<< Chapter < Page
  Dspa   Page 1 / 1
Chapter >> Page >

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 !

Questions & Answers

a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
how did I we'll learn this
Noor Reply
f(x)= 2|x+5| find f(-6)
Prince Reply
f(n)= 2n + 1
Samantha Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Dspa' conversation and receive update notifications?