SHOW: x(at) is a periodic signal with period T/a (a>0)

Yes, this is correct! You have shown that x_a(t) is also periodic with a period of T/a, as desired. So now you can use the same logic to show that x_b(t) is also periodic with a period of bT. Great job summarizing the content!
  • #1
VinnyCee
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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:
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  • #2
Anyone verify this?
 
  • #3
Yep, it's right!
 
  • #4
No, it's not right!

VinnyCee said:

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]
You can't assume that- that's what you are trying to prove!

[tex]x_a(t)\,=\,x_a\left(t\,+\,\frac{T}{a}\right)[/tex]

But from here on you are okay. You aren't using "[itex]x_a(t)= x_a(t + T/a)[/itex]" you are using a(x+ T)= a(x) which is your hypothesis.

[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?
 

1. What is a periodic signal?

A periodic signal is a type of signal that repeats itself over a specific time interval. This means that the signal has a regular pattern and the same sequence of values repeats after a certain period of time.

2. What does 'T/a' represent in the formula?

'T/a' represents the period of the signal divided by a constant value, which results in the time interval between each repetition of the signal. This constant value, 'a', can be any positive number greater than 0.

3. How is the period of a periodic signal calculated?

The period of a periodic signal can be calculated by finding the time interval between two consecutive repetitions of the signal. This can be done by measuring the distance between two peaks or two zero-crossings of the signal.

4. Can a periodic signal have a variable period?

Yes, a periodic signal can have a variable period. This means that the time interval between each repetition of the signal can change over time. In this case, the signal is known as a non-stationary periodic signal.

5. What are some examples of periodic signals?

Some examples of periodic signals include the daily rise and fall of the tides, the rotation of the Earth on its axis, and the oscillations of a pendulum. In technology, periodic signals are commonly used in audio signals, such as music or speech, and in electrical signals, such as AC current.

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