Digital Signal Processing. An Introduction

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	An Introduction
	
	1.1 INTRODUCTION
	1.2 APPLICATIONS OF DIGITAL SIGNAL PROCESSING
	1.3 SIGNALS
	1.4 CLASSIFICATION OF SIGNALS
	1.5 SIGNAL PROCESSING SYSTEMS
	1.6 SIGNAL PROCESSING
	1.7	ADVANTAGES OF DIGITAL SIGNAL PROCESSING OVER
	ANALOG SIGNAL PROCESSING
	1.8	ELEMENTS OF DIGITAL SIGNAL PROCESSING SYSTEM
	EXERCISES
	2.1	INTRODUCTION
	2.2	DISCRETE-TIME SIGNALS
	T
	0 < n < да
	Transformation of the Dependent Variable (Signal Amplitude):
	2.3
	DISCRETE-TIME SYSTEMS
	EXAMPLE 2.	1
	EXAMPLE 2.	2
	SOLUTION:
	EXAMPLE 2.	3
	EXAMPLE 2.4
	= S A =
	EXAMPLE 2.5
	= SI A = 1+1A+1 Al2+...
	2.4 CONVOLUTION OF TWO DISCRETE-TIME SIGNALS
	EXAMPLE 2.6
	t
	t
	t
	t
	t
	t
	EXAMPLE 2.7
	A i
	EXAMPLE 2.8
	7	^ 1 - A J
	EXAMPLE 2.9
	2.5 INVERSE SYSTEMS
	= I — I u (n) 13 )
	13 )
	2.6 CORRELATION OF TWO DISCRETE-TIME SIGNALS
	EXAMPLE 2.12
	t
	t
	t
	t
	t
	t
	EXAMPLE 2.13
	2.7 SIGNALS AND VECTORS
	x = c 1 x + x 1
	v ■ x = |x||u\\ cos 0
	v ■ x = 0.
	d Ге t 2 2 / xj 1 d Г гt
	И2 = lxl2 +1 x ’
	=	J^\\х(t)|2 dt + J
	EXAMPLE 2.14
	—
	2.8 REPRESENTATION OF SIGNALS ON ORTHOGONAL
	BASIS
	SAMPLING OF CONTINUOUS-TIME SIGNALS
	= A cosl 2n max-n + 0 I
	s I	Fs J
	EXAMPLE 2.15
	EXAMPLE 2.16
	F 3 = 2П
	2.10 RECONSTRUCTION OF A SIGNAL FROM ITS SAMPLE VALUES
		no. T
	j s(nTs )
	EXERCISES
	NUMERICAL EXERCISES
	9.
	tt
	t
	13 J	14 J
	3.1	INTRODUCTION
	3.2	DEFINITION OF THE z -TRANSFORM
	3.3 REGION OF CONVERGENCE (ROC)
	t
	= s ( -2 ) z2 + s (-1) z1 + s ( 0 ) z0 + s (1) z-1 + s ( 2 ) z 2
	t
	EXAMPLE 3.2
			=z -° + z-1 + z-2 + ... + z -x=
	1 - Az-1 - B-1 z + AB-1
	B - ABz-1 - z + A ~ B+A - z - ABz-
	PROPERTIES OF z -TRANSFORM
	EXAMPLE 3.3
	\"	[1, n > 0\"
	EXAMPLE 3.4
	t
	Ans(n) < \' > S f-^ IA )
	EXAMPLE 3.5
	Z [ u(-n)] = S ( z 1) = —^--1
	EXAMPLE 3.6
	EXAMPLE 3.7
	t
	t
	EXAMPLE 3.8
	= S[ Az-1 ] n
		+
	j = -~^
	3.5	SOME COMMON z -TRANSFORM PAIRS
	3.6	THE INVERSE z-TRANSFORM
	dz = <
	EXAMPLE 3.9
	EXAMPLE 3.10
			= У (z - z)	-
				+	+
	EXAMPLE 3.11
	2-3z-1+z-2
	2 - 3 z-1
	1.3 -1.7	42 .
	1 ч— z	ч— z	+....
	24
	7 J - 3 z-3
	7
	-4
	15 3
	4
	3-17-
	+2
	2 z2 + 6 z3 +14 z4 +...
	- 4 z2
	-18 z2 +12 z3
	30z3 - 28z4
	EXAMPLE 3.12
	/з	- 6 z -2 -12 z 43
	+ + +
	+ 6 z -2 + 12 z 43
	- 6 z -2 +12 z 43
	+ + +
	+ 6 z ~2 + 12 z 43
	- 8 z -3 -16 z -4
	+ + +
	6 z -2 + 20 z 43 + 16 z -4
	EXAMPLE 3.13
	14 )
	EXAMPLE 3.14
	z 3 X ;t - 4 •
	a a a 11
	a2 =
	+ Z-1
	3.7 SYSTEM FUNCTION
	EXAMPLE 3.15
	3.8 POLES AND ZEROS OF RATIONAL z -TRANSFORMS
	EXAMPLE 3.16
	3.9 SOLUTION OF DIFFERENCE EQUATIONS USING
	z-TRANSFORM
	EXAMPLE 3.17
	Y (z) [1 - Az-1 ] = A +1—r
	() 1 - Az-1 (1 - z-1 )(1 - Az-1)
	A A A L1 - A ) L1 - A ) Y (n) =		 +	+ + 	f-
	I (1 - A) I
	1 - A L -
	EXAMPLE 3.18
	EXAMPLE 3.19
	3.10 ANALYSIS OF LINEAR TIME-INVARIANT (LTI) SYSTEMS
	IN THE z -DOMAIN
	EXAMPLE 3.20
	p TT.
	\\ 3	)
	EXAMPLE 3.2	1
	I 4 )
					or	—— =
	(1 - 0.5z 1) 1 -72z 1 + z 2
	+ (	jp	A +
	I 1 - e 4 z-1 I
	14	J
	Z h(n)|	X
	EXAMPLE 3.2	2
						■
	\\ 2 J
	EXAMPLE 3.2	3
	EXAMPLE 3.2	4
	I 1 -1 z-1 II 1 -1 z-1 I
	(1 ^ n , ч y(n) = I — I u(n)
	\\ 2 J
	EXAMPLE 3.25
	EXAMPLE 3.26
	z
	a1 =
	a =—1—
	h (n) = Z-1 [ H ( z )]
	D *
	+		T
	EXERCISES
	NUMERICAL EXERCISES
	4.1 INTRODUCTION TO DISCRETE-TIME FOURIER
	TRANSFORM (DTFT)
	4.2 DEVELOPMENT OF THE DISCRETE-TIME FOURIER
	TRANSFORM (DTFT)
	4.3 CONVERGENCE OF THE DTFT
	EXAMPLE 4.	1
	EXAMPLE 4.	2
	EXAMPLE 4.3
	z
		s(ew)=y e-jwn =
	4.4 FOURIER TRANSFORM OF DISCRETE-TIME PERIODIC
	SIGNALS
	S(e )= X 2pAkd w~~n~
	EXAMPLE 4.4
	- 2nm)+	X 2%d (w + w0
	“	7	Q-n-
	^
	7
	2 p A	f	2 p A
	— + pd w +	, -p < w < p
	EXAMPLE 4.5
	4.5 PROPERTIES OF THE DTFT
	If	s(n) < DTFT > {s (ew)}
	if	«(n) < DTFT s s (ew)
	EXAMPLE 4.6
	S ( e )=(1 - e - jw ) + p ^ d (w - wp k )
	if	s(n) < DTFT s s (ew)
	EXAMPLE 4.7
	[1 - e-w ]
	(kw \'1 к 2 )
	= e4w I sin ( 5w ) , sin (w)
	ds (ejw)	d Г-A
	jdS (ejw)	“
	jdS ( e jw )
	EXAMPLE 4.8
	EXAMPLE 4.9
	1p
	1 1 1 1 	\"
	EXAMPLE 4.10
	s (ew )=td^
	S A 7
	a 2 =7	T
	1 - A1\' B J
	Y(ejw)=
	f_B-\'
					J- +A	—
	( A ^
	= X ■ n) t J si(ej )
	EXAMPLE 4.11
	p
	4.6 TABULATION OF PROPERTIES OF DTFT
	4.7 TABULATION OF DTFT PAIRS
	Discrete-time signal s (n)
	4.8 DUALITY
	мn)=7т X s2(-k)j0n
	EXAMPLE 4.12
	Ak = <
	Ak =
	p
	p
	4.9 DISCRETE-TIME LTI SYSTEMS CHARACTERIZED BY LINEAR CONSTANT-COEFFICIENT DIFFERENCE
	EQUATIONS
	EXAMPLE 4.14
	Y(ejw)
	H (ew )=id??	(4)
	EXAMPLE 4.15
	H (ejw ) = rj- ( ) s (ew)
	I 4 JI
	a,
	= -2
	:	1 ”4
	.,. .(1^. J1^n ,.
	12 J	12 J
	EXAMPLE 4.16
	7
	y (ew)
	EXERCISES
	NUMERICAL EXERCISES
	M ...	.
	w+—
	I 2 Л1
	H1 (j )=	1	1 1
	12 J
	An . .
	V 2 J
	5.1 INTRODUCTION
	5.2 DEFINITION OF DFT
	EXAMPLE 5.1
	EXAMPLE 5.2
	EXAMPLE 5.3
	5.3 THE DFT AS A LINEAR TRANSFORMATION TOOL
	sN =
	■ s(0) ’
	EXAMPLE 5.4
	5.4 PROPERTIES OF DFT
	Step in computation of circular convolution.
	EXAMPLE 5.5
	EXAMPLE 5.6
	5.5	TABULATION OF PROPERTIES OF DFT
	5.6	RELATIONSHIP BETWEEN DFT AND z -TRANSFORM
	LINEAR CONVOLUTION USING DFT
	EXAMPLE 5.7
	5.8 PITFALLS IN USING DFT
	EXAMPLE 5.8
	( N ^
	\\ 2 )
	= - eJ	+ — eJ	eJ + e J
	EXAMPLE 5.9
	EXAMPLE 5.1	0
	EXAMPLE 5.1	1
	b.	SI	K\\ = S *|—+ K
	\\ 2 J
	11 2 J
	EXAMPLE 5.13
	|Д m ))	1Д m ))
	(I F J)	(I M ))
	EXERCISES
	NUMERICAL EXERCISES
	t
	t
	6.1 INTRODUCTION
	6.2 GOERTZEL ALGORITHM
	(1 - WKz-1 )(1 - WKz-1)
	.
	(2 p K A l~
	6.3 FAST FOURIER TRANSFORM ALGORITHMS
	EXAMPLE 6.1
	n	„ N \\
	o/n 1	V	/ / .	. N ^
	V 2 2
	(N -11 12	)
	IZ 4	/ 4	( . N ^
	EXAMPLE 6.2
	EXERCISES
	7.1 INTRODUCTION
	EXAMPLE 7.1
	7.2 MAJOR FACTORS INFLUENCING OUR CHOICE OF
	SPECIFIC REALIZATION
	7.3 NETWORK STRUCTURES FOR IIR SYSTEMS
	EXAMPLE 7.2
	EXAMPLE 7.3
	EXAMPLE 7.5
							1	—	1
	NETWORK STRUCTURE FOR FIR SYSTEMS
						H (z) = T1
	=z
	+x
	EXAMPLE 7.	6
	EXAMPLE 7.	7
	Yh(°)
	Yh(2)	Yh<6) Yh<7)
	EXAMPLE 7.	8
	or	Y(z) [1 - b z-1 ] = Y‘(z) [z-1 - b ]
	EXAMPLE 7.9
	(1+ 55
	55
	55
	(1 — a z ) (1 — b z )
	1 - a z = 0
	EXAMPLE 7.10
	EXAMPLE 7.11
	EXAMPLE 7.12
	EXERCISES
	NUMERICAL EXERCISES
	8.1 INTRODUCTION
	Review of Analog Filter Design
	Z-1 [ H (z) = Z-1 ]
	1 + e Q
	8.2 MAJOR CONSIDERATIONS IN USING DIGITAL FILTERS
	8.3	COMPARISON BETWEEN DIGITAL AND ANALOG FILTERS
	8.4	COMPARISON BETWEEN IIR AND FIR DIGITAL FILTERS
	8.5	REALIZATION PROCEDURES FOR DIGITAL FILTERS
	8.6 NOTCH FILTERS
	8.7 COMB FILTERS
	8.8 ALL-PASS FILTERS
	8.9 DIGITAL SINUSOIDAL OSCILLATORS
	8.10 DIGITAL RESONATORS
	H ( z) =
	EXERCISES
	9.1 INTRODUCTION
	9.2 APPROXIMATION OF IIR DIGITAL FILTERS FROM
	ANALOG FILTERS
	EXAMPLE 9.1
					1
	Step II.
	S
	S
	= V\" Rm
	9.2.3.1 Derivations of Formula for Bilinear Transformation Method
	У (nTB)- У (nTs - Ts) = —[s(nTs- Ts) + s(nTs)] 2
	Ts к z +1J
	9.2.3.2 Properties of Mapping of Bilinear Transformation
	2 ( z -1 ^ s = I	I
	) + w2
	Case III:
	(2/Ts + s )
					. .	_	.	, (0T \'
	I 2 J
	9.2.3.3 Warping Effect
	9.2.3.4 Influence of the Warping Effect on the Amplitude Response of
	a Digital Filter
	9.2.3.5 Influence of the Warping Effect on the Phase Response of
	a Derived Digital Filter
	EXAMPLE 9.2
						1
						1
	s
	EXAMPLE 9.3
	Cs v	CsJ
					+
	of h ( nTs) = Z |j
	EXAMPLE 9.4
	( z — 1 ^ I — 1 + 0.1
	I z +1)
	I	1 + 0.1
	I z +1)
	s = I 	 I
	T I z +1J
	( T j
	1 + | ~ 15
	< T j
	1 + (T J) (s + jQ)
	z =	(T ^
	T ^	fQTл
	=1	2 JI 2 >
	( AA ‘ + B2) + jB( A + A ‘)
	B2 -	B2 - B2
	C=
	D=
	, ^1T 1 +-^-s-
	EXAMPLE 9.6
	o=^3
	1 + I -x-
	(9)2 +(^/3)2 \\ 84
	I 7 I
	(s +1)2 (s2 + s +1)
	4l	1 +1 Ml	1^ + 4l	1 +1
	^ 1 +1) J [ ^ z +1 )^ z +1)
	к 2 J
	I 2 J
	Pole Locations for Chebyshev Filters
			r = 1
	( M1 n - M -1/ n ^
	R = 1	1
	I 2 J
	EXAMPLE 9.7
	R =	=
	I 2	) L	2	J
	= - P1 X - P 2 X - P 2
	(S - P1 )(S - P2 )(S - P2*)
				—	1	1
	2. Design using Bilinear Transformation Method
	Q = 2 tan | w2 | = 2 tan | — x — p | = 0.65
	Q2 = 2tan| w2- | = 2tan| — x — p | = 1.02
	( z -1 YI2	Г ( z -1 ^1	-
	I z +1)
	H ( ep1) = H ( e 02p ) = 0.9421-164°
	ECN [Ji j
	1 + ECN IO j
	=	^7^^ d •• l--Q c
	i+e 2 CN f l J
	E=
	N=
	9.3 FREQUENCY TRANSFORMATION
	s	>Q s(O2 -QJ
	H(s) = Hp fo/2 +^^^
	(O2 -Al)
	s	>O C
	/4	(	s (О2-О,)С
	H (s) = H I О- -M	1 I
	EXAMPLE 9.8
	CI s J
	EXAMPLE 9.9
	Q C QC
	s + Q C
	Q C QC
	C C +Q C
	QCs + (s + QC ) = s + QC \'
	= 1 f ( e - w )|
	Transformation
	Design Parameters
	A=
	z 2 - A1 z 1 + A 2
	EXERCISES
	s = — I	I
	Ts I z +1J
	NUMERICAL EXERCISES
	10
	10.1	INTRODUCTION
	10.2	PROPERTIES OF FIR DIGITAL FILTERS
	ф(а} = -та = tan
	IA J
	^h(nTs)sin(aT - anTs) = 0
	0(0) = ^ h ( nTs) sin(aT - anTs)
	S h(mTs )
	+ £ /г(пТ5)е^и[(^1)/21т\'
	j27i(nTs)[e->[f(^1)/2)’\"ffi
	®\\	n |T
	10.3 DESIGN OF FIR DIGITAL FILTERS USING FOURIER
	SERIES METHOD
	h (nTs )=_L j h (eaT) e^da
	( N -1 ^	( N -1 ^
	I —-— l> n >1 —-— I
	Ways to reduce Gibb’s oscillations
	10.3.2.1. Rectangular Window Function
	w ( nTs )=<
	—-— I < n <1 —-—
	Spectrum of Rectangular Window Function
	M=
	I e s - s I
	I	2j	J
	10.3.2.2. Hann and Hamming Window Functions
	a + (1 - a)cos	, for-I	l< n <1-
	<	N -1	^ 2 J ^
	= awR (nTs) + (1 - a)
	(nT ) = awR (nT ) + N 1-O- J(A-1) nwR (nT )
	+N 12a 1( A)-nwR (nTs)
	f 12Г^( A-1) wR (nTs)
	к 2 у
	WH (e- ) = aW, (e-) + ^W, (ej■\"-1)\"■ )
	WH(e )
	2n A N
	। 1 - a A
				+1	I
	I 2 J .
	2 n 2n A 1
	wT +	I—
	10.3.2.3. Blackman Window Function
	2 ) I 2 )
	EXAMPLE 10.1
	h I I + 2 > h I	- n I cos naT
	I 2 J	nS I 2	1	’
	Delay = т = ^ j T = ^	| T, = 3 T
	h I I + 2 > h I	- n I cos na
	nn
	nn
	3n
	2n
	n
	10.3.2.4. Kaiser Window Function
	wk (nTs ) = <
					I < n <1
	4
	t |
	Ъ J
	a	2a
	2a
	1 - Y
	H (ejT) = <
	\" <|a|< \'2
	aC =
	Op + aa
	D=
	EXAMPLE 10.2
	2n
	nn
	nn
	N > ^D +1
	f F0.W	f„. ( N - 1 U.,f N - 11
	10.4 DESIGN OF FIR DIGITAL FILTER BASED ON NUMERICAL-ANALYSIS FORMULAE
	Gregory-Newton Forward Difference Formula
	Gregory-Newton Backward Difference Formula
	1 +	8 2 +			-
	m Г f _ T 4 Г _ T 4
	+— 8s I nT —- I + 8s I nT —s- I
	m ( m2 -1) ( m2
	+ 85sI nT + -
	) ) f f T 4 f f T
	f T 4
	8s I nTs + \"2“ 1 = s (nTs + Ts )- s (nTs )
	= d\'S (nTs + mTs )
	y(nTs) =
	EXAMPLE 10.3
	y(nTs) =
	— 1) ( m2 2(Z5)
	+ d I nT
	= 71- <
	T 4 .( ds I nT I + d I nT
	s- I + d3s I nT + -
	. I T
	+ d I nT
	1 t f T 4 t f T 4
	+	 d5sI nT —- | + d5sI nT +■— I
	T A
	8s I nTs +T 1 = s (nTs + Ts )- s (nTs )
	N 2 7
	TA
	N 2 7
	.( _ T A Г _ T A
	3s \\nTs + Ф +83s\\nTs -TH = s(nTs + Ts)-s(
	8 8s I nT + 7T
	( t t A Г t t s I nT + T + — I — s I nT + T + —
	Г	t AT
	sI nTs — Ts +	I — sI nTs
	N	2 7 N
	Г _	3 T A Г _ T
	s | nT +   I — s | nT + -
	Г _ T A Г _	3 T
	—s | nT + - I — s | nT +	-
	„ Г T A „ Г T
	s| nT +		 | — s| nT + L
	t A Г
	s- I + s I nT
	„з Г	t A „з Г t
	83s| nT + - I + 83s| nT — -
	3 T A . Г T A . Г
	+   | — 8s| nT + L I — 8s| nT
	2 7 N s 2) N s
	+ 8s I nTs
	+^T [ s ( nTs + 3 Ts) — 4 s ( nTs + 2 Ts) + 5 s ( nTs + Ts) — 5 s ( nTs — Ts)
	or	y (nTs ) = 601t [ s (nTs + 3 Ts) — 9 s ( nTs + 2 Ts) + 45 s ( nTs + Ts)
	10.5 DESIGN OF OPTIMAL LINEAR-PHASE FIR DIGITAL
	FILTERS USING M-CLELLAN-PARKS METHOD
	f Ss	к Л
	10.6 FINITE WORD LENGTH EFFECTS IN DIGITAL FILTERS
	H (z) = — z-a
	z - a
	IM <£
	I ■ j
	Requirements for low coefficient sensitivity
	Reduction of Product Round-off Error
	Granular Limit Cycles
	Overflow Limit Cycles
	EXERCISES
	NUMERICAL EXERCISES
	a < a < a
	0, ac <|a|< a
	a  = E
	13.15 A COMPUTER VOICE RESPONSE SYSTEM
	EXERCISES
	14
	14.1	INTRODUCTION
	14.2	APPLICATIONS AND ADVANTAGES OF RADARS
	Advantages of Using Radar
	14.3 LIMITATIONS OF USING RADAR
	14.4 CHIRP z -TRANSFORM (CZT) ALGORITHM
	S (zk ) = £ s(n) [AW-k ] n
	nk = ~[ n2 + k2 —( k — n )2 ]	(14.7)
	EXAMPLE 14.1
	14.5 RADAR SYSTEM AND RADAR PARAMETERS
	, „ , vT	,
	2
	14.6 RADAR SIGNAL DESIGN AND AMBIGUITY FUNCTIONS
	= £ S J 5 ( nTs + Т ) s *( nT ) s *( mTs + T ) s ( mTs ) n=-X m=-X -X
	14.7 AMBIGUITY FUNCTIONS OF CHIRPS AND
	SINUSOIDAL PULSES
	Z ,<
	“ I n 1
	“I n 1
	14.8	AMBIGUITY FUNCTION OF A CW PULSE
	14.9	AMBIGUITY FUNCTIONS OF A BURST
	14.10 OTHER SIGNALS
	2v
	T
	14.11 AIRBORNE SURVEILLANCE RADAR FOR AIR TRAFFIC
	CONTROL (ATC)
	14.12 LONG-RANGE DEMONSTRATION RADAR (LRDR)
	14.13 DIGITAL MATCHED FILTER FOR A
	HIGH-PERFORMANCE RADAR (HPR)
	\\ 2 J	\\ 2 J
	EXAMPLE 14.2
	EXERCISES
	A
	C
	D
	E
	F
	G
	H
	I
	L
	M
	N
	P
	R
	S
	T
	U
	V
	Z