Even and Odd Signals: Exploring Questions

In summary: The reason is because when you time shift x(-t) it shifts away from the t-axis and when you time shift x(t-3) it shifts towards the -ve t-axis.
  • #1
tina_singh
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1)-why is x(t)+x(-t) always even??..no matter if x(t) even or odd?

2)-when we talk about unit step function...u(t)..and we add..u(t)+u(-t)..the value of both is 1 at t=0..so does'n't that gets added twice??..and it becomes 2 at t=0...

3)when we have x(-t) and we time shift it say x(-t-3) it shifts toward the -ve t axis.. where as x(t-3) the function is shifted on the + axis..why is it so??

i would be really greatful if you can help me out with the above 3 doubts..
 
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  • #2


1) x(t)+x(-t) = x(t)+x(t) for even functions and if x(t) is even then x(t) + x(t) would be even.
As for odd functions the definition of an odd function is for an f(x), -f(x)=f(-x) and therefore x(t)+x(-t) = 0 for an odd function which is technically even.

2) Yes I believe so assuming you define your step function as u(t) = 1 if t≥0 and 0 otherwise

3) For x(t), a shift to x(t-3) would be a shift to the right. Likewise if you wanted to shift x(-t) to the right you would need to have x(-(t-3)) =x(-t+3) and NOT x(-t-3). Just be careful and use parentheses because when you shift you are substituting the independent variable not just throwing a "-3" in there somewhere.
 
  • #3


tina_singh said:
1)-why is x(t)+x(-t) always even??..no matter if x(t) even or odd?

Define F(t) = x(t) + x(-t). F will be even if F(-t) = F(t). Does that work? Does it matter what the formula for x(t) is?

2)-when we talk about unit step function...u(t)..and we add..u(t)+u(-t)..the value of both is 1 at t=0..so does'n't that gets added twice??..and it becomes 2 at t=0...

The value at a single point usually doesn't matter because u(t) is usually used in integration. Sometimes u(t) isn't even defined at 0 because of this; it is just defined as u(t) = 0 for t < 0 and u(t) = 1 for t > 0.
3)when we have x(-t) and we time shift it say x(-t-3) it shifts toward the -ve t axis.. where as x(t-3) the function is shifted on the + axis..why is it so??

i would be really greatful if you can help me out with the above 3 doubts..

Both x(t-3) and x((-t) - 3) are shifted to the right.
 

1. What is the difference between even and odd signals?

Even and odd signals are two types of signals that are commonly used in signal processing. The main difference between them is their symmetry. Even signals are symmetric about the y-axis, meaning that they have the same value at equal distances on either side of the y-axis. Odd signals, on the other hand, are anti-symmetric about the y-axis, meaning that they have opposite values at equal distances on either side of the y-axis.

2. How can even and odd signals be identified?

Even and odd signals can be identified by analyzing their mathematical expressions or graphs. An even signal can be identified if it satisfies the condition f(-t) = f(t), while an odd signal can be identified if it satisfies the condition f(-t) = -f(t). In other words, if the function remains unchanged when the independent variable is replaced by its negative, it is an even or odd signal.

3. What are some real-life examples of even and odd signals?

Even and odd signals can be observed in various real-life situations. For example, a square wave is an odd signal as it has alternating positive and negative values. Similarly, a sine wave is an even signal as it is symmetric about the y-axis. Other examples include the sound waves produced by musical instruments, where the even and odd components of the signal determine the timbre or quality of the sound.

4. How are even and odd signals used in signal processing?

Even and odd signals are commonly used in signal processing for different purposes. Even signals have the property of having all even harmonics, while odd signals have only odd harmonics. This property is utilized in various electronic circuits, such as filters and amplifiers, to achieve specific frequency responses. Additionally, even and odd signals are also used in digital signal processing to analyze and manipulate signals in the time and frequency domains.

5. Can a signal be both even and odd?

No, a signal cannot be both even and odd at the same time. This is because the conditions for even and odd signals are mutually exclusive. If a signal satisfies the condition for being even, it cannot satisfy the condition for being odd and vice versa. However, a signal can be a combination of even and odd components, such as a triangle wave, which has both even and odd components in its Fourier series representation.

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