The Pictorial Waveform Imaging Computer Science Essay

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The pictorial representation of the form or shape of a wave, obtained by plotting the amplitude of the wave with respect to time. There are an infinite number of possible waveforms (see illustration). One such waveform is the square wave, in which a quantity such as voltage alternately assumes two discrete values during repeating periods of time. Other waveforms of particular interest in electronics are the sine wave and rectified sine wave, the saw tooth wave and triangular wave, and the arbitrary wave-a recurrent waveform which takes on an arbitrary shape over one complete cycle; this shape is then repeated in successive cycles

Waveform means the shape and form of a signal such as a wave moving in a solid, liquid or gaseous medium.

In many cases the medium in which the wave is being propagated does not permit a direct visual image of the form. In these cases, the term 'waveform' refers to the shape of a graph of the varying quantity against time or distance. An instrument called an oscilloscope can be used to pictorially represent the wave as a repeating image on a CRT or LCD screen.

By extension of the above, the term 'waveform' is now also sometimes used to describe the shape of the graph of any varying quantity against time.

Some examples of wave forms as follows

Triangular Waveforms

Square Waveforms

Sign Waves

Saw tooth Waveforms

Now we are going to briefly describe each of the waveforms.

Sign Waves

A wave having a form which, if plotted, would be the same as that of a trigonometric sine or cosine function. The sine wave may be thought of as the projection on a plane of the path of a point moving around a circle at uniform speed. It is characteristic of one-dimensional vibrations and one-dimensional waves having no dissipation.

The sine wave is the basic function employed in harmonic analysis. It can be shown that any complex motion in a one-dimensional system can be described as the superposition of sine waves having certain amplitude and phase relationships. The technique for determining these relationships is known as Fourier analysis

Basic Sign Wave

The terms defined below are needed to describe sine waves and other waveforms precisely:

PERIOD (T)

The period is the time taken for one complete cycle of a repeating waveform. The period is often thought of as the time interval between peaks, but can be measured between any two corresponding points in successive cycles.

FREQUENCY

This is the number of cycles completed per second. The measurement unit for frequency is the hertz, Hz. 1 Hz = 1 cycle per second.

PHASE

A phase is one part or portion in recurring or serial activities or occurrences logically connected within a greater process, often resulting in an output or a change. If two sine waves have the same frequency and occur at the same time, they are said to be in phase. On the other hand, if the two waves occur at different times, they are said to be out of phase. When this happens, the difference in phase can be measured in degrees, and is called the phase angle, .

AMPLITUDE

The amplitude is the power of a signal. The greater the amplitude, the greater the energy carried.

HARMONICS OF SIGN WAVE

First of all we discuss that what harmonics is. A harmonic is a signal or wave whose frequency is an integral (whole-number) multiple of the frequency of some reference signal or wave. The term can also refer to the ratio of the frequency of such a signal or wave to the frequency of the reference signal or wave.

Nearly all signals contain energy at harmonic frequencies, in addition to the energy at the fundamental frequency. If all the energy in a signal is contained at the fundamental frequency, then that signal is a perfect sine wave. If the signal is not a perfect sine wave, then some energy is contained in the harmonics.

In this first plot, we see the fundamental-frequency sine-wave of 50 Hz by itself. It is nothing but a pure sine shape, with no additional harmonic content. This is the kind of waveform produced by an ideal AC power source: (Figure below)

Next, we see what happens when this clean and simple waveform is combined with the third harmonic (three times 50 Hz, or 150 Hz). Suddenly, it doesn't look like a clean sine wave any more: (Figure below)

Sum of 1st (50 Hz) and 3rd (150 Hz) harmonics approximates a 50 Hz square wave.

The rise and fall times between positive and negative cycles are much steeper now, and the crests of the wave are closer to becoming flat like a squarewave. Watch what happens as we add the next odd harmonic frequency: (Figure below)

Sum of 1st, 3rd and 5th harmonics approximates square wave.

The most noticeable change here is how the crests of the wave have flattened even more. There are more several dips and crests at each end of the wave, but those dips and crests are smaller in amplitude than they were before. Watch again as we add the next odd harmonic waveform to the mix: (Figure below)

Sum of 1st, 3rd, 5th, and 7th harmonics approximates square wave.

Here we can see the wave becoming flatter at each peak. Finally, adding the 9th harmonic, the fifth sine wave voltage source in our circuit, we obtain this result: (Figure below)

Sum of 1st, 3rd, 5th, 7th and 9th harmonics approximates square wave.

The end result of adding the first five odd harmonic waveforms together (all at the proper amplitudes, of course) is a close approximation of a square wave. The point in doing this is to illustrate how we can build a square wave up from multiple sine waves at different frequencies, to prove that a pure square wave is actually equivalent to a series of sine waves. When a square wave AC voltage is applied to a circuit with reactive components (capacitors and inductors), those components react as if they were being exposed to several sine wave voltages of different frequencies, which in fact they are.

Representation of Sine Waves in Matlab

The basic MATLAB command for generating sinusoidal signal is:

A*sin (w0*t + phi), Where phi is the phase shift angle in radians.

MATLAB CODE OF SINE WAVE

OUTPUT OF SINE WAVE

A=4;

w0=20*pi;

phi=pi/6;

t=0:0.0001:.5;

sine=A*sin(w0*t + phi);

plot(t,sine)

Fig. Output of Sine Wave

SQUARE WAVE

The square wave is a periodic waveform consisting of instantaneous transitions between two levels. The square wave is sometimes also called the Rademacher function. The square wave illustrated above has period 2 and levels and 1/2. Other common levels for square waves include and (digital signals).

HARMONICS OF SQUARE WAVE

In contrast to the sawtooth wave, which contains all integer harmonics, the square wave contains only odd integer harmonics.

Using Fourier series we can write an ideal square wave as an infinite series of the form.

For a reasonable approximation to the square-wave shape, at least the fundamental and third harmonic needs to be present, with the fifth harmonic being desirable. These bandwidth requirements are important in digital electronics, where finite-bandwidth analog approximations to square-wave-like waveforms are used. (The ringing transients are an important electronic consideration here, as they may go beyond the electrical rating limits of a circuit or cause a badly positioned threshold to be crossed multiple times.)

REPRESENTATION OF SQUARE WAVE IN MATLAB

MATLAB CODE OF SQUARE WAVE

OUTPUT OF SQUARE WAVE

A=0.5;

w0=10*pi;

rho=50;

t=0:0.00001:1;

sq=A*square(w0*t,rho);

plot(t,sq)

axis([0 1 -1.1 1.1])

TRIANGULAR WAVE

A triangle wave is a non-sinusoidal waveform named for its triangular shape.

Triangular wave pictured in time domain.

HARMONICS OF TRIANGULAR WAVE

Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of The harmonic number as opposed to just the inverse), and so its sound is smoother than a square wave and is nearer to that of a sine wave

It is possible to approximate a triangle wave with additive synthesis by adding odd harmonics of the fundamental, multiplying every (4n−1)th harmonic by −1 (or changing its phase by π), and rolling off the harmonics by the inverse square of their relative frequency to the fundamental.

This infinite Fourier series converges to the triangle wave:

REPRESENTATION OF TRIANGULAR WAVE IN MATLAB

MATLAB CODE OF TRIANGULAR WAVE

OUTPUT OF TRIANGULAR WAVE

fs = 10000;

t = -1:1/fs:1;

x1 = tripuls(t,20e-3);

subplot(211),plot(t,x1), axis([-0.1 0.1 -0.2 1.2])

xlabel('Time (sec)');

ylabel('Amplitude');

title('Triangular Periodic Pulse')

SAWTOOTH WAVE

A waveform that increases linearly with time for a fixed interval, returns abruptly to the original level, and repeats the process periodically, producing a shape resembling the teeth of a saw.

HARMONICES OF SAWTOOTH WAVE

A sawtooth wave's sound is harsh and clear and its spectrum contains both even and odd harmonics of the fundamental frequency. Because it contains all the integer harmonics, it is one of the best waveforms to use for synthesizing musical sounds, particularly bowed string instruments like violins and cellos, using subtractive synthesis.

A sawtooth can be constructed using additive synthesis. The infinite Fourier series

converges to an inverse sawtooth wave. A conventional sawtooth can be constructed using

In digital synthesis, these series are only summed over k such that the highest harmonic, Nmax, is less than the Nyquist frequency (half the sampling frequency). This summation can generally be more efficiently calculated with a Fast Fourier transform. If the waveform is digitally created directly in the time domain using a non-bandlimited form, such as y = x - floor(x), infinite harmonics are sampled and the resulting tone contains aliasing distortio

REPRESENTATION OF SAWTOOTH IN MATLAB

MATLAB CODE OF SAWTOOTH

OUTPUT OF SAWTOOTH WAVE

fs = 10000;

t = 0:1/fs:1.5;

x1 = sawtooth(2*pi*50*t);

subplot(211),plot(t,x1), axis([0 0.2 -1.2 1.2])

xlabel('Time (sec)');

ylabel('Amplitude');

title('Sawtooth Periodic Wave')

EQUIPMENTS USED IN LABORATORY TO PERFORM CH-05

OSCILOSOPE

FUNCTION GENERATOR

SPECTRUM ANALYSER

OSCILOSCOPE

An oscilloscope is a laboratory instrument commonly used to display and analyze the waveform of electronic signals. In effect, the device draws a graph of the instantaneous signal voltage as a function of time.

A typical oscilloscope can display alternating current (AC) or pulsating direct current (DC) waveforms having a frequency as low as approximately 1 hertz (Hz) or as high as several megahertz (MHz). High-end oscilloscopes can display signals having frequencies up to several hundred gigahertzes (GHz). The display is broken up into so-called horizontal divisions (hor div) and vertical divisions (vert div). Time is displayed from left to right on the horizontal scale. Instantaneous voltage appears on the vertical scale, with positive values going upward and negative values going downward.

There are two types of oscilloscope. Which are as follows.

Analogue Oscilloscope

Digital Oscilloscope

Analogue Oscilloscope

An analogue oscilloscope works by directly applying a voltage being measured to an electron beam moving across the oscilloscope screen. The voltage deflects the beam up and down proportionally, tracing the waveform on the screen. This gives an immediate picture of the waveform.

Fig. 20 Analog Oscilloscope Block Diagram

Digital Oscilloscope

A digital oscilloscope samples the waveform and uses an analogue-to-digital converter (or ADC) to convert the voltage being measured into digital information. It then uses this digital information to reconstruct the waveform on the screen.

Fig. 21 Digital Oscilloscope Block Diagram

FUNCTION GENERATOR

A function generator is a piece of electronic test equipment or software used to generate electrical waveforms. These waveforms can be either repetitive, or single-shot in which case some kind of triggering source is required (internal or external).

Another type of function generator is a sub-system that provides an output proportional to some mathematical function of its input; for example, the output may be proportional to the square root of the input. Such devices are used in feedback control systems and in analog computers.

TYPES OF FUNCTION GENERATOR

There are two types of function Generator.

Analogue

Digital

ANALOGUE FUNCTION GENERATOR

Analog function generators use a voltage controlled oscillator (VCO) to generate a triangular waveform of variable frequency. Sinusoidal waveforms and square waves are generated from this.

DIGITAL FUNCTION GENERATOR

Digital generators use a digital to analogue converter (DAC) to generate a wave shape from values stored in memory. Normally such generators only offer sine and square waves up to the maximum generator frequency. Triangle waves and other waveforms are limited to a much lower frequency.

SPECTRUM ANALYSER

A spectrum analyzer or spectral analyzer is a device used to examine the spectral composition of some electrical, acoustic, or optical waveform. It may also measure the power spectrum.

There are analog and digital spectrum analyzers:

An analog spectrum analyzer uses either a variable band-pass filter whose mid-frequency is automatically tuned (shifted, swept) through the range of frequencies of which the spectrum is to be measured or a super heterodyne receiver where the local oscillator is swept through a range of frequencies.

A digital spectrum analyzer computes the discrete Fourier transform (DFT), a mathematical process that transforms a waveform into the components of its frequency spectrum.

Some spectrum analyzers (such as Tektronix's family of "real-time spectrum analyzers") use a hybrid technique where the incoming signal is first down-converted to a lower frequency using super heterodyne techniques and then analyzed using fast Fourier transformation (FFT) techniques.

Types of Spectrum Analyzers

Analog Spectrum Analyzer

An analog spectrum analyzer uses either a variable band pass filter whose mid-frequency is automatically tuned (shifted, swept) through the range of frequencies of which the spectrum is to be measured or a super heterodyne receiver where the local oscillator is swept through a range of frequencies.

Digital Spectrum Analyzer

A digital spectrum analyzer computes the Fast Fourier transform (FFT), a mathematical process that transforms a waveform into the components of its frequency spectrum.

Some spectrum analyzers use a hybrid technique where the incoming signal is first down-converted to a lower frequency using super heterodyne techniques and then analyzed using FFT techniques.

9. PRACTICAL RESULTS

Initial Set-up: f = 10 KHz SINE WAVE

Oscilloscope

Spectrum Analyzer

Explanation

Explanation

T = 50 µsec * 2 = 100 µsec

A = 50mV * 4 = 200 mV

f (calculated) = 1/T = 10 KHz

When frequency is initially set up at 10KHz, it is observed that spectrum is wider.

Change to: f = 5 KHz SINE WAVE

Oscilloscope

Spectrum Analyzer

Explanation

Explanation

T = 50 µsec * 4 = 200 µsec

A = 50mV * 4 = 200 mV

f (calculated) = 1/T = 5 KHz

If frequency is decreased to half than time would become twice, provided amplitude remains constant.

If frequency is decreased than bandwidth of the channel should also be decreased. For e.g., f = 2 KHz, T = 0.5 msec. When f = 4 KHz than T = 0.25 msec. In this way bandwidth becomes less, because bandwidth is defined as " differences of highest frequency and lowest frequency."

From the figure, it is observed that spectrum became sharper when frequency is decreased, provided amplitude is constant.

Initial Set-up: f = 10 KHz SQUARE WAVE

Oscilloscope

Spectrum Analyzer

Explanation

Explanation

T = 50 µsec * 2 = 100 µsec

A = 50mV * 4 = 200 mV

f (calculated) = 1/T = 10 KHz

When frequency is initially set up at 10KHz, it is observed that spectrum is wider.

Change to: f = 5 KHz SQUARE WAVE

Oscilloscope

Spectrum Analyzer

Explanation

Explanation

T = 50 µsec * 4 = 200 µsec

A = 50mV * 4 = 200 mV

f (calculated) = 1/T = 5 KHz

If frequency is decreased to half than time would become twice, provided amplitude remains constant.

From the figure, it is observed that spectrum became sharper when frequency is decreased, provided amplitude is constant.

Initial Set-up: f = 10 KHz TRIANGULAR WAVE

Oscilloscope

Spectrum Analyzer

Explanation

Explanation

T = 50 µsec * 2 = 100 µsec

A = 50mV * 5 = 250 mV

f (calculated) = 1/T = 10 KHz

When frequency is initially set up at 10KHz, it is observed that spectrum is wider.

Change to: f = 5 KHz TRIANGULAR WAVE

Oscilloscope

Spectrum Analyzer

Explanation

Explanation

T = 50 µsec * 4 = 200 µsec

A = 50mV * 5 = 250 mV

f (calculated) = 1/T = 5 KHz

If frequency is decreased to half than time would become twice, provided amplitude remains constant.

From the figure, it is observed that spectrum became sharper when frequency is decreased, provided amplitude is constant.

Initial Set-up: f = 10 KHz SAWTOOTH WAVE

Oscilloscope

Spectrum Analyzer

Explanation

Explanation

T = 50 µsec * 2 = 100 µsec

A = 50mV * 5 = 250 mV

f (calculated) = 1/T = 10 KHz

When frequency is initially set up at 10KHz, it is observed that spectrum is wider.

Change to: f = 5 KHz SAWTOOTH WAVE

Oscilloscope

Spectrum Analyzer

Explanation

Explanation

T = 50 µsec * 4 = 200 µsec

A = 50mV * 5 = 250 mV

f (calculated) = 1/T = 5 KHz

If frequency is decreased to half than time would become twice, provided amplitude remains constant.

From the figure, it is observed that spectrum became sharper when frequency is decreased, provided amplitude is constant.

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