Solving The Homogeneous Equation What does it mean?

Clearly y h n depends on the INITIAL CONDITIONS of the system T . Linearity Time-Invariance Causality will each depend on these conditions. In this course, we will emphasize the simplest case, when T is "initially at rest" with "zero initial conditions." we will get LTI and causal solutions. (although possibly at the expense of stability Solve y n a y n 1 x n where a 1 for x n δ n . Step 1: Particular Solution Assume n 0 and "zero initial conditions" y p 0 δ 0 1 a y p -1 0 1 y p 1 δ 1 0 a y p 0 1 a y p 2 δ 2 0 a y p 1 a a 2 y p n a n where n 0

Step 2: Homogeneous Solution If x n 0 , then y h n a y h n 1 0 y h n a y h n 1 A solution is given by y h n c a n for all n .

Step 3: Reconcile y n y n p y h n a n u n c a n How to pick c ? Need auxilliary conditions.

If we desire a causal system, then c 0 and y n a n u n

If we desire an anticausal system then choose c -1 , so y n a n u n This does not assume "system initially at rest!"

Notes Solution 1 was causal and stable. Solution 2 was anticausal and unstable. In general, linearity, time-invariance, and causality of a system implemented as a DE will depend on the auxilliary conditions. If we assume that the system is initially at rest ("zero initial conditions"), then it will be LINEAR, TIME-INVARIANT, and CAUSAL. Note: Setting input x δ 0 impulse and setting initial conditions all 0 and solving for y p yields y p h as the impulse response of this LSI system. Frequency Response of a "wire"

Impulse Response:

so Frequency Response: H 𝔽 H δ 0 1 N n N 1 0 h n 2 N k n 1 N n N 1 0 δ 0 n 2 N k n δ 0 n 1 n 0 0 1 N Flat