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S Q1 Periodic Signals A small portion of a causal piecewise continuous periodic signal 2 (t) = 2(t+ nT) is shown in Figure Q1 below. x(t) (1 marks А b marks wered T T 3T t[ms] narks wered anges - T 4. 4 4 marks swered -A marks swered Figure Q1: A Piecewise-continuous periodic waveform (t) = 2(t+ nT) For the rest of this question, the period of the signal T = 100 ms, A = 9 V, and b = 4. 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() Properties of a periodic signal For the signal shown in Figure Q1

Q1(i) Properties of a periodic signal For the signal shown in Figure Q1 a) Give the frequency of the periodic signal (t) in kHz. Round your answer to 3 decimal places arks ered arks vered arks b) What is the fundamental angular frequency Ro rads+of the periodic signal (t)? vered ngos. | Round your answer to 3 significant figures. arks wered arks overed c) Define the symmetry of the periodic signal z(t). Odd symmetry Neither odd nor even symmetry Even symmetry d)

d Does the periodic signal r(t) have half-wave symmetry *(t) = 2(t-T/2)? No Submit part 1 mark Unanswered Q1(ii) Signal synthesis If the zt) shown in Figure QI is causal, complete the below to synthesize the signal over one period (0<t <T) using the unit step function us (t). (t)+ (Hot- -ul )) You should simplify the result using the numerical values of A=9v, b = 4 and T = 0.1 s. Submit part 4 marks Unanswered Qi(iii) Exponential Fourier series For a periodic signal z(t) = 2(t+nT), the exponential Fourier series approximation for æ(t) is defined as Σ Okeht

f Match the properties of (t) with the properties of the Exponential Fourier series coefficients 1 at (t)e ju dt. The coefficients C are imaginary. The coefficients Ce are real The coefficients C, for k even k = 0, +2, +4,...) are zero. The coefficients C are complex e The periodic signal has even symmetry f(t) = f(-t). The periodic signal has odd symmetry f(t) = -f(-+). The periodic signal has half-wave symmetry f(t) = -ff-T/2). The periodic signal has neither odd nor even symmetry O o Submit part 4 marks Unanswered Compute the exponential Fourier series coefficient (mean voltage) 1 Co = Lacey z(t) dt for the signal shown in Figure 21. Co = Round your answer to 3 Significant figures. V. Submit part 2 marks

Q1(iv) Line spectra and their applications The exponential Fourier series coefficients Cik > 0. kodd, for the signal shown in Figure Q1 are given by C = A(6+1) kom k = 1,5,9,... rks A(b +1) k = 3, 7, 11,... kon or, more compactly rks red Ce=(-1) (***) A(6+1) komt ks. The coefficients Ct = 0 for keven. red It will be convenient to pre-compute the constant A(b + 1) bm -ks red so that G-002 (1 -ks red h) What is y for the signal shown in Figure Q1? Round your answer to 3 significant figures. Su Use the following definitions, the value you computed for Co and the expressions y and for Ck given above, to answer the questions that follo Power Spectrum

Use the following definitions, the value you computed for Co and the expressions y and for Che given above, to answer the questions that fol 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 IC #2 = 10;1? = |Ck/? eks ed Note: The absolute value of a complex number c = a + jb is given by cl = v(a+jb) (a - jb) = CC Va+62 Parseval's Theorem -ks red Parseval's theorem states that the power in a periodic signal is given by: 1 -ks PE * L* 1500)? d = Ś ICH red le-00 i) ks red Compute the power (PW) in the first seven harmonics of the signal shown in Figure 21. P= w S ks ed RMS Power By a similar argument, the RMS power is given by: PRMS - [ 15(e)? dt = Σ ( C2.

RMS Power By a similar argument, the RMS power is given by: 1 PRMS = . Vt lineal ) ΣC. 1) 00 j) Compute the RMS power (PRMS W) in the first seven harmonics of the signal shown in Figure Q1. PRMS = w Sub Un Total Harmonic Distortion (THD) THD is defined as the ratio of the RMS value for all the harmonics fork > 1 (the distortion) to the RMS of the fundamental which is (2/C12: ΣIC 에 |CZ| 2 S THD = 100% X k) Compute the percentage THD in the first seven harmonics of the signal shown in Figure 01. THD = % Sub Un