Contains classification of signals and systems

1. Signals and Systems
Classification of Signals and Systems
Presented by
Dr.S.Vairaprakash, Associate Professor/ECE
Ramco Institute of Technology,Rajapalayam

2. Signals and Systems
• Based on Nature & Characterisitics in the time domain the signals may be classified as
(i) Continuous timesignals
(ii) Discrete time signals
• Continuous time signals
The signals that are defined for every instant of time are known as continuous time signals. It is
denoted as x(t).
+A
t

3. Discrete time signals
The signals that are defined at discrete instant of time are known as discrete time signals.
The discrete time signals are continuous in amplitude and discrete in time. It is denoted as x(n).

4. System
• A system may be defined as a set of element (or) functional blocks (or) physical device which are connected
together and produces an output in response to an input signal.
Mathematically the functional relationship between Input and Output may be written as y(t) = H[x(t)]
Symbolically, x(t)  y(t)
where, x(t)  Input signal
y(t)  Output signal
H  System operator Ex: Audio and Video amplifiers
System
x(t) y(t)

5. Classification of CT and DT signals
Both CT and DT signals are classified into several types
(i) Deterministic and Random signals
(ii) Periodic and Aperiodic signals
(iii) Even and odd signals
(iv) Causal and Non-causal signals
(v) Energy and power signals

6. Deterministic and Random signals
A Deterministic signal can be completely represented by a mathematical equation at any time.
The nature and amplitude of such signals at any time can be predicted.
Example: sinusoidal signal, exponential signal
The signal cannot be predicted at any time. The signal whose characteristics are random in nature is called
random signals. It cannot be represented by mathematical expression.
Example: noise signals
Periodic and Aperiodic signals
A continuous time signal x(t) is said to be periodic if and only if
x(t + T) = x(t) for all t

7. (i) x(t) = 3cos5t + 2sin t
Period, T1
= 2
sec ond
5
T =
2
 2 second
2

The ratio of two periods,
T1 
2
5 


T2 2 5
is a irrational number

8. Even and odd signals
A continuous time signal x(t) is said to be even signal if it satisfies the condition
x(–t) = x(t) for all t
Example: Acos ωt
A continuous time signal x(t) is said to be odd signal if it satisfies the condition
x(–t) = –x(t) for all t

9. Causal and Non-causal signals
• A continuous time signal x(t) is said to be causal if x(t) = 0 for t < 0.
• Acontinuous time signal x(t) is said to be anticausal if x(t) = 0 for t > 0.
Energy and Powersignals
• For energy signals, the energy will be finite (or) constant i.e., (0 < E <  ) and average power will be zero.
• For power signals, the average power is finite (or) constant i.e., (0 < P <  ) and energy will be infinite.

10. Basic operations on signals
The basic operations on signals are
(i) Time Scaling
(ii) Time Reversal
(iii) Time shifting

11. Time Scaling:
The time scaling of a signal x(t) can be accomplished by replacing t by at in it.
It is expressed as
y(t) = x( b t)
where b  Scaling factor
If b < 1, then the signal expands.
If b > 1, then the signal compresses.
b 
1
2
x(t) x(t/2) x(2t)
–1 0 1 t –2 –1 0 1 2 t –0.5 0 0.5 t
b  2

12. Time shifting
The time shifting of a signal x(t) can be represented by y(t) = x(t – t0 )
If t0 > 0. For all values of t0 then a signal is said to be positive (right sided) shifted signal. The
shifting delays the signal.
If t0 < 0. For all values of t0 then a signal is said to be negative (left sided) shifted signal. The shifting advances
the signal.
Example:
2
1
x(t)
2
1
x(t)
0
Problems
1 2 3 t 0 1 2 3 4 5 t –2 –1 0 1 t
x(t)

13. Time Reversal
The time reversal of a signal x(t) can be obtained by folding the signal about t = 0. It is denoted by x(–t).
Example:
x(–t)
x(t)
1
–4 –3 –2 –1 0

14. Classifications of systems
• Static and dynamic systems.
• Linear and Non-Linear systems.
• Time variant and Time invariant systems.
• Causal and Non-causal systems.
• Stable and unstable systems.

15. Static and Dynamic systems
A system is said to be static if the output of the system depends only on present input.
Example:
y(t) = x(t)
y(n) = n x2(n)
A system is said to be dynamic if the output of the system depends on past and future values of inputs.
Example:
y(t) = x(t – 2)
y(n) = x(n) + x(n + 1)

16. Linear and Non-Linear systems
The response to a weighted sum of input signals is equal to the weighted sum of the outputs corresponding to
each of the individual input signal.
H[ax1(t) + bx2(t)] = aH[x1(t)] + bH[x2(t)]
H[ax1(n) + bx2(n)] = aH[x1(n)] + bH[x2(n)]
Time variant and Time invariant system
Asystem is said to be time invariant if its input-output characterisitics does not change with time. If a time shift
in the input results in a corresponding time shift in the output.
•
x(t-T) =y(t-T)
x(n-k) = y(n-k)

17. Problem: y(n) = n2 x(n)
For an input x1(n), the corresponding output is y1(n), then
y1(n) = n2 x1(n) (1)
For an input x2(n), the corresponding output is y2(n), then
y2(n) = n2 x2(n) (2)
Adding Equation (1) & (2)
y1(n) + y2(n) = n2 x1(n) + n2 x2(n) (3)
Substitute y(n) = y1(n) + y2(n) and
x(n) = x1(n) + x2(n) in given system
y1(n) + y2(n) = n2[x1(n) + x2(n)] (4)
From equations (3) & (4)
(3) = (4)
Therefore the system is linear system.

18. Problem: Given y(t) = x(–t)
• The output due to input delayed by T seconds is, y(t, T) = x(–t – T)
• The output delayed by T seconds is, y(t – T) = x[–(t – T)]
= x[–t + T]
i.e., the delayed output is not equal to the output due to delayed input. Therefore the system is time variant
system.

19. Stable and unstable systems
• A system is said to be BIBO (Bounded Input Bounded Output) stable if and only if every bounded input
produces a bounded output.
• Let the input signal x(t) be finite (bounded) i.e., |x(t)| < Mx < ∞ for all t
• If output signal y(t) is also finite (bounded)
i.e., |y(t)| < My < ∞ for all t
where Mx, My are positive real number
The system gives unbounded output for bounded input is called unstable system.
 | h(t) |dt < 
