Linear Time Invariant System [LTIS]:
x1(t)
y1(t)
x2(t)
y2(t)
superposition is valid
ax1(t)+ bx2(t)
ay1(t) + by2(t)
ax1(t)
x1(t)
x2(t)
y1(t)
T
ax1(t)
∑
y2(t)
ay2(t)
T
+
bx2(t)
+
by2(t)
bx2(t)
If x1(t) is applied, it gives y1(t) for x2(t)=0
If x2(t) is applied, it gives y2(t) for x1(t)=0
And,
If x1(t) and x2(t) both are applied, that results in y1(t) and y2(t)
In case of AM;
M(t)* C(t)= y(t)
 If C(t)=0, then for M(t) only, y(t)=0
 If M(t)=0, then for C(t) only, y(t)=0
 But for both M(t) and C(t), output is y(t)=M(t)*C(t)
Considering C(t) be constant,
|M1(t)|----M1(t)*C(t) and |M2(t)|----M2(t)*C(t)
Then, M(t)= M1(t) + M2(t)
Y(t) = { M1(t) + M2(t)}*C(t)=M1(t)*C(t) + M2(t)*C(t)
Hence superposition principle is satisfied.
Time Invariant:
 Behavior of the system does not change with time
 If system is time variant then the delayed input will produce delayed output
for the same value and the delay is same.
 Most of the system are generally time invariant.
Eg.
If x(t)--------y(t)
x(t-t0)--------y(t-t0)
x(t)
x(t-t0)
Signal Property:
1. Continuous and Discrete Signals
CT:
x(t)---------analog
t------------continuous/analog
DT:
x[n]---------continuous
n-------------discrete
Digital:
x[n]----------discrete
n--------------discrete
2. Power and Energy Signals:
Power signal:
 Energy is infinite, power is finite
 P∞=finite, E∞=∞
 Dealt with harmonics
Energy signal:
 Power is infinite, energy is finite
 E∞=∞, P∞=0
3. Periodic and Non-periodic:
Periodic:
 x(t)=x(t+T) ; T=1,2,3,….
 Every signal may have zero period or infinite period
Non-periodic:
 Above condition not satisfied ie, x(t)≠x(t+T)
x[n]=x[n+kn]
ejωt
T=2П/ω
ej ωn ej ωn ej ωn
= *
ωN=2Пk
N=2Пk/ω
ej(ω+2П)t , then T=2П/(ω+2П)
4. Deterministic or Random:
Deterministic:
 At any value of time, value of the system can be known
Random:
 Value is uncertain at any value of time t
 Speech is random signal
Any system is whether or not:
 Causal
 Stable
 Linear
 Memory
 Invertible
 Time invariant
 Observable (internal description with view point of output)
 Controllable (internal description with view point of input)
LTI System Analysis/Characteristics:
Linear and time invariant property is satisfied
CT
ax1 (t) +bx2 (t) ----------ay1 (t) +by2 (t)
x1(t-t0)---------y1(t-t0)
x2(t-t0)----------y2(t-t0)
DT
ax1 [n] + bx2 [n] -------ay1 [n] +by2 [n]
x1[n-n0]-------y1[n-n0]
x2[n+n0]------y2[n+n0]

Concept of linearity gives the decomposition of signals that allows to simplify the
complexity of signals.
In terms of Impulse: (δ and shifted δ)
DT system
δ[n+5]
δ[0]
δ[n-5]
-5
5
x[0]
x[n]
N
5
x[n]*δ[n-5]=x[5]*δ[n-5]
In general,
x[1] δ[n-1]
x[2] δ[n-2]
…..
….
…..
Hence,
x[n] = ……….+ x[-2]δ[n+2]+x[-1] δ[n+1]+x[0] δ[n]+x[1] δ[n-1]+……….
X[n]= ∑ x[k] δ[n-k]
δ[n]
system
h[n]
impulse response
Time invariant system:
δ[n-1]-------h[n-1]
δ[n-k]-------h[n-k]
y[n]= …….+ x[-2]h[n-2] + x[-1]h[n-1] + x[0]h[n] + x[1]h[n+1] + ……………
y[n]= ∑ x[k]h[n-k]


If system is LTI and its impulse response is known, we can obtain the output for
any kind of inputs.
Conversely, impulse response completely characterizes the LTI system.
y[n]= ∑ x[k]h[n-k] but how to compute????
We know,
y(t)=∫ x(‫*)ﺡ‬h(t-‫(ﺡ‬d‫ﺡ‬
tietulor onilnrgetni
x[n]=x[k]
n
h[n]
h[-k]
k
k
h[n-k]
k
x[k]h[n-k]
Steps:






Write the system representation
Flip h[k]
Shift (delay) by ‘n’
Multiply x[k]h[n-k]
Add
Get y[n]