Signals and systems Module 2 part 1.pptx

RAT 306, Dr Sreepriya.S, Associate Professor, Department of RA, ASIET
1
SIGNALS AND SYSTEM
MODULE 2

RAT 306, Dr Sreepriya.S, Associate Professor, Department of RA, ASIET 2
SYLLABUS
 Frequency domain representation of continuous time signals
 Continuous time Fourier series and its properties.
 Continuous time Fourier transform and its properties.
 Relation between Fourier and Laplace transforms.
 Analysis of LTI systems using Laplace and Fourier transforms.
 Concept of transfer function, Frequency response, Magnitude and
phase response
 Sampling of continuous time signals, Sampling theorem for low pass
signals, aliasing

3
FREQUENCY DOMAIN
REPRESENTATION OF CONTINUOUS
TIME SIGNALS
Continuous time Fourier series and its properties.
Continuous time Fourier transform and its properties.

FREQUENCY DOMAIN ANALYSIS

FREQUENCY DOMAIN ANALYSIS OF SIGNALS
 study the various frequency components present in the signal, magnitude and
phase of various frequency components.
 Laplace Transform
 Fourier Transform Continuous time systems
 Fourier Series
 Z transform
 Discrete Fourier Transform Discrete time systems
• The graphical plots of magnitude and phase as a function of frequency are also drawn.
• The plot of magnitude versus frequency is called magnitude spectrum and the plot of phase versus
frequency is called phase spectrum.
• In general, these plots are called frequency spectrum

FREQUENCY DOMAIN ANALYSIS OF SIGNALS
 The Fourier representation of signals can be used to perform
frequency domain analysis of signals.
 The French mathematician Jean Baptiste Joseph Fourier (J.B.J. Fourier)
has shown that any periodic non-sinusoidal signal can be
expressed as a linear weighted sum of harmonically related
sinusoidal signals.
 This leads to a method called Fourier series in which a periodic
signal is represented as a function of frequency.
 The Fourier representation of periodic signals has been extended to
non-periodic signals by letting the fundamental period T tend to
infinity, and this Fourier method of representing non-periodic signals
as a function of frequency is called Fourier transform.

7
FOURIER SERIES
Any periodic signal represented as a weighted
superposition of sinusoidal or complex exponentials

FOURIER SERIES: INTRODUCTION
Depending upon how the signal is represented
Two type
Trigonometric Fourier series- linear weighted combination
of trigonometric function (sinusoidal)
Exponential Fourier series- Combination of exponential
function

FOURIER SERIES: INTRODUCTION
For Fourier Series to exist for a periodic signal, it must
satisfy Dirichlet’s Condition
In each period,
1. Single-valued property:The function x(t) must be a single-
valued function.
2. Finite peaks:The function x(t) has only a finite number of
maxima and minima.
3. Finite Discontinuities:The function x(t) has a finite number
of discontinuities.
4. Absolute Integrability:The function x(t) is absolutely
integrable over one period

FOURIER SERIES: EXPLANATION OF DIRICHLET CONDITION:
1. Single-valued property:The function x(t) must be a single-
valued function.

Finite peaks:The function x(t) has only a finite number of
maxima and minima.

Finite Discontinuities:The function x(t) has a finite number of
discontinuities.

Absolute Integrability:The function x(t) is absolutely integrable
over one period

TRIGONOMETRIC
FOURIER SERIES
 A sinusoidal signal is
a periodic signal with
period , can be
represented as
infinite sum of sine
and cosine functions.

EXPONENTIAL
FOURIER SERIES
The signal x(t) is
represented as a
weighted sum of
complex exponential
functions
More convenient and
compact, hence widely
used.

EXPONENTIAL
FOURIER SERIES
The exponential form of Fourier series
of a periodic signal x(t) with period T is
given by
Where , Cn is the Fourier coefficient,T is
the fundamental period
The Fourier coefficients are given as

CONVERSION
 Exponential to Trignometric
coefficients
 )
 Trignometric to Exponential

18
PROPERTIES OF FOURIER
SERIES

LINEARITY PROPERTY

TIME SHIFTING PROPERTY

TIME REVERSAL PROPERTY

TIME SCALING

Signals and systems Module 2 part 1.pptx

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