- #1
VinnyCee
- 489
- 0
Homework Statement
If x(t) is a periodic signal with period T, show that x(at), a > 0, is a periodic signal with period [itex]\frac{T}{a}[/itex], and [itex]x\left(\frac{t}{b}\right)[/itex], b > 0, is a periodic signal with period bT.
Homework Equations
HINT: Define [tex]x_a(t)\,=\,x(at)[/tex] and [tex]x_b(t)\,=\,x\left(\frac{t}{b}\right)[/tex]. Show that [tex]x_a\left(t\,+\,T_a\right)\,=\,x_a(t)\,\forall\,t\,\in\,\mathbb{R}[/tex] and [tex]x_b\left(t\,+\,T_b\right)\,=\,x_b(t)\,\forall\,t\,\in\,\mathbb{R}[/tex], where [tex]T_a\,=\,\frac{T}{a}[/tex] and [tex]T_b\,=\,bT[/tex].
The Attempt at a Solution
I take the hint, and define
[tex]x_a(t)\,=\,x(at)[/tex]
Now, I assume that [itex]x_a(t)[/itex] is periodic, with a period [itex]\frac{T}{a}[/itex]
[tex]x_a(t)\,=\,x_a\left(t\,+\,\frac{T}{a}\right)[/tex]
[tex]x_a\left(t\,+\,\frac{T}{a}\right)\,=\,x\left[a\left(t\,+\,\frac{T}{a}\right)\right]\,=\,x\left(at\,+\,T\right)[/tex]
[tex]\therefore\,x_a\left(t\,+\,\frac{T}{a}\right)\,=\,x_a(t)\,\forall\,t\,\in\,\mathbb{R}[/tex]
Does this seem right?
Last edited: