Complex Variables - PowerPoint Presentation

Slide1

Complex Variables

& Transforms 232

Presentation No.1

Fourier Series & Transforms

Group

A

Uzair

Akbar

Hamza

Saeed

Khan

Muhammad Hammad

Saad Mahmood

Asim Javed

Sumbul Bashir

Mona

Ali

Zaib

Maria Aftab

Hafiz

Muhammad Abdullah

Bin Ashfaq

Behlol Nawaz

Bee-5ASlide2

Fourier Series and Transforms

MATH 232 presentationSlide3

Contents

Fourier Series & Transforms in Signals & Systems:

Introduction

Impulse Response

LTI Systems

Convolution Integral

Applications of Fourier Series & Transforms:

Finding Time Domain Output from Impulse Response

Radar System

Modulation

Digital Recording

Image Compression & AnalysisSlide4

Fourier Series & Transforms in Signals & Systems

MATH 232 PRESENTATIONSlide5

Introduction

Fourier

series representation can be used to construct any periodic signal in discrete time and essentially all periodic continuous-time signals of practical importance

The response of an LTI system to a complex exponential signal is particularly simple to express in terms of the frequency response of the system

.

Furthermore, as a result of the superposition property for LTI systems, we can express the response of an

LTI

system to a linear combination of complex exponentials with equal ease. Slide6

Impulse Response

The

impulse response describes the reaction of the system as a function of time. Impulse function contains all frequencies. The impulse response defines the response of a linear time-invariant system for all frequencies.

Depends

on whether the system is modeled in discrete or continuous time.

Modeled

as a Dirac delta function for continuous-time systems.

The

Dirac delta represents the limiting case of a pulse made very short in time while maintaining its area or integral. In Fourier analysis theory, such an impulse comprises equal portions of all possible excitation frequencies.Slide7

LTI Systems

Any LTI system can be characterized in the frequency

Linearity means that the relationship between the input and the

output of the system is a linear

map.

Time

invariance means that whether we apply an input

to the

system now or T seconds from now, the output will

be identical

except for a time delay of the T

seconds.

LTI

system can be characterized entirely by a

single function

called the system's impulse response. The

output of

the system is simply the convolution of the input to

the system

with the system's impulse response.

domain

by the system's transfer function, which is the Laplace transform of the system's impulse response.

The output of the system in the frequency domain is the product of the transfer function and the transform of the input.Slide8

Convolution Integral

A

convolution is an integral that

expresses

the amount of overlap of

one

function as it is shifted over

another

function. It therefore "blends"

one

function with another. If

and

are piecewise

continuous functions, then their convolution integral is given in the time domain as:

Convolution

in the time domain is equivalent to multiplication in the frequency

domain:

 Slide9

Applications of Fourier Series & Transforms

MATH 232 PRESENTATIONSlide10

Finding Time

Domain Output from Impulse Response

First by using the Fourier Transform,

is converted to the frequency domain representation

. Similarly

, we find

from

.

So now, the output signal in the frequency domain

is

.

The time domain output

can then be found by taking the inverse Fourier Transform of

.

Knowing the impulse response of a system,

we can

find

the transfer

function; the Fourier

transform of the impulse response

.

And

since all possible input signals are just the sum of sinusoids, we can easily find the output of an LTI system due to any input signal.Slide11

Filtering

In

signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies and not others in order to suppress interfering signals and reduce background noiseSlide12

Radar System

Radar systems use the linear time invariant theory for its operation. A transmitted signal reflects back to the receiver with a time shift. As the system is time invariant, the received output is a time shifted

version of the known output which can be analyzed and used accordingly.

The RADAR's receiver has to ensure that the signal it receives back is its own and not background noise. For noise cancellation, the incoming signal is split into component frequencies with the Fourier transform and all the irrelevant bands of frequencies are cut off.Slide13

Modulation

Once

at the

receiver

gives back the original information

signal, the

filtering of the original signal also requires its division into its oscillatory components using Fourier series.

The

process of Amplitude Modulation uses

convolution along with Fourier transform.

So the information signal is convoluted with a carrier

wave; a high frequency cosine wave.

This is necessary

, because

to transmit a radio wave of a certain frequency,

an antenna

of a particular size and characteristics has to be built

.Slide14

Digital Recording

An

incoming audio signal is fed into what is known as an Analogue-to-Digital (A-D) converter. This A-D converter takes a series of measurements of the signal at regular intervals, and stores each one as a number. The resultant long series of numbers is then placed onto some kind of storage medium, from which it can be retrieved. Playback is essentially the same process in reverse: a long series of numbers is retrieved from a storage medium, and passed to what is known as a Digital-to-Analogue (D-A) converter. The D-A converter takes the numbers obtained by measuring the original signal, and uses them to construct a very close approximation of that signal, which can then be passed to a loudspeaker and heard as sound.

The

generic name for this system is Pulse Code Modulation (PCM

).

So what an MP3 encoder does is it breaks the PCM signal (amplitudes in time domain) into its contributing frequencies. Then, its algorithm determines which frequencies to cut off and which to retain, based on different factors, some mentioned here. The result is that now lesser information has to be stored. The sound can then be played by a software that can decode MP3.Slide15

Image Compression & Analysis

Superposition of a lot of these can produce a proper image. Hence, an image can be represented by such Fourier series, and analyzed. However, to describe a complete image, the Fourier series should be in both vertical and horizontal dimensions.

An

image can be split into sub-components

, and those that

have very little contribution to the image are cut

off. As

an example of breaking image into frequencies, lets consider a black and

white pictures

. The patterns shown can be captured in a single Fouier term that

encodes

Spatial frequency

Magnitude (positive or negative)

The phase   Slide16

Questions?Slide17

END