for Communication Systemsfor Communication Systems
 
Multirate processing arises in many fieldsMultirate processing arises in many fields of digital signal processingof digital signal processing
Digital audio sampling fre!uency con"ersionDigital audio sampling fre!uency con"ersion #$2 %&'( ))*1%&'( )+%&',( sharp cut-off of FIR#$2 %&'( ))*1%&'( )+%&',( sharp cut-off of FIR filter( .filter( .
Signal processing for digital communicationsSignal processing for digital communications sym/ol rate processing( /it rate processing(sym/ol rate processing( /it rate processing( sample rate processing( .sample rate processing( .
Speech processing $ speech codec #dapti"eSpeech processing $ speech codec #dapti"e Multi Rate,( fractionnal pitch estimation( ***Multi Rate,( fractionnal pitch estimation( ***
..
 
Multirate Processing 2 of 2Multirate Processing 2 of 2
In"ol"es to actions on the digital signalIn"ol"es to actions on the digital signal
Donsampling resampling donards theDonsampling resampling donards the digital signal in the digital domain*digital signal in the digital domain*
3psampling resampling upards the digital3psampling resampling upards the digital signal in the digital domain*signal in the digital domain*
M e e!" Retain one sample o"erRetain one sample o"er M M and discardand discard
thethe M-1 M-1 others( e"eryothers( e"ery M M samples*samples*
 
∑
=
 
 
nti-aliasing Filternti-aliasing Filter
M x(n) y(m)&#',
4 x(m) y(n)
n
∑
  ∞
−∞
−
∞
−∞
− ,#,#,#
 
3psampling 2 of 23psampling 2 of 2 Interpolating FilterInterpolating Filter
o/le identity for upsamplingo/le identity for upsampling
4 x(m) y(n)&#',
Decimation Case 1 of )Decimation Case 1 of )
M&#', M7#'M,
4et4et n=lM+k n=lM+k 
 
 
=
 
Decimation Case 2 of )Decimation Case 2 of )
M
are discarded*are discarded*
M&#',
8ime8ime
M8M8 ee
Polyphase Implementation of FIR FiltersPolyphase Implementation of FIR Filters
Decimation Case $ of )Decimation Case $ of ) 3sing no/le identity3sing no/le identity
M 70#',
M8M8 ee

 
70#',
71#',
7M-1#',
Fe Fe6M
* Commutator runs atCommutator runs at F F  ee ,. ,. t each input sample only one component is computed and accu-t each input sample only one component is computed and accu-
mulated ith the result of the pre"ious one* 8he result is output hen the last componentmulated ith the result of the pre"ious one* 8he result is output hen the last component
  is reached and accumulator is reset* 8his spreads the processing load o"eris reached and accumulator is reset* 8his spreads the processing load o"er MT  MT  ee**
8ime8ime
M8M8 ee
6M6M
4 &#', R#'4,4
4et4et n=mL+L-1-k n=mL+L-1-k 
 
−
=
−
=
 
4 &#',
4Fe
88ee6464

*  L-1 L-1 multiplications /y 0 o"ermultiplications /y 0 o"er L L
For each filter e"aluation*For each filter e"aluation*
*  N  N  filter length*filter length*
Fe
Interpolation Case $ of 5Interpolation Case $ of 5
3sing no/le identity3sing no/le identity
R 0#',
R 1#',
R M-1#',
* t each output sampling instant(t each output sampling instant(
 
R 0#',
R 1#',
R M-1#',
Fe 4Fe
6464
88ee6464
* For each output sampling instant one polyphase component is computed*For each output sampling instant one polyphase component is computed*
 
Polyphase Implementation of FIR FiltersPolyphase Implementation of FIR Filters
Interpolation Case 5 of 5Interpolation Case 5 of 5 4inear Periodically :arying 8ime system4inear Periodically :arying 8ime system
hh00
 z  z -1-1
hh L-1 L-1
 z  z -1-1
hh00
hh2L-12L-1
Case StudyCase Study
Shaping filters for a ;PS< modem Shaping filters for a ;PS< modem 7mitter interpolation case*7mitter interpolation case*
Recei"er decimation caseRecei"er decimation case
7fficient lgorithm Implementation 7fficient lgorithm Implementation ood ordering of computations(ood ordering of computations(
7fficient memory organi'ation and management*7fficient memory organi'ation and management*
 
 its
A- 
fre!uencyfre!uency
Sym/olSym/ol
fre!uencyfre!uency
SampleSample
fre!uencyfre!uency
 
 
7mitter 2 of )7mitter 2 of )
4et F4et Fee>1?F>1?Fss #1? sample 6 sym/ol,#1? sample 6 sym/ol,
Define a raised cosine filter ithDefine a raised cosine filter ith ? sym/ols length*? sym/ols length*
Roll@off 0*5Roll@off 0*5
Matla/ commandMatla/ command h=RCOSFIR(0.5,3,16,1);h=RCOSFIR(0.5,3,16,1);
* In red ideal interpolating filterIn red ideal interpolating filter
* In /lue actual RC filterIn /lue actual RC filter
1? &#',
7mitter $ of )7mitter $ of )
 
−
k    z k  LmLh z  R
  }
  }
=
=
=

 
h(16)
h(32)
h(48)
h(64)
h(80)
h(96)
h(1)
h(17)
h(33)
h(49)
h(65)
h(81)
h(97)
h(15)
h(31)
h(47)
h(63)
h(79)
h(95)
h(111)
CoefficientsCoefficients
 x(0)
 x(1)
 x(2)
 x(3)
 x(4)
 x(5)
 x(6)
Sym/olsSym/ols
1515thth samplesample
11stst samplesample
22ndnd samplesample
9hen coefficient pointer reaches this address a ne9hen coefficient pointer reaches this address a ne
sym/ol ill /e input at the net output sample periodsym/ol ill /e input at the net output sample period
R=flipud(reshape(h,8,12));
ed 
7mitter #C calla/le,7mitter #C calla/le, .se$&.se$& +$oefs2++$oefs2+
 $o-p $o-p   .se&.se&   1616   ;u-er of pol/phase $o-poe&;u-er of pol/phase $o-poe&
$oefs2$oefs2   .i$lude.i$lude +$oefpol/2.i$++$oefpol/2.i$+
$oefsfi$oefsfi
$oefsie$oefsie .se& $oefsfi$oefs2.se& $oefsfi$oefs2
filfil   .se&.se&   $oefsie$o-p$oefsie$o-p   ;pol/phase $o-poe& le4&h;pol/phase $o-poe& le4&h
filuffiluf   .use$&.use$& +fil&re2+,fil+fil&re2+,fil   ;da&a uffer;da&a uffer
.&e&.&e&
 7firii& 7firii&
SS   9$oefs2,!(aduf)9$oefs2,!(aduf)   ;poi&er &o $urre& $oefs poi&er;poi&er &o $urre& $oefs poi&er
S: S:   9filuf,R29filuf,R2   ;eroed ii&ial uffer $odi&io;eroed ii&ial uffer $odi&io
R<R<   9fil19fil1
SS   ,!R2 ,!R2
AA   9@ar,A<9@ar,A<
S: S:   9$oefsie,B9$oefsie,B
 :DA:  :DA:   aduf,R2aduf,R2   ;$urre& $oefs poi&er;$urre& $oefs poi&er
S: S:   91,R091,R0
S: S:   9filuf,R39filuf,R3   ;s/-ol uffer;s/-ol uffer
SS   ,!R3 ,!R3   ;e% sa-ple (4uess hold duri4 16 sa-ples);e% sa-ple (4uess hold duri4 16 sa-ples)
R<ER<E   ,9fil1 ,9fil1   ;$o-pu&e oe pol/phase $o-poe&;$o-pu&e oe pol/phase $o-poe&
 :C :C   !R20*,!R3, !R20*,!R3, 
 :D:A :D:A   R2,aduf R2,aduf   ;sa@e e% $urre& $oefs poi&er;sa@e e% $urre& $oefs poi&er
SF SF   ,16 ,16
SF SF   ,1 ,1   ;ou&pu& of RCF $a e 4rea&er &ha 1 ;ou&pu& of RCF $a e 4rea&er &ha 1
C:<: C:<:   Gaduf,9$oefs2Gaduf,9$oefs2   ;&es& if dela/ s/-ols is eeded ;&es& if dela/ s/-ols is eeded 
BCBC   ed,Ced,C   ;Hu-p if o& e$essar/;Hu-p if o& e$essar/
 :R  :R   !R3(2)!R3(2)
R<R<   9fil29fil2
A>A> !R3!R3
eded
R>R>
FFee 1? %h' 1? %h'
FFss 1 %h' 1 %h'
φ : π/4
f> Ff> F ss6+>125 &'6+>125 &'
Sym/ol outputSym/ol output Sample outputSample output
 
 
Recei"er 2 of 2Recei"er 2 of 2
Recei"er structure is !uite similar( eceptRecei"er structure is !uite similar( ecept thatthat
7ach polyphase component has its on delay tap7ach polyphase component has its on delay tap
7ach polyphase output has to /e accumulated for7ach polyphase output has to /e accumulated for  M M polyphase computations and accumulator is outputedpolyphase computations and accumulator is outputed e"erye"ery M  M  input sample and reset*input sample and reset*
70#',
71#',
7M-1#',
Follo on cti"itiesFollo on cti"ities
4a/oratory 10 for the 8MS$20C5)1? DS< 4a/oratory 10 for the 8MS$20C5)1? DS< 
Illustrates the effects of decimation and anti-Illustrates the effects of decimation and anti- aliasing filters*aliasing filters*
4a/oratory 11 for the 8MS$20C5)1? DS< 4a/oratory 11 for the 8MS$20C5)1? DS< 
Illustrates the effects of interpolation and anti-Illustrates the effects of interpolation and anti- imaging filters*imaging filters*
pplication A for the 8MS$20C5)1? DS< pplication A for the 8MS$20C5)1? DS< 
 
ReferenceReference