Digital transformation of the sampling rate of signals, or
signal processing with different sampling rates in the system.
Applications
CD to DAT format change, for example.
oversampling
converters; which reduce performance requirements onanti-aliasing or reconstruction filters
bandwidth of individual channels is much less than theoverall bandwidth
Eyes and
ears are not as sensitive to errors in higher frequencybands, so many coding schemes split signals into different
frequency bands and quantize higher-frequency bands withmuch less precision.
This procedure is motivated by an analog-based method: one
conceptually simple method to change the sampling rate is tosimply convert a digital signal to an analog signal and
resample it! (
[link] )
Digital implementation of rate-changing according to this
formula has three problems:
Infinite sum: The solution is to truncate. Consider
$sinc(t)\approx 0$ for
$t< {t}_{1}$ ,
$t> {t}_{2}$ : Then
$m{T}_{1}-n{T}_{0}< {t}_{1}$ and
$m{T}_{1}-n{T}_{0}> {t}_{2}$ which implies
$${N}_{1}=\lceil \frac{m{T}_{1}-{t}_{2}}{{T}_{0}}\rceil $$$${N}_{2}=\lfloor \frac{m{T}_{1}-{t}_{1}}{{T}_{0}}\rfloor $$$${x}_{1}(m)=\sum_{n={N}_{1}}^{{N}_{2}} {x}_{0}(n){sinc}_{{T}^{\prime}}(m{T}_{1}-n{T}_{0})$$
This is essentially lowpass filter design using a boxcar
window: other finite-length filter design methods could beused for this.
Lack of
causality : The solution is to delay by
$\max\{\left|N\right|\}$ samples. The mathematics of the analog portions
of this system can be implemented digitally.
So we have an all-digital formula for
exact digital-to-digital rate changing!
Cost of computing
${sinc}_{{T}^{\prime}}(m{T}_{1}-n{T}_{0})$ : The solution is to precompute the table of
$\mathrm{sinc}(t)$ values. However, if
$\frac{{T}_{1}}{{T}_{0}}$ is not a rational fraction, an infinite number of
samples will be needed, so some approximation will have tobe tolerated.
Rate transformation of any rate to any other rate can be
accomplished digitally with arbitrary precision (if somedelay is acceptable). This method is used in practice in
many cases. We will examine a number of special cases andcomputational improvements, but in some sense everything
that follows are details; the above idea is the centralidea in multirate signal processing.
Useful references for the traditional material (everything
except PRFBs) are
Crochiere and Rabiner,
1981 and
Crochiere and Rabiner,
1983 . A more recent tutorial is
Vaidyanathan ; see also
Rioul
and Vetterli . References to most of the original papers
can be found in these tutorials.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
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
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?