Presentation on theme: "Applications of DSP Imaging Medical Imaging Bandwidth compression"— Presentation transcript:
1Applications of DSP Imaging Medical Imaging Bandwidth compression sGraphicSpectrum AnalysisArray ProcessorsControl and GuidanceRadar
2Reason for Processing of signals Signals are carriers of informationUseful and unwantedExtracting, enhancing, storing and transmitting the useful informationHow signals are being processed?---Analog Signal ProcessingDigital Signal Processing
3DSPPrF: antialiasing filteringPoF: smooth out the staircase waveform
4Comparison of DSP over ASP -AdvantagesDeveloped Using Software on Computer;Working Extremely Stable;Easily Modified in Real Time ;Low Cost and Portable;Flexible
5Comparison of DSP over ASP Contd… -DisdvantagesLower Speed and Lower FrequencyCan not be used at Higher frequencySkilled manpower is requiredWeak Signals can not be able to process
6The two categories of DSP Tasks Signal Analysis:Measurement of signal propertiesSpectrum(frequency/phase) analysisTarget detection, verification, recognitionSignal FilteringSignal-in-signal-out, filterRemoval of noise/interferenceSeparation of frequency bands
7Digital Filter Specification Digital Filter designed to pass signal components of certain frequencies without distortion.The frequency response should be equal to the signal’s frequencies to pass the signal. (passband)The frequency response should be equal to zero to block the signal. (stopband)
8Basic Filter Types Low pass filters High Pass filters Band pass filtersBand reject filters
10Digital Filter Specification Contd… The magnitude response specifications are given some acceptable tolerances.
11Digital Filter Specification Contd… Transition band is specified between the passband and the stopband to permit the magnitude to drop off smoothly.In PassbandIn StopbandWhere δp and δs are peak ripple values, ωp are passband edge frequency and ωs are stopband edge frequency
12Digital Filter Specification Contd… Digital filter specification are often given in terms of loss function,A(ω) = -20 log10 |G(ejω)|Loss specification of a digital filterPeak passband ripple, αp = -20 log10 (1 – δp) dBMinimum stopband attenuation, αs = -20 log10 (δs) dB
13Digital Filter Specification Contd… The magnitude response specifications may be given in a normalized form.
14Digital Filter Specification Contd… In practice, passband edge frequency and stopband edge frequency are specified in HzFor digital filter design, normalized bandedge frequencies need to be computed from specifications in Hz using
15Digital Filter Specification Contd… Example - Let kHz, kHz, and kHzThen
16Digital Filter TypeObjective of digital filter design is to develop a causal transfer function meeting the frequency response specification.For IIR digital filter design
17Digital Filter Type Contd… For FIR digital filter designThe degree N of H(z) must be small, for a linear phase, FIR filter coefficient must satisfy the constraint
19FIR Filter Design by Window function technique Simplest FIR the filter design is window function techniqueAn ideal frequency response may expresswhere
20FIR Filter Design by Window function technique Contd… To get this kind of systematic causal FIR to be approximate, the most direct method intercepts its ideal impulse response!
21FIR Filter Design by Window function technique Contd… 1.Rectangular window2.Triangular window (Bartett window)1.main lobe越窄 resolution越高 side lobe 越低越好2.統計上常用 resolution降一半 main lobe 變寬(trade off)Main lobe變寬(trade off) side lobe降一半
22FIR Filter Design by Window function technique Contd… 1.Rectangular window2.Triangular window (Bartett window)
23FIR Filter Design by Window function technique Contd… 3.HANN window4.Hamming window1.In fact, the length of window is M-12.main lobe和HANN差不多但side lobe降了10dB3.Hamming 常用在語音處理
24FIR Filter Design by Window function technique Contd… 3.HANN window4.Hamming window
25FIR Filter Design by Window function technique Contd… 5.Kaiser’s window6.Blackman window有參數可調,能得適當的組合
26FIR Filter Design by Window function technique Contd… 5.Kaiser’s window6.Blackman window
27Window Table Type of the window Transition Bandwidth Minor Lobe attenuation in dBRectangular4π/M-21Triangu;ar8π/M-26Hanning-44Hamming-53Blackmann12π/M-74Kaiservariable
28Filter Design by Windowing Simplest way of designing FIR filtersMethod is all discrete-time no continuous-time involvedStart with ideal frequency responseChoose ideal frequency response as desired responseMost ideal impulse responses are of infinite lengthThe easiest way to obtain a causal FIR filter from ideal isMore generally351M Digital Signal Processing
29Rectangular Window Narrowest main lob 4/(M+1) Sharpest transitions at discontinuities in frequencyLarge side lobs-21 dBLarge oscillation around discontinuitiesSimplest window possible
30Bartlett (Triangular) Window Medium main lob8/MSide lobs-25 dBHamming window performs betterSimple equation
31Hanning Window Medium main lob 8/M Side lobs -44 dB Hamming window performs betterSame complexity as Hamming
32Hamming Window Medium main lob Good side lobs Simpler than Blackman -53 dBSimpler than Blackman
33Blackman Window Large main lob Very good side lobs Complex equation -73 dBComplex equation
34Lowpass filter Desired frequency response Corresponding impulse response
38FIR Filter Design Procedure Step1:Draw the response of the given problem.Step2:Convert the Analog frequencies in to the Digital frequenciesStep3:Calculate the Transition Band width.Step4:Calculate the order of the filter by equating the calculated Transitions band width to the transition band width in the table.
39FIR Filter Design Procedure Contd… Step5:Calculate the Ʈ parameter Ʈ =(M-1)/2Step6:Choose the Window to be used by considering the attenuation.Step7:Calculate ht(n)Step8:Calculate w(n) for the choosen window.
40FIR Filter Design Procedure Contd… Step9:Then calculate h(n)=ht(n) x w(n)Step10: For verifying the design use the equation for calculating the magnitude response and the frequency response.
41Table: Frequency Response ѠØAttenuation20log0.2π0.4π0.6π0.8ππ
42Kaiser Window Filter Design Method Parameterized equation forming a set of windowsParameter to change main-lob width and side-lob area trade-offI0(.) represents zeroth-order modified Bessel function of 1st kind
43Determining Kaiser Window Parameters Given filter specifications Kaiser developed empirical equationsGiven the peak approximation error or in dB as A=-20log10 and transition band widthThe shape parameter should beThe filter order M is determined approximately byAfter the kaiser window design follow the same procedure for the filter design
45IIR Filter DesignThe transfer function of the IIR Filters will be of the form
46Commonly used analog IIR filters Butterworth filterChebyshev filters
47Butterworth filtersIt is governed by the magnitude squared response
48Butterworth filters-Properties The response is maximally flat at the originMagnitude square is having a value of 0.5 at the cutoff frequencyIt is a monotonically decreasing function beyond the cutoff frequency.
49Butterworth Polynomial OrderButterworth polynomial1S+12S2+√2 S+13(S2+S+1)(S+1)This Polynomial may be obtained by finding the roots for n is odd and evenThen by considering the left half side poles the butterworth polynomial may be constructed
50Butterworth filter design Step1:Find the order of the filter n=log[(10(k1/10)-1)/ (10(k2/10)-1)]/2log(Ω1/Ω2) Step2:Obtain the normalised transfer function Hn(s)=1/Bn(s) Step3:By substituting the value of s from the analog transformation Table the actual filter transfer function may be obtained
52Chebyshev filter design- Some Prelimnaries Relative linear scaleThe lowpass filter specifications on the magnitude-squared response are given byWhere epsilon is a passband ripple parameter, Omega_p is the passband cutoff frequency in rad/sec, A is a stopband attenuation parameter, and Omega_s is the stopband cutoff in rad/sec.
54Design Procedure for chebyshev filters Step1: Calculate the order of the filter\Whereg-[(A2-1)/ϵ2]1/2Ωr= Ω2/Ω1n=log[(g+(g2-1)1/2 ]/log(Ωr/(Ωr2-1)1/2A=10-K2/20Step2:Obtained the normalised transfer function Hn(s)=k/(Sn+bn-I Sn-1+…+b1S+b0)
56Analog to Digital Conversion Impulse Invariance TransformationBilinear Transformation
57Impulse Invariance method The most straightforward of these is the impulse invariance transformationLet be the impulse response corresponding to , and define the continuous to discrete time transformation by settingWe sample the continuous time impulse response to produce the discrete time filter
58Impulse Invariance method contd… The impulse invariance transformation does map the -axis and the left-half s plane into the unit circle and its interior, respectivelyRe(Z)Im(Z)1S domainZ domain
59Impulse invariance method contd… is expanded a partial fraction expansion to produceWe have assumed that there are no multiple polesAnd thusThe impulse invariant transformation is not usually performed directly in the form of (2.1)the parameters of H(z) may be obtained directly from H(s)
60Impulse invariance method contd… Hence it is sufficient if we substitute
61Impulse invariance method contd… Example:Expanding in a partial fractionexpansion, it produceThe impulse invariant transformationyields a discrete time design with thesystem function
62Bilinear transformation method The most generally useful is thebilinear transformation.To avoid aliasing of the frequency response as encountered with the impulse invariance transformation.We need a one-to-one mapping from the s plane to the z plane.The problem with the transformation is many-to-one.
63Bilinear transformation method Contd… We could first use a one-to-one transformation from to , which compresses the entire s plane into the stripThen could be transformed to z bywith no effect from aliasing.優點:To avoid aliasing of the frequency response缺點:It is nonlinear between discrete-time frequency and continuous-time frequency.s domains’ domain
64Bilinear transformation method Contd… Hence by using this equation a digital transfer function may be obtained
65Bilinear transformation method Contd… The discrete-time filter design is obtained from the continuous-time design by means of the bilinear transformationUnlike the impulse invariant transformation, the bilinear transformation is one-to-one, and invertible.Feb.2008DISP Lab
70IntroductionThe major factors that influence the choice of a specific structureComputational complexityrefers to the number of arithmetic operations (multiplications, divisions, and additions) required to compute an output value y(n) for the system.Memory requirementsrefers to the number of memory locations required to store the system parameters, past inputs, past outputs, and any intermediate computed values.Finite-word-length effects in the computationsrefers to the quantization effects that are inherent in any digital implementation of the system, either in hardware or in software.
71IIR Filter Structures The characteristics of the IIR filter IIR filters have Infinite-duration Impulse ResponsesThe system function H(z) has poles in可参考教材P75～76The order of such an IIR filter is called N if aN≠0
72Direct formIn this form the difference equation is implemented directly as given. There are two parts to this filter, namely the moving average part and the recursive part (or the numerator and denominator parts). Therefore this implementation leads to two versions: direct form I and direct form II structures
79Conclusions Discussed about the FIR filter design IIR Filter design Realization of structures
80ReferencesB. Jackson, Digital Filters and Signal Processing, Kluwer Academic Publishers 1986Dr. DePiero, Filter Design by Frequency Sampling, CalPoly State UniversityW.James MacLean, FIR Filter Design Using Frequency SamplingMaurice G.Bellanger, Adaptive Digital Filters second edition, Marcel dekker 2001Feb.2008DISP Lab
81References Lawrence R. Rabiner, Linear Program Design of Finite Impulse Response Digital Filters, IEEE 1972 Terrence J mc Creary, On Frequency Sampling Digital Filters, IEEE 1972Feb.2008DISP Lab