Unit - IV
Fourier method of Waveform analysis

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A sine wave

a sine wave, cannot be decomposed into simpler signals. A composite periodic analog signal is composed of multiple sine waves.

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A composite periodic signal

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Fourier analysis is a tool that changes a time domain signal to a frequency domain signal and vice versa.

Fourier Analysis

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Fourier Series

Every composite periodic signal can be represented with a series of sine and cosine functions.The functions are integral harmonics of the fundamental frequency f of the composite signal.Using the series we can decompose any periodic signal into its harmonics.

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Fourier Transform

Fourier Transform gives the frequency domain signal of a nonperiodic time domain signal.

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A periodic function f (t) satisfies

where n is an integer and T is the period of the function.

According to the Fourier theorem, any practical periodic function of frequency 0 can be expressed as an infinite sum of sine or cosine functions that are integral multiples of 0. Thus, f (t) can be expressed as

TRIGONOMETRIC FOURIER SERIES

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The Fourier series of a periodic function f (t) is a representation that resolves f (t) into a dc component and an ac component comprising an infinite series of harmonic sinusoids.

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where 0 = 2/T is called the fundamental frequency in radians per second. The sinusoid sin n0t or cos n0t is called the nth harmonic of f (t); it is an odd harmonic if n is odd and an even harmonic if n is even. Equation is called the trigonometric Fourier series of f (t). The constants an and bn are the Fourier coefficients. The coefficient a0 is the dc component or the average value of f (t).

The harmonic frequency n is an integral multiple of the fundamental frequency 0, i.e., n = n0.

A major task in Fourier series is the determination of the Fourier coefficients a0, an, and bn. The process of determining the coefficients is called Fourier analysis.

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The plot of the amplitude An of the harmonics versus n0 is called the amplitude spectrum of f(t); the plot of the phase n versus n0 is the phase spectrum of f(t). Both the amplitude and phase spectra form the frequency spectrum of f (t).

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P.1 Determine the Fourier series of the waveform shown in Fig. Obtain the amplitude and phase spectra.

Our goal is to obtain the Fourier coefficients a0, an, and bn

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First, we describe the waveform as

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Finally, let us obtain the amplitude and phase spectra for the signal in Fig.

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P.2 Find the Fourier series of the square wave in Fig. Plot the amplitude and phase spectra.

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SYMMETRY CONSIDERATIONS

three types of symmetry:

(1) even symmetry, (2) odd symmetry, (3) half-wave symmetry.

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even symmetry

A function f (t) is even if its plot is symmetrical about the vertical axis; that is,

f (t) = f (t)

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(2) Odd Symmetry

A function f (t) is said to be odd if its plot is antisymmetrical about the

vertical axis:

f (t) = f (t)

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(3) Half-Wave Symmetry

A function is half-wave (odd) symmetric if

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P.3 Determine the Fourier series for the half-wave rectified cosine function shown in Fig.

This is an even function so that bn = 0. Also, T = 4, 0 = 2/T = /2.

Over a period,

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CIRCUIT APPLICATIONS

Many circuits are driven by nonsinusoidal periodic functions. To find the steady-state response of a circuit to a nonsinusoidal periodic excitation requires the application of a Fourier series, ac phasor analysis, and the superposition principle. The procedure usually involves three steps.

S t e p s f o r A p p l y i n g F o u r i e r S e r i e s :

1. Express the excitation as a Fourier series.

2. Find the response of each term in the Fourier series.

3. Add the individual responses using the superposition principle.

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This shows that in average-power calculation involving periodic voltage and current, the total average power is the sum of the average powers in each harmonically related voltage and current.

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P.4

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There are many important nonperiodic functionssuch as a unit step or an exponential functionthat can not be represented by a Fourier series. The Fourier transform allows a transformation from the time to the frequency domain, even if the function is not periodic.

The Fourier Transform decomposes any function into a sum of sinusoidal basis functions. Each of these basis functions is a complex exponential of a different frequency. The Fourier Transform therefore gives us a unique way of viewing any function - as the sum of simple sinusoids.

FOURIER TRANSFORM

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Inverse Fourier transform

Fourier transform of f (t) and is represented by F().

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SINGULARITY FUNCTIONS

Singularity functions are functions that either are discontinuous or have discontinuous derivatives.

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The derivative of the unit step function u(t) is the unit impulse function (t), which we write as

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P.5

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This is a property of the impulse function known as the sampling or sifting property.

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