# 5.16 Digital signal processing problems  (Page 5/6)

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## Digital filtering

A digital filter has an input-output relationship expressed by the difference equation $y(n)=\frac{x(n)+x(n-1)+x(n-2)+x(n-3)}{4}$ .

1. Plot the magnitude and phase of this filter's transfer function.
2. What is this filter's output when $x(n)=\cos \left(\frac{\pi n}{2}\right)+2\sin \left(\frac{2\pi n}{3}\right)$ ?

## Detective work

The signal $x(n)$ equals $\delta (n)-\delta (n-1)$ .

1. Find the length-8 DFT (discrete Fourier transform) of this signal.
2. You are told that when $x(n)$ served as the input to a linear FIR (finite impulse response) filter, the output was $y(n)=\delta (n)-\delta (n-1)+2\delta (n-2)$ . Is this statement true? If so, indicate why andfind the system's unit sample response; if not, show why not.

A discrete-time, shift invariant, linear system produces an output $y(n)=\{1, -1, 0, 0, \dots \}$ when its input $x(n)$ equals a unit sample.

1. Find the difference equation governing the system.
2. Find the output when $x(n)=\cos (2\pi {f}_{0}n)$ .
3. How would you describe this system's function?

## Time reversal has uses

A discrete-time system has transfer function $H(e^{i\times 2\pi f})$ . A signal $x(n)$ is passed through this system to yield the signal $w(n)$ . The time-reversed signal $w(-n)$ is then passed through the system to yield the time-reversed output $y(-n)$ . What is the transfer function between $x(n)$ and $y(n)$ ?

## Removing “hum”

The slang word “hum” represents power line waveforms that creep into signals because of poor circuitconstruction. Usually, the 60 Hz signal (and its harmonics) are added to the desired signal. What weseek are filters that can remove hum. In this problem, the signal and the accompanying hum have been sampled;we want to design a digital filter for hum removal.

1. Find filter coefficients for the length-3 FIR filter that can remove a sinusoid having digital frequency ${f}_{0}$ from its input.
2. Assuming the sampling rate is ${f}_{s}$ to what analog frequency does ${f}_{0}$ correspond?
3. A more general approach is to design a filter having a frequency response magnitude proportional to the absolute value of a cosine: $\left|H(e^{i\times 2\pi f})\right|\propto \left|\cos (\pi fN)\right|$ . In this way, not only can the fundamental but also its first few harmonics be removed. Select theparameter $N$ and the sampling rate so that the frequencies at which the cosine equals zerocorrespond to 60 Hz and its odd harmonics through the fifth.
4. Find the difference equation that defines this filter.

## Digital am receiver

Thinking that digital implementations are always better, our clever engineer wants to design a digital AM receiver. The receiverwould bandpass the received signal, pass the result through an A/D converter, perform all the demodulationwith digital signal processing systems, and end with a D/A converter to produce the analog message signal.Assume in this problem that the carrier frequency is always a large even multiple of the message signal's bandwidth $W$ .

1. What is the smallest sampling rate that would be needed?
2. Show the block diagram of the least complex digital AM receiver.
3. Assuming the channel adds white noise and that a $b$ -bit A/D converter is used, what is the output's signal-to-noiseratio?

## Dfts

A problem on Samantha's homework asks for the 8-point DFT of the discrete-time signal $\delta (n-1)+\delta (n-7)$ .

#### Questions & Answers

I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
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
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
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
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 .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily