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Q1 Periodic Signals A small portion of a causa/ plecewise continuous periodic signal x (t) = x(+ nT) is shown in Figure Q1 below. x(t) [ А т 3T т 4 T 4 t[ms] F」 4 A b b Figure Q1: A Plecewise-continuous periodic waveform X(t) = x(t + nT). : -x1) (1 + ). For the rest of this question, the period of the signal T = 200 ms, A = 8V, and b = 2. = b Answer all parts of this question. The use of mathematical software, such as MATLAB, Is permitted. , Round all numerical answers to 3 significant figures unless otherwise advised. Q1(i) Properties of a periodic signal For the signal shown in Figure Q1 a) Give the frequency of the periodic signal x (1) in kHz. Round your answer to significant figures. Submit part 1 mark Unanswered b) What is the fundamental angular frequency 2, rad s of the periodic signal x() to three significant figures? Round your answer to 3 significant figures. Submit part 1 mark Unanswered c) Define the symmetry of the periodic signal x(t). O Even symmetry O Neither odd nor even symmetry O odd symmetry Submit part I mark Unanswered d) Does the periodic signal x() have half-wave symmetry x(t) = -xit-T/2)? O Yes ONO Submit part lil 1 mark Unanswered Q1(ii) Signal synthesis e) If the signal x(t) shown in Figure Q1 is causal, complete the expression given below to synthesize the signal over one period (O SIST) using the unit step function wo (1). x(t) = 20(1+C! ( ut- ) )) You should simplify the result using the numerical values of A = 80,=2 and T = 0.2 s. ) Submit part 4 marks Unanswered

Q1(iii) Exponential Fourier series For a periodic signal x(t) = x(1 + nT), the exponential Fourier series approximation for x() is defined as 30) = § Celke f) Match the properties of x(t) with the properties of the Exponential Fourier series coefficients 6 = X(t)e-12 ( dt. The coefficients care real. The coefficients CA are imaginary. The coefficients are complex The coefficients C, fork even k = 0, +2, +4,...) are zero. 0 0 O O The periodic signal has even symmetry f(t) = f(-1). The periodic signal has odd symmetry f(t) = -f(-1). The periodic signal has half-wave symmetry f(1) = -1-T/2). The periodic signal has neither odd nor even symmetry 0 O O O Submit part 4 marks Unanswered g) Compute the exponential Fourier series coefficient (mean voltage) = / Cos x(1) di for the signal shown in Figure Q1. Co = Round your answer to 3 significant figures. V. Submit part 2 marks Unanswered Q1(iii) Line spectra and their applications The exponential Fourier series coefficients Cek > 0, k odd, for the signal shown in Figure Q1 are given by . C = Ab+1) kbx k = 1,5,9,... с. Ab+1) kbx k= 3, 7, 11,... or, more compactly, C = (-1)(A(+1) kbx The coefficients C = 0 for k even. It will be convenient to pre-compute the constant Alb + 1) bx so that C = (-1) h) What is y for the signal shown in Figure Q1? Round your answer to 3 significant figures. Submit part 1 mark Unanswered Use the following definitions, the value you computed for Co and the expressions y and forgiven above, to answer the questions that follow. Power Spectrum The power spectrum of a periodic signal is the sequence of average powers in each complex harmonic is given by Cl. For real periodic signals the power spectrum is a real even sequence as |C| = 101 = 1C/?. Note: The absolute value of a complex numberc = a + jb is given by Icl = Vla+jb) (a - b) = vec* = Va+. Parseval's Theorem Parseval's theorem states that the power in a periodic signal is given by: p=4 lirojë dt = 16.12

Power Spectrum The power spectrum of a periodic signal is the sequence of average powers in each complex harmonic is given by C. For real periodic signals the power spectrum is a real even sequence as Cx = 111 =1C1 Note: The absolute value of a complex numberc = a + jb is given by el = va+ b)(a - b) = Vect = V2? +. Parseval's Theorem Parseval's theorem states that the power in a periodic signal is given by: P= Odt IC i) Compute the power (Pw) in the first seven harmonics of the signal shown in Figure Q1. Round your answer to significant figures. Submit part 2 marks Unanswered RMS Power By a similar argument, the RMS power is given by: PRMs = VES Lít) dt = Σ IC i) Compute the RMS power (PMS W) in the first seven harmonics of the signal shown in Figure Q1. Round your answer to 3 significant figures Submit part 1 mark Unanswered Total Harmonic Distortion (THD) THD is defined as the ratio of the RMS value for all the harmonics fork > I (the distortion) to the RMS of the fundamental which is 2007: ΣIC, THD = 100% X CA k) Compute the percentage THD in the first seven harmonics of the signal shown in Figure Q1. Round your answer to significant figures. Submit part 2 marks Unanswered Submit all parts 20 marks