Homework #1 Fourier analysis

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.2/2
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?
9. 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.
10. 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)