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5 Q3 - Fourier Transforms for Circuit Analysis This question is concerned with the analysis of circuits in the frequency domain. Answer all parts of this question. Use of mathematical software, such as MATLAB, is permitted, but the answers expected will be numbers or mathematical expressions. Enter all numbers rounded to 3 significant figures unless otherwise advised. arks ered Q3(i) Fourier Transform Properties arks ered a Which of the following integrals represents the Fourier transform? arks wered nges. *(t) exp(-jk2ot) dt T 00 17 L =(t) exp(-swt) dt | a(t) exp(+ st) dt (r)exp(t – r) dt arks vered 00 .00 arks vered inges b) Match the Fourier transform property to the example F(jw) shown below. F*(-6) 2nf (jw) 1 ju + 2 F(W) + F(0)8(W) jw Time integration { float

b) Match the Fourier transform property to the example F(jw) shown below. 1 F*(-jw) 2. f(-jw) F(W) + F(0)8(W) jw jw + 2 Time integration: I f(r) at Symmetry: F(t) arks wered Conjugate of a time function: f'(t) 0 0 0 o o o Time/frequency scaling: exp(26) 0 marks wered marks wered anges marks wered Q3(ii) Fourier transform of a system A first-order low-pass filter with input (t) and output y(t) and cut-off frequency 50 rads- has impulse response: h(t) = 50 exp(-50t) uo(t) marks wered where uo(t) is the unit step or Heaviside function. manges. C Give an expression for the transfer function of the low-pass filter H(3), 50/(S+50) 50 S + 50

d Give the system function of the low-pass filter H(jw). 50 50/(w+50) jw + 50 Submitp 10 d Unans Ks ed Q3(iii) Step response of a system e) -ks red 105 The Fourier transform of the unit-step function is given by the following transform pairs 1 Ut) (W) + jw rks ered Determine the response y. (t) of the filter h(t) when the inputti(t= 6uo(t). Hint use the Fourier transforms of h(t) and 21(t) and the convolution property to find Yi (jw) then inverse Fourier transform the result. arks cred ges Show steps You will lose 1 mark) Answer: Submit 4 Unan Q3(iv) Delayed response

S The Fourier transform of the unit-step function is given by the following transform pair: 1 uo(t) = m(W) + ju Determine the response y: (t) of the filter h(t) when the input 21(t) =6uo(t). Hint use the Fourier transforms of h(t) and a (t) and the convolution property to find Yi(jw) then inverse Fourier transform the marks swered Show steps You will lose I mark.) Answer: marks swered marks swered hanges Q3(iv) Delayed response marks swered f) Give the response of the system is initially at rest and then subject to a delayed input zit-6). marks nswered Changes Submit all parts 10 marks Created using Numbas, developed by Newcastle University