Please answer Q3 in reference to the information provided in Q1 and Q2.
Question 1 Develop the Fourier Series expression for the given periodic waveform of triangular wave from the data given in Figure Q1. A general expression up to n=is expected. Assume that the peak-to-peak amplitude, A=4 V. The period is 100 ms. Time Domain Frequency Domain a. Pulse T+ a = - Ad 2A sin (nad) 7171 b=0 (d - 0.27 in this example) dk/T 0 -0 of 27 3f 4f Sf of b. Square ao -0 2A 1 TT sin IM HHH 0 04 Of 2f3f 4f 5f 6f b 0 (all even harmonies are zero) 1=0 c. Triangle = IMA 1 do = 0 4A (nnt) b=0 (all even harmonies are zero) 0 t=0 of 2 3f 4f 5f 6f Figure Q1 (20 marks) Question 2 Create with the assistance of MATLAB and its associated toolboxes a discrete-time domain plot with noise added of the ideal triangular wave from Q1. You may assume that the wave was sampled at fs = 200 Hz with n=8192 discrete-time samples and that you may use the MATLAB function, randn to add noise to the triangular wave. Figure Q2 serves as a guide to construct the time-domain plot. (a) Compose the MATLAB codes used to generate both the discrete-time noisy triangular wave as well as the single sided frequency magnitude plots. (10 marks) (6) Prepare the discrete-time and log magnitude single-sided frequency plots from the MATLAB codes of part (a). (10 marks) f1 3f1 5f1 7f1 9f1 + with Noise Noisy Triangular Wave Figure Q2 Question 3 Implement a comb filter utilizing the concept of a parallel bank of IR filters with the Elliptic Filter Template utilizing the concept of Figure Q3 to remove the noise from the noisy triangular wave constructed in Q2. The filter specifications for each of the "bristles of the comb filter should use the following: Centre frequencies: 10, 30, 50, 70 and 90 Hz Bandwidth: 1 Hz Stopband: #1 Hz from centre frequency Passband Ripple: 0.5 dB Stopband Attenuation: 60 dB (a) For each of the passbands specified, construct the MATLAB code used to obtain the desired specifications. (10 marks) (b) Set up the combined impulse response output from the five filters in the bank using a 8192 sample unit sample response, showing the impulse response plot in discrete-time for a 40 second duration. The MATLAB code is required. (10 marks) c) Use the designed parallel filter bank to filter the noisy triangular wave of Q2. You will need to provide the discrete-time as well as the log magnitude single sided spectrum plots of the filtered noisy triangular wave. The MATLAB code is required. You may use "edit plot to provide the clearest possible plot indicating the effectiveness of the filtering process of the parallel comb filter (20 marks) bank HIR BPF - 10 I IZ Test Frequency and Time Domain Charactensson Fitered Signal Test noisy input's time and frequency domain characteristics TIR BPF - 30 Hz Z IIR BPF 50 Hz Filtered Output IIR BPF - 70 Hz Noise Source TIR BPF 90 HZ Figure Q3 Question 1 Develop the Fourier Series expression for the given periodic waveform of triangular wave from the data given in Figure Q1. A general expression up to n=is expected. Assume that the peak-to-peak amplitude, A=4 V. The period is 100 ms. Time Domain Frequency Domain a. Pulse T+ a = - Ad 2A sin (nad) 7171 b=0 (d - 0.27 in this example) dk/T 0 -0 of 27 3f 4f Sf of b. Square ao -0 2A 1 TT sin IM HHH 0 04 Of 2f3f 4f 5f 6f b 0 (all even harmonies are zero) 1=0 c. Triangle = IMA 1 do = 0 4A (nnt) b=0 (all even harmonies are zero) 0 t=0 of 2 3f 4f 5f 6f Figure Q1 (20 marks) Question 2 Create with the assistance of MATLAB and its associated toolboxes a discrete-time domain plot with noise added of the ideal triangular wave from Q1. You may assume that the wave was sampled at fs = 200 Hz with n=8192 discrete-time samples and that you may use the MATLAB function, randn to add noise to the triangular wave. Figure Q2 serves as a guide to construct the time-domain plot. (a) Compose the MATLAB codes used to generate both the discrete-time noisy triangular wave as well as the single sided frequency magnitude plots. (10 marks) (6) Prepare the discrete-time and log magnitude single-sided frequency plots from the MATLAB codes of part (a). (10 marks) f1 3f1 5f1 7f1 9f1 + with Noise Noisy Triangular Wave Figure Q2 Question 3 Implement a comb filter utilizing the concept of a parallel bank of IR filters with the Elliptic Filter Template utilizing the concept of Figure Q3 to remove the noise from the noisy triangular wave constructed in Q2. The filter specifications for each of the "bristles of the comb filter should use the following: Centre frequencies: 10, 30, 50, 70 and 90 Hz Bandwidth: 1 Hz Stopband: #1 Hz from centre frequency Passband Ripple: 0.5 dB Stopband Attenuation: 60 dB (a) For each of the passbands specified, construct the MATLAB code used to obtain the desired specifications. (10 marks) (b) Set up the combined impulse response output from the five filters in the bank using a 8192 sample unit sample response, showing the impulse response plot in discrete-time for a 40 second duration. The MATLAB code is required. (10 marks) c) Use the designed parallel filter bank to filter the noisy triangular wave of Q2. You will need to provide the discrete-time as well as the log magnitude single sided spectrum plots of the filtered noisy triangular wave. The MATLAB code is required. You may use "edit plot to provide the clearest possible plot indicating the effectiveness of the filtering process of the parallel comb filter (20 marks) bank HIR BPF - 10 I IZ Test Frequency and Time Domain Charactensson Fitered Signal Test noisy input's time and frequency domain characteristics TIR BPF - 30 Hz Z IIR BPF 50 Hz Filtered Output IIR BPF - 70 Hz Noise Source TIR BPF 90 HZ Figure Q3
