Defining a Signal. periodic, bounded finite etc.

[Evo8]

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,

 

[rude man]

Not sure either, but I would certainly start with the Euler relation.

 

[Evo8]

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)?

 

[rude man]

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?

 

[Evo8]

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?

 

[rude man]

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?

 

[Evo8]

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...

 

[rude man]

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.

 

[rude man]

If you like that chapter (30) looks like you can access the entire book for free at

http://www.analog.com/en/processors...nt/content/scientist_engineers_guide/fca.html

 

[Evo8]

Thanks for the help guys