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ELEC270 Fourier Synthesis of Periodic Waveforms

创建日期：2024-08-15 15:40:26

所属分类：电气工程electric engineering

标签： ELEC270

Fourier Synthesis of Periodic Waveforms

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Instructions:

Read this script before attempting the experiment.

The Pre-Lab Questions should be answered before the lab day. They are

available online and worth 10%.

The script questions should be answered while carrying out the experimental

procedure.

The report template should be completed with the graphs and answers to the

questions and submitted before the deadline.

1 Objectives

This experiment is aimed at:

Studying quantitatively Fourier components of some basic periodic waveforms.

Verifying that periodic waveforms can be synthesised by the superposition of sinusoids

having harmonically related frequencies.

Studying the effect of filters on waveforms.

Investigating the effect of phase on sound.

2 Apparatus

A computer with Matlab

Students of the University of Liverpool can download a copy of Matlab for use in their personal

computers/laptops from the following link:

https://www.liverpool.ac.uk/csd/software/software-downloads/#matlab

3 Introduction

In the fields of communications, signal processing and in electrical engineering more generally, a

signal is any time-varying or spatial-varying quantity. A graph of the amplitude of this signal

plotted against time produces a waveform (an example is shown in Figure 1). Perhaps the most

familiar type of waveforms is the sinusoid shown in Figure 1 (b). Here, the voltage (V ) at any

time (t) is given by the equation:

Figure 1: Examples of waveforms.

V (t) = Vosin(2πf t + φ) (1)

where Vo is the amplitude of the sinusoid, f is the frequency and φ is the phase angle measured

in radians.

It may be seen from the figure that this sinusoidal waveform is periodic with period T, i.e.

the waveform repeats itself after every T seconds, or V (t + nT) = V (t) for all t, where n is an

integer. The frequency f, measured in Hertz, and the period T are related by T = 1/f. The

phase angle determines the starting point of the sine wave at t = 0.

Analysis:

Many real-life signals are periodic but non-sinusoidal. Examples are shown in Figure 2. Fourier

analysis is based on a theorem which states that periodic waveforms may be expressed as the

sum of infinite simple sines and cosines (plus possibly a constant). If T is the period of a

periodic signal f(t), which satisfies certain conditions (normally satisfied by signals of practical

interest), then f(t) can be expressed as the sum of sine and cosine functions called a Fourier

series that is uniquely defined by constants known as Fourier coefficients. Hence:

Figure 2: Non-sinusoidal periodic waveforms.

f(t) = ao +

X∞

n=1

an cos(2πnfot) + X∞

n=1

bn sin(2πnfot) (2)

where fo = 1/T and is termed as the fundamental frequency, ao, an and bn (for n = 1, 2, 3, . . . )

are the Fourier coefficients that can be calculated from the following expressions:

f(t) sin(2πnfot)dt (5)

Equation 2 may also be written in an alternative form:

f(t) = co +

X∞

n=1

cn cos(2πnfot + φn) (6)

) and tan φn = ?bn/an.

Note that f(t) has been expressed as the sum of a constant (DC) voltage co and sinusoids of

frequencies fo, 2fo, 3fo, 4fo and so on. The sinusoid at frequency fo, i.e. c1 sin(2πfot + φ1),

is called the fundamental component of f(t). The other sinusoids are called harmonic components; the component at frequency 2fo being the second harmonic; the component at 3fo being

the third harmonic and so on.

Representing a periodic function f(t) as a series of sinusoids of specific amplitudes is referred

to as Fourier Analysis.

Example: Fourier analysis of a triangular wave.

It is required to analyse a triangular wave (Figure 3) using Fourier analysis. First, we need to

establish an expression for the time domain waveform f(t). It can be concluded easily that f(t)

can be written as:

Figure 3: Triangular wave.

(7)

Now, using the definition Equations 3, 4 and 5, we can find that:

The spectrum of f(t) (frequency domain view) contains an infinite number of frequency components, hence, the bandwidth of f(t) is infinite. A practical system would restrict f(t) to a finite

bandwidth by removing all frequency components above certain upper limit. This would distort

the shape of the triangular waveform. Fortunately, the first few terms in the Fourier series of a

triangular wave are the most important since the amplitudes, an, decrease rapidly as n increases.

Synthesis:

The addition of a series of harmonically-related sinusoids in order to generate a periodic waveform is referred to as Fourier synthesis, and is the reverse process of analysis.

To synthesise a periodic wave with a good approximation, the sine and cosine waves with the

proper amplitudes (as defined by the coefficients) must be electronically generated and combined up to the highest possible (and practical) value of n. The larger the value of n for which

sine and cosine wave signals are generated the more nearly the synthesised waveform matches

the desired waveform. Figure 4 is an example of successive approximation of a sawtooth wave

by adding harmonics with amplitude inversely proportional to the harmonic number. The resultant waveform at each stage of addition is shown at right.

Figure 4: Successive approximation of sawtooth signal.

Fourier synthesis is used in many practical applications. For example, it is used extensively

in electronic music applications to generate waveforms that mimic the sounds of familiar musical instruments. Although many musicians can create very clear-tones on their instruments,

all instruments have their own characteristic voice. Voices are differentiated by the shape of

their associated sound waves. All instrument voices display an array of frequencies which sum

together to produce a wave of characteristic shape. Fourier synthesis is also employed in laboratory instruments known as function (signal) generators that are used to generate different

waveforms for various purposes like electronic and communication systems test.

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