5.1 SYLLABUS EC2312 DIGITAL SIGNAL PROCESSING
3 104
1. INTRODUCTION
9
Classification of systems: of systems: Continuous, discrete, linear, causal, stable, dynamic, recursive, time variance; classification of signals: continuous and discrete, energy and power; Mathematical representation of signals; of signals; spectral density; sampling techniques, quantization, quantization error, Nyquist rate, aliasing effect. Digital signal representation. 2. DISCRETE TIME SYSTEM ANALYSIS
9
!transform and its properties, its properties, inverse z!transforms; difference equation " equation " #olution #olution by by z! transform, application to discrete systems ! #tability analysis, frequency response " onvolution " onvolution " $ourier $ourier transform transform of discrete of discrete sequence " sequence " Discrete Discrete $ourier series. $ourier series. 3. DISCRETE FOURIER TRANSFORM & COMPUTATION
9
D$% properties, D$% properties, magnitude and phase and phase representation ! Computation of D$% of D$% using $$% algorithm " algorithm " D&% D&% ' D&$ ! $$% using radi( ) " *utterfly " *utterfly structure. 4. DESIGN OF DIGITAL FILTERS
9
$&+ ' &&+ filter realization " arallel ' cascade forms. $&+ design: -indowing %echniques " Need Need and choice of windows of windows " " inear inear phase phase characteristics. &&+ design: &&+ design: /nalog filter design filter design ! *utterworth and Chebyshev appro(imations; digital design using impulse invariant and bilinear transformation ! -arping, prewarping ! $requency transformation. 5. DIGITAL SIGNAL PROCESSORS
9
&ntroduction " /rchitecture " $eatures " /ddressing $ormats " $unctional modes ! &ntroduction to Commercial rocessors TOTAL : 45 PERIODS TEXT BOOKS
0. 1.2. roa3is and D.2. Manola3is, 4Digital #ignal rocessing rinciples, /lgorithms and /pplications5, earson 6ducation, New 6ducation, New Delhi, )778 9 &. ). #.. Mitra, 4Digital #ignal rocessing " rocessing " / / Computer *ased Computer *ased /pproach5, %ata Mc2raw ill, New ill, New Delhi, )770.
5.2 SHORT UESTIONS AND ANS!ERS
UNIT"I " SIGNALS & SYSTEMS 1.
D#$%# S%'(). / #ignal is defined as any any physical physical quantity that varies with time, space or or any any other independent other independent variables.
2.
D#$%# ( *+*,#-. / #ystem is a physical device
3.
!(, (/# ,# *,#* %)# % %'%,() *%'() /#**%'
>.
G%# *-# ()%(,%* $ DSP
5.
#peech processing " #peech processing " #peech #peech compression ' decompression for for voice voice storage system Communication " Communication " 6limination 6limination of of noise noise by by filtering and echo cancellation. *io!Medical " *io!Medical " #pectrum #pectrum analysis of of 6C2,662 6C2,662 etc.
!/%,# ,# )(**%$%(,%* $ DT S%'()*.
6.
Converting the analog signal to digital signal, this is is performed performed by by /9D converter rocessing Digital signal signal by by digital system. Converting the digital signal to analog signal, this is performed is performed by by D9/ converter.
6nergy ' ower ower signals signals eriodic ' /periodic signals 6ven ' ?dd signals.
!(, %* ( E#/'+ ( P7#/ *%'() E#/'+ *%'(): / finite energy signal is is periodic periodic sequence, which has a finite energy energy but but zero average power. average power. P7#/ *%'(): /n &nfinite energy signal with finite average average power power is is called a power power signal. signal.
8.
!(, %* D%*/#,# T%-# S+*,#-* %he function of of discrete discrete time systems is to to process process a given input sequence to generate output sequence. &n &n practical practical discrete time systems, all signals are digital signals, and operations on such signals also lead to digital signals. #uch discrete time systems are called digital filter.
.
!/%,# ,# (/%;* )(**%$%(,%* $ D%*/#,#"T%-# *+*,#-*.
9.
inear ' Non linear system Causal ' Non Causal system #table ' @n stable system #tatic ' Dynamic systems
D#$%# L%#(/ *+*,#-
/ system is said to to be be linear system if if it it satisfies #uper position principle. position principle. et us consider consider (0
%he impulse response of a system consist of infinite number of samples are called &&+ system ' the impulse response of a system consist of finite number of samples are called $&+ system.
15. !(, (/# ,# >(*% #)#-#,* ;*# , *,/;, ,# >)? %('/(- $ %*/#,# ,%-# *+*,#- %he basic elements used to construct the bloc3 diagram of discrete time #ystems are /dder, Constant multiplier '@nit delay element. 16. !(, %* ROC % @"T/(*$/- %he values of z for which z " transform converges is called region of convergence <+?C=. %he z!transform has an infinite power series; hence it is necessary to mention the +?C along with z!transform.
18. L%*, (+ $;/ /#/,%#* $ @"T/(*$/-.
inearity %ime #hifting $requency shift or $requency translation %ime reversal
1. !(, (/# ,# %$$#/#, -#,* $ #();(,%' %#/*# ",/(*$/-
artial fraction e(pansion ower series e(pansion Contour integration <+esidue method=
19. D#$%# *(-)%' ,#/#-. / continuous time signal can be represented in its samples and recovered bac3 if the sampling frequency $s ≥ )*. ere 4$s5 is the sampling frequency and 4*5 is the ma(imum frequency present in the signal. 20. C#? ,# )%#(/%,+ ( *,(>%)%,+ $ '<=
#ince square root is nonlinear, the system is nonlinear. /s long as (
21. !(, (/# ,# /#/,%#* $ );,% 0. ). 8.
Commutative property (
UNIT"II DISCRETE TIME SYSTEM ANALYSIS 1. D#$%# DTFT.
et us consider the discrete time signal (
%he conditions are H &f (
3. L%*, ,# /#/,%#* $ DTFT. 1. eriodicity 2. inearity 3. %ime shift 4. $requency shift 5. #caling 6. Differentiation in frequency domain 7. %ime reversal 8. Convolution 9. Multiplication in time domain 10. arseval5s theorem 4. !(, %* ,# DTFT $ ;%, *(-)#
%he D%$% of unit sample is 0 for all values of w. 5. D#$%# DFT. D$% is defined as E
%he %widdle factor is defined as -Ne!G) 9N 8. D#$%# @#/ (%'.
%he method of appending zero in the given sequence is called as ero padding. . D#$%# %/;)(/)+ ## *#;##.
/ #equence is said to be circularly even if it is symmetric about the point zero on the circle. (
14. D#$%# ROC. %he value of for which the transform converged is called region of convergence. 15. F% @ ,/(*$/- $ <=1234 ( Ez!8. 0B)9zB89z)B>9z8. 16. S,(,# ,# );,% /#/,+ $ @ ,/(*$/-. %he convolution property states that the convolution of two sequences in time domain is equivalent to multiplication of their transforms. 18. !(, ,/(*$/- $ <"-= *y time shifting property A/ ,(%%' %#/*# @ ,/(*$/-. 1. &nverse z transform can be obtained by using 2. artial fraction e(pansion. 3. Contour integration 4. ower series e(pansion 5. Convolution. 20. O>,(% ,# %#/*# ,/(*$/- $ X<=1"(JJJ(J
2iven E
UNIT"III DISCRETE FOURIER TRANSFORM AND COMPUTATION
1.
!(, %* DFT
&t is a finite duration discrete frequency sequence, which is obtained by sampling one period of $ourier transform. #ampling is done at N equally spaced points over the period e(tending from w7 to ). 2.
D#$%# N %, DFT.
%he D$% of discrete sequence (
!(, %* DFT $ ;%, %-;)*# <=
%he D$% of unit impulse P
4.
L%*, ,# /#/,%#* $ DFT.
inearity, eriodicity, Circular symmetry, symmetry, %ime shift, $requency shift, comple( conGugate, convolution, correlation and arseval5s theorem. 5.
S,(,# L%#(/%,+ /#/,+ $ DFT.
D$% of linear combination of two or more signals is equal to the sum of linear combination of D$% of individual signal. 6.
!# ( *#;## %* ())# %/;)(/)+ ##
%he N point discrete time sequence is circularly even if it is symmetric about the point zero on the circle. 8.
!(, %* ,# %,% $ ( *#;## , ># %/;)(/)+
/n N point sequence is called circularly odd it if is antisymmetric about point zero on the circle.
. !+ ,# /#*;), $ %/;)(/ ( )%#(/ );,% %* , *(-# Circular convolution contains same number of samples as that of (
9.
!(, %* %/;)(/ ,%-# *%$, $ *#;##
#hifting the sequence in time domain by 405 samples is equivalent to multiplying the sequence in frequency domain by - N3l 07. !(, %* ,# %*((,('# $ %/#, -;,(,% $ DFT $or the computation of N!point D$%, N) comple( multiplications and NAN!0 Comple( additions are required. &f the value of N is large than the number of computations will go into la3hs. %his proves inefficiency of direct D$% computation. 11. !(, %* ,# 7(+ , /#;# ;->#/ $ (/%,-#,% #/(,%* ;/%' DFT -;,(,% Number of arithmetic operations involved in the computation of D$% is greatly reduced by using different $$% algorithms as follows. 0. +adi(!) $$% algorithms. !+adi(!) Decimation in %ime $$% algorithm. 12. !(, %* ,# -;,(,%() -)#%,+ ;*%' FFT ()'/%,- 0. ).
Comple( multiplications N9) log) N Comple( additions N log) N
13. H7 )%#(/ $%),#/%' %* # ;*%' FFT Correlation is the basic process of doing linear filtering using $$%. %he correlation is nothing but the convolution with one of the sequence, folded. %hus, by folding the
sequence h #/ $ -;),%)%(,%* ### % ,# ();)(,% $ DFT ( FFT 7%, 64"%, *#;##.
%he number of comple( multiplications required using direct computation is N)S>)>7US. %he number of comple( multiplications required using $$% is N9) log) N S>9)log )S>0U). #peed improvement factor >7US90U))0.88 18. !(, %* ,# -(% ((,('# $ FFT
$$% reduces the computation time required to compute discrete $ourier transform. 1. C();)(,# ,# ;->#/ $ -;),%)%(,%* ### % ,# ();)(,% $ DFT ;*%' FFT ()'/%,- 7%, ;*%' FFT ()'/%,- 7%, 32"%, *#;##.
$or N!point D$% the number of comple( multiplications needed using $$% algorithm is N9) log) N. $or N8), the number of the comple( multiplications is equal to 8)9)log )8)0SFTV7. 19. !(, %* FFT
%he fast $ourier transforms <$$%= is an algorithm used to compute the D$%. &t ma3es use of the #ymmetry and periodically properties of twiddles factor -N to effectively reduce the D$% computation time. &t is based on the fundamental principle of decomposing the computation of the D$% of a sequence of length N into successively smaller discrete $ourier transforms. %he $$% algorithm provides speed!increase factors, when compared with direct computation of the D$%, of appro(imately S> and )7T for )TS! point and 07)>!point transforms, respectively. 20. H7 -(+ -;),%)%(,%* ( (%,%* (/# /#;%/# , -;,# N"%, DFT ;*%' /#%"2 FFT
%he number of multiplications and additions required to compute N!point D$% using redi(! ) $$% are N log) N and N9) log) N respectively.
21. !(, %* -#(, >+ /(%"2 FFT %he $$% algorithm is most efficient in calculating N!point D$%. &f the number of output points N can be e(pressed as a power of ), that is, N)M, where M is an integer, %hen this algorithm is 3nown as radi(!s $$% algorithm.
22. !(, %* ( #%-(,%"%",%-# ()'/%,- Decimation!in!time algorithm is used to calculate the D$% of a N!point #equence. %he idea is to brea3 the N!point sequence into two sequences, the D$%s of which can be combined to give the D$% of the original N!point sequence. &nitially the N!point sequence is divided into two N9)!point sequences (e point D$%s. %his process is continued till we left with )!point D$%. %his algorithm is called Decimation!in!time because the sequence (#,7## DIF ( DIT ()'/%,-* D%$$#/##*: 0. the ). the
$or D&%, the input is bit reversal while the output is in natural order, whereas for D&$, input is in natural order while the output is bit reversed. %he D&$ butterfly is slightly different from the D&% butterfly, the difference being that comple( multiplication ta3es place after the add!subtract operation in D&$. S%-%)(/%,%#*: *oth algorithms require same number of operations to compute the D$%. *ot algorithms can be done in place and both need to perform bit reversal at some place during the computation. 24. !(, (/# ,# ()%(,%* $ FFT ()'/%,-* 0. ). 8.
inear filtering Correlation #pectrum analysis
25. !(, %* ( #%-(,%"%"$/#;#+ ()'/%,- &n this the output sequence E <= is divided into two N9) point sequences and each N9) point sequences are in turn divided into two N9> point sequences.
UNIT"I " DESIGN OF DEGITAL FILTER 1= D#$%# IIR $%),#/
&&+ filter has &nfinite &mpulse +esponse. 2= !(, (/# ,# (/%;* -#,* , #*%' IIR $%),#/*
/ppro(imation of derivatives &mpulse invariance *ilinear transformation.
3= !% $ ,# -#,* +; /#$#/ $/ #*%'%' IIR $%),#/* !+
*ilinear transformation is best method to design &&+ filter, since there is no aliasing in it. 4= !(, %* ,# -(% />)#- $ >%)%#(/ ,/(*$/-(,%
$requency warping or nonlinear relationship is the main problem of bilinear transformation. 5= !(, %* /#7(/%'
rewarping is the method of introducing nonlinearly in frequency relationship to compensate warping effect. 6= S,(,# ,# $/#;#+ /#)(,%*% % >%)%#(/ ,/(*$/-(,% Ω )
tan
% 8= !#/# ,# Ω (%* $ *")(# %* -(# % ")(# % >%)%#(/ ,/(*$/-(,%
%he GΩ a(is of s!plane is mapped on the unit circle in z!plane in bilinear transformation = !#/# )#$, ( *%# ( /%', ( *%# (/# -(# % ")(# % >%)%#(/ ,/(*$/-(,% eft hand side !! &nside unit circle +ight hand side! outside unit
9= !(, %* ,# $/#;#+ /#**# $ B;,,#/7/, $%),#/ *utterworth filter has monotonically reducing frequency response. 10= !% $%),#/ (/%-(,% (* /%)#* % %,* /#**# Chebyshev appro(imation has ripples in its pass band or stop band. 11= C( IIR $%),#/ ># #*%'# 7%,;, (()' $%),#/* Kes. &&+ filter can be designed using pole!zero plot without analog filters
12= !(, %* ,# ((,('# $ #*%'%' IIR F%),#/* ;*%' )#"#/ ),* %he frequency response can be located e(actly with the help of poles and zeros.
13= C-(/# ,# %'%,() ( (()' $%),#/. Digital flter i) Operates on digital samples of the signal. ii) It is governed ! linear di"eren#e e$%ation. iii) It #onsists of adders& m%ltipliers and dela!s implemented in digital logi#. iv) In digital 'lters the 'lter #oe(#ients are designed to satisf! the desired fre$%en#! response.
Analog flter i) Operates on analog signals. ii) It is governed ! linear di"eren#e e$%ation. iii) It #onsists of ele#tri#al #omponents lie resistors& #apa#itors and ind%#tors. iv) In digital 'lters the appro*imation prolem is solved to satisf! the desired fre$%en#! response.
14= !(, (/# ,# ((,('#* ( %*((,('#* $ %'%,() $%),#/* A(,('#* $ %'%,() $%),#/* igh thermal stability due to absence of resistors, inductors and capacitors. &ncreasing the length of the registers can enhance the performance characteristics li3e accuracy, dynamic range, stability and tolerance. %he digital filters are programmable. Multiple(ing and adaptive filtering are possible. D%*((,('#* $ %'%,() $%),#/* %he bandwidth of the discrete signal is limited by the sampling frequency. %he performance of the digital filter depends on the hardware used to implement the filter. 15= !(, %* %-;)*# %(/%(, ,/(*$/-(,% %he transformation of analog filter to digital filter without modifying the impulse response of the filter is called impulse invariant transformation.
16= O>,(% ,# %-;)*# /#**# $ %'%,() $%),#/ , //#* , ( (()' $%),#/ 7%, %-;)*# /#**# (<,= 0.5 #"2, ( 7%, ( *(-)%' /(,# $ 1.0?H ;*%' %-;)*# %(/%(, -#,. 18= H7 (()' )#* (/# -(# , %'%,() )#* % %-;)*# %(/%(, ,/(*$/-(,%
&n impulse invariant transformation the mapping of analog to digital poles are as follows, %he analog poles on the left half of s!plane are mapped into the interior of unit circle in z!plane. %he analog poles on the imaginary a(is of s!plane are mapped into the unit circle in the z!plane. %he analog poles on the right half of s!plane are mapped into the e(terior of unit circle in z!plane. 1= !(, %* ,# %-/,(# $ )#* % $%),#/ #*%'
%he stability of a filter is related to the location of the poles. $or a stable analog filter the
poles should lie on the left half of s!plane. stable digital filter the poles should lie r e f s n a r t e h t$or a e n i m r e t e d o t d e t a l u c l a c e b inside the unit circle in the z!plane. o t s a h s r e t e m a r a p f o r e b m u n e g r a l / . v . c Ω y c n e u q e r f f f o t u c e h t t a = ε B 0 < √ 9 0 f o e u l a v a s a h )
e s n o p s e r e d u t i n g a m d e z i l a m r o n e h % . v i . d n a p o t s e h t n i g n i s a e r c e d#","# 19= !+ ( %-;)*# %(/%(, ,/(*$/-(,% %* b , *%#/# , ># y l l a c i n o t o n o m d n a d n a b s s a p n i e l p p i r iany u q estrip s i e s n o p s e r e d u t i n g a m e h % . i i i for values of s! &n impulse invariant transformation of width )W9% in the s!plane plane in the range <)3!0=9% ≤ X . e <)3!0= W9% is i l l mapped into the entire z!plane. %he left half ≤ n a l p ! s n i e s p e a n o e i l s e l o p e h % . i i of each strip in s!plane is mapped into the interior of unit circle in l l / z!plane, right half of . n g i s e d e l o p . i each strip in s!plane is mapped into the e(terior of unit circle in z!plane . n o i t c n u f and the imaginary a(is of each strip in s!plane is mapped on the unit circle in z!plane. ence the impulse r e f s n a r t e h t e n i m r e t e d o t d e t a l u c l a c invariant transformation is many!to!one. e b o t s a h s r e t e m a r a p w e f y l n ? . v . c Ω y c n e u q e r f
20= !(, %* B%)%#(/ ,/(*$/-(,% f f o t u c e h t t a ) √ 9 0 f o e u l a v a s a h
e s n o p s e r e d u t i n g a m d e z i l a m r o n e h % . v i %he bilinear transformation is conformal mapping that transforms the s!plane to z!plane. &n . Ω in z!plane, %he left this mapping the imaginary a(is of s!plane is mapped into the unit circle f o n o i t c n u f g n i s a e r c d y l l a c i n o t o n o m half of s!plane is mapped into interior of unit circle in e z!plane and the right half of s!plane is mapped into e(terior of unit circle in z!plane. mapping is a one!to!one d n a n i g i r o%he e h t t*ilinear a t a l f y l l a m i ( a m mapping and it is accomplished when s i e s n o p s e r e d u t i n g a m e h % . i i i . e n a l p ! s n i e l c r i c a n o e i l s e l o p e h % . i i 21= H7 ,# /#/ $ ,# $%),#/ ($$#,* ,# $/#;#+ /#**# $ B;,,#/7/, $%),#/. . n g i s e d e l o p l l / . i 1 " # 0 + T # * + > # C %he magnitude response of butterworth filter is shown in figure, from which it can be , / 2 7 / # , , ; B observed that the magnitude response approaches the ideal response as the order of the filter is increased. 22= !/%,# ,# /#/,%#* $ C#>+*# ,+# 1 $%),#/*.
%he magnitude response is equiripple in the passband and monotonic in the stopband. %he chebyshev type!0 filters are all pole designs. %he normalized magnitude function has a value of at the cutoff frequency Ωc. . e r o m of N increases. %he magnitude response approaches the ideal response as the value e r a s r e t l i f + & & n i e s i o n f f o d n u o r e h % . s r e t l i f f o d n i 3 o t d e t i m i l y l l a u s u , y t i l i b i ( e l f s s e . y l e v i s r u c e r d e z i l a e r e b n a c s r e t l i f + & & s a h p r a e n i l e v a h t o n o d s r e t l i f e s e h % 23= C-(/# ,# B;,,#/7/, . e ( C#>+*# T+#"1 $%),#/*. . d e s u t o n s i 3 c a b d e e f e s u a c e b y l n i a m , s r e t l i f + & $ n i e r e v e s s s e l e r a e s i o n f f o d n u o r o t e u d s r o r r 6 . e s n o p s e r e d u t i n g a m r i e h t f o e p a h s e h t l o r t n o c o t y t i l i b i ( e l f r e t a e r 2 . y l e v i s r u c e r ! n o n d n a y l e v i s r u c e r d e z i l a e r e b n a c s r e t l i f + & $ . e s a h p r a e n i l y l t c e f r e p e v a h o t d e n g i s e d y l i s a e e b n a c s r e t l i f e s e h % . > . 8 . ) . 0
22. !(, %* FIR $%),#/*
. 2 N . S
%he specifications of the desired filter will be given in terms of ideal frequency response d(*# %-;)*# /#**#
*ased on impulse response the filters are of two types 0. &&+ filter ). $&+ filter %he &&+ filters are of recursive type, whereby the present output sample depends on the present input, past input samples and output samples. %he $&+ filters are of non recursive type, whereby the present output sample depends on the present input, and previous output samples. 24. !(, (/# ,# %$$#/#, ,+#* $ $%),#/ >(*# $/#;#+ /#**# %he filters can be classified based on frequency response. %hey are &= ow pass filter ii= igh pass filter iii= *and pass filter iv= *and reGect filter. 25. D%*,%';%* >#,7## FIR ( IIR $%),#/*.
26. !(, (/# ,# ,#%;#* $ #*%'%' FIR $%),#/*
%here are three well!3nown methods for designing $&+ filters with linear phase. %hese are 0= windows method )= $requency sampling method 8= ?ptimal or minima( design. 28. S,(,# ,# %,% $/ ( %'%,() $%),#/ , ># (;*() ( *,(>)#.
/ digital filter is causal if its impulse response h)#
$&+ filter is always stable because all its poles are at origin. 29. !(, (/# ,# /#/,%#* $ FIR $%),#/
0. ). 8. >. 30.
$&+ filter is always stable. / realizable filter can always be obtained. $&+ filter has a linear phase response. H7 (*# %*,/,% ( #)(+ %*,/,%* (/# %,/;#
%he phase distortion is introduced when the phase characteristics of a filter is not linear within the desired frequency band.
%he delay distortion is introduced when the delay is not constant within the desired frequency range. 31. !/%,# ,# *,#* %)# % FIR $%),#/ #*%'.
Choose the desired
32. !(, (/# ,# ((,('#* $ FIR $%),#/*
inear phase $&+ filter can be easily designed. 6fficient realization of $&+ filter e(ist as both recursive and nonrecursive structures. $&+ filters realized nonrecursively are always stable. %he roundoff noise can be made small in nonrecursive realization of $&+ filters.
33. !(, (/# ,# %*((,('#* $ FIR $%),#/*
%he duration of impulse response should be large to realize sharp cutoff filters. %he non!integral delay can lead to problems in some signal processing applications.
34. !(, %* ,# ##**(/+ ( *;$$%%#, %,% $/ ,# )%#(/ (*# (/(,#/%*,% $ ( FIR $%),#/ %he necessary and sufficient condition for the linear phase characteristic of an $&+ filter is that the phase function should be a linear function of w, which in turn requires constant phase and group delay.
35. !(, (/# ,# %,%* , ># *(,%*$%# $/ *,(, (*# #)(+ % )%#(/ (*# FIR $%),#/* %he conditions for constant phase delay /+6 hase delay, Y )# ,+#* $ %-;)*# /#**# $/ )%#(/ (*# FIR $%),#/*
%here are four types of impulse response for linear phase $&+ filters #ymmetric impulse response when N is odd. #ymmetric impulse response when N is even. /ntisymmetric impulse response when N is odd. /ntisymmetric impulse response when N is even.
3. L%*, ,# 7#))"?7 #*%' ,#%;#* $ )%#(/ (*# FIR $%),#/*.
%here are three well!3nown design techniques of linear phase $&+ filters. %hey are $ourier series method and window method $requency sampling method. ?ptimal filter design methods.
39. !(, %* G%>>* #-# </ G%>>* O*%))(,%= &n $&+ filter design by $ourier series method the infinite duration impulse response is truncated to finite duration impulse response. %he abrupt truncation of impulse response introduces oscillations in the passband and stopband. %his effect is 3nown as 2ibb5s phenomenon )# (/(,#/%*,%* $ ,# $/#;#+ /#**# $ 7%7 $;,% %he desirable characteristics of the frequency response of window function are %he width of the mainlobe should be small and it should contain as much of the total energy as possible. %he sidelobes should decrease in energy rapidly as w tends to W.
42. !/%,# ,# /#;/# $/ #*%'%' FIR $%),#/ ;*%' $/#;#+"*(-)%' -#,.
Choose the desired
43. !(, (/# ,# /(7>(? % FIR $%),#/ #*%' ;*%' 7%7* ( $/#;#+ *(-)%' -#, H7 %, %* #/-#
%he $&+ filter design using windows and frequency sampling method does not have recise control over the critical frequencies such as w p and ws. %his drawbac3 can be overcome by designing $&+ filter using Chebyshev appro(imation technique.&n this technique an error function is used to appro(imate the ideal frequency response, in order to satisfy the desired specifications. 44. !/%,# ,# (/(,#/%*,% $#(,;/#* $ /#,(';)(/ 7%7.
%he mainlobe width is equal to >W9N. %he ma(imum sidelobe magnitude is "08d*. %he sidelobe magnitude does not decrease significantly with increasing w.
45. L%*, ,# $#(,;/#* $ FIR $%),#/ #*%'# ;*%' /#,(';)(/ 7%7.
%he width of the transition region is related to the width of the mainlobe of window spectrum. d > > s i n o i t a u n e t t a 2ibb5s oscillations are noticed in the passband . * and stopband. %he attenuation in the stopband is constant and cannot be d n a b p o t s m u m i n i m e h tvaried. w o d n i w
g n i n n a h g n i s u d e n g i s e d r e t l i f + & $ n & = v i 46. !+ G%>>* *%))(,%* (/# ##)# % /#,(';)(/ 7%7 ( 7 %, ( ># . w g n i s a e r c n i h t i w s e s a e r c e d e d u t i n g a m #)%-%(,# / /#;# e b o l e d i s e h t m u r t c e p s w o d n i w n & = i i i . * d 0 8 " s iare m u r t c e p o d n i w transitions from 0 %he 2ibb5s oscillations in rectangular window due to s w the sharp n i e d u t i n g a m e b o l e d i s m u m i ( a m e h % = i i to 7 at the edges of window sequence. %hese oscillations can be eliminated or reduced by replacing sharp transition by N 9 W V s i m u rthe t c e p s gradual transition. %his is the motivation for development w o d n i w n i e b o l n i a m f oof triangular h t d i w e h % = i and cosine windows. . * d ) ) s i n o i t a u n e t t a d n a b p o t s m u m i n i m e h t w o d n i w r a l u g n a t c e r 48. L%*, ,# (/(,#/%*,%* $ FIR $%),#/* #*%'# ;*%' 7%7*. g n i s u d e n g i s e d r e t l i f + & $ n & = v i %he width of the transition band depends on the type . of window. w g n i s a e r c n i %he width of the transition band h can be made narrow by increasing . * d > > s i n o i t u n e t a d n a b p o t s the value of N t i w s e s a e r c e d y l a t h g i l t s e d u t i n g a m where N is the length of the window sequence. m u m i n e h t e w o n i w g n i m m a h e b o l e d i s e h t i m u r t c p s d w o d n i w n & = i i i %he attenuation in the stop band is fi(ed for a given window, e(cept in case of aiser g n i s u d e n g i s s e d u r e i f + $ n & = i . * d 8 0 " i m r t l c e p s & w o d n i v w window where it is variable. . t n a t s n o c i a m e e d u t i h n g a m n i e d u t i n g a m e b o l e d i s n m u m i r ( a m e % = i i e b o l e d i s e h t m u r t c e p s w o d n i w n & e = i i s i N 9 W > s i m u r t c p . * 0 > " s m u r t c e s w o d n i w w o d n i w n i d e b o l n i i a m f o h t p d i w e h % = i n i e d u t i n g a m e b o l e d i s 7 m u m i ! ( a m e % = i i 2 3 & % ' & % h & & ( H N 9 7 W V s i m u r t c e p s 7 2 3 & % / ( ) ; ' & ( , 4 # R w o d n i w n i e b o l n i a m f o h t d i w e h % = i 4. C-(/# ,# /#,(';)(/ 7%7 ( (%' 7%7. . * d ) ) s i n o i t a u n e t t a d n a b p o t s m u m i n i m e h t w o d n i w r a l u g n a t c e r g n i s u d e n g i s e d r e t l i f + & $ n & = v i . w g n i s a e r c n i h t i w s e s a e r c e d y l t h g i l s e d u t i n g a m e b o l e d i s e h t m u r t c e p s w o d n i w n & = i i i . * d 8 0 " s i m u r t c e p s w o d n i w n i e d u t i n g a m e b o l e d i s m u m i ( a m e h % = i i N 9 W > s i m u r t c e p s w o d n i w n i e b o l n i a m f o h t d i w e h % = i 7 2 3 & % ! ' & % ( H 7 2 3 & % 7 / ( ) ; ' & ( , 4 # R 49. C-(/# ,# /#,(';)(/ 7%7 ( (--%' 7%7.
50. !/%,# ,# (/(,#/%*,% $#(,;/#* $ (%' 7%7 *#,/;-.
%he mainlobe width is equal to VW9N.
%he ma(imum sidelobe magnitude is ">0d*. %he sidelobe magnitude remains constant for increasing w. . Y f o e u l a v e h t 51. !(, %* ,# -(,#-(,%() />)#%)# % ,# #*%' $ 7%7 n o s d n e p e d d n a e l b a i r a v s i n o i t a u n e t t a $;,% d n a b p o t s m u m i n i m e h t w o d n i w %he mathematical problem involved in the design of window function)# $#(,;/#* $ K(%*#/ !%7 s i e b o l n i a m f o*#,/;-. 3 a e p o t t c e p s e r h t i w e d u t i n g a m e b o l e d i s m u m i ( a m e h % = i i %he width of the mainlobe and the pea3 sidelobe are variable. . N %he parameter Y in the aiser -indow function is an independent variable that can be ' Y f o s e u l a v e h t n o s d n e p e d m u r t c e p s varied to control the sidelobe levels with respect to mainlobe pea3. w o d n i w n i e b o l n i a m f o h t d i w e h % = i %he width of the mainlobe in the window spectrum can be varied by varying the length . * d > > s i n o i t a u n e t t a N of the window sequence. d n a b p o t s m u m i n i m e h t w o d n i w g n i m m a h g n i s u d e n g i s e d r e t l i f + & $ n & = v i . t n a t s n o c s n i a m e r e d u t i n g a m e b o l e d i s e h t m u r t c e p s w o d n i w n & = i i i . * d 0 > " s i m u r t c e p s w o d n i w n i e d u t i n g a m e b o l e d i s m u m i ( a m e h % = i i 53. C-(/# ,# (--%' 7%7 ( K(%*#/ 7%7. N 9 W V s i m u r t c e p s w o d n i w n i e b o l n i a m f o h t d i w e h % = i w o d n i r e s i a ; w o d n i g n i m m a :
UNIT " DIGITAL SIGNAL PROCESSOR
1.
!/%,# */, ,#* '##/() ;/*# DSP /#**/*
2eneral!purpose digital signal processors are basically high speed microprocessors with hard ware architecture and instruction set optimized for D# operations. %hese processors ma3e e(tensive use of parallelism, arvard architecture, pipelining and dedicated hardware whenever possible to perform time consuming operations . 2. !/%,# ,#* *#%() ;/*# DSP /#**/*. %here are two types of special; purpose hardware.
B/%#$)+ #)(% (>;, H(/(/ (/%,#,;/#.
%he principal feature of arvard architecture is that the program and the data memories lie in two separate spaces, permitting full overlap of instruction fetch and e(ecution. %ypically these types of instructions would involve their distinct type. 0. &nstruction fetch ). &nstruction decode 8. &nstruction e(ecute 4. B/%#$)+ #)(% (>;, -;),%)%#/ (;-;)(,/.
%he way to implement the correlation and convolution is array multiplicationMethod.$or getting down these operations we need the help of adders and multipliers. combination of these accumulator and multiplier is called as multiplier accumulator.
5. !(, (/# ,# ,+#* $ MAC %* ((%)(>)#
0. ).
%here are two types M/C5# available Dedicated ' integrated #eparate multiplier and integrated unit
6.
!(, %* -#(, >+ %#)%# ,#%;#
%he pipeline technique is used to allow overall instruction e(ecutions to overlap. %hat is where all four phases operate in parallel. *y adapting this technique, e(ecution speed is increased. 8.
!(, (/# $;/ (*#* ((%)(>)# % %#)%# ,#%;#
%he four phases are
V. I ( "%#)%# -(%# ,# %*,/;,% $#, ## ( ##;,# ,(?# 30 * 45 * ( 25 * /#*#,%#)+. D#,#/-%# ,# %/#(*# % ,/;';, %$ ,# %*,/;,% 7#/# %#)%#. /ssume a Tns pipeline overhead in each stage and ignore other delays. %he average instruction time is 87 nsB>T ns B )T ns 077 ns 6ach instruction has been completed in three cycles >T ns F 8 08Tns %hroughput of the machine %he average instruction time9Number of M9C per instruction 077908T 7.[>7[ *ut in the case of pipeline machine, the cloc3 speed is determined by the speed of the slowest stage plus overheads. &n our case is >T ns B T ns T7 ns %he respective throughput is 0779T7 ).77 %he amount of speed up the operation is 08T9T7 ).[ times 9. A**;-# ( -#-/+ (#** ,%-# $ 150 * -;),%)%(,% ,%-# $ 100 * (%,% ,%-# $ 100 * ( #/#( $ 10 * (, #( %# *,('#. D#,#/-%# ,# ,/;';, $ MAC /fter getting successive addition and multiplications
%he total time delay is 0T7 B 077 B 077 B T 8TT ns #ystem throughput is 098TT ns. 10.!/%,# 7 ,# (-# $ ,# (/#**%' -#*. Direct addressing. &ndirect addressing. *it!reversed addressing. &mmediate addressing. i. #hort immediate addressing. ii. ong immediate addressing. Circular addressing. 11.!(, (/# ,# %*,/;,%* ;*# $/ >)? ,/(*$#/ % C5X P/#**/*
%he *DD, *D and *D instructions use the *M/+ to point at the source or destination space of a bloc3 move. %he M/DD and M/D# also use the *M/+ to address an operand in program memory for a multiply accumulator operation
0).B/%#$)+ #)(% (>;, ,# #%(,# /#'%*,#/ (/#**%' -#*. %he dedicated!registered addressing mode operates li3e the long immediate addressing modes, e(cept that the address comes from one of two special!purpose memory! mapped registers in the C@: the bloc3 move address register <*M/+= and the dynamic bit manipulation register
13. B/%#$)+ #)(% (>;, >%,"/##/*# (/#**%' -#
&n the bit!reversed addressing mode, &NDE specifies one!half the size of the $$%. %he value contained in the current /+ must be equal to )n!0, where n is an integer, and the $$% size is )n. /n au(iliary register points to the physical location of a data value. -hen we add &NDE t the current /+ using bit reversed addressing, addresses are generated in a bit! reversed fashion. /ssume that the au(iliary registers are eight bits long, that /+) represents the base address of the data in memory <7007 7777)=, and that &NDE contains the value 7777 0777). 14. B/%#$)+ #)(% (>;, %/;)(/ (/#**%' -#.
Many algorithms such as convolution, correlation, and finite impulse response <$&+= filters can use circular buffers in memory to implement a sliding window; which contains the most recent data to be processed. %he 4CT( supports two concurrent circular buffer operating via the /+s. %he following five memory!mapped registers control the circular buffer operation. 0. C*#+0! Circular buffer 0 start register. ). C*#+)! Circular buffer ) start +egister, 8. C*6+0! Circular buffer 0 end register >. C*6+)! Circular buffer ) end register T. C*C+ ! Circular buffer control register. 15. !/%,# ,# (-# $ (/%;* (/, $ C5X (/7(/#.
0. ). 8.
Central arithmetic logic unit
>. T.
Memory!mapped registers. rogram controller.
0S. !/%,# */, ,#* (>;, (/%,-#,% )'% ;%, ( (;-;)(,/. %he 8)!bit general!purpose /@ and /CC implement a wide range of arithmetic and logical functions, the maGority of which e(ecute in a single cloc3 cycle. ?nce an operation is performed in the /@, the result is transferred to the /CC, where additional operations, such as shifting, can occur. Data that is input to the /@ can be scaled by the prescaler. %he following steps occur in the implementation of a typical /@ instruction: 0. Data is fetched from memory on the data bus, ). Data is passed through the prescaler and the /@, where the arithmetic is performed, and 8. %he result is moved into the /CC. %he /@ operates on 0S!bit words ta3en from data memory or derived from immediate instructions. &n addition to the usual arithmetic instructions, the /@ can perform *oolean operations, thereby facilitating the bit manipulation ability required of high!speed controller. ?ne input to the /@ is always supplied by the /CC. %he other input can be transferred from the +62 of the multiplier, the /CC*, or the output of the prescaler. /fter the /@ has performed the arithmetic or logical operation, the result is stored in the /CC.
18. !/%,# */, ,#* (>;, (/())#) )'% ;%,. %he parallel logic unit <@= can directly set, clear, test, or toggle multiple bits in control9status register pr any data memory location. %he @ provides a direct logic operation path to data memory values without affecting the contents of the /CC or the +62. 1. !(, %* -#(, >+ (;%)%(/+ /#'%*,#/ $%)# %he au(iliary register file contains eight memory!mapped au(iliary registers ;, %/;)(/ /#'%*,#/* % C5X. %he 4CT( devices support two concurrent circular buffers operating in conGunction with user!specified au(iliary register. %wo 0S!bit circular buffer start registers
