Is this system time variant or time invariant

Question

I'm studying DSP from a particular book.

There is an exercise that examines the properties of some given systems.

The system I'm referring to is:

y(n) = x(n) + nx(n + 1)

If we delay the input by k samples the aforementioned system becomes:

y(n, k) = x(n - k) + nx(n - k + 1)

Now if we delay the output by k samples it becomes:

y(n - k) = x(n - k) + (n - k)x(n - k + 1)

So as one can see y(n - k) is not equal to y(n, k) therefore it is a time variant system.

However the book classifies it a time invariant system.

Am I missing something?

Answer 1

I think your book is wrong; in fact, having n outside an indexing operation is a strong hint that the system is time-variant. Sometimes a quick numerical example can give you confidence that your equations and conclusions are correct. Say x[n] = 0 except for x[0] = 1. In this case,

y[-1] = x[-1] + (-1)*x[0] = 0 - 1 = -1
y[0] = x[0] + 0*x[1] = 1 + 0 = 1

Now shift the input so that x[10] = 1:

y[9] = x[9] + 9*x[10] = 0 + 9 = 9
y[10] = x[10] + 10*x[11] = 1 + 0 = 1