⑀ ∿ Signals and Systems ∿ ⑀

A signal 𝗫 that interceeds into a system 𝗛 produces an answer 𝗬

On a time measured signal system that intercedence is called Convolution.

y(t) = x(t) ∗ h(t) = ∫ x(𝜏)h(t-𝜏) d𝜏 ,  𝜏:{-∞,∞}
y[n] = x[n] ∗ h[n] = ∑ x[k]h[n-k],  k:{-∞,∞} 

Signals

Systems

Delta System/Signal

The identity unit. Satisfies

x = x ∗ 𝛿 

Linearity

A Linear sistem is such as giben

x → h = y

Then it is such as:

a·x → h = a·y
x1 + x2 → h = y1 + y2

Therefor

∑ ai·xi → h = ∑ ai·yi

Cascading Systems

Associative and commutative

x ∗ h1 ∗ h2 = (x ∗ h1) ∗ h2 = x ∗ (h1 ∗ h2) = x ∗ (h2 ∗ h1) = x ∗ h2 ∗ h1  

Invertibility

x ∗ h ∗ hᵢ = x

Properties

Time Invariance

hk(t) = h0(t-k

Memoryless

h(t-τ)

Nonzero only at t=τ:

h(t) = k δ(t)
h[n] = k δ[n]

y(t) = k x(t)
y[n] = k x[n]