# 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.

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LITNING
scanning tunneling microscope
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what is Nano technology ?
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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anyone know any internet site where one can find nanotechnology papers?
research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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absolutely yes
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
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how can I make nanorobot?
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what is fullerene does it is used to make bukky balls
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.