# 5.16 Digital signal processing problems

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## Sampling and filtering

The signal $s(t)$ is bandlimited to 4 kHz. We want to sample it, but it has been subjected to various signal processingmanipulations.

1. What sampling frequency (if any works) can be used to sample the result of passing $s(t)$ through an RC highpass filter with $R=10\mathrm{k\Omega }$ and $C=8\mathrm{nF}$ ?
2. What sampling frequency (if any works) can be used to sample the derivative of $s(t)$ ?
3. The signal $s(t)$ has been modulated by an 8 kHz sinusoid having an unknown phase: the resultingsignal is $s(t)\sin (2\pi {f}_{0}t+\phi )$ , with ${f}_{0}=8\mathrm{kHz}$ and $\phi =?$ Can the modulated signal be sampled so that the original signal can be recovered from the modulated signal regardless of the phase value $\phi$ ? If so, show how and find the smallest sampling rate that can be used; if not,show why not.

## Non-standard sampling

Using the properties of the Fourier series can ease finding a signal's spectrum.

1. Suppose a signal $s(t)$ is periodic with period $T$ . If ${c}_{k}()$ represents the signal's Fourier series coefficients, what are the Fourier seriescoefficients of $s(t-\frac{T}{2})$ ?
2. Find the Fourier series of the signal $p(t)$ shown in [link] .
3. Suppose this signal is used to sample a signal bandlimited to $\frac{1}{T}\mathrm{Hz}$ . Find an expression for and sketch the spectrum of the sampled signal.
4. Does aliasing occur? If so, can a change in sampling rate prevent aliasing;if not, show how the signal can be recovered from these samples.

## A different sampling scheme

A signal processing engineer from Texas A&M claims to have developed an improved sampling scheme. He multiplies the bandlimited signal by the depicted periodic pulse signal to perform sampling ( [link] ).

1. Find the Fourier spectrum of this signal.
2. Will this scheme work? If so, how should ${T}_{S}$ be related to the signal's bandwidth? If not, why not?

## Bandpass sampling

The signal $s(t)$ has the indicated spectrum.

1. What is the minimum sampling rate for this signal suggested by the Sampling Theorem?
2. Because of the particular structure of this spectrum, one wonders whether a lower sampling ratecould be used. Show that this is indeed the case, and find the system that reconstructs $s(t)$ from its samples.

## Sampling signals

If a signal is bandlimited to $W$ Hz, we can sample it at any rate $\frac{1}{{T}_{s}}> 2W$ and recover the waveform exactly. This statement of the Sampling Theorem can be taken to mean that allinformation about the original signal can be extracted from the samples. While true in principle, you do haveto be careful how you do so. In addition to the rms value of a signal, an important aspect of a signal isits peak value, which equals $\max\{\left|s(t)\right|\}$ .

1. Let $s(t)$ be a sinusoid having frequency $W$  Hz. If we sample it at precisely the Nyquist rate, how accurately do thesamples convey the sinusoid's amplitude? In other words, find the worst case example.
2. How fast would you need to sample for the amplitude estimate to be within 5% of the truevalue?
3. Another issue in sampling is the inherent amplitude quantization produced by A/D converters. Assume themaximum voltage allowed by the converter is ${V}_{\mathrm{max}}()$ volts and that it quantizes amplitudes to $b$ bits. We can express the quantized sample $Q(s(n{T}_{s}))$ as $s(n{T}_{s})+\epsilon (t)$ , where $\epsilon (t)$ represents the quantization error at the ${n}^{\mathrm{th}}()$ sample. Assuming the converter rounds, how large is maximum quantization error?
4. We can describe the quantization error as noise, with apower proportional to the square of the maximum error. What is the signal-to-noise ratio of thequantization error for a full-range sinusoid? Express your result in decibels.

#### Questions & Answers

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Eke Reply
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Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
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William
currently
William
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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Crow Reply
what about nanotechnology for water purification
RAW Reply
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Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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