Defining a Signal. periodic, bounded finite etc.

Click For Summary

Discussion Overview

The discussion revolves around defining a signal represented by the expression x(k) = j^-k u(k), where participants explore its properties, including periodicity, boundedness, finiteness, and power. The context includes theoretical considerations and mathematical reasoning related to signal processing.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about defining the signal due to the presence of the imaginary unit j and seeks clarification on its properties.
  • Another participant suggests starting with Euler's relation to analyze the signal.
  • A participant attempts to manipulate the signal and questions how Euler's identity applies, noting that substituting values leads to different forms of the signal.
  • There is a discussion about the periodic nature of the signal, with one participant asserting it is periodic and providing a sequence of values (+1, -j, -1, +j, +1, ...).
  • Some participants express uncertainty about the implications of the imaginary unit on the existence of the signal and its properties.
  • One participant mentions the assumption that k represents integers and acknowledges a lack of familiarity with Euler's identity.
  • Another participant references a textbook chapter that may provide further insights into the signal's properties and suggests that the complex Fourier series could help determine its power.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the properties of the signal. While some assert it is periodic, others express uncertainty and explore different interpretations without resolving the disagreements.

Contextual Notes

Participants acknowledge limitations in their understanding of Euler's identity and its application to the signal, as well as the implications of the imaginary unit on the signal's properties.

Who May Find This Useful

Individuals interested in digital signal processing, particularly those exploring the properties of complex signals and their mathematical representations.

Evo8
Messages
168
Reaction score
0
Im having a little trouble about how to go about defining this signal. It has a sqrt(-1) in it raised to a power so this is where i get confused. No doubt my poor algebra skills may be holding me back from understanding this problem.

The signal is x(k)=j^-k u(k)

I need to determine:
A. Whether or not the signal is periodic or aperiodic. If periodic what is the period?
B. Is it bounded or unbounded?
C. Finite or Infinite?
D. Calculate the power of x(k)

Im starting off with A. I know the definition of a periodic signal is if I can replace "k" with "K+N" and get the same signal. The value of N that achieves this is my period. I don't even know how to go about that with the j in there. j=square root(-1).

For a bounded signal I am not sure how to really go about this one either but since there is no "bounds" defined for k i would say its bounded with a bound of 1? I am not really sure on this though.

C. I would say this is a finite signal? I think because there are no operators to make the signal reach infinity?

D. I haven't attempted this yet.

I feel ill be able to deal with this problem a little bit better if i fully understand what to do with the imaginary number.

Any ideas?

Thanks,
 
Physics news on Phys.org
Not sure either, but I would certainly start with the Euler relation.
 
rude man said:
Not sure either, but I would certainly start with the Euler relation.

Good point. I think your on to something here. I just can't really see how eulers identity fully applies.

However since I've posted this question I've tried taking x(k)=j^-k*u(k) --> x(k)=1/j^-k. If k=2 then we get x(k)=-1u(k)?
 
Evo8 said:
Good point. I think your on to something here. I just can't really see how eulers identity fully applies.

However since I've posted this question I've tried taking x(k)=j^-k*u(k) --> x(k)=1/j^-k. If k=2 then we get x(k)=-1u(k)?

How about j = exp(jπ/2)? Can you take it from there?
 
Im not really sure if that helps me or not. I don't see it anyway. There is still a j in there. Thats exp(j*pi/2) correct?
 
Evo8 said:
Im not really sure if that helps me or not. I don't see it anyway. There is still a j in there. Thats exp(j*pi/2) correct?

Correct.

Just because there is still a j in there doesn't mean the signal doesn't exist, does it? What about phasing?

BTW I assume k are integers?
 
Yes K are integers. I think the signal does exist. I am not super familiar with eulers identity (unfortunately) But it seems like it could be on the right track...I don't know...
 
Your signal is certainly periodic: +1, -j, -1, +j, +1, ...

Is this a course in digital signal processing? If so, doesn't your textbook cover this business?

Take a look at the appropriate section of this:

http://www.analog.com/static/imported-files/tech_docs/dsp_book_Ch30.pdf

According to that, cos(wt + φ) --> exp(-jφ). I assume the complex Fourier series for this signal will also enable you to determine its power.

I'm no expert in this field myself, unfortunately. Hopefully I did kick-start you.
 
  • #10
Thanks for the help guys
 

Similar threads

Engineering Reconstruct a signal by determining the N Fourier Coefficients
  • · Replies 6 ·
Replies
6
Views
5K
Engineering Recursive implementation of an FIR Filter
  • · Replies 2 ·
Replies
2
Views
2K
Comp Sci How is "bounded waiting" achieved in here?
  • · Replies 4 ·
Replies
4
Views
2K
How can I prove this discrete signal is periodic?
  • · Replies 8 ·
Replies
8
Views
2K
Problem finding the output voltage using Laplace transform
  • · Replies 2 ·
Replies
2
Views
2K
Power of first three harmonics of periodic signal
  • · Replies 1 ·
Replies
1
Views
1K
Fourier series neither odd nor even
  • · Replies 3 ·
Replies
3
Views
6K
How to use the window functions on a signal in MATLAB?
  • · Replies 2 ·
Replies
2
Views
3K
Engineering Diff gain of a push pull degenerated differential pair
  • · Replies 6 ·
Replies
6
Views
3K
Sampling Continuous-Time Signal
  • · Replies 5 ·
Replies
5
Views
2K