discrete-time signal

[logistic_guy]

A discrete-time signal \(\displaystyle x(n)\) is defined as

\(\displaystyle x(n) = \begin{cases} 1 + \frac{n}{3}, & -3 \leq n \leq - 1 \\1, & 0 \leq n \leq 3 \\0, & \text{elsewhere}\end{cases}\)

\(\displaystyle \bold{(a)}\) Determine its values and sketch the signal \(\displaystyle x(n)\).
\(\displaystyle \bold{(b)}\) Sketch the signals that result if we:
\(\displaystyle \ \ \ \ \ \ \ \ \bold{1.}\) First fold \(\displaystyle x(n)\) and then delay the resulting signal by four samples.
\(\displaystyle \ \ \ \ \ \ \ \ \bold{2.}\) First delay \(\displaystyle x(n)\) by four samples and then fold the resulting signal.
\(\displaystyle \bold{(c)}\) Sketch the signal \(\displaystyle x(-n + 4)\).
\(\displaystyle \bold{(d)}\) Compare the results in parts \(\displaystyle \bold{(b)}\) and \(\displaystyle \bold{(c)}\) and derive a rule for obtaining the signal \(\displaystyle x(-n + k)\) from \(\displaystyle x(n)\).
\(\displaystyle \bold{(e)}\) Can you express the signal \(\displaystyle x(n)\) in terms of signals \(\displaystyle \delta(n)\) and \(\displaystyle u(n)\)?

 

[logistic_guy]

\(\displaystyle \bold{(a)}\) Determine its values and sketch the signal \(\displaystyle x(n)\).

\(\displaystyle x(n) = \bigg\{\cdots ,0,\frac{1}{3},\frac{2}{3},\textcolor{red}{1},1,1,1,0, \cdots\bigg\}\)

 

[logistic_guy]

\(\displaystyle \bold{(b)}\) Sketch the signals that result if we:
\(\displaystyle \ \ \ \ \ \ \ \ \bold{1.}\) First fold \(\displaystyle x(n)\) and then delay the resulting signal by four samples.

\(\displaystyle x(-n + 4) = \left\{\cdots,\textcolor{red}{0},1,1,1,1,\frac{2}{3},\frac{1}{3},0,\cdots\right\}\)

 

[logistic_guy]

\(\displaystyle \bold{(b)}\) Sketch the signals that result if we:
\(\displaystyle \ \ \ \ \ \ \ \ \bold{2.}\) First delay \(\displaystyle x(n)\) by four samples and then fold the resulting signal.

\(\displaystyle x(-n-4) = \left \{\cdots,0,1,1,1,1,\frac{2}{3},\frac{1}{3},0,\textcolor{red}{0},\cdots\right \}\)

 

[logistic_guy]

\(\displaystyle \bold{(c)}\) Sketch the signal \(\displaystyle x(-n + 4)\).

I will leave this as an exercise for the audience to sketch.

\(\displaystyle \bold{(d)}\) Compare the results in parts \(\displaystyle \bold{(b)}\) and \(\displaystyle \bold{(c)}\) and derive a rule for obtaining the signal \(\displaystyle x(-n + k)\) from \(\displaystyle x(n)\).

To get \(\displaystyle x(-n+k)\), first fold \(\displaystyle x(n)\), then shift the folded to the right if \(\displaystyle k > 0\) or to the left if \(\displaystyle k < 0\).

 

[logistic_guy]

\(\displaystyle \bold{(e)}\) Can you express the signal \(\displaystyle x(n)\) in terms of signals \(\displaystyle \delta(n)\) and \(\displaystyle u(n)\)?

Yes, we can.

\(\displaystyle x(n) = \textcolor{blue}{\frac{1}{3}\delta(n + 2) + \frac{2}{3}\delta(n + 1) + u(n) - u(n - 4)}\)