11. • The system is memoryless if it doesn’t need memory to store information from
past or future.
• The system is memoryless arbitrarily if the output at any time t0 depend on the
input at that time not past, not future.
• A System that is not memoryless is said to have memory.
• Although simple, a memoryless system is not very flexible. Since its current
output value cannot rely on past or future values of the input.
Static (Memoryless) and Dynamic
(Memory) system
11
16. Causal and Non-Causal System
Causal System:
A system is said to be causal if the response of a system at any instant of time depends only
on the present input, past input and past output but does not depend upon the future
input and future output
Ex: y(t) = 3x(t) + x(t-1)
Non-Causal system:
A system is said to be non-causal if the response of a system at any instant of time depends
on the future input and also on the present input, past input , past output.
Ex: y(t) = x(t+2) + x(t-1)
y(t) = x(-t) + x(t+4)
A memoryless system is always causal, although the converse is not necessarily true.
Note: online system (e.g. telephone) is causal
offline system (e.g. music) is non-causal
16
19. Invertibility
19
The System is invertible if the input signals can be generated from the output signals
Mathematically, the system is invertible if every element of output is corresponding to
only one element of input. (It is one to one)
If output is corresponding to more input so it is not invertible. 1
4
5
3
2
9
X(t) Y(t)
20. Examples cont.,
20
Ex: Y (t) = sin (x(t))
Input is an argument of sine function
X(t) = 0 y=sin (0) =0
X(𝝅) = 𝝅 y= sin (𝝅) =0
We get y for two inputs so it is not one to one so it is not invertible
Ex: Y (t) = 3 x(t) + 5
Y (t) – 5 = 3 X(t)
X(𝐭) =
𝐲 𝐭 −𝟓
𝟑
The system is invertible since the input signal is generated from the output
21. Time-invariant and Time-variant
Systems
21
• A system is said to be time invariant if the time shifts in the input signals
results in corresponding time shift in the output signal
• The input and output characteristics do not change with time. So if you repeat
the same test over time, you will get same output
• For a continuous time system
f[x(t1-t2)] = y(t1-t2)
• For a discrete time system,
F [x(n-k)] = y (n-k)
• If the above relation does not satisfy, then the system is said to be a time
variant system
• Practically speaking. Compared to time-varying systems. Time-invariant
systems are much easier to design and analyze, since their behavior does not
change with respect to time.
22. Time-invariant and Time-variant
Systems cont.,
22
System Delay by t0
X(t) Y(t) Y(t-t0
)
Delay by t0 System
X(t-t0) Y(t)
Y(t)= Y(t-t0)
Y(t)≠ Y(t-t0)
Time-invariant System
Time-variant System
Time Invariant Test
1- Shift input by T Y(t)=x(t-t0)
2- Shift output by T y(t-t0)
3- If Y(t)= Y(t-t0)
So the system is time invariant
23. Examples
23
Ex: Determine whether the following system is
time invariant or not:
Y (t) = x(2t)
Solution:
1- Y(t)= x(2t-T)
2- Y(t-T) = x(2(t-T))= x(2t-2T)
Y(t) ≠ Y(t-T)
Hence the system is not time invariant
Ex: Is the system is time invariant?
Y (t) = sin (x(t))
Solution:
1- Y(t)= sin (x(t-T))
2- Y(t-T) = sin (x(t-T))
Y(t)= Y(t-T)
Hence the system is time invariant
Time Invariant Test
1- Shift input by T Y(t)=x(t-T)
2- Shift output by T Y(t-T)
3- If Y(t)= Y(t-T)
So the system is time invariant
24. Examples cont.,
24
Ex: Determine whether the following system is
time invariant or not:
Y (t) = t x(t)
Solution:
1- Y (t) = t x(t-T)
2- Y(t-T) = (t-T) x(t-T)
Y (t) ≠ Y(t-T)
Hence the system is not time invariant
Ex: Determine whether the following system is time invariant or not
Y (t) = sin (x(t))
Solution:
1- Y (t) = sin (x(t-T))
2- Y(t-T) = sin (x(t-T))
Y (t) = y(t-T)
Hence the system is time invariant
Time Invariant Test
1- Shift input by T Y(t)=x(t-T)
2- Shift output by T Y(t-T)
3- If Y(t)= Y(t-T)
So the system is time invariant
26. Linear and Non-linear System
26
• A system is said to be linear if it satisfy the superposition principle.
• Superposition principle depends on two laws
- Law of Additivity
- Law of homogeneity
1- Law of additivity (LoA)
It states that the weighted sum of input signal be equal to the weighted sum of output
signal corresponding to each of the individual input signal.
1- Apply X1(t) y1(t)
2- Apply X2(t) y2(t)
3- Apply (X1(t) + X2(t)) if output is (y1(t) + y2(t)) so the system follow the LoA
2- Law of Homogenity (LoH)
1- Apply k Y(T)
2- Apply k X(t)
If k x(t) = k Y(t) so if follow the law of homogeneity
If it satisfy 1 and 2 so the system is linear
27. Examples
27
Ex: Determine whether the following system is linear or not:
Y (t) = x(sint)
Solution:
1- Law of Additivity
Y1(t) = X1(sin t)
Y2(t) = X2(sin t)
Y1(t) + Y2(t) = X1(sin t) + X2(sin t)
X1(t) +X2(t) system X1(sin t) + X2(sin t)
X1(t)+X2(t) = Y1(t) + Y2(t) so the system follow the law of Additivity
2- Law of Homogeneity
K Y(t) = k X (sin t)
K X(t) system k X(sin t)
K X(t) = k Y(t) so the system follow the law of Homogeneity
Since it follows Law of Additivity and Law of Homogeneity so it follow the law of
superposition so the system is linear
28. Stable and Non-Stable System
28
• Stability is very critical in reality because if the system is not stable, it will
be out of control.
• A system is said to be stable (Bounded input bounded output (BIBO stable)
when every bounded input produces bounded output. We mean by
bounded that the signal is limited to a finite range.
• Otherwise the system is not stable
Bounded signal Not Bounded signal
29. Ex : Determine whether the following system is stable or not
Y(t) =x(2t)
Solution:
If x(t) is bounded
X(2t) is bounded
Since 2 is time scaling function which just change signal in time direction not
amplitude direction
So y(t) is bounded so the system is stable
Ex :
Y(t) =x(t)/t
Solution:
At t 0 y=x(t)/0 Infinity
Y(t) is not bounded so the system is not stable
Examples
