Digital Processing in a Nutshell Vol. 1

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<p>300189 800077 5 ID: 12395631www.lulu.comDigital Signal Processing in a Nutshell (Volume I)Jan Allebach, Krithika ChandrasekarDigital Signal Processing in aNutshell (Volume I)Jan P Allebach Krithika Chandrasekar2Contents1 Signals 51.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Signal Types . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Signal Characteristics . . . . . . . . . . . . . . . . . . . . . 71.4 Signal Transformations . . . . . . . . . . . . . . . . . . . . . 101.5 Special Signals . . . . . . . . . . . . . . . . . . . . . . . . . 141.6 Complex Variables . . . . . . . . . . . . . . . . . . . . . . . 171.7 Singularity functions . . . . . . . . . . . . . . . . . . . . . . 231.8 Comb and Replication Operations . . . . . . . . . . . . . . 272 Systems 292.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 System Properties . . . . . . . . . . . . . . . . . . . . . . . 322.4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.5 Frequency Response . . . . . . . . . . . . . . . . . . . . . . 443 Fourier Analysis 513.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Continuous-Time Fourier Series (CTFS) . . . . . . . . . . . 513.3 Continuous-Time Fourier Transform . . . . . . . . . . . . . 573.4 Discrete-Time Fourier Transform . . . . . . . . . . . . . . . 704 Sampling 794.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Analysis of Sampling . . . . . . . . . . . . . . . . . . . . . . 794.3 Relation between CTFT and DTFT . . . . . . . . . . . . . 874.4 Sampling Rate Conversion . . . . . . . . . . . . . . . . . . . 9034 CONTENTS5 Z Transform (ZT) 995.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Derivation of the Z Transform . . . . . . . . . . . . . . . . . 995.3 Convergence of the ZT . . . . . . . . . . . . . . . . . . . . . 1035.4 ZT Properties and Pairs . . . . . . . . . . . . . . . . . . . . 1075.5 ZT,Linear Constant Coecient Dierence Equations . . . . 1085.6 Inverse Z Transform . . . . . . . . . . . . . . . . . . . . . . 1145.7 General Form of Response of LTI Systems . . . . . . . . . . 1246 Discrete Fourier Transform (DFT) 1296.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . 1296.2 Derivation of the DFT . . . . . . . . . . . . . . . . . . . . . 1296.3 DFT Properties and Pairs . . . . . . . . . . . . . . . . . . . 1326.4 Spectral Analysis Via the DFT . . . . . . . . . . . . . . . . 1346.5 Fast Fourier Transform (FFT) Algorithm . . . . . . . . . . 1376.6 Periodic Convolution . . . . . . . . . . . . . . . . . . . . . . 1427 Digital Filter Design 1477.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . 1477.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.3 Finite Impulse Response Filter Design . . . . . . . . . . . . 1547.4 Innite Impulse Response Filter Design . . . . . . . . . . . 1658 Random Signals 1778.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . 1778.2 Single Random Variable . . . . . . . . . . . . . . . . . . . . 1778.3 Two Random Variables . . . . . . . . . . . . . . . . . . . . 1818.4 Sequences of Random Variables . . . . . . . . . . . . . . . . 1848.5 Estimating Distributions . . . . . . . . . . . . . . . . . . . . 1918.6 Filtering of Random Sequences . . . . . . . . . . . . . . . . 1958.7 Estimating Correlation . . . . . . . . . . . . . . . . . . . . . 203Chapter 1Signals1.1 Chapter OutlineIn this chapter, we will discuss:1. Signal Types2. Signal Characteristics3. Signal Transformations4. Special Signals5. Complex Variables6. Singularity Functions7. Comb and Replication Operators56 CHAPTER 1. SIGNALS1.2 Signal TypesThere are three dierent types of signals:1. Continuous-time (CT)2. Discrete-time (DT)3. Digital (discrete-time and discrete-amplitude)The dierent types of signals are shown in Figure 1.2(a) below.Figure 1.2(a) Dierent types of signals.Comments The CT signal need not be continuous in amplitude The DT signal is undened between sampling instances Levels need not be uniformly spaced in a digital signal We make no distinction between discrete-time and digital signals intheory The independent variable need not be time1.3. SIGNAL CHARACTERISTICS 7 x(n) may or may not be sampled from an analog waveform, i.e.xd(n) = xa(nT) T- sampling intervalThroughout this book, we will use the subscripts d and a onlywhen necessary for clarity.Equivalent Representations for DT signalsFigure 1.2(b) Equivalent representations for DT signals.1.3 Signal CharacteristicsSignals can be:1. Deterministic or random2. Periodic or aperiodicFor periodic signals, x(t) = x(t +nT), for some xed T and for allintegers n and times t.3. Right-sided, left-sided, or two-sidedFor right-sided signals, x(t) = 0 only when t t0.If t0 0, the signal is causal.For left-sided signals, x(t) = 0 only when t t0.If t0 0, the signal is anticausal.For 2-sided or mixed causal signals, x(t) = 0 for some t &lt; 0, andx(t) = 0 for some t &gt; 0.4. Finite or innite durationFor nite duration signals, x(t) = 0 only for &lt; t1 t t2 &lt; .8 CHAPTER 1. SIGNALSMetricsMetrics CT signal DT signalEnergy Ex = _ |x(t)|2dt Ex =</p> <p>n|x(n)|2Power Px = limT12T_ TT|x(t)|2dt Px = limN12N + 1N</p> <p>n=N|x(n)|2Magnitude Mx = maxt|x(t)| Mx = maxn|x(n)|Area Ax = _ |x(t)|dt Ax =</p> <p>n|x(n)|Average value xavg = limT12T_ TTx(t)dt xavg = limN12N + 1N</p> <p>n=Nx(n)Comments If signal is periodic, we need only average over one period, e.g. ifx(n)=x(n+N) for all nPx = 1Nn0+N1</p> <p>n=n0|x(n)|2, for any n0 root-mean-square (rms) valuexrms = _PxExamplesExample 1Figure 1.3(a) Signal for Example 1.x(t) =_ e(t+1)/2, t &gt; 10, t &lt; 11.3. SIGNAL CHARACTERISTICS 9The signal is:1. deterministic2. aperiodic3. right-sided4. mixed causalMetrics1. Ex = _1|e(t+1)/2|2dt, let s = t + 1Ex = _0 esds = es|0 = 12. Px = 03. Mx = 14. Ax = 25. xavg = 0Example 2Figure 1.3(b) Signal for Example 2.x(n) = Acos(n/4)The signal is:1. deterministic2. periodic (N = 8)10 CHAPTER 1. SIGNALS3. 2-sided4. mixed causalMetrics1. Ex = 2. Px = 187</p> <p>n=0|Acos(n/4)|2Px = A2167</p> <p>n=0[1 + cos(n/2)]Px = A223. xrms = A24. Mx = A5. Ax is undened6. xavg = 01.4 Signal TransformationsContinous-Time CaseConsider1. Reectiony(t) = x(t)1.4. SIGNAL TRANSFORMATIONS 112. Shiftingy(t) = x(t t0)3. Scalingy(t) = x(ta)4. Scaling and ShiftingExample 1y(t) = x(ta t0)What does this signal look like?center of pulse: ta t0 = 0 t = at012 CHAPTER 1. SIGNALSleft edge of pulse: ta t0 = 12 t = a(t0 12)right edge of pulse: ta t0 = 12 t = a(t0 + 12)Example 2y(t) = x_tt0a_What does this signal look like?center: (t t0)/a = 0 t = t0left edge of pulse: (t t0)/a = 1/2 t = t0a/2right edge of pulse: (t t0)/a = 1/2 t = t0 +a/2Discrete-Time CaseConsider1.4. SIGNAL TRANSFORMATIONS 131. Reectiony(n) = x(n)2. Shiftingy(n) = x(n n0)Note that n0 must be an integer. Non-integer delays can only bedened in the context of a CT signal, and are implemented byinterpolation.3. Scalinga. downsamplery(n) = x(Dn), D Zb. upsamplery(n) =_ x(n/D), n/D Z, D Z0, else14 CHAPTER 1. SIGNALS1.5 Special SignalsUnit stepa. CT caseu(t) =_ 1, t &gt; 00, t &lt; 0b. DT caseu(n) =_ 1, n 00, elseRectanglerect(t) =_ 1, |t| &lt; 1/20, |t| &gt; 1/21.5. SPECIAL SIGNALS 15Triangle(t) =_ 1 |t|, |t| 10, elseSinusoidsa. CTxa(t) = Asin(at +)where A is the amplitudea is the frequency is the phaseAnalog frequencya_radianssec_ = 2fa_cyclessec_16 CHAPTER 1. SIGNALSPeriodT_ seccycle_ = 1fab. DTxd(n) = Asin(dn +)digital frequencyd_radianssample_ = 2_ cyclessample_Comments1. Depending on , xd(n) may not look like a sinusoid2. Digital frequencies 1 = 0 and 2 = 0 + 2k are equivalentx2(n) = Asin(2n +)= Asin[(0 + 2k)n +]= x1(n), for all n3. xd(n) will be periodic if and only if d = 2(p/q) where p and q areintegers. In this case, the period is q.1.6. COMPLEX VARIABLES 17Sincsinc(t) = sin(t)(t)1.6 Complex VariablesDenitionCartesian coordinatesz = (x, y)Polar coordinatesz = RR = _x2+y2x = Rcos ()18 CHAPTER 1. SIGNALS = arctan_yx_y = Rsin ()Alternate representationdene j = (0, 1) = 1(/2)then z = x +jyAdditional notationRe {z} = xIm{z} = y|z| = RAlgebraic OperationsAddition (Cartesian Coordinates)z1 = x1 +jy1z2 = x2 +jy2z1 +z2 = (x1 +x2) +j(y1 +y2)1.6. COMPLEX VARIABLES 19Multiplication (Polar Coordinates)z1 = R11z2 = R22z1z2 = R1R2(1 +2)Special Examples1. 1 = (1, 0) = 102. 1 = (1, 0) = 11803. j2= (190)2= 1180 = 120 CHAPTER 1. SIGNALSComplex Conjugatez = x jy = R()Some useful identitiesRe {z} = 12[z +z]Im{z} = 1j2[z z]|z| =zzComplex ExponentialTaylor series for exponential function of real variable xex=k=0</p> <p>xkk!Replace x by jej= 1 +j + j22! + j33! + j44! + j55!= (1 22! + 44! ...) +j( 33! + 55! ...)= cos +j sin 1.6. COMPLEX VARIABLES 21|ej| =_cos2 + sin2 = 1ej= arctan_sin cos _ = Alternate form for polar coordinate representation of complex numberz = R = RejMultiplication of two complex numbers can be done using rules formultiplication of exponentials:z1z2 = (R1ej1)(R2ej2) = R1R2ej(1+2)Complex exponential signalCT x(t) = Aej(t)(t) = at + instantaneous phased(t)dt = a_radsec_ instantaneous phasor velocity22 CHAPTER 1. SIGNALSDT x(n) = Aej(dn+)d is the phasor velocity in_radianssample_.Exampled = /3 = /6Aliasing1. It refers to the ability of one frequency to mimic another2. For CT case, ej1t= ej2tfor all t 1 = 23. In DT, ej1n= ej2nif 2 = 1 + 2k1.7. SINGULARITY FUNCTIONS 23Consider1 = 76 2 = 56x1(n) = ej1nx2(n) = ej2nIn general, if 1 = + and 2 = ( ), thenx1(n) = ej1n= ej(+)n= ej[(+)n2n]= ej[(+)n]= ej()n= x2(n)1.7 Singularity functionsCT Impulse FunctionLet (t) = 1rect( t)Note that_ (t)dt = 1.What happens as 0?1 &gt; 2 &gt; 324 CHAPTER 1. SIGNALSIn the limit, we obtain(t) = lim0(t)(t) =_ 0, t = 0, t = 0and_ (t)dt = 1We will use impulses in three ways:1. to sample signals2. as a way to decompose signals into elementary components3. as a way to characterize the response of a class of systems to anarbitrary input signalSifting propertyConsider a signal x(t) multiplied by (t t0)for some xed t0:1.7. SINGULARITY FUNCTIONS 25From the mean value theorem of calculus,_ x(t)(t t0)dt = x()for some which satisest0 2 t0 + 2As 0, t0 and x() x(t0)So we have_ x(t)(t t0)dt = x(t0)provided x(t) is continuous at t0.EquivalenceBased on sifting property,x(t)(t t0) x(t0)(t t0)Indenite integral of (t)Let u(t) =_ t()d26 CHAPTER 1. SIGNALSLet 0,u(t) =_ t()dWhen we dene the unit step function, we did not specify its value att=0. In this case, u(0) = 0.5More general form of sifting property_ bax()( t0)d =_ x(t0), a &lt; t0 &lt; b0, t0 &lt; a or b &lt; t0DT Impulse Function (unit sample function)(n) =_ 1, n = 00, else1.8. COMB AND REPLICATION OPERATIONS 27Sifting Propertyn2</p> <p>n=n1x(n)(n n0) =_ x(n0), n1 n0 n20, elseEquivalencex(n)(n n0) = x(n0)(n n0)Indenite sum of (n)u(n) =n</p> <p>m=(m)1.8 Comb and Replication OperationsSifting property yields a single sample at t0. Consider multiplying x(t) byan entire train of impulses:Denexs(t) = combT[x(t)]= x(t)</p> <p>n(t nT)=</p> <p>nx(t)(t nT)=</p> <p>nx(nT)(t nT)28 CHAPTER 1. SIGNALSArea of each impulse = value of sample there.The replication operator similarly provides a compact way to expressperiodic signals:xp(t) = repT[x0(t)] =</p> <p>nx0(t nT)Note that x0(t) = xp(t)rect_ tT_Chapter 2Systems2.1 Chapter OutlineIn this chapter, we will discuss:1. System Properties2. Convolution3. Frequency Response2.2 SystemsA system is a mapping which assigns to each signal x(t) in the inputdomain a unique signal y(t) in the output range.2930 CHAPTER 2. SYSTEMSExamples1. CT y(t) =_ t+1/2t1/2x()dlet x(t) = sin(2ft)y(t) =_ t+1/2t1/2sin(2ft)dt= 12f cos(2ft)|t+1/2t1/2= 12f {cos[2f(t + 1/2)] cos[2f(t 1/2)]}use cos( ) = cos cos sin sin y(t) = 12f {[cos(2ft) cos(f) sin(2ft) sin(f)][cos(2ft) cos(f) + sin(2ft)sin(f)]}y(t) = sin(f)f sin(2ft)= sinc(f)sin(2ft)Look at particular values of ff = 0 : x(t) 0 y(t) 0f = 1 : x(t) = sin(2t)sinc(1) = 0y(t) 02.2. SYSTEMS 312. DT y(n) = 13[x(n) +x(n 1) +x(n 2)]Eect of lter on output:1. widened,smoothed, or smeared each pulse2. delayed pulse train by one sample time32 CHAPTER 2. SYSTEMS2.3 System PropertiesNotation:y(t) = S[x(t)]A. Linearitydef. A system S is linear(L) if for any two inputs x1(t) and x2(t) and anytwo constants a1 and a2, it satises:S[a1x1(t) +a2x2(t)] = a1S[x1(t)] +a2S[x2(t)]Special cases:1. homogeneity (let a2 = 0)S[a1x1(t)] = a1S[x1(t)]2. superposition (let a1 = a2 = 1)S[x1(t) +x...</p>

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