Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Experiment 3

Fourier Synthesis of Periodic Waveforms

Report template

Part A (30 Marks)

A.1) Provide the Matlab code required to synthesise the waveform in equation 12 and the resulting waveform. [5 marks]

A.2) Explain the features of the resulting waveform (peak-to-peak amplitude, symmetry, ripple, etc.) [5 marks]

A.3) How do you think the waveform would look if an unlimited number of harmonics was available (i.e. n goes to ∞)? To support your answer, provide a couple of figures along with their associated code. [5 marks]

A.4) Referring to Equation 3, what is the value of a_o? [5 marks]

A.5) Find an expression (in terms of n) for a_n and b_n. [5 marks]

A.6) Plot the spectrum (frequency domain view) of f(t) using c_n = Ö(a_n² + b_n²). Provide the figure and the Matlab code used to obtain it (you can use the stem function from Matlab to plot the frequency components). [5 marks]

Part B (20 Marks)

B.1) Write f(t) in Fourier synthesis form, i.e. as in Equation 2. [4 marks]

B.2) Calculate the first 10 sinewave coefficients (i.e. b₁, b₂, … , b₁₀). [4 marks]

B.3) Synthesise the first 10 harmonics of this waveform and plot the result (provide your Matlab code as well). [4 marks]

B.4) Plot in the same figure the original and synthesised sawtooth waveforms (provide your Matlab code). Compare the resulting waveform with what you expected to see and discuss the results. [4 marks]

B.5) If the number of harmonics is reduced to 5, comment on the changes that will be observed practically. [4 marks]

Part C (15 Marks)

C.1) Synthesise the waveforms below. Plot the resulting waveforms and provide the Matlab code used to obtain the plots as well. [1, 1, 1 and 2 marks, respectively]

C.2) Comment on your results for each waveform. [3, 3, 2 and 2 marks, respectively]

Part D (15 Marks)

D.1) Synthesise and plot this square wave (provide your Matlab code as well). Calculate the percentage overshoot of the synthesised waveform (compared with the ideal waveform) at the discontinuity. How does this compare with the expected limit of 17.9%? [2 marks]

D.2) What is the name of this overshoot? Explain it. [2 marks]

D.3) Give the Fourier series in each case for the resulting waveform if the above square wave is used as an input for:

D.3) (a) A low-pass filter with gain and phase responses as given in Figures 8 and 10 respectively. [1 mark]

D.3) (b) A low-pass filter with gain and phase responses as given in Figures 8 and 11 respectively. [1 mark]

D.3) (c) A band-pass filter with gain and phase responses as given in Figures 9 and 10 respectively. [1 mark]

D.4) Synthesise the above waveforms and draw the obtained waveforms for filters (a), (b) and (c), providing the Matlab code as well. [3 marks]

D.5) What is the fundamental frequency of the output from filter (c)? Why? [5 marks]

Part E (10 Marks)

E.1) For the wave in equation 15, listen to harmonics 1, 2 and 3 individually then as a chord. Plot the chord waveform as well. Provide the Matlab code used to listen to the harmonics/chord and plot the chord. [3 marks]

E.2) Alter the phase of the third harmonic in equation 15 by 90° and repeat the tasks in the point above. [3 marks]

E.3) Discuss the effect of altering the phase of the third harmonic, both on the sound and plot of the chord. Do the sound or plot change as you alter the phase? Why? [2 marks]

E.4) What does the above tell you about the human ear? [2 marks]