- #1
JonathanT
- 18
- 0
So I'm trying to find the DTFT of the following; where u(n) is the unit step function.
[itex]u \left( n \right) =\cases{0&$n<0$\cr 1&$0\leq n$\cr}[/itex]
I want to find the DTFT of
[itex]u \left( n \right) -2\,u \left( n-8 \right) +u \left( n-16 \right)[/itex]
Which ends up being a piecewise defined function looking like
[itex]u \left( n \right) -2\,u \left( n-8 \right) +u \left( n-16 \right) = \cases{1&$0\leq n$\ and \ $n\leq 7$\cr -1&$8\leq n$\ and \ $n\leq 15$\cr}[/itex]
With the function zero elsewhere.
I plug this into the formula for a DTFT and get the following:
[itex]1+{{\rm e}^{-i\omega}}+{{\rm e}^{-2\,i\omega}}+{{\rm e}^{-3\,i\omega}}
+{{\rm e}^{-4\,i\omega}}+{{\rm e}^{-5\,i\omega}}+{{\rm e}^{-6\,i\omega
}}+{{\rm e}^{-7\,i\omega}}-{{\rm e}^{-8\,i\omega}}-{{\rm e}^{-9\,i
\omega}}-{{\rm e}^{-10\,i\omega}}-{{\rm e}^{-11\,i\omega}}-{{\rm e}^{-
12\,i\omega}}-{{\rm e}^{-13\,i\omega}}-{{\rm e}^{-14\,i\omega}}-{
{\rm e}^{-15\,i\omega}}[/itex]
This should be correct, however, it is very ugly and I'm looking for a better form for my answer. I cannot reduce the summation using a harmonic series because the coefficient |a| = 1.
I can keep it in summation form but I feel like I'm missing an easy step that can simplify this.
Thanks for any help you can offer.
[itex]u \left( n \right) =\cases{0&$n<0$\cr 1&$0\leq n$\cr}[/itex]
I want to find the DTFT of
[itex]u \left( n \right) -2\,u \left( n-8 \right) +u \left( n-16 \right)[/itex]
Which ends up being a piecewise defined function looking like
[itex]u \left( n \right) -2\,u \left( n-8 \right) +u \left( n-16 \right) = \cases{1&$0\leq n$\ and \ $n\leq 7$\cr -1&$8\leq n$\ and \ $n\leq 15$\cr}[/itex]
With the function zero elsewhere.
I plug this into the formula for a DTFT and get the following:
[itex]1+{{\rm e}^{-i\omega}}+{{\rm e}^{-2\,i\omega}}+{{\rm e}^{-3\,i\omega}}
+{{\rm e}^{-4\,i\omega}}+{{\rm e}^{-5\,i\omega}}+{{\rm e}^{-6\,i\omega
}}+{{\rm e}^{-7\,i\omega}}-{{\rm e}^{-8\,i\omega}}-{{\rm e}^{-9\,i
\omega}}-{{\rm e}^{-10\,i\omega}}-{{\rm e}^{-11\,i\omega}}-{{\rm e}^{-
12\,i\omega}}-{{\rm e}^{-13\,i\omega}}-{{\rm e}^{-14\,i\omega}}-{
{\rm e}^{-15\,i\omega}}[/itex]
This should be correct, however, it is very ugly and I'm looking for a better form for my answer. I cannot reduce the summation using a harmonic series because the coefficient |a| = 1.
I can keep it in summation form but I feel like I'm missing an easy step that can simplify this.
Thanks for any help you can offer.