Al-Nahrain University College of Engineering Department of Electronic and Communications Engineering Communication Systems I Laboratory Report Third Year Class First Semester Experiment No.1 Experiment Title: Signals and Spectra Group No.3 Test Bench No.3 Participant(s) Name(s) Ali Hassan, Ahmed Zeyad,Riyam Hameed and Rafal Fawzi Academic Year 3rd Date of Experiment 2019/10/24 Report Due Date 2019/10/28 Report Submission Date 2020/02/6 Grading Particulars Maximum Mark Mark Scored Objectives, Apparatus Procedure Lab paper Discussion Promptness, Neatness and Organization Total Mark Instructor Name and Signature Date Communication Systems I - Laboratory Experiment 1 Signals and Spectra Objective To be aware of Fourier series and Fourier Transform techniques, and to practice their applications in communication theory, so you will be able to understand the next Experiments . Theory Procedure Part I: Fourier Series Techniques 1. First step was opening MATLAB then went to the command window and typed Fourier Series. This would bring up the application software that could be used to demonstrate many concepts and properties of the Fourier series expansion. And the one that we’ve already work on. 2. For each signal we’ve typed monitor its waveform and line spectra and sketched them down in our report. 3. I had changed the number of harmonics to be displayed in line spectra and used in the truncated FS expansion of the different types of signal, then reported the observations. 4. Had note the plots for amplitude and phase spectra, And Stated what kind of symmetry is present in each type of spectrum, also we had figured out why. Part II: Fourier Transform Techniques 1. First step was opening MATLAB then went to the command window and typed Fourier Transform. This would bring up the application software that could be used to demonstrate many concepts and properties of the Fourier transform. And the basic signal used was a rectangular unit pulse. 2. I’ve had Introduced a certain time delay in the function and noticed what happened to the amplitude spectra, and never forget to explain why. 3. We’ve been Introduced different scaling factors and commented on what we had observing . 4. Had Introduce a frequency shift and observed the resultant signal waveform and its spectra. So we’ve noticed and commented what we were observing. Discussion 1-State the symmetry properties associated with each type of signal considered, both in time and frequency domains. Solution: n displaying or specifying the Fourier transform of a real-valued time function it is necessary to display the transform only for positive values of w. A second important property is that of time and frequency scaling, spe- cifically that a linear expansion (or contraction) of the time axis in the time domain has the effect in the frequency domain of a linear contraction (expan- sion). In other words, linear scaling in time is reflected in an inverse scaling in frequency. 2. Give a brief explanation of the Gibbs phenomenon and the signal(s) with which it can be observed in the experiment. Solution: Fourier Series and Gibbs Phenomenon Overview In this experiment you work with the Fourier series representation of periodic continuous-time signals and learn about Gibbs phenomenon. The Fourier series representation of a periodic signal, with period T=1/fo, is defined by where the complex Fourier series coefficients, also expressed in polar form, are computed via the integral formula where T is the fundamental period of the signal. The DC component of the signal is equal to the first Fourier series coefficient and is simply the average value of the signal over one period. The sinusoidal components of the signal that occur at multiples of the fundamental frequency are called harmonics. In general, for well-behaved (continuous) periodic signals, a sufficiently large number of harmonics can be used to approximate the signal reasonably well. For periodic signals with discontinuities, however, such as a periodic square wave, even a large number of harmonics will not be sufficient to reproduce the square wave exactly. This effect is known as Gibbs phenomenon and it manifests itself in the form of ripples of increasing frequency and closer to the transitions of the square signal. An illustration of Gibbs phenomenon is shown in the figure below. The figure shows the result of adding one, three, five, seven, and nine harmonics. In all cases, and regardless of the number of harmonics, it is observed that the overshoot of the ripples has a constant magnitude (around 18%). In this experiment we study the Fourier series representation of two periodic signals, a triangular waveform and a square waveform. Both have periods equal to 2 seconds. It will be seen that Gibbs' effect is significantly more pronounced in the square wave case.