synthesis of wave forms

8/10/2019 Synthesis of Wave Forms

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Formal Report

Four ier Analysis

And

Synthesis of Wave forms



Learning objectives

This experiment will on completion, help the student to understand:

(a) The physical meaning of signal spectrum

(b) That each different waveform has a different spectrum (or spectral density)(c) That certain standard results exist for standard waveforms(d) The effects of spectrum limitation on the transmission of signals

Abstract

This experiment explores Fourier analysis. Fourier series is used to convert a

time-domain signal to frequency domain. Fourier series decomposes any

periodic function or periodic signal into the sum of a set of simple oscillating

functions, namely sines and cosines. This experiment is concerned with the

analysis of a signal and the exploration of the properties of the signal. In thisexperiment, we are going to study the basic waveforms such as sinusoidal,

square, saw tooth and triangular waves. We will use Fourier series to observe

and study the effects of limiting the bandwidth of real signal, the amount of

bandwidth needed to support a binary representation of an analogue signal,

quantisation of analogue signals and unusual signals. A special java basedapplication will be used to perform the experiment.

Introduction

Fourier series were introduced by Joseph Fourier for the purpose of solving the

heat equation in a metal plate. The Fourier series has many applications in

electrical engineering, vibration analysis, acoustics, optics, signal processing,

image processing, quantum mechanics, and econometrics etcThis experiment focuses on the analysis and synthesis of signals and Fourier

Transform. A synthesized signal will be generated which will be an

approximation of the input signal. The synthesized signal will look more andmore like the input signal as the number of terms will increase. Using Fourier

series a real signal in time domain will be characterised in frequency domain

which will be observed by the frequency domain graph. This experiment willalso demonstrate the effect of varying the number of terms, Quantizing, Phase

Shifting and Clipping the wave.

Background Theory



A signal can be regarded as the variation of any measurable quantity that

conveys information concerning the behaviour of a related system . A system is

a means of processing a signal. Signals come in many shapes and forms and

can be classified as; Analogue/ Continuous Time Signals and Digital/ DiscreteTime Signals.

Analogue or Continuous Time Signal

A continuous time signal can be represented mathematically as a function of a

continuous time variable. It is defined at all time (t).

Digital or Di screte Time Signal

A discrete time signal is defined only at particular set of instants of time.



A discrete time signal is often derived from a continuous time signal

by sampling it at uniform rate.

Time Domain

It is a term used to describe the analysis of a physical quantity whenthe magnitude of a signal is compared with respect to time

Frequency Domain It is a term used to describe the analysis of a signal with respect to frequency.



In Fourier series, the frequency domain can be graphically represented by two

different components:

(i) Sine Component: The component of the signal in which there is no phase

shift observed or phase shift is equal to zero degrees as shown in the figure

below in red colour. In Fourier series, sine components are represented by ‘bn’.

(ii) Cosine Component: The component of the signal in which there is an

observed phase shift of 90 degrees as illustrated by the blue colour wave. In

Fourier series, cosine components are represented by ‘an’ .

Certain properties such as rectification, quantization, phase-shifting and

clipping are tested on the signal and output is observed. The testing will becarried on analogue signals such Square wave, Saw tooth wave and triangular

wave.



Quantization

It is a process of approximating the continuous range of values by a finite set of

discrete values. In this process, a PAM (pulse amplitude modulated) or a

sampled signal gets converted to a digital signal by comparing it with steps. The

step size depends on the amplitude of the sampled signal.



Clipping

The processes by which a particular part of the signal can be limited once it’s

exceed its threshold value. In this experiment, it will increase the amplitude but,

it will clip the signal if goes out of range. When a clip function is applied to a

sine wave, jagged edges will be observed as shown in the figure below

Rectification

It is a process by which an AC (alternating current) signal can be converter toits equivalent pulsating DC (direct current) form. With the application of

rectification in this experiment, only the positive component of the signal can be

observed. The negative component of the signal will get waived off.

Types of Signals

Periodic Signals

Signals which repeat itself after a particular period are termed as periodic

signals. Square wave, sine wave and triangular waves are examples of periodic signals.

If the original real periodic signal in the time domain, x(t), is defined by:

x(t) = x(t + n.T)

Where, T is the fundamental period of x(t) and ‘n’ is any integer.

Aperiodic SignalsSignals which do not have any relation with respect to time are termed as

aperiodic signals.

Random Signals

Signals that cannot be characterized by a limited number of precise measuresand does not have continuous relation with time or frequency. Noise is an

example of a random signal.



ODD AND EVEN FUNCTIONS:

Odd Function

If x(t)= -x(-t) the signal is odd, eg a Sine wave.

Even Function

If x(t)= x(-t) the signal is even, eg a cosine wave.

Note:

ODD* ODD = EVEN EVEN * EVEN = EVEN

EVEN * ODD = ODD

ODD * EVEN = ODD

ORHTHOGONALITY

When the product of two signals integrates or averages to zero over a specified

time interval, the signal can be defined as orthogonal over the particular

interval.



Discrete signals

If the two signals averages to zero over the period T, then those two signals are

orthogonal in that interval(T).

Continuous signals If the product of two signals integrates to zero over the period T, then those two signals are ORTHOGONAL in that interval (T).

Time Domain Representation

Output signal to the input signal is represented as a function of time is called

Time Domain Representation.

FOURIER SERIES

Fourier series can be defined as sum of the sine and cosine components present

in a signal. It is method of expressing a function of a signal in terms of the sum

of its projections (amplitude and frequencies) onto a set of basic functions.

Fourier series help us move from digital to analogue. Any periodic signal x (t),

which has time period of T can be represented by the approximate sum of sine

and cosine components.



Description of the Experiment

Type in the following URL: ‘ http://www.falstad.com/fourier/ ’.

An external JAVA applet window will be opened.

Figure

The figure shows the Applet used for this experiment. It represents signals in

time domain and frequency domain. In the Applet, the real signal being observed and analysed is represented as a

white line. The menu on the right hand side allows you to choose which signal

type you want to look at in detail. The red line represents the synthesised signal,

i.e. the signal which results when you have only a limited number of spectra of

the original signal available to combine them into the resulting, desired signal

(e.g. a received signal at the system’s receiver, or a signal at a system’s output).The number of line spectra is set by the “slider bar” labelled NUMBER OF

TERMS at the bottom right of the menu.

The command buttons on the right represent the types of waves. The user can

select any one of them by simply clicking on them. Below the command buttons



are two sliders which represent number of terms and Playing frequency. They

can be set according to requirement by simply moving them right or left.

A) To use Four ier Analysis to observe the effect of l imiting the

bandwidth of a real signal .Select the square wave option.

1. Move the slider to the far left hand side.

2. The red line becomes flat

3. Note a single white dot under the cosines represented by a0. Magnitude is displayed when the cursor is placed on the dot.

Reason

As the number of terms becomes “0” the curve becomes a straight line. Fourier series a0 + ∑an cos(nwt) + ∑bn sin(nwt). Since n=0 therefore 2

nd

and the 3rd

component of the series would be zero, the only component

remaining is a0 which is component of cosine and thus the only a white

dot is visible under cosines.4.

Slowly increase the number of terms by moving the slider to the

right.

Note that the value remains 0 for dots under Cosines whereas it is

maximum for sines at n=1 and “0” at even no. of terms i.e n=0,2,4... As

the generalised formula for sine components bn states that bn=0 for evenvalues of “n” and bn=4V/nπ for odd values of n.



5.

The first five values are observed under and later confirmed by

calculation.

Terms(n) Sine Value(bn) Cosine Value(an)

1 1.27324 0

2 0 0

3 0.4244 0

4 0 0

5 0.25463 0

Calculations

a0 = 1/T –

a0 = 1/T

= 1/T –

= 1/T (0)

a0 = 0



For n=1

N=1



For n=2

For n=3b3 = 0.424

For n=4

b4 =0



For n=5



7)Implications

The synthesized signal tries to take the approximate shape of the original

signal depending on the no of terms. As we increase the number of terms, the

synthesized signal looks more and more like the original signal. At maximumvalue of n in the applet i.e n=159, the synthesized signal looks exacltly like theoriginal signal.

N=159



Part ii) Repeat the above experiment with rectified square wave option.

Click the clear button on the right

Select the square wave option and Rectify

Term(n) Sine(bn) Cosine(an)

1 0.636618 0

2 0 0

3 0.212206 0

4 0 0

5 0.127323 0



For n=1



For n=5



Part iii)

Press Clear button Square Wave and move the no of terms slider to the

halfway.

Then press phase shift button from the right hand menu ten times but slowly and

notice after each click the change in original signal.

Explanation By pressing the phase shift button the original wave gets shifted 18˚. Therefore

by pressing the phase shift button ten times the original wave gets shifted 180˚.

When we apply 180˚ phase shift to sine wave it becomes inverted e.g sin90=1 &

sin270= -1. The magnitude of the wave remains unchanged, only the polaritychanges. At each click value of both sine and cosine components change

i . Part 4 ( Saw tooth wave form for F our ier Analysis)Click the clear button and then the saw tooth option from the right side

menu. 1. Reduce the no. of terms to minimum possible by moving the

slider to the far left.

2.

The red line(Synthesized wave) becomes flat because the n=0.



Note only 1 white dot is visible under cosines in frequency domain plot. Explanation

When n=0, only ao component of the fourier series exist while an

and bn become zero. ao is the component of cosine therefore there

is a white dot under cosines.3.

By placing the cursor on the sine and cosine components

displays their magnitudes.

4.

Note that as we increase the number of terms and newcomponents appear, the sine ones have non-zero values and the

cosine ones have zero values, similar to the case with square

wave. This is because sine waves square waves; triangular and sawtooth waves all represent AC signals and obey Fourier

series likewise.

Keep on increasing the no. of terms until the synthesized signal

looks approxiamately like the original signal. As we increase the

terms the synthesized signal looks more and more like the original

signal.



At n=35 the synthesized signal looks approxiamtely like the

original signal and the frequency of the highest frequency terms is7725Hz. The least bandwidth required here is 7940 Hz to transmit

the signal, which is the same as square wave at 35 terms.

At n=60 the synthesized signal looks approximately like the

original signal and at n=159 it looks exactly like the original signal.



5.

Move the number of terms slider to half way and then press

phase shift button 10 times slowly and note on each click the

change in original signal.

Explanation

6. When the phase shift is clicked 10 times the sine component of

the wave inverts. The original wave gets phase shifted at eachclick without any change in magnitude.

7.

Move the slider so far to the left that only the 1st harmonic

appears. The synthesized wave takes the shape of the sine wave.

Now again press phase shift 10 times and note that the wave

gets phase shifted and sine component inverted.

Part B) Use of F our ier Series to decide on Bandwidth

needed to binary representation of analogue signal .

1. Click the Clear button and then the SAW TOOTH

Option; this would create a Saw tooth wave.

Keep on increasing the no. of terms until the synthesized signal looks approximately the same as original signal.

Here, the value of number of terms (n) is considered to be

60.



2.

Click the quantize button from the right side menu and

observe the change to the sawtooth signal. Upon

pressing the quantise button the wave acquires a staircase like shape.



Observation :

3.

When the quantize button is pressed for the first time,

the sawtooth waveform gets transformed to a staircase

waveform of step size 17 as shown in fig (a).

When the quantize button is pressed for the second time, the staircase waveformof size 17 gets converted to size 9 as shown in fig (b).

When the quantize button is pressed the third time, the waveform getstransformed into a bigger staircase waveform of step size 5 as shown in fig (c).

4) Fs >= 2fmax

Given: f= 1HzThe minimum sampling frequency required is 2 Hz.

Click on the CLEAR option and select COSINE.

Move the no. terms slider unless the synthesized signal looks like

the original signal. Press the quantise button 3 times. The cosine wave changes to a

step like shape waveform.



Sampling frequency fs>=2fmax

If f=1 Hz, then fs=2Hz.

6. Click the Clear button and then the Sawtooth again.

Increase the number of terms until the synthesized signal looks

approximately like the original signal.

Again synthesize the waveform 3 times

Now we have quantised sawtooth waveform, with 5 quantisation levels

and 5 samples per period. For 5 levels, 3 bits per sample is required.

= M M = Quantisation Level.

N = No. of bits

=5

n = log5

n= log5/log2

n=2.32 rounding off to n=3

7) Assume your coding scheme uses 1’s and 0’s with equal probability (this maynot be the case in real systems), and assume the sawtooth has a fundamental



frequency of f=1Hz. What bit rate B do you need to transmit the bit stream

representing the coded quantised sawtooth?

k=3 bits

F=1 Hz

fs>=2f

fs = 2 Hz

B = fs*k

B=2*3=6 bits/s

8) A fundamental frequency is required and to get that a square wave is needed.

Square wave because of its infinite precision can be used as ideal wave.

9) Considering fs to be the minimum sampling frequency,

fq=fs/2 = 0.5 Hz.

11) Press the clear button. Select sawtooth wave. Press quantize button twice. It

is observed that there are nine quantization levels instead of 5.

12) Design a digital communications system which transmits this quantized

sawtooth as a stream of bits. There are 9 levels, so how many bits k will you

need per sample?

A: Consider k to be the number of binary bits to be transmitted, L as the

quantization level.Given: L = 9

=9

n = log9/log4

Since the value of k is in decimals, round it off to next digit)

n = 4

To transmit a sawtooth wave with 9 quantization levels 4 bits will be required.

It is observed that with the increase in quantization level, the bit rate required

to transmit the signal increase which in turn increases the cost

Part C) Unknown Signals

Click the CLEAR option in the menu, and then the TRIANGLE option.

1.

Increase the number of terms until you have a very

good representation of the original signal. At n=60

synthesized wave looks like the original wave.



2.

Now press the CLIP button 15 times slowly and

observe the change in original signal and its spectra.



Note that the edge of the waves starts getting jagged when the

CLIP button is pressed. The amplitude of the waveform increaseswhile the frequency remains constant.

3. Now reduce the number of terms to n= 12 so that a

handful of harmonics are left.

Original Signal

Synthesized Signal at hand full of harmonics

With fewer spectra’s , the number of harmonic components reduces. The

harmonics components are not sufficient to represent the original white signal

as shown in the above fig.

Discussion and Conclusion

After doing this experiment, I completely understand the importance and proper

application of Fourier series. Fourier series helps us understand better the

behaviour of different types of signals. By increasing the number of terms a

synthesized signal can be made to look exactly like the original signal. Withvariations in time domain we also experience variations in frequency domain.

With the help of Fourier series we can improve the quality of signals.

The sawtooth signal has similarities to square wave as both represent A.Cvoltages. Phase shift of 180˚ C to these waves causes the sine components to



invert or change their polarity to negative. Fourier series also help us to decide

on bandwidth requirement and observe and work on quantized signals and

sample their frequencies.

We also learned that when a triangular wave is clipped, it starts to look more

and more like a square wave as the clipping goes on. FOURIER SERIES helpsengineer’s to calculate the required harmonic, Phase shift, Amplitude of Sine aswell as Cosine and the zero frequency components. It can analyze and process

images, it can efficiently compress images and video streams, and it can assist

in visual pattern recognition, where a complex pattern may be efficiently and

concisely described using a set of Fourier terms.

REFERENCES: Lecture notes by John Schormans, Dept of electronic engineering QMUL

Lab sheet 1 of ELE-374, Dept of electronic engineering QMUL

Signal and Systems by Meade and Dillon

Signals and System by Simon Haykins

synthesis of wave forms

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