# determine the impulse response of given second order difference equation

$y[n]-3y[n-1]-4y[n-2]=x[n]$

i solved it as follows :

1.first calculated natural-response(ZIR i.e,zero input response)

by using characteristic equation it came out as

$y_n[n]=[c_1(-1)^{n}+c_2(4)^{n}]$ here i couldn't be able to calculate $c_1$ and $c_2$ because there were no initial conditions given in the question.

2.And,Then calculated forced-response(ZSR i.e,with all initial conditions zero) and ,since we know it consists of two parts

homogeneous solution and particular solution i.e,($y_h[n] +y_p[n]$) i calculated both of parts separately as follows: first $y_p[n]=0$ because input is $\delta [n]$ and secondly $y_h[n]$ =$[C_1(-1)^{n}+C_2(4)^{n}$ (here i was able to calculate values of $C_1$ and $C_2$ as $\dfrac{-1}{5}$ and $\dfrac{6}{5}$ respectively by using zero initial conditions i.e, $y[-1]=y[-2]=0$,$x(0)=1$ ,as per definition of forced response)

3.and finally i added all solutions to get total solution i.e, $y[n]= y_n[n]+y_h[n]+y_p[n]$ and got answer as $y_n[n]=[c_1(-1)^{n}+c_2(4)^{n}-0.2(-1)^{n}+1.2(4)^{n}+0]$ but answer in the book is given as $y [n]=[-0.2(-1)^{n}+1.2(4)^{n}]$ so my problem is why didn't they incorporated natural response term in final answer . thus, my questions are

1.does "finding impulse response in numerical" means finding only "forced response" and leaving natural response.terms.unlike we do in other input cases(like $x_\left[n\right]=\left[\dfrac{1}{2}\right]^n$) where we also calculate natural responses(where of course initial conditions are provided so that we can calculate $c_1$ and $c_2$ as well)...... is impulse input an exceptional case of inputs?

2.same they have done for questions regarding 'step response'

i haven't read z-transform and generating function method to solve difference equations so please do not give answer by using those techniques.

## 1 Answer

The impulse response of a system is its zero-state response to an impulse at the input. If the system is linear and time-invariant (LTI), then the system's response to any input signal can be described in terms of the impulse response. The impulse response only depends on the structure of the system, and it cannot depend on any initial conditions. In fact, a system with non-zero initial conditions is not linear because a part of the output (the zero-input response) does not depend on the input signal but solely on the initial conditions and the system itself.

• do you mean since system is LTI that is reason why they have not incorporated ZIR (zero input response) terms because by doing so it will make system non linear...... ....but,...can we say same about step response too that it also can't depend on the initial conditions like impulse response to ensure if system is LTI . – user33321 Jan 18 '18 at 15:24
• @veereshpandey: The step response is just the integral of the impulse response, so the same applies. – Matt L. Jan 18 '18 at 16:39