- What answer should Samantha obtain?
- As a check, her group partner Sammy says that he
computed the inverse DFT of her answer and got
$\delta (n+1)+\delta (n-1)$ . Does Sammy's result mean that Samantha's
answer is wrong?
- The homework problem says to lowpass-filter the
sequence by multiplying its DFT by
$$H(k)=\begin{cases}1 & \text{if $k=\{0, 1, 7\}$}\\ 0 & \text{otherwise}\end{cases}()$$ and then computing the inverse DFT. Will this
filtering algorithm work? If so, find the filteredoutput; if not, why not?
Stock market data processing
Because a trading week lasts five days, stock markets
frequently compute running averages each day over theprevious five trading days to smooth price fluctuations.
The technical stock analyst at the Buy-Lo--Sell-Hibrokerage firm has heard that FFT filtering techniques
work better than any others (in terms of producing moreaccurate averages).
- What is the difference equation governing the
five-day averager for daily stock prices?
- Design an efficient FFT-based filtering
algorithm for the broker. How much data should beprocessed at once to produce an efficient algorithm?
What length transform should be used?
- Is the analyst's information correct that FFT
techniques produce more accurate averages than anyothers? Why or why not?
Echoes
Echoes not only occur in canyons, but also in auditoriums and telephone circuits.
In one situation where the echoed signal has been sampled, the input signal
$x(n)$ emerges as
$x(n)+{a}_{1}x(n-{n}_{1})+{a}_{2}x(n-{n}_{2})$ .
- Find the difference equation of the system that models the production of echoes.
- To simulate this echo system, ELEC 241 students are asked to write the most efficient (quickest) program that has the same input-output relationship.
Suppose the duration of
$x(n)$ is 1,000 and that
${a}_{1}=\frac{1}{2}$ ,
${n}_{1}=10$ ,
${a}_{2}=\frac{1}{5}$ , and
${n}_{2}=25$ .
Half the class votes to just program the difference equation while the other half votes to program a frequency domain approach that exploits the speed of the FFT.Because of the undecided vote, you must break the tie.
Which approach is more efficient and why?
- Find the transfer function and difference equation of the system that suppresses the echoes.
In other words, with the echoed signal as the input, what system's output is the signal
$x(n)$ ?
Digital filtering of analog signals
RU Electronics wants to develop a filter that would be
used in analog applications, but that is implementeddigitally. The filter is to operate on signals that
have a 10 kHz bandwidth, and will serve as a lowpassfilter.
- What is the block diagram for your filter
implementation? Explicitly denote which componentsare analog, which are digital (a computer performs
the task), and which interface between analog anddigital worlds.
- What sampling rate must be used and how many
bits must be used in the A/D converter for theacquired signal's signal-to-noise ratio to be at
least 60 dB? For this calculation, assume thesignal is a sinusoid.
- If the filter is a length-128 FIR filter (the
duration of the filter's unit-sample response equals128), should it be implemented in the time or
frequency domain?
- Assuming
$H(e^{i\times 2\pi f})$ is the transfer function of the digital filter, what
is the transfer function of your system?
Signal compression
Because of the slowness of the Internet, lossy signal
compression becomes important if you want signals to bereceived quickly. An enterprising 241 student has
proposed a scheme based on frequency-domain processing.First of all, he would section the signal into
length-
$N$ blocks, and
compute its
$N$ -point DFT.
He then would discard (zero the spectrum) at
half of the frequencies, quantize
them to
$b$ -bits, and send
these over the network. The receiver would assemble thetransmitted spectrum and compute the inverse DFT, thus
reconstituting an
$N$ -point
block.
- At what frequencies should the spectrum be
zeroed to minimize the error in this lossycompression scheme?
- The nominal way to
represent a signal digitally is to use simple
$b$ -bit quantization of
the time-domain waveform. How long should a sectionbe in the proposed scheme so that the required
number of bits/sample is smaller than that nominallyrequired?
- Assuming that effective
compression can be achieved, would the proposedscheme yield satisfactory results?