[SOLVED] Major Understanding Issue in Discrete-Time Signals

[dzafar]

Hello there,

I am reading this book "Discrete-Time Signal Processing" by Alan V. Oppenheim and Ronald W. Schafer. Here it says,



Below is my understanding. Can someone please explain, where I am going wrong?



Also, please answer the questions in the image below



Thanks

 

[nomigoraya]

A is amplitude and yeas your interpretation is correct.

 

[dzafar]

Thanks nomigoraya! What about the question in the second image? Thanks

 

[CataM]

You have already answered that simple question.
u[5]= 0+0+0+0.....+1+0+0+0+0+0 = 1 Where is the problem ?

u[12454352]=0+0+0+0+0+....1+0+0+0+0+0+0.... = 1

u[1324583425893245738984237441394]=0+0+0+0+0+....1+0+0+0+0+0+0.... = 1

Do you see it now ?

What you have written in the question is wrong because it is a sum of values i.e. it is 0+0+0+.....+1+0+0+0.... that "1" at delta[0]=1 makes the sum to be "1".

 

[dzafar]

Umm.. okay! The problem that I am having is that at n = 5, u[5] = 1 whereas, impulse[5] = 0!
So for n = 5, they are not equal!
I understand that impulse function is with a summation. But that just means it's 1 at n = 0.

Q1: So does it not matter what the impulse is at n = 5?!

Q2: If so, we can then just remove the summation and say:
u[n] = impulse[0] for any index n?!

Also, if they just mean the maginitude of u[n] equals the magnitude of the summation of impulses,

Q3: Why don't they use the magnitude symbol:
|u[n]| = |summation of impulses|?

Q4: Also, why does the summation stop at n?! We can easily have it upto positive infinity (as the magnitude will still be 1)?!

Thanks! I hope someone can answer my doubts in the same order (of questions asked) so that I have a better understanding.

Thanks

 

[CataM]

Q1: So does it not matter what the impulse is at n = 5?!

No, it does not.

Q2: If so, we can then just remove the summation and say:
u[n] = impulse[0] for any index n?!

Only for "n" higher or equal to 0, otherwise u[n]=0.

Also, if they just mean the maginitude of u[n] equals the magnitude of the summation of impulses,

Q3: Why don't they use the magnitude symbol:
|u[n]| = |summation of impulses|?

In the book is written the same, u[n]=summation of impulses, isn't it ? I see no reason for the magnitude symbol here... magnitude symbol makes sense in complex numbers... but I do not see any complex number here. The module symbol on the other hand (i.e. |-3|=3) does not make sense either because there are no negative values in this case.
The author simply gave good mathematical definition. You can remember it however you want e.g. as I have answered to Q2 but it would be useless when applying demonstrations of formulas or transforms because correct mathematical definitions can be used to prove other stuff.

Q4: Also, why does the summation stop at n?! We can easily have it upto positive infinity (as the magnitude will still be 1)?!

To be a finite sum and hence more easily apply transforms and convolution and etc..