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Lecture5 - Digital Filters
5.1 Digital filtering
Digital filtering canbe implementedeither in hardwareor software; in the firstcase,thenumericalprocessoriseitheraspecial-purposechipor it is assembledoutof asetof digital integratedcircuitswhichprovidetheessentialbuilding blocksofa digital filtering operation– storage,delay, addition/subtractionandmultiplica-tion by constants.On theotherhand,a general-purposemini-or micro-computercanalsobeprogrammedasa digital filter, in which casethenumericalprocessoris thecomputer’sCPUandmemory.
Filtering may be appliedto signalswhich are then transformedback to theanaloguedomainusingadigital to analogueconvertoror workedwith entirelyinthedigital domain(e.g. biomedicalsignalsaredigitised,filteredandanalysedtodetectsomeabnormalactivity).
5.1.1 Reasonsfor usinga digital rather than an analoguefilter� The numericalprocessorcan easily be (re-)programmedto implementanumberof differentfilters.� Theaccuracy of a digital filter is dependentonly on the round-off error inthearithmetic.Thishastwo advantages:
– the accuracy is predictableandhencethe performanceof the digitalsignalprocessingalgorithmis known a priori .
– The round-off error canbe minimizedwith appropriatedesigntech-niquesandhencedigital filters canmeetvery tight specificationsonmagnitudeandphasecharacteristics(whichwouldbealmostimpossi-ble to achieve with analoguefilters becauseof componenttolerancesandcircuit noise).
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� Thewidespreaduseof mini- andmicro-computersin engineeringhasgreatlyincreasedthenumberof digital signalsrecordedandprocessed.Powersup-ply andtemperaturevariationshave no effect on a programmestoredin acomputer.� Digital circuitshaveamuchhighernoiseimmunity thananaloguecircuits.
5.1.2 The samplingprocess
This is performedby ananalogueto digital converter(ADC) in whichthecontin-uousfunction
�������is replacedby a“discretefunction”
��� �, whichis definedonly
at���� �
, with������������
. We thenceonly needconsiderthedigitisedsampleset
���� �andthesampleinterval
�.
Aliasing
Consider�����������! #"���$ %'&( � (onecycle every 4 samples)andalso
���������)�! #"*�,+-$%.&( �(3 cyclesevery 4 samples)asshown in Fig. 5.1. Note that theresultantsamplesare the same. This result is referredto asaliasing. The mostpopularsolutionto this problemis to precedethe digital filter by an analogue low-passfilter toensurethat any frequency above
�0/#�0�Hz is adequatelysuppressed.(This LPF
is known asan“anti-aliasing”filter andis shown to theleft of theA/D converteron theblockdiagramof page46). Without ananti-aliasingfilter,
��1�23/4�Hz at the
outputcould have beenproducedeitherby��152#/4�
Hz or��156#/4�
Hz at the input.With the anti-aliasingfilter, any signalat
�7186#/0�Hz will be suppressedprior to
A/D conversion,i.e.��1�23/4�
comingout mustrepresent��1�23/4�
goingin. Sincewecannotbuild perfectanti-aliasingfilters,however, digital filterswill normallyhavetheirpassbandwell below
�0/��4�Hz.
5.1.3 Reconstructionof output waveform
The dataat the outputof the D/A convertermustbe interpolatedin orderto re-constructthefilteredsignal.Usuallythis takestheform of stepor one-pointinter-polation(in which thevalueat thesampletime is assumedto bethevalueof thefunctionuntil the next sampletime), followedby analoguelow-passfiltering asshown in Fig. 5.2. Theoutputlow-passfilter is sometimesknown asa recoveryfilter.
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0 10 20 30 40 50 60 70 80 90 100−1
−0.5
0
0.5
1
0 10 20 30 40 50 60 70 80 90 100−1
−0.5
0
0.5
1
Figure5.1: Aliasing.
5.2 Intr oduction to the principles of digital filtering
We canseethat the numericalprocessingis at the heartof the digital filteringprocess.How canthearithmeticmanipulationof a sequenceof numbersproducea “filtered” versionof that sequence?Considerthe noisy signal of figure 5.3,togetherwith its sampledversion:
Onewayto reducethenoisemightbeto try andsmooththedata.For example,wecouldtry a polynomialfit usinga least-squarescriterion. If wechoose,say, tofit aparabolato everygroupof 5 pointsin thesequence,then,for everypoint,wewill makeaparabolicapproximationto thatpointusingthevalueof thesampleatthatpoint togetherwith thevaluesof the4 nearestsamples(thisformsaparabolicfilter), asin Fig. 5.4
9 �� �:�);�<>=?@;BAC=? % ; %where9 �� � = valueof parabolaateachof the5possiblevaluesof
D�FEHGI� ��GJ���K�����#�K� Land
;�<*��;BA��K; %are the variablesusedto fit eachof the parabolaeto 5 input data
points.
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samples
DAC
LPF
Figure5.2: Reconstructionusinga DAC andLPF.
-2 -1 0 1 2
Figure5.3: Noisydata.
We obtaina fit by finding a parabola(coefficients;�<
,;BA
and; %
) which bestapproximatesthe5 datapointsasmeasuredby theleast-squareserrorE:
M �N;�<��K;�A,�K; % �O� %PQKR:S % ��T>�� �UGV��;�<>=?@;BAC=?% ; % �W� %
Minimizing theleast-squareserrorgives:X MX ;�< �Y�7� X MX ;BA �Z�7�and
X MX ; % �Y�andthus:
[ ;�<=\�*�3; % � QKR %PQKR:S % T>�� �
�*�3;BA�� Q]R %PQKR:S % T>�� �62
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-2 -1 0 1 2
parabolic fit
centre point, k=0
Figure5.4: Parabolicfit.
�*�3;�<>=^2B_H; % � QKR %PQ]R:S % % T>��H�
which thereforegives:
;�<�� �2 [ �`Ga24T>�bGa�4��=V�0�0T>�bGc�!� =\�064T>�d�B� =V�*�4T>�b�e�fGg2BT>���4���;BAh� ��*� �`GI�4T>�iGa�4�UGjT>�bGc�!�7=kT>�b�!�H=?�4T>���4�W�
; % � ���_ �N�4T>�iGI�0�UGjT>�bGc�!�UGg�4T>�d�B�UGlT>�i�!�7=^�4T>���0���Thecentrepointof theparabolais givenby:
9 �� �Om QKR <K�Y;*<>=^@;BAn=^ % ; % m QKR <��Z;*<Thus,theparabolacoefficient
;�<givenabove is theoutputsequencenumber
calculatedfrom a setof 5 input sequencespoints. The outputsequenceso ob-tainedis similar to theinput sequence,but with lessnoise(i.e. low-passfiltered)becausetheparabolicfiltering providesa smoothedapproximationto eachsetoffive datapointsin thesequence.Fig. 5.5 shows this filtering effect. Themagni-tuderesponse(which we will re-considerlater) for the5-pointparabolicfilter isshown below in Fig. 5.6.
Thefilter whichhasjustbeendescribedis anexampleof anon-recursivedig-ital filter, which aredefinedby thefollowing relationship(known asa difference
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0 10 20 30 40 50 60 70 80 90 100−0.5
0
0.5
1
1.5
Figure5.5: Noisydata(thin line) and5-pointparabolicfiltered(thick line).
0 100 200 300 400 500 600 7000
0.2
0.4
0.6
0.8
1
f
|H(z
)| fsamp
/ 2
Figure5.6: Frequencyresponseof 5-pointparabolicfilter.
equation):
o �� �p� qP r R <tsr T>��uGkvw�
wherethe srcoefficientsdeterminethefilter characteristics.Thedifferenceequa-
tion for the5-pointsmoothingfilter, therefore,is:
o �� �p� �2 [ �xGy2BT>��J=^�4� =\�0�4T>��c=\�!� =\�064T>�� �7=V�0�0T>��uGz�!�fGj2BT>��{G|�4�W�This is a non-causalfilter sincea given outputvalue o �� � dependsnot only onpreviousinputs,but alsoonthecurrentinput
T>�� �, theinput
T>��}=k�!�andtheinputT>��~={�4�
. Theproblemis solvedby delayingthecalculationof theoutputvalueo �� �(thecentrepoint of the parabola)until all the 5 input valueshave beensampled(i.e. adelayof
�4�where
�= samplingperiod),ie:
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o �� �:� �2 [ �`Ga2BT>�� �7=\�0�4T>��{Gz�e� =V�060T>��uGg�4�H=V�0�4T>��{Gk2B�fGg2BT>��{Gj_��W�It is of importanceto notethat the equationo ��H����� s
r T>���G�vw�represents
a discreteconvolution of the input datawith the filter coefficients; hencethesecoefficientsconstitutethe impulseresponseof thefilter.
Proof:
LetT>�� �����
, exceptatl���
, whereT>���3�c���
. Then o ��H�t��� rsr T>���G?vw���
s Q T>���3� (all termszeroexceptwhenva��
). This is equalto s Q sinceT>���3�J���
.Thereforeo ���3�I� s < ; o �`�0�a� s A ; etc
1�1!1. As thereis a finite numberof s ’s, the
impulseresponseis finite. For this reason,non-recursive filters arealsocalledFinite-Impulse Response(FIR) filters.
As we will see,we mayalsoformulatea digital filter asa recursive filter; inwhich,theoutput o �� � is alsoa functionof previousoutputs:
o �� �p� qP r R <�sr T>��{Gkvw�H= �P r R A��
r o ��uGjv��Beforewecandescribemethodsfor thedesignof bothtypesof filter, weneed
to review theconceptof the � -transform.
5.3 The � -transform
The � -transformis importantin digital filtering becauseit describesthesamplingprocessandplaysa role in thedigital domainsimilar to thatof theLaplacetrans-form in analoguefiltering.
TheLaplacetransformof a unit impulseoccurringat time�J�� �
is � S Q (�� .Considerthe discretefunction
����H�to be a successionof impulses,for example
of area�����3�
occurringat�c���
,���`�0�
occurringat�c���
, etc1�1�1
. TheLaplacetransformof thewholesequencewouldbe:�>� �w;0�t�Z���w�#�C=?���`�0� � S (�� =^���N��� � S % (�� =V1!1�1*=^����H� � S Q (��Thesuffix � denotesthetransformof thediscretesequence,notof thecontinuous�������
.
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Let usreplace� (�� by anew variable� , andrename��� �N;0�
as� � � � :� � � �h�Z���w�3�C=^���x�*� � S A =?���w�#� � S % =\1�1�1]����H� � S Q
For many functions,theinfinite seriescanberepresentedin “closedform”, ingeneralastheratioof two polynomialsin � S A .5.3.1 Examplesof the z-transform� stepfunction
��� �= 0 for
��?�,���� �
= 1 for����
� � � �h�F��= � S A = � S % =\1�1�1*= � S Q =V1�1!1� � � �h� ��'G � S A
by summationof ageometricprogression6.� Decayingexponential�������t� � S � & G:� ����H�:� � S �*Q (� � � �t�F��= � S � ( � S A = � S � % ( � S % =V1�1�1*= � S �*Q ( � S Q =V1�1�1
� � � �h� ��'G � S � ( � S A� Sinewaves
�������h�V�! �"f�h��G:� ��� �:�\�! #"p3�h�\� �K� Q�� ( G � S � Q�� (�� � � �h� �� � ��'G � � � ( � S A =
��'G � S � � ( � S A �� � � �h� �'Gk�e #"��h� � S A�'G|���! �"7�h� � S A = � S %
6strictly valid only if �K�4�3���� ¢¡66
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5.3.2 The PulseTransfer Function
This is thenamefor ( � -transformof output)/(� -transformof input).Let the impulseresponse,for exampleof anFIR filter, be s < at
�c���, s A at�O�\�
,1�1�1 s
rat�O�Vv��
withv>�Y�
to £ .Let ¤ � � � bethe � -transformof thissequence:
¤ � � �t� s <�= s A � S A = s % � S% =\1�1�1�= s
r� Sr=V1!1�1 s q � S q
Let ¥ � � � beaninput:
¥ � � �h�\T>�d�B�7=kT>�b�!� � S A =gT>���4� � S % =\1�1�10=kT>�� � � S Q =V1!1�1Theproduct¤ � � � ¥ � � � is:
¤ � � � ¥ � � �t�¦� s <�= s A � S A =\1�1�1 sr� Sr=V1�1!1 s q � S q �!��T>���4�7=gT>�i�!� � S
A =V1!1�1�T>�� � � S Q �in which thecoefficientof � S Q is:
s <]T>�� �7= s A`T>��uGz�!� =\1�1�1 sr T>��uGkvw�H=V1�1�1 s q T>��§G £ �
This is nothing else than the value of the output sampleat�^� �
. Hencethe whole sequenceis the � -transformof the output,say ¨ � � � , where ¨ � � ���¤ � � � ¥ � � � . Hencethe pulsetransferfunction, ¤ � � � , is the � -transformof theimpulseresponse.
For non-recursivefilters:
¤ � � �t� qP r R < sr� Sr
For recursivefilters:
¨ � � �t� qP r R < sr� Sr¥ � � �C= �P r R A �
r� Sr¨ � � �
¤ � � �h� ¨ � � �¥ � � � �� s
r� Sr
�'G|� �r� Sr
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5.3.3 © -planepole-zero plot
Let � � � �ª( , where�
= samplingperiod.Since;I�V«¬=®��
, we have:
� � ��¯ ( � � � (If«°�Z�
, thenm � md�±�
and � � � � � ( �V�e #"��h�²=¢�"�³µ´��h� , i.e. theequationofacircleof unit radius(theunit circle) in the � -plane.
Thus,the imaginaryaxis in the;-plane(
«?���) mapsonto theunit circle in
the � -planeandthe left half of the;-plane(
«Y���) onto the interior of the unit
circle.We know that all the polesof ¤ �N;0� must be in the left half of the
;-plane
for a continuousfilter to bestable.We canthereforestatetheequivalentrule forstability in the � -plane:
For stability all polesin the � -planemust be inside the unit circle.
5.4 Frequencyresponseof a digital filter
This canbeobtainedby evaluatingthe(pulse)transferfunctionon theunit circle( i.e. � � �K� � ( ).
ProofConsiderthegeneralfilter
o �� �p� ¶P r R <tsr T>��uGkvw�
NB: A recursive typecanalwaysbeexpressedasaninfinite sumby dividing out:
eg., for ¤ � � ��� s <�}G � A � S A�
wehave o �� �:� ¶P r R < s <�1 �rA T>��{Gkvw�
Let inputbeforesamplingbe�e #"0���h��=�·#�
, sampledat�h�V���¸�'��1!1�1C�� �
. ThereforeT>�� �:�Z�e #"*�W�� �g=|·#�O� A%#E �K�K¹ �BQ (�º@»`¼ = � S ��¹ �BQ (�º@»`¼ Lie.o �� �U� �� ¶P r R <�s
r� �]½ � ¾ QeS
r5¿ (�º@»xÀ = �� ¶P r R <tsr� S �]½ � ¾ QeS
r5¿ (�º@»xÀ
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� �� � �K¹ �BQ (�º@»`¼ ¶P r R < sr� S � �
r ( = �� � S �K¹ �BQ (�º@»`¼ ¶P r R < sr� � �r (
Now¶P r R <ts
r� S � �
r ( � ¶P r R <tsr � � � � ( � S
r
But¤ � � � for thisfilter is¶P r R <�s
r� Sr
andso¶P r S <�s
r� S � �
r ( � ¤ � � �`Á�� � � � (Let ¤ � � �xÁ�� � � � ( �\ � �xà .Then
¶P r R <�sr� � �r ( �Z � S ��à (complex conjugate)
Henceo ��H�:� A% �K��¹ �BQ (�º@»`¼  �K��à = A% � S ��¹ �BQ (�º@»`¼  � S �xÃor o �� �p�VÂÄ�! #"����� �j=^·'=^ÅÆ� when
T>�� �Æ�V�e #"0�W�� �k=|·#�Thus
Âand
Årepresentthegainandphaseof thefrequency response.i.e. the
frequency response(asacomplex quantity)is
¤ � � �*m Á R@ÇWȪÉeÊExample
Considerthe5-pointparabolicfilter.
o �� �:� �2 [ �`Ga2BT>�� �7=\�0�4T>��{Gz�e� =V�060T>��uGg�4�H=V�0�4T>��{Gk2B�fGg2BT>��{Gj_��W�hence
¨ � � �t� �2 [ �xGy2}=V�0� � S A =\�06 � S % =V�0� � S + Gg2 � SHË � ¥ � � �69
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thus
¤ � � � Á R@ǵÈ�É,Ê � �2 [ �`Ga2}=\�0� � S � � ( =V�*6 � S % � � ( =V�0� � S + � � ( Gk2 � SHË � � ( �
¤ � � � � ( �O� �2 [ � S % � � ( �`�06'=?�B_h�e #"7�h��GjÌh�e #"p�0�h�a�
Thereforem ¤ � � � � ( �*mB� �2 [ m5�06�=?�4_t�e #"��h�^GkÌO�! #"p�0�h�§m
When�h�z�Y�7��m ¤ � �K� � ( �0mB�Í�
andm ¤ � �K� � ( �*mB�Y��156��36
when�h��Î)��1�_3ÏBÐ
, i.e. ÑÑNÒ �Y��15�B_Ó ¤ � �K� � ( �t�FGI�0�h�i.e. linear phase(true for any FIR filter with palindromiccoefficients); all fre-quenciesdelayedby
�4�(asexpected!)
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