Instructions

Before working these problems, watch this video:

• Complex exponential signals video (33 min)

Here are some related slides:

• Complex exponential signals slides

Note: These slides end with "assignments" pages. Ignore those problems and do the problems below.

Abbreviations

• CT = continuous time
• DT = discrete time
• CE = complex exponential
• LTI = linear, time-invariant

Problems

1. Define and sketch the three types of DT CE signals: everlasting, causal, and finite (windowed).
2. When a CT CE with frequency $F=440$$F=440$ Hz is sampled at $\frac{1}{T}=8000$$\frac{1}{T}=8000$ samples/second, what is the frequency $f$$f$ of the resulting DT CE signal?
3. If a CT CE is reconstructed from DT CE with frequency $f=0.26257$$f=0.26257$ cycles/sample using a sample rate of $\frac{1}{T}=6$$\frac{1}{T}=6$ Gigasamples/second, what is the resulting frequency $F$$F$ in Hertz? (Interesting fact: When GPS was created, the frequency $F$$F$ became extremely important for navigation.)
4. What is the angular frequency $\omega$$\omega$ associated with the cyclic frequency $f=0.26257$$f=0.26257$ cycles/sample?
5. Let $x(t)=e^{jt}$$x(t) = e^{jt}$ and $x[n]=e^{jn}$$x[n] = e^{jn}$. (a) Explain why $x(t)$$x(t)$ is periodic but $x[n]$$x[n]$ is not. (b) What are the frequencies of $x(t)$$x(t)$ and $x[n]$$x[n]$? (c) What is the period of $x(t)$$x(t)$?
6. Let $x[n]=e^{j2\pi fn}$$x[n] = e^{j2\pi fn}$ where $f=\frac{213}{355}$$f=\frac{213}{355}$. (a) Explain why $x[n]$$x[n]$ is periodic. (b) What is the period of $x[n]$$x[n]$?

Note: Matlab has a function called factor that might be useful for this problem. Use Matlab's built-in help system to find out what this function does.

7. Do the following in Matlab. (a) Plot $e^{j2\pi 0.1t}$$e^{j2\pi 0.1 t}$ and $e^{j2\pi 1.1t}$$e^{j2\pi 1.1 t}$ for $0\le t\le 10$$0 \leq t \leq 10$ on the same axis. (Hint: Use t=[0:0.01:10]; to generate the time samples.) (b) On the same axis add $e^{j2\pi 0.1n}$$e^{j2\pi 0.1 n}$ and $e^{j2\pi 1.1n}$$e^{j2\pi 1.1 n}$ as stem plots. (Hint: Use Matlab's stem function instead of the plot function. Use n=[0:10]; to generate the time samples.) (c) Explain why $F=0.1$$F=0.1$ Hz and $F=1.1$$F=1.1$ Hz give different CT CE signals while $f=0.1$$f=0.1$ cycles/sample and $f=1.1$$f=1.1$ cycles/sample give the same DT CE sequence. (d) Draw the unit circle on the complex plane. Show the point $e^{j2\pi 0.1}=e^{j2\pi 1.1}$$e^{j2\pi 0.1} = e^{j2\pi 1.1}$ and use the fact $(e^{j2\pi 0.1}{)}^n=e^{j2\pi 0.1n}=(e^{j2\pi 1.1}{)}^n=e^{j2\pi 1.1n}$$(e^{j2\pi 0.1})^n = e^{j2\pi 0.1 n} = (e^{j2\pi 1.1})^n = e^{j2\pi 1.1 n}$ to explain frequency aliasing which is that $e^{j2\pi fn}=e^{j2\pi (f+k)n}$$e^{j2\pi f n} = e^{j2\pi (f+k) n}$ for all $n$$n$, where $k$$k$ is any integer.
8. Find a frequency alias for $f=37.8$$f=37.8$ cycles/sample in the fundamental interval assuming: (a) $0\le f<1$$0 \leq f < 1$ is the fundamental interval (b) $-\frac{1}{2}\le f<\frac{1}{2}$$-\frac{1}{2} \leq f < \frac{1}{2}$ is the fundamental interval
9. Why is $f=\pm \frac{1}{2}$$f=\pm \frac{1}{2}$ cycles/sample the highest frequency in discrete time?
10. Which is a higher frequency $f=26.4$$f=26.4$ cycles/sample or $38.9$$38.9$ cycles/sample? (Hint: Compare their aliased frequencies using $-\frac{1}{2}\le f<\frac{1}{2}$$-\frac{1}{2} \leq f < \frac{1}{2}$ as the fundamental interval.)
11. Explain why ${\displaystyle \frac{\sin (\pi n)}{\pi n}=\delta [n]}$$\displaystyle\frac{\sin (\pi n)}{\pi n} = \delta[n]$. Include a sketch in your explanation.
12. Prove by integration that $e^{j2\pi 10t}$$e^{j2\pi 10 t}$ and $e^{j2\pi 30t}$$e^{j2\pi 30 t}$ are orthogonal over the time interval $0\le t<0.2$$0 \leq t < 0.2$ seconds.
13. Prove by summation that $e^{j\frac{2\pi 2n}{9}}$$e^{j\frac{2\pi 2 n}{9}}$ and $e^{j\frac{2\pi 4n}{9}}$$e^{j\frac{2\pi 4 n}{9}}$ are orthogonal over $0\le n<9$$0 \leq n < 9$. (Hint: Be really careful with the summation limits. What is the upper limit? 8 or 9? Why?)
14. Suppose the everlasting DT CE sequence

is applied to a DT LTI system with frequency response

What is the resulting output signal $y[n]$$y[n]$ if (a) $f=\frac{2}{3}$$f=\frac{2}{3}$ (b) $f=\frac{1}{2}$$f=\frac{1}{2}$

Policy note: If an assignment asks you to do something in Matlab, then write your code in a Matlab script (.m file) and attach a copy of your code when you turn in the assignment. Apply this rule throughout the semester.