{align} x(-t)= \begin{cases} 2(-t) & \text{if}\ -t>0 \\ -(-t) & \text{if}\ -t<0 \\ 0 & \text{if}\ -t=0 \end{cases} \end{align}

On simplifying,

(3)
\begin{align} x(-t)= \begin{cases} -2t & \text{if}\ t<0 \\ t & \text{if}\ t>0 \\ 0 & \text{if}\ t=0 \end{cases} \end{align}
So, the corresponding plot is as follows-
ch1.1.5.1.jpg

Now,

(4)
\begin{align} \frac{1}{2}\big[x(t)+x(-t)\big]= \begin{cases} -\frac{3}{2}t & \text{if}\ t<0 \\ -\frac{3}{2}t & \text{if}\ t>0 \\ 0 & \text{if}\ t=0 \end{cases} \end{align}

Next,

(5)
\begin{align} x(1-2t)= \begin{cases} 2(1-2t) & \text{if}\ 1-2t>0 \Rightarrow t<\frac{1}{2}\\ -(1-2t) & \text{if}\ 1-2t<0 \Rightarrow t>\frac{1}{2}\\ 0 & \text{if}\ 1-2t=0 \Rightarrow t=\frac{1}{2} \end{cases} \end{align}

Generalizing the above, suppose

(6)
\begin{align} x(t)= \begin{cases} g_1(t) & \text{if}\ h_1(t)>0 \\ g_2(t) & \text{if}\ h_2(t)>0 \\ . & \text{.}\ \\ . & \text{.}\ \\ g_N(t) & \text{if}\ h_N(t)>0\\ g_{N+1}(t) & \text{if}\ h_{N+1}(t)>0\\ . & \text{.}\ \\ . & \text{.}\ \\ g_{N+M}(t) & \text{if}\ h_{N+M}(t)=0\\ \end{cases} \end{align}

What is $x(f(t)$?

(7)
\begin{align} x(f(t))= \begin{cases} g_1(f(t)) & \text{if}\ h_1(f(t))>0 \\ g_2(f(t)) & \text{if}\ h_2(f(t))>0 \\ . & \text{.}\ \\ . & \text{.}\ \\ g_N(f(t)) & \text{if}\ h_N(f(t))>0\\ g_{N+1}(f(t)) & \text{if}\ h_{N+1}(f(t))>0\\ . & \text{.}\ \\ . & \text{.}\ \\ g_{N+M}(f(t)) & \text{if}\ h_{N+M}(f(t))=0\\ \end{cases} \end{align}

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