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Changing the Sampling Rate-Digital Signal Processing-Lecture Slides, Slides of Digital Signal Processing

COMSATS Institute of Information Technology (CIIT)Digital Signal Processing

This lecture is part of lecture series delivered by Dr Muhammad Fasih Uddin Butt for Digital Signal Processing course at COMSATS Institute of Information Technology. Its main points are: Changing, Sampling, Rate, Continuous-time, Signal, Interpolation, Downsampling, Frequency, Domain, Representation

Typology: Slides

2011/2012

Uploaded on 07/06/2012

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Changing the Sampling Rate
Quote of the Day
There is no branch of mathematics, however
abstract, which may not someday be applied to
the phenomena of the real world.
Nicolai Lobachevsky
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck,
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Changing the Sampling Rate-Digital Signal Processing-Lecture Slides and more Slides Digital Signal Processing in PDF only on Docsity!

Changing the Sampling Rate

Quote of the Day

There is no branch of mathematics, however

abstract, which may not someday be applied to

the phenomena of the real world.

Nicolai Lobachevsky

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck,

Changing the Sampling Rate

  • A continuous-time signal can be represented by its samples as
  • We can use bandlimited interpolation to go back to the

continuous-time signal from its samples

  • Some applications require us to change the sampling rate
    • Or to obtain a new discrete-time representation of the same

continuous-time signal of the form

  • The problem is to get x’[n] given x[n]
  • One way of accomplishing this is to
    • Reconstruct the continuous-time signal from x[n]
    • Resample the continuous-time signal using new rate to get x’[n]
    • This requires analog processing which is often undersired

x  n  xcnT 

x'  n  xc nT ' whereT  T'

Frequency Domain Representation of Downsampling

  • Recall the DTFT of x[n]=xc(nT)
  • The DTFT of the downsampled signal can similarly written as
  • Let’s represent the summation index as
  • And finally



 

k

c

j

T

2 k

T

X j T

X e





 

r

c r

c

j d MT

2 r

MT

X j MT

T'

2 r

T'

X j T'

X e

r ikM where - k   and 0 i M



  

M 1

i 0 r

c

j d MT

2 i

T

2 k

MT

X j T

M

X e

 

  

  

 

M 1

i 0

M

2 i M

j j d X e M

X e

Frequency Domain Representation of Downsampling: No Aliasing

Increasing the Sampling Rate by an Integer Factor: Upsampling

  • We increase the sampling rate of a sequence interpolating it
  • This is accomplished with a sampling rate expander
  • We obtain xi[n] that is identical to what we would get by

reconstructing the signal and resampling it with T’=T/L

  • Upsampling consists of two steps
    • Expanding
    • Interpolating

xi  n  xn /L  xcnT /L

 

 

   ^ 



k

e xk n kL 0 else

xn/L n 0 , L, 2 L,... x n

Frequency Domain Representation of Expander

  • The DTFT of xe[n] can be written as
  • The output of the expander is frequency-scaled



 



 



   

k

j Lk j L

n

j n

k

j Xe e xk n kL e xk e Xe

Interpolator in Time Domain

  • xi[n] in a low-pass filtered version of x[n]
  • The low-pass filter impulse response is
  • Hence the interpolated signal is written as
  • Note that
  • Therefore the filter output can be written as

 

 

n/ L

sin n/L hi n 

 

   

   

 

   

   k

i n kL /L

sin n kL /L x n xk

 

h n 0 n L, 2L,...

h 0 1

i

i

   

x (^) i  n  xn /L  xcnT /L  xcnT ' forn 0,L,2L,...

Changing the Sampling Rate by Non-Integer Factor

  • Combine decimation and interpolation for non-integer factors
  • The two low-pass filters can be combined into a single one