# To find the impulse repsonse using the difference equation

A causal linear time-invariant filter has transfer function

a) Denote the input signal by x[n] and the output signal by y[n]. Find the difference equations for the filter.

f) Find the impulse response of the system. (Eliminate j (the imaginary unit) from your answer.)

I used the partial fractions method and obtained the following solution.

However it was time consuming. Is there a way that I can simplify this difference equation to get the impulse response of the IIR filter to get the same solution. Any help will be appreciated(as I am not aware of solving by difference equations)

There is no real shortcut for computing the impulse response. Partial fractions is the standard way to do it. However, in the case of a second-order transfer function as in your example, the result can be found in tables of $$\mathcal{Z}$$-transform pairs. E.g., if you use the last two correspondence of this table, you can pretty quickly write down the result.
By the way, your result can't be correct (at least not for negative $$n$$), because you know that the system is causal, so its impulse response must be zero for $$n<0$$.