# 4.12 Unknown signal delay  (Page 2/2)

We have argued the doubtfulness of assuming that the noise is white in discrete-time detection problems. The approach forsolving the colored noise problem is to use spectral detection. Handling the unknown delay problem in this way is relativelystraightforward. Since a sequence can be represented equivalently by its values or by its DFT, maximization can becalculated in either the time or the frequency domain without affecting the final answer. Thus, the spectral detector'sdecision rule for the unknown delay problem is (from this equation )

$\max\{ , \sum_{k=0}^{L-1} \frac{\Re (\overline{R(k)}S(k)e^{-\left(\frac{i\times 2\pi k}{L}\right)})}{{}_{k}^{2}}-\frac{1}{2}\frac{\left|S(k)\right|^{2}}{{}_{k}^{2}}\}\underset{{}_{0}}{\overset{{}_{1}}{}}$
where, as usual in unknown delay problems, the observationinterval captures the entire signal waveform no matter what the delay might be. The energy term is a constant and can beincorporated into the threshold. The maximization amounts to finding the best linear phase fit to the observations' spectrumonce the signal's phase has been removed. A more interesting interpretation arises by noting that the sufficient statistic isitself a Fourier Transform; the maximization amounts to finding the location of the maximum of a sequence given by $\Re (\sum_{k=0}^{L-1} \frac{\overline{R(k)}S(k)}{{}_{k}^{2}}e^{-\left(\frac{i\times 2\pi k}{L}\right)})$ The spectral detector thus becomes a succession of two Fourier Transforms with the final result determined by themaximum of a sequence!

Unfortunately, the solution to the unknown-signal-delay problem in either the time or frequency domains is confounded when twoor more signals are present. Assume two signals are known to be present in the array output, each of which has an unknown delay: $r(l)={s}_{1}(l-{}_{1})+{s}_{2}(l-{}_{2})+n(l)$ . Using arguments similar to those used in the one-signal case, the generalized likelihood ratio test becomes $\max\{{}_{1}, , {}_{2} , \sum_{l=0}^{L-1} r(l){s}_{1}(l-{}_{1})+r(l){s}_{2}(l-{}_{2})-{s}_{1}(l-{}_{1}){s}_{2}(l-{}_{2})\}\underset{{}_{0}}{\overset{{}_{1}}{}}^{2}\ln +\frac{{E}_{1}+{E}_{2}}{2}$ Not only do matched filter terms for each signal appear, but also a cross-term between the two signals. It is this latterterm that complicates the multiple signal problem: if this term is not zero for all possible delays, a non-separable maximization process results and both delays mustbe varied in concert to locate the maximum. If, however, the two signals are orthogonal regardless of the delay values, thedelays can be found separately and the structure of the single signal detector (modified to include matched filters for eachsignal) will suffice. This seemingly impossible situation can occur, at least approximately. Using Parseval's Theorem, thecross term can be expressed in the frequency domain. $\sum_{l=0}^{L-1} {s}_{1}(l-{}_{1}){s}_{2}(l-{}_{2})=\frac{1}{2\pi }\int_{-\pi }^{\pi } {S}_{1}()\overline{{S}_{2}()}e^{i({}_{2}-{}_{1})}\,d$ For this integral to be zero for all ${}_{1}$ , ${}_{2}$ , the product of the spectra must be zero. Consequently, if the two signals have disjoint spectralsupport, they are orthogonal no matter what the delays may be.

We stated earlier that this situation happens "at least approximately." Why the qualification?
Under these conditions, the detector becomes $(l, , \max\{{}_{1} , (r(l), {s}_{1}(D-1-l))\})+(l, , \max\{{}_{2} , (r(l), {s}_{2}(D-1-l))\})\underset{{}_{0}}{\overset{{}_{1}}{}}$ with the threshold again computed independently of the received signal amplitudes.
Not to be boring, but we emphasize that ${E}_{1}$ and ${E}_{2}$ are the energies of the signals ${s}_{1}(l)$ and ${s}_{2}(l)$ used in the detector, not those of their received correlates ${A}_{1}{s}_{1}(l)$ and ${A}_{2}{s}_{2}(l)$ .
${P}_{F}=Q(\frac{}{\sqrt{({E}_{1}+{E}_{2})^{2}}})$ This detector has the structure of two parallel, independently operating, matched filters, each of which is tuned to thespecific signal of interest.

Reality is insensitive to mathematically simple results. The orthogonality condition on the signals that yielded therelatively simple two-signal, unknown-delay detector is often elusive. The signals often share similar spectral supports,thereby violating the orthogonality condition. In fact, we may be interested in detecting the same signal repeated twice (or more) within the observation interval.Because of the complexity of incorporating inter-signal correlations, which are dependent on the relative delay, theidealistic detector is often used in practice. In the repeated signal case, the matched filter is operated over the entireobservation interval and the number of excursions above the threshold noted. An excursion is defined to be a portion of the matched filter's output that exceeds thedetection threshold over a contiguous interval. Because of the signal's non-zero duration, the matched filter's response tojust the signal has a non-zero duration, implying that the threshold can be crossed at more than a single sample. When onesignal is assumed, the maximization step automatically selects the peak value of an excursion. As shown in lower panels of this figure , a low-amplitude excursion may have a peak value less than anon-maximal value in a larger excursion. Thus, when considering multiple signals, the important quantities are the times atwhich excursion peaks occur, not all of the times the output exceeds the threshold.

This figure illustrates the two kinds of errors prevalent in multiple signal detectors. In the left panel, we find two excursions, the firstof which is due to the signal, the second due to noise. This kind of error cannot be avoided; we never said that detectorscould be perfect! The right panel illustrates a more serious problem: the threshold is crossed by four excursions, all of which are due to a single signal. Hence, excursions must besorted through, taking into account the nature of the signal being sought. In the example, excursions surrounding a largeone should be discarded if they occur in close proximity. This requirement means that closely spaced signals cannot bedistinguished from a single one.

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
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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