# Homework 3, ECE438, Fall 2014, Prof. Boutin

Hard copy due in class, Monday September 22, 2014.

## Presentation Guidelines

• Write only on one side of the paper.
• Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
• Staple the pages together.
• Include a cover page.

## Question 1

Let x(t) be a continuous-time signal and let y[n]=x(nT) be a sampling of that signal with period T>0. We would like to interpolate the samples (i.e., "connect the dots") in order to try to recover x(t).

a) Derive a formula for a band-limited interpolation of the samples (i.e., an expression for a continuous signal z(t) in terms of the samples y[n]). (Do not simply write down the formula; show how to derive it.)

b) Show that your interpolation is equal to the original signal at all sample points.

c) Under what circumstances is your interpolation equal to the original signal x(t)? Explain.

## Question 2

Again, we consider a continuous-time signal x(t) and a sampling y[n]=x(nT) of that signal.

a) Write a formula for a piece-wise constant interpolation of the samples.

b) Derive the relationship between the Fourier transform of the interpolation you wrote in 2a) and the Fourier transform of x(t). (Do not simply write down the formula; show how to derive it.)

c) Is the interpolation you wrote in 2a) band-limited? Explain.

## Question 3

Let

$x(t)=7 \text{sinc } ( \frac{t-5}{2} ).$

a) Obtain the Fourier transform X(f) of the signal and sketch the graph of |X(f)|.

b) What is the Nyquist rate $f_0$ for this signal?

c) Let

$T = \frac{1}{3 f_0}.$

Write a mathematical expression for the Fourier transform $X_s(f)$ of

$x_s(t)= \text{ comb}_T \left( x(t) \right).$

Sketch the graph of $|X_s(f)|$.

d) Let

$T = \frac{1}{5 f_0}.$

Write a mathematical expression for the Fourier transform ${\mathcal X}_d(\omega)$ of $x_d[n]= x(nT)$ and sketch the graph of $|{\mathcal X}_d(\omega)|$.

## Discussion

You may discuss the homework below.

• write comment/question here

Anyone get number 3???

No, I was actually having a lot of trouble with this one as well.

Want to meet up to work on it?

MSEE 189, right now?

Deal. C u there

I'm here, now what?

I'm on my way. <(*o*<) <(^__^)> (>*o*)>

Umm... Nevermind, I don't think I need your help buddy. Thanks though!!

## Alumni Liaison

Sees the importance of signal filtering in medical imaging

Dhruv Lamba, BSEE2010