Discrete-Time Waveforms: Properties & Conditions

[RaduAndrei]

I don't know if I write in the right section or not.

I saw that every discrete-time waveform can be written as a linear combination of almost any other discrete-time waveform.

What are the conditions imposed on this other discrete-time waveform?
For example, the unit step function obeys these conditions.

PS: the waveforms are real not complex

 

[BvU]

Well, one of the conditions on this other discrete-time waveform is that any discrete-time waveform can be written as a linear combination of these base waveforms.
In other words that your base waveforms form a complete basis. Just like in linear algebra. There are rules for those bases.

Must say I don't follow your "For example, the unit step function obeys these conditions" ? But I can imagine a time series built up from step functions.

 

[RaduAndrei]

The unit step function is used in LTI systems. If you know the response of the LTI system to the unit step function, then you can calculate the output to any input function.
You just write the input function as a sum of unit step functions. Then, by homogeneity,additivity and time invariance you can calculate the output just by knowing the response to the unit step function.

In other words, you must write the discrete-time waveform that you want to apply to the system as a linear combination of unit step functions.

A first condition is obviously that this other discrete waveform must be non-zero. The unit step function, for example, is non-zero for n>=0. But there are other conditions as well.

 

[BvU]

To make the expansion unique, the base vectors must be linearly independent. (That covers your on-zero).
And there must be N of them if the space has N dimensions (they must span the space).
And that's about it, not much more.

But now I'm repeating the statements in the link I gave.

 

[RaduAndrei]

Thanks.