# Carrier phase modulation

 Page 1 / 1
A description of phase shift keying in which information is conveyed through the phase of the modulated signal.

## Phase shift keying (psk)

Information is impressed on the phase of the carrier. As data changes from symbol period to symbol period, the phase shifts.

$\forall m, m\in \{1, 2, , M\}\colon {s}_{m}(t)=A{P}_{T}(t)\cos (2\pi {f}_{c}t+\frac{2\pi (m-1)}{M})$

Binary ${s}_{1}(t)$ or ${s}_{2}(t)$

## Representing the signals

An orthonormal basis to represent the signals is

${}_{1}(t)=\frac{1}{\sqrt{{E}_{s}}}A{P}_{T}(t)\cos (2\pi {f}_{c}t)$
${}_{2}(t)=\frac{-1}{\sqrt{{E}_{s}}}A{P}_{T}(t)\sin (2\pi {f}_{c}t)$

The signal

${S}_{m}(t)=A{P}_{T}(t)\cos (2\pi {f}_{c}t+\frac{2\pi (m-1)}{M})$
${S}_{m}(t)=A\cos \left(\frac{2\pi (m-1)}{M}\right){P}_{T}(t)\cos (2\pi {f}_{c}t)-A\sin \left(\frac{2\pi (m-1)}{M}\right){P}_{T}(t)\sin (2\pi {f}_{c}t)$

The signal energy

${E}_{s}=\int_{()} \,d t$ A 2 P T t 2 2 f c t 2 m 1 M 2 t 0 T A 2 1 2 1 2 4 f c t 4 m 1 M
${E}_{s}=\frac{A^{2}T}{2}+\frac{1}{2}A^{2}\int_{0}^{T} \cos (4\pi {f}_{c}t+\frac{4\pi (m-1)}{M})\,d t\approx \frac{A^{2}T}{2}$
(Note that in the above equation, the integral in the last step before the aproximation is very small.)Therefore,
${}_{1}(t)=\sqrt{\frac{2}{T}}{P}_{T}(t)\cos (2\pi {f}_{c}t)$
${}_{2}(t)=-\sqrt{\frac{2}{T}}{P}_{T}(t)\sin (2\pi {f}_{c}t)$

In general,

$\forall m, m\in \{1, 2, , M\}\colon {s}_{m}(t)=A{P}_{T}(t)\cos (2\pi {f}_{c}t+\frac{2\pi (m-1)}{M})$
and ${}_{1}(t)$
${}_{1}(t)=\sqrt{\frac{2}{T}}{P}_{T}(t)\cos (2\pi {f}_{c}t)$
${}_{2}(t)=\sqrt{\frac{2}{T}}{P}_{T}(t)\sin (2\pi {f}_{c}t)$
${s}_{m}=\left(\begin{array}{c}\sqrt{{E}_{s}}\cos \left(\frac{2\pi (m-1)}{M}\right)\\ \sqrt{{E}_{s}}\sin \left(\frac{2\pi (m-1)}{M}\right)\end{array}\right)$

## Demodulation and detection

${r}_{t}={s}_{m}(t)+{N}_{t}\text{for some}m\in \{1, 2, , M\}$

We must note that due to phase offset of the oscillator at the transmitter, phase jitter or phase changes occur because of propagation delay.

${r}_{t}=A{P}_{T}(t)\cos (2\pi {f}_{c}t+\frac{2\pi (m-1)}{M}+)+{N}_{t}$

For binary PSK, the modulation is antipodal, and the optimum receiver in AWGN has average bit-error probability

${P}_{e}=Q(\sqrt{\frac{2({E}_{s})}{{N}_{0}}})=Q(A\sqrt{\frac{T}{{N}_{0}()}})$
${r}_{t}=(A{P}_{T}(t)\cos (2\pi {f}_{c}t+))+{N}_{t}$
The statistics
${r}_{1}=\int_{0}^{T} {r}_{t}\cos (2\pi {f}_{c}t+\stackrel{}{})\,d t=(\int_{0}^{T} A\cos (2\pi {f}_{c}t+)\cos (2\pi {f}_{c}t+\stackrel{}{})\,d t)+\int_{0}^{T} \cos (2\pi {f}_{c}t+\stackrel{}{}){N}_{t}\,d t$
${r}_{1}=(\frac{A}{2}\int_{0}^{T} \cos (4\pi {f}_{c}t++\stackrel{}{})+\cos (-\stackrel{}{})\,d t)+{}_{1}$
${r}_{1}=(\frac{A}{2}T\cos (-\stackrel{}{}))+\int_{0}^{T} (\frac{A}{2}\cos (4\pi {f}_{c}t++\stackrel{}{}))\,d t+{}_{1}(\frac{AT}{2}\cos (-\stackrel{}{}))+{}_{1}$
where ${}_{1}=\int_{0}^{T} {N}_{t}\cos ({}_{c}t+\stackrel{}{})\,d t$ is zero mean Gaussian with $\mathrm{variance}\approx \frac{^{2}{N}_{0}T}{4}$ .

Therefore,

$\langle {P}_{e}\rangle =Q(\frac{2\frac{AT}{2}\cos (-\stackrel{}{})}{2\sqrt{\frac{^{2}{N}_{0}T}{4}}})=Q(\cos (-\stackrel{}{})A\sqrt{\frac{T}{{N}_{0}}})$
which is not a function of  and depends strongly on phase accuracy.
${P}_{e}=Q(\cos (-\stackrel{}{})\sqrt{\frac{2{E}_{s}}{{N}_{0}}})$
The above result implies that the amplitude of the local oscillator in the correlator structure does not play a role inthe performance of the correlation receiver. However, the accuracy of the phase does indeed play a major role. Thispoint can be seen in the following example:

${x}_{{t}^{}}=-1^{i}A\cos (-(2\pi {f}_{c}{t}^{})+2\pi {f}_{c})$
${x}_{t}=-1^{i}A\cos (2\pi {f}_{c}t-2\pi {f}_{c}{}^{}-2\pi {f}_{c}+{}^{})$

Local oscillator should match to phase  .

what is variations in raman spectra for nanomaterials
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
yes that's correct
Professor
I think
Professor
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 did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!