Complex Variables& Transforms 232Presentation No.1

Fourier Series & Transforms

Group AUzair Akbar

Hamza Saeed KhanMuhammad Hammad

Saad MahmoodAsim Javed

Sumbul BashirMona Ali ZaibMaria Aftab

Hafiz Muhammad Abdullah Bin AshfaqBehlol Nawaz

Bee-5A

Fourier Series and Transforms

MATH 232 PRESENTATION

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 & Analysis

Fourier Series & Transforms in Signals & SystemsMATH 232 PRESENTATION

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.

Impulse ResponseThe 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.

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.

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:

Applications of Fourier Series & TransformsMATH 232 PRESENTATION

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.

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 noise

Radar SystemRadar 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.

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.

Digital RecordingAn 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.

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

Questions?

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