Utah State University
Department of Electrical and Computer Engineering
ECE 3640 - Discrete-time Signals & Systems

Instructions

Before working these problems, watch these videos:

Integrating complex exponentials video (15 min)
Summing complex exponentials video (21 min)

Here are some slides on these subjects:

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

Some Useful Identities

Here are a few identities from calculus and trigonometry.

\begin{aligned} (1) & \int_{a}^{b} e^{\pm j 2 π F t} d t & = {\frac{e^{\pm j 2 π F t}}{\pm j 2 π F} |}_{t = a}^{t = b} = \frac{e^{\pm j 2 π F b} - e^{\pm j 2 π F a}}{\pm j 2 π F} \\ (2) & \int_{a}^{b} e^{\pm j 2 π F t} d t & = \frac{\sin (π F (b - a))}{π F} e^{\pm j π F (b + a)} \\ (3) & \sum_{n = M}^{N - 1} e^{\pm j 2 π f n} & = \frac{e^{\pm j 2 π f M} - e^{\pm j 2 π f N}}{1 - e^{\pm j 2 π f}} \\ (4) & \sum_{n = M}^{N - 1} e^{\pm j 2 π f n} & = \frac{\sin (π f (N - M))}{\sin (π f)} e^{\pm j π f (N + M - 1)} \\ (5) & e^{j b} - e^{j a} & = [e^{j \frac{b - a}{2}} - e^{- j \frac{b - a}{2}}] e^{j \frac{b + a}{2}} = 2 j \sin (b - a) e^{j \frac{b + a}{2}} \end{aligned}

Problems

$x(t) = e^{-j2\pi F t}$ $X(F) = \int_S x(t) dt$ . For each problem:

$S$ $\ref{eq:ceint1}$ ).
Put your answers in sinc-function form.
$|X(F)|$ $20 \log_{10}|X(F)|$ $F \in [-2, 2]$ .

$S = [-5,5]$
$S = [0,10]$
$S = [-10,-5] \cup [5,10]$

Work out the integral in problem 3 in two ways.

$X(F) = \int_{-10}^{-5} x(t) dt + \int_{5}^{10} x(t) dt$ $\ref{eq:ceint1}$ ) and then simplify.

$X(F) = \int_{-10}^{10} x(t) dt - \int_{-5}^{5} x(t) dt$ $\ref{eq:ceint2}$ ) to write down the answer.

Try to get your answer in part (a) to match your answer in part (b). Also explain why the difference of integrals in part (b) is equivalent to the sum of integrals in part (a).

$x[n] = e^{-j2\pi fn}$ $X(f) = \displaystyle\sum_{n\in S} x[n]$ . For each problem:

$S$ $\ref{eq:cesum1}$ ).
Put your answers in sinc-function form.
$|X(f)|$ $20 \log_{10} |X(f)|$ $f \in [-2,2]$ .

$S = [-5, 5]$ (integers between and including -5 and 5)
$S = [0, 10]$
$S = [-10, -5]\cup [5, 10]$

Work out the sum in problem 6 in two ways.

$X(f) = \displaystyle\sum_{n=-10}^{-5} x[n] + \sum_{n=5}^{10} x[n]$ $\ref{eq:cesum1}$ ).

$X(f) = \displaystyle\sum_{n=-10}^{10} x[n] - \sum_{n=-4}^{4} x[n]$ $\ref{eq:cesum2}$ ) to write down the answer.

Try to get your answer in part (a) to match your answer in part (b). Also explain why the difference of sums in part (b) is equivalent to the sum of sums in part (a).

Policy note 1: 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.

Policy note 2: Combine your work into a single PDF file and upload to Canvas.

One way to create a PDF file showing your work, plots, and code would be to typeset your work using LaTeX. There are lots of ways to do this including Overleaf (online) or installing a LaTeX distribution on your computer (MiKTex for PCs, MacTex for Apple ... check out this URL https://www.latex-project.org/get/). WYSIWYG editors such as Typora (https://typora.io/) let you write in Markdown (which is a lightweight markup language) providing LaTeX mathematics, figure inclusion, and code fences for code. All of these methods output/export PDF files.
A low-tech but perfectly acceptable way to produce a PDF file is to work out your math on paper and then scan the pages into a PDF file. Then use a PDF editor to interleave your work with the plots and include your code.
Another option is to pull everything together in an MS Word document and export as PDF.