- Determine Laplace transform of .
- Determine if y(t)=cos(x(t)) is linear, time-invariant, and/or causal.
- Determine if is periodic; if periodic, calculate the period.
- Compute the inverse Laplace transform of .
- For the circuit shown below, find the input/output differential equation.

- Watch video entitled “Module 5 – Fourier Transform in MATLAB”
- Perform activity 1 below for the lab assignment using MATLAB.
- Include answers for Problems and include MATLAB coding along with any output plots that support solutions into a Word document entitled “Lab5_StudentID”. Where your student id is substituted in the file name.
- Upload file “Lab5_StudentID”.

**Activity 1:**

A continuous time function is shown below in figure 1. This signal is a sinc function defined as y(t) = sinc(t). The Fourier transform of this signal is a rectangle function.

- Use the function linspace to create a vector of time values from -5 ≤ t ≤ 5. Next, plot the function shown in figure 1 using the sinc function for y(t) = sinc(t).
- Using Matlab and the command fft, show that the Fourier transform pair is indeed a rectangle function. Use the command fftshift to center your plot. Don’t forget that the Fourier transform is complex, with both magnitude and phase. Your result should be the same as figure 2. Show both your m-file code and plot.

Matlab tip: The following commands are useful when working with the Fourier transform:

abs gives the magnitude of a complex number (or absolute value of a real number)

angle gives the angle of a complex number, in radians

[Note: The fft command does not give the exact transform for a continuous time signal, which we have in this case. For instance, the magnitude will not be correct. However, in order to obtain the general shape including relative magnitudes, it can be quite useful.]

- Using the same time values, plot the continuous time function defined as y(t) = sinc(2t).
- Plot the transform pair for this signal.

**Questions:**

- What is the cause of the “ringing” seen on top of the rectangular pulse shown in figure 2?
- In step 3 above, the sinc function gets compressed by a factor of 2, as seen by comparing the graphs in the time domain. What happened to the rectangular pulse in the frequency domain? What property does this represent?

Figure 1

Figure 2

- Read Chapters 5 and section 8.2.5 in the text Signals and Systems Using MATLAB.
- Download and review the supplemental questions.
- Work the below five homework problems.
- Submit homework solutions via Assignment Upload Tool. Show all work for full credit.

- A continuous time signal x(t) has the Fourier transform where b is a constant. Determine the Fourier transform for .
- A continuous time signal x(t) has the Fourier transform where b is a constant. Determine the Fourier transform for
- Compute the Fourier transform for
- Compute the inverse Fourier transform for
- A signal with the highest frequency component at 10 kHz is to be sampled. To reconstruct the signal, the sampling must be done at a minimum frequency of?

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