1
$\begingroup$

I'm using the following definition for a linear time invariant digital filter:

"A digital filter L that transforms an input sequence $\{ x_{t} \}$ into an output sequence $\{ y_{t} \}$ is called a linear time invariant (LTI) digital filter if it has the following three properties:

  1. $L \{ \{ \alpha x_{t} \} \} = \alpha L \{ \{ x_{t} \} \} $

  2. $L \{ \{ x_{t,1} + x_{t,2} \} \} = L \{ \{ x_{t,1} \} \} + L \{ \{ x_{t,2} \} \}$

  3. If $L \{ \{ x_{t} \} \} = \{ y_{t} \}$ then $L \{ \{ x_{t + \tau} \} \} = \{ y_{t + \tau} \}$

Here $\tau$ is integer valued, and the notation $\{ x_{t+\tau} \}$ refers to the sequence whose t-th element is $x_{t+\tau}$"

I'm confused about the 3rd property. Surely if you have any linear filter $L \{ \{ x_{t} \} \} = \{ y_{t} \}$, then setting $t = t + \tau$ gives property 3?

Does there exist a linear filter that obeys properties 1. and 2. but not 3.?

$\endgroup$
2
$\begingroup$

Consider the filter $L \{ \{ x_{t} \} \} =\{ t \, x_{t} \}$. It complies with 1 and 2, but not 3. It is a linear filter with time-dependent static gain $t$, hence "L" but not "TI".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.