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Interpolation: by an integer factor l

Interpolation means increasing the sampling rate, or filling in in-between samples. Equivalent to sampling abandlimited analog signal L times faster. For the ideal interpolator,

X 1 ω X 0 L ω ω L 0 L ω
We wish to accomplish this digitally. Consider [link] and [link] .
y m X 0 m L m 0 ± L ± 2 L 0
The DTFT of y m is
Y ω m ω y m ω m n x 0 n ω L n n x n ω L n X 0 ω L
Since X 0 ω is periodic with a period of 2 , X 0 L ω Y ω is periodic with a period of 2 L (see [link] ).
By inserting zero samples between the samples of x 0 n , we obtain a signal with a scaled frequency response that simply replicates X 0 ω L times over a 2 interval!

Obviously, the desired x 1 m can be obtained simply by lowpass filtering y m to remove the replicas.

x 1 m y m h L m
Given H L m 1 ω L 0 L ω In practice, a finite-length lowpass filter is designed using any of the methods studied so far ( [link] ).

Interpolator block diagram

Decimation: sampling rate reduction (by an integer factor m)

Let y m x 0 L m ( [link] )

That is, keep only every L th sample ( [link] )
In frequency (DTFT):
Y ω m y m ω m m x 0 M m ω m n M m n x 0 n k δ n M k ω n M ω ω M n x 0 n k δ n M k ω n DTFT x 0 n DTFT δ n M k
Now DTFT δ n M k 2 k 0 M 1 X k δ ω 2 k M for ω as shown in homework #1, where X k is the DFT of one period of the periodic sequence. In this case, X k 1 for k 0 1 M 1 and DTFT δ n M k 2 k 0 M 1 δ ω 2 k M .
DTFT x 0 n DTFT δ n M k X 0 ω 2 k 0 M 1 δ ω 2 k M 1 2 μ X 0 μ 2 k 0 M 1 δ ω μ 2 k M k 0 M 1 X 0 ω 2 k M
so Y ω k 0 M 1 X 0 ω M 2 k M i.e. , we get digital aliasing .( [link] )
Usually, we prefer not to have aliasing, so the downsampler is preceded by a lowpass filter to remove all frequencycomponents above ω M ( [link] ).

Rate-changing by a rational fraction l/m

This is easily accomplished by interpolating by a factor of L , then decimating by a factor of M ( [link] ).

The two lowpass filters can be combined into one LP filterwith the lower cutoff, H ω 1 ω L M 0 L M ω Obviously, the computational complexity and simplicity of implementation will depend on L M : 2 3 will be easier to implement than 1061 1060 !

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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5
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