Signal Processing question Is the system y[n] = x[n] - x[n-1] time invariant?

[seang]

Homework Statement

Is the system y[n] = x[n] - x[n-1] time invariant?

Homework Equations

The Attempt at a Solution

I say no, but there's one thing I don't understand. I think if you shifted the input, and then ran those samples through the system, you'd get x[n-n0] - x[n-1].

If you ran the signal through the system first, then shifted the output, you'd get x[n-n0] - x[n-n0-1].

This is my thinking, is it incorrect?

 

[asd1249jf]

Homework Statement

Is the system y[n] = x[n] - x[n-1] time invariant?

Homework Equations

The Attempt at a Solution

I say no, but there's one thing I don't understand. I think if you shifted the input, and then ran those samples through the system, you'd get x[n-n0] - x[n-1].

If you ran the signal through the system first, then shifted the output, you'd get x[n-n0] - x[n-n0-1].

This is my thinking, is it incorrect?

As for the answer, the system is time invariant since the system does not explicitly depend on one of the inputs.

As for your method of approaching this, even if you shift the input and run the samples through the system, you'd get the same output as running the signal through the system first.

Here's a formal proof you can look at to see why it is true.

http://en.wikipedia.org/wiki/Time-invariant

 

[wildman]

146kok is correct.

Your error comes in the first answer where you shifted the input and then ran it through the equation. What the equation says is that y is equal to the present input plus the input just before the present input. If the present input is x(n-n0) then the input just before the present input has to be x(n-n0-1). This agrees with your second answer so it is time invariant.