# Isn't every linear filter a linear time invariant (LTI) filter?

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.?

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".