Transformation of signals defined piecewise

### 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-

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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page revision: 2, last edited: 25 Jan 2017 17:47