Signal Representation Using Fourier Series 2a4u65

FACULTY OF ELECTRONICS AND COMPUTER ENGINEERING BENG 3211 ELECTRONIC ENGINEERING LAB 3 (BENT 3733) LABORATORY SESSION 2 LAB 2: SIGNAL REPRESENTATION USING FOURIER SERIES

Prepared by: WAN NURUL FATIHAH BINTI WAN ARDAN

Prepared for: DR. JUWITA BINTI MOHD SULTAN

TITLE

: Signal representation using Fourier Series

OBJECTIVE

: The objectives of this laboratory exercise are to: 1. To express a periodic function in of Fourier series 2. To the Fourier series of the signal in MATLAB

APPARATUS

: MATLAB software

THEORY

: Fourier series is always used for solving infinite type of

mathematical series that include trigonometric functions. Its also can be applied in physics and electronic to shows the periodic function using waveform. This Fourier series shows a periodic signal of sine waves and cosine wave that have sinusoidal signal with different period of frequency. This infinite set of sine and cosine waves and DC signal can be disintegrated into periodic signal. The Fourier analysis of the amplitude with multiple frequency can be change to constituent sinusoidal that known as frequency spectrum.

In Fourier series, by adding the cosine and sine frequency of

f, 2f,

3f and etc the equation can be formed. The variable of a1,a2, a3, etc can represent the amplitude of cosine waves while b1,b2,b3 and etc show the amplitude of sine waves. In a nut shell, the coefficient of ‘a’ and ‘b’ was the real and imaginary part of frequency spectrum. The ao coefficient were usually know to shows the value of DC for the time domain waveform. For the time domain, it can be in many form of waves such as square, triangular or sine waves. The frequency spectrum is usually continuous but zero at all frequency except the harmonics. For the frequency domain

signal, usually it show two spectrum that in discrete which is amplitude spectrum and phase spectrum that used frequency to determined the harmonic frequency spectrum.

Time domain signal

Frequency domain signal

From the figure of frequency domain signal that shows the Fourier series of square wave in time domain signal, the Fourier series will take the periodic signal of x(t) with period T and frequency , ωo that will expressed as an infinite sum of sine and cosine function.

Then, mathematically we can described the Fourier series as below :

Where ao, an and bn are the coefficient of the Fourier series and can be calculated as :

DC component

Fourier coefficient

Fourier coefficient

when ωo is fundamental frequency and ղωo is harmonic frequency.

Beside, the Fourier series also can be compute in amplitude phase as shown below

Where the amplitude and phase angle can be calculate as

By using MATLAB, we can shows the amplitude and phase spectra from the frequency spectrum of x(t) and it will help to decide which frequency that's good for approximation of periodic function. By using MATLAB, we can shows the amplitude and phase spectra from the frequency spectrum of x(t) and it will help to decide which frequency that's good for approximation of periodic function

PROCEDURE :

3.1 Manually (Pre Lab) 1) The Fourier coefficients was calculated as below :

2) The trigonometric Fourier series is writes as below :

3) The signal up to 5th harmonic for amplitude and phase spectrum was draw and the plotted

RESULT : 3.2 Using MATLAB Signal for harmonic of N1=99

Figure 3.2.1 : The signal for the harmonic of 99 respectively was plotted. This is the first experiment for amplitude and phase spectrum using MATLAB

Signal for harmonic of N1=5

Figure 3.2.2 : The amplitude and phase spectrum was reproduced for 5th harmonic based on the calculated Fourier coefficients in Pre lab

Signal for harmonic of N1=21

Figure 3.2.3 : The amplitude and phase spectrum plot for maximum harmonic of 21

Signal for harmonic of N1=191

Figure 3.2.4 : Show the amplitude and phase spectrum plot for maximum harmonic of 191

DISCUSSION :

From this experiment, a square signal was given in this signal representation using Fourier series. First, we calculated the Fourier coefficient and find out the trigonometric Fourier series. After that, we tried to plotted the amplitude and phase spectra of the signal up to 5th harmonic. From this calculation, we assumed that the signal are in even symmetry because the value of bn = 0. Next, we used the MATLAB to observed the signal of amplitude and phase spectrum. By using the coding given, we reproduce the original signal up to 5th harmonic using N1=5. Then, we repeated again the experiments up to the maximum harmonic value of N1=21 and N1=191 to differentiate every waveform and make our understanding in Fourier series more better.

For this experimental result, we analyzed and observe the amplitude and phase spectrum for every value of N1. As we observe the waveform, the signal tends to reconstructed slightly like the original waveform as the number of harmonic we used gets more higher. But, there is a ripple in every waveform at discontinuities. The ripple did become more narrow as the number of harmonic we used get more higher, but when we tried to maximum number of harmonic the ripple doesn’t remove and remain stay a little bit at waveform. From that, we seen that the width slowly become closer to zero and we can understand that the signal reconstructed is not exact as its roughly follow like the original. This signal was fluctuates and it called as Gibbs Phenomenon and happens when the Fourier series has large oscillation around jump discontinuities.

Finally, we clearly see that as the number of harmonic used increase, the value at every point of phase and amplitude spectrum also increased. The higher the frequency, the signal became more resemble to original waveform.

CONCLUSION :

As conclusion, we learned how to express a periodic function in of Fourier series. Besides, we understood that every harmonic we used will affected the waveform in phase and amplitude spectrum. In addition, we knew how to analyzed the spectra pattern and and realized that the important of this Fourier series. We also able to configured the type of distortion happen and the harmonic selected. Other than that, we understood how to use the MATLAB software and able to differentiate every resulted. Lastly, we able to verified the Fourier series of this signal in MATLAB and learned how to solved the problem we faced when using it.

REFERENCES : 1) https://en.wikibooks.org/wiki/Signals_and_Systems/Fourier_Series 2) http://www.dspguide.com/ch13/4.htm 3) https://www.seas.upenn.edu/~kassam/tcom370/n99_2B.pdf