# SIGNAL ANALYSIS

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Introduction

Engineering Lab Report AIMS 2 OBJECTIVES 2 PART 1: 2 PART 2: 2 EQUIPMENT 2 PROCEDURE 3 PART 1: 3 THEORY 4 FILTERS: 4 Fig 1 Ideal Filter 4 Fig 2 Typical Filter 5 Fig 3 Typical 2-stage Filter 5 TRANSFER FUNCTIONS AND THE LAPLACE TRANSFORM: 6 FOURIER ANALYSIS: 8 Components of a sine function: 8 RESULTS 9 LOW PASS FILTER 9 Table of Results: 9 Graph of Results: 9 Bode Plot for the Low Pass Filter: 10 HIGH PASS FILTER 10 Table of Results: 10 Graph of Results: 11 Bode Plot for the High Pass Filter: 11 BAND PASS FILTER 12 Table of Results: 12 Graph of Results: 13 Bode Plot for the Band Pass Filter: 13 STOP BAND FILTER 14 Table of Results: 14 Graph of Results: 15 Bode Plot for the Band Stop Filter: 15 2-STAGE LOW PASS FILTER 16 Table of Results: 16 Graph of Results: 16 Bode Plot for the 2-StageLow Pass Filter: 17 RESULTS FOR PART 2 17 SQUARE WAVE: 17 RAMP FUNCTION: 18 HALF WAVE RECTIFIER: 20 FULL WAVE RECTIFIER: 20 MODULATED SINE WAVE: 21 MODULATED SQUARE WAVE 21 DISCUSSION 22 CONCLUSION 23 APPENDICES 24 SIGNAL ANALYSIS AIMS Through experimentation of signal analysis, an understanding will be gained of a signal's behaviour when passing through filter systems of various orders. This also includes the understanding of the mathematical representation of signals and filter systems in both time and frequency domains with the use of Fourier analysis (series and transforms) and the Laplace transform. OBJECTIVES Part 1: The main objectives of this exercise are to enable students: 1. To construct simple analogue passive filter circuits 2. To measure performance of low pass, high pass, band pass and band stop (notch) filters and compare the practical results to theoretically computed results. 3. To get an appreciation of multistage filters. Part 2: Observe four different waveforms in both time and frequency domains and compare the spectrum with that of there calculated values. ...read more.

Middle

The gain i.e. the ratio between the output and the input is represented by the system model. The transformed representation of the system model is referred to as the Transfer Function. E.g. The Low Pass filter circuit on the previous page can be written as: Which when rearranged gives the transfer function of: If the Transfer Function for a filter is calculated, then it is possible to calculate the cut off frequency of the filter by the use of simple mathematics. When the denominator of a Transfer function is equal to zero, it is known as a 'pole'. By calculating the poles of a filters Transfer function, the cut off frequency is obtained. As illustrated below: Now substituting S for j?, we get: Fourier Analysis: Fourier, a French mathematician discovered that any periodic function could be represented by the summation of a series of Sine waves. This method is known as Fourier Series and can be obtained by the following formula. The Fourier Coefficients ao, an and bn can be found with the following formulas: Each sine or cosine in the Fourier series represents the harmonic content of the signal. The first in the series is the first harmonic or fundamental frequency. This will usually have the largest amplitude and the lowest frequency. Components of a sine function: The Process of the Fourier Transform allows a signal/function to be represented as a spectra. This is obtained by using the following formula: RESULTS Below are the results for part 1. Low Pass Filter R = 470 ? C = 0.33?F The Laplace Transform for this filter is 1/(RCs + 1). Taking the poles to zero gives a cut off frequency of 1026Hz. Table of Results: Frequency (Hz) Output Voltage (v) Output Decibels(dB) 250 7.7 -0.33 500 7 -1.16 750 6.3 -2.07 1500 4.4 -5.19 2000 3.5 -7.18 3000 2.5 -10.1 4000 1.9 -12.49 5000 1.6 -13.98 Graph of Results: Bode Plot for the Low Pass Filter: High Pass Filter R = 47 ? ...read more.

Conclusion

These must be converted to frequency in Hertz by dividing by 2?. For the first harmonic this gives a frequency value of 1000Hz and for the second harmonic a value of 3000Hz. Again this corresponds to the graph. This method was then repeated for the Ramp Function and again the results of the Fourier series correspond to the graph. The results of this clearly show that indeed a function can be represented by a summation of sine waves but also that the Fourier series can provide the harmonic content of any periodic function. The modulated signal demonstrates that a signal can be multiplied by another signal to produce a new signal combining both sets of harmonics. The Modulated ~Sine wave demonstrated that if we have a frequency ? and modulate this with a carrier frequency ?c, we actually get a symmetry of the carrier frequency ?c about the frequency ?. In the case of the Modulated Sine wave the frequency ?c was 500Hz and the carrier frequency ? was 30kHz. If we look at the graph we can see that about the 30kHz spike, are two other spikes that are 500Hz away from the 30kHz one. The smaller spikes on the graph are noise. This symmetry is also supported in the maths of the Fourier Transform (in the appendices). The text in green demonstrates the reason why this symmetry exists. Conclusion All of the results from this experiment have met the requirements of the objectives of this report and also proves the theory section of the report. The lab has proved to be a valuable learning experience that has joined both the mathematical representation of filter circuits and signals in a visualised manor, which has allowed an understanding and interest to be gained. Errors occurring in this lab are due to component values and tolerances, mistakes in the mathematics and human error in obtaining readings and measurements. Appendices ?? ?? ?? ?? ASSIGNMENT 1&2 LAB REPORT BEng Electronic Engineering Year '2' Page 1 ...read more.

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