Homework #1: Experiencing your first Fourier analysis Solution

Description

The goal of this homework is to warm up and give you a chance to practice the Fourier analysis via programming. After going through our DSP course, you may come back and think about any you learn can be applied to this homework.

Given HW1_samplecodes.m

1. Go through HW1_samplecodes.m, run the codes and then illustrate how the continuous-time Fourier transform (CTFT) is implemented in the digital domain. Is the magnitude spectrum is the same as what you expect?

2. Given F0 = 5 MHz, assign the values of 1, 2, 4, and 8, respectively, to the variable iFNpointRatio. Please tell the change in the magnitude spectrum as the variable iFNpointRatio goes from 1 to 8, and elaborate what you observe.

3. Given F0 = 5 MHz and the variable iFNpointRatio = 1, assign the values of 1, 2, 3, and 4, respectively, to the variable iFRangeRatio. Please tell the change in the magnitude spectrum as the variable iFRangeRatio goes from 1 to 4, and elaborate what you observe. Show the magnitude spectrum with the normalized frequency axis and elaborate what you observe.

4. Given the variables iFNpointRatio = 1 and iFRangeRatio = 1, assign the values of 0, 1, 5, 10, 15, 19, and 20, respectively, to the variable iF0Ratio and plot the discrete-time sinusoid x[n] with these iF0Ratios as a function of time n. Please tell the change in the rate of oscillation of the discrete-time sinusoid as iF0Ratio goes from 0 to 20, and elaborate what you observe. In your elaboration, you may try to convert F0 to normalized frequency. (Hint: see Figure 4.6 in the textbook)

5. From 2, 3, and 4, what is the working frequency range in the continuous-time domain for the provided CTFT codes?

6. Line 48 in HW1_samplecodes.m, Given the variables iFNpointRatio = 1, iFRangeRatio = 1, and iF0Ratio = 1, change Npoint2 = Npoint1*iFRangeRatio to Npoint2 = Npoint1*iFRangeRatio, Npoint2 = Npoint1*iFRangeRatio -5, Npoint2 = Npoint1*iFRangeRatio -20, show the magnitude spectrum with the normalized frequency axis or absolute frequency axis and elaborate what you observe.

7. In the provided codes, the variable t_axis starts from 0 s. What would happen for Fourier analysis if t_axis does not starts from 0 s.

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8. Can the CTFT be implemented with positive sign of the phase in the complex exponential term (i.e., change exp(-j …) to exp(+j …) for Fourier analysis?

8. Change x[n] from discrete-time cosine to sine and then compare the magnitude and phase spectra of F0 = 5 MHz and F0 = -5MHz. Elaborate your findings.

8. Given a MATLAB data file – PPG.mat where a raw PPG data and Fs (in Hz) used to acquire the PPG data are stored, perform Fourier analysis over (a) one single PPG wavelet (i.e., one PPG cycle) and (b) the whole PPG signal, and then elaborate which ((a) or (b)) magnitude spectrum provides the spectral information needed for PPG front end circuit design (e.g., pre-amplifier, filter)

(Note that the PPG data was acquired by a digital oscilloscope)

Notice:

1. You can find some PPG details at https://en.wikipedia.org/wiki/Photoplethysmogram#Photoplethysmograph

2. Please hand in your solution files to the LMS elearning system, including your word file of the detailed solutions, the associated Matlab codes, and all the related materials. It would be nice that you can put your “KEY” code segment with comments side by side along with your answer in the word file if needed.

3. Name your solution files “EE3660_HW1_StudentID.doc” and “EE3660_HW1_StudentID.m”, and archive them as a single zip file: EE3660_HW1_StudentID.zip.

4. The first line of your word or Matlab file should contain your name and some brief description, e.g., % EE 3660 王小明 u9612345 HW1 MM/DD/2017

5. If you need raw data file instead of MATLAB data, please let me know ASAP.

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