6. 6
Let’s Start with this…
Classification in Periodic Phenomenon
Periodicity in time (Phenomenon comes to you)
• Use Frequency
• Fix the Position of object
Periodicity in space (You goes to Phenomenon)
• Use Period
• Fix the time and measure how the pattern is
distributed in space
Fourier Analysis is associated with symmetry
7. 1807, Periodic signal could be
represented by sinusoidal series
Why sinusoidal?
They are orthogonal.
The shape of the signal will not vary, when
used in LTI System.
They are smooth
Has finite power
Violates none of our criteria for real-world
signals.
8. Periodic Aperiodic
8
Not all phenomenon are periodic.
Aperiodic deals with long period T ∞
A signal that repeats
its pattern over a
period
can be represented by
a mathematical
equation
Deterministic signals
Power signals
Eg: Pendulum activity
A signal that doesn’t
repeats its pattern
over a period
cannot be represented
by a mathematical
equation
Random signals
Energy Signals
Eg: Chaotic signal
9. 9
Why can’t we use Fourier series for
Aperiodic signal?
Fourier Series (FS) exists only for periodic signals.
Fourier Transform (FT) is derived from FS, i.e. FT is the
envelope of the FS. Hence, the frequency domain becomes
more finer for a aperiodic signal.
Moreover, being able to express a periodic signal as a discrete sum of frequencies is a
stronger statement than expressing it as a continuous sum via the inversion formula.
Representing Aperiodic signal over finite interval -> loss of
information
10. 10
Linear Time-Invariant (LTI) System
Satisfies 2 conditions
Linearity – Superposition Principle
Time Shifting – Noether’s Theorem
LTI system works because of Dirac delta function
Linearity Time Shifting
After an hour
LTI
LTI
LTI
LTI
12. Significance of Time Domain and Frequency Domain
12
Any signal can be represented by sum of sinusoidal signal
of different frequencies
Time Domain or Spatial Domain
representation of signal (1KHZ)
Frequency Domain representation of
signal (1KHZ)
LTI
Noise
Input Signal
Ripples
14. F.T Analysis : Break the signal or functions into simpler
constituent parts
F.T Synthesis : Reassemble a signal from its
constituent parts
Both Procedures are accomplished by linear operation
(Integral and Series)
14
Fourier transform Procedures
Fourier Analysis is a part of Linear System
15. Continuous Time FT
Magnitude : Determines the contribution of
each component
Phase : Determines which components
are present
16. Discrete Time FT
• Representation of the sequence in terms of
the complex exponential sequence 𝑒𝑗𝜔𝑛
𝐹(ω) = 𝑛=−∞
∞ 𝑓(𝑛) 𝑒−𝑗𝜔𝑛
𝐹 𝜔 = 𝐹𝑟𝑒(ω) + j 𝐹𝑖𝑚𝑔(ω)
Where,
Magnitude, |F(ω)| 2=|Fre(ω)| 2+|Fimg(ω)| 2
Phase , θ(ω) = arg F(ω)
17. Dirichlet’s Condition for existence of FT
The signal should have
• Finite number of maxima and minima over any
finite interval
• Finite number of discontinuities over any finite
interval
• Absolutely integrable
17 Its only a sufficient condition not necessary
18. Properties of FT
▪ http://fourier.eng.hmc.edu/e101/lectures/hand
out3/node2.html
▪ https://www.tutorialspoint.com/signals_and_s
ystems/fourier_transforms_properties.htm
18
If
and
19. Properties of CTFT
Linearity
Time Shifting
Frequency Shifting
Time Reversal
Time Scaling
Differentiation
Convolution
19
Consider, If
and
Helps to turn differential
equation into algebraic
equation