# confused in practical example of shift invariance?

I am reading book of w smith ,"The scientist and engineer guide to digital signal processing" 2 edition I am trying to understand topic of shift invariance from theory given on pg 92

I have attached screen shot and i have indicated the example with in red marks.

I am confused in whether shift in variance ,author is talking about is in regard to which system?telephone line or digital filter? What is our target system in this example that needs to be shift invariant?Is it telephone line or digital filter?

They talk about the shift-invariance (or variance) of the digital filter. In general, the response of a shift-invariant system is independent of a shift of the input signal. The corresponding output is just shifted by the same amount as the input. So if $$y[n]$$ is the response to an input $$x[n]$$, then the response to a shifted input $$x[n-n_0]$$ is $$y[n-n_0]$$. If the independent variable is time, then such systems are also called "time-invariant".

Naturally, an adaptive system cannot be shift (time)-invariant, because it changes its characteristics over time, and consequently, it will react differently to the same input applied at different times.

The example in the text refers to the fact that not all problems can be solved by shift-invariant systems only; in certain situations one needs adaptive systems.

• Sir Can you please let me know difference between adaptive systems and shift invariant system?If our system uses adaptive system, it will be no longer obeying LTI (Linear Time invariant) systems theory and rules?am i right?? – engr Apr 7 '19 at 6:42
• @engr: A shift (or time) invariant system is non-adaptive, so it doesn't change its behavior, whereas an adaptive system can change its behavior. An adaptive system is not LTI, as you expected. – Matt L. Apr 7 '19 at 9:47