Applications of DSP Imaging Medical Imaging Bandwidth compression sGraphic Spectrum Analysis Array Processors Control and Guidance Radar

Reason for Processing of signals Signals are carriers of information Useful and unwanted Extracting, enhancing, storing and transmitting the useful information How signals are being processed?--- Analog Signal Processing Digital Signal Processing

DSP PrF: antialiasing filtering PoF: smooth out the staircase waveform

Comparison of DSP over ASP -Advantages Developed Using Software on Computer; Working Extremely Stable; Easily Modified in Real Time ; Low Cost and Portable; Flexible

Comparison of DSP over ASP Contd… -Disdvantages Lower Speed and Lower Frequency Can not be used at Higher frequency Skilled manpower is required Weak Signals can not be able to process

The two categories of DSP Tasks Signal Analysis: Measurement of signal properties Spectrum(frequency/phase) analysis Target detection, verification, recognition Signal Filtering Signal-in-signal-out, filter Removal of noise/interference Separation of frequency bands

Digital 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)

Basic Filter Types Low pass filters High Pass filters Band pass filters Band reject filters

Digital Filter Specification 4 Types

Digital Filter Specification Contd… The magnitude response specifications are given some acceptable tolerances.

Digital Filter Specification Contd… Transition band is specified between the passband and the stopband to permit the magnitude to drop off smoothly. In Passband In Stopband Where δp and δs are peak ripple values, ωp are passband edge frequency and ωs are stopband edge frequency

Digital 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 filter Peak passband ripple, αp = -20 log10 (1 – δp) dB Minimum stopband attenuation, αs = -20 log10 (δs) dB

Digital Filter Specification Contd… The magnitude response specifications may be given in a normalized form.

Digital Filter Specification Contd… In practice, passband edge frequency and stopband edge frequency are specified in Hz For digital filter design, normalized bandedge frequencies need to be computed from specifications in Hz using

Digital Filter Specification Contd… Example - Let kHz, kHz, and kHz Then

Digital Filter Type Objective of digital filter design is to develop a causal transfer function meeting the frequency response specification. For IIR digital filter design

Digital Filter Type Contd… For FIR digital filter design The degree N of H(z) must be small, for a linear phase, FIR filter coefficient must satisfy the constraint

FIR FILTERS

FIR Filter Design by Window function technique Simplest FIR the filter design is window function technique An ideal frequency response may express where

FIR 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!

FIR Filter Design by Window function technique Contd… 1.Rectangular window 2.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

FIR Filter Design by Window function technique Contd… 1.Rectangular window 2.Triangular window (Bartett window)

FIR Filter Design by Window function technique Contd… 3.HANN window 4.Hamming window 1.In fact, the length of window is M-1 2.main lobe和HANN差不多但side lobe降了10dB 3.Hamming 常用在語音處理

FIR Filter Design by Window function technique Contd… 3.HANN window 4.Hamming window

FIR Filter Design by Window function technique Contd… 5.Kaiser’s window 6.Blackman window 有參數可調,能得適當的組合

FIR Filter Design by Window function technique Contd… 5.Kaiser’s window 6.Blackman window

Window Table Type of the window Transition Bandwidth Minor Lobe attenuation in dB Rectangular 4π/M -21 Triangu;ar 8π/M -26 Hanning -44 Hamming -53 Blackmann 12π/M -74 Kaiser variable

Filter Design by Windowing Simplest way of designing FIR filters Method is all discrete-time no continuous-time involved Start with ideal frequency response Choose ideal frequency response as desired response Most ideal impulse responses are of infinite length The easiest way to obtain a causal FIR filter from ideal is More generally 351M Digital Signal Processing

Rectangular Window Narrowest main lob 4/(M+1) Sharpest transitions at discontinuities in frequency Large side lobs -21 dB Large oscillation around discontinuities Simplest window possible

Bartlett (Triangular) Window Medium main lob 8/M Side lobs -25 dB Hamming window performs better Simple equation

Hanning Window Medium main lob 8/M Side lobs -44 dB Hamming window performs better Same complexity as Hamming

Hamming Window Medium main lob Good side lobs Simpler than Blackman -53 dB Simpler than Blackman

Blackman Window Large main lob Very good side lobs Complex equation -73 dB Complex equation

Lowpass filter Desired frequency response Corresponding impulse response

Highpass filter Corresponding impulse response

Bandpass Filter The Impulse Response is

Bandreject Filter The Impulse Response is

FIR Filter Design Procedure Step1:Draw the response of the given problem. Step2:Convert the Analog frequencies in to the Digital frequencies Step3: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.

FIR Filter Design Procedure Contd… Step5:Calculate the Ʈ parameter Ʈ =(M-1)/2 Step6:Choose the Window to be used by considering the attenuation. Step7:Calculate ht(n) Step8:Calculate w(n) for the choosen window.

FIR 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.

Table: Frequency Response Ѡ Ø Attenuation 20log 0.2π 0.4π 0.6π 0.8π π

Kaiser Window Filter Design Method Parameterized equation forming a set of windows Parameter to change main-lob width and side-lob area trade-off I0(.) represents zeroth-order modified Bessel function of 1st kind

Determining Kaiser Window Parameters Given filter specifications Kaiser developed empirical equations Given the peak approximation error  or in dB as A=-20log10  and transition band width The shape parameter  should be The filter order M is determined approximately by After the kaiser window design follow the same procedure for the filter design

IIR FILTER

IIR Filter Design The transfer function of the IIR Filters will be of the form

Commonly used analog IIR filters Butterworth filter Chebyshev filters

Butterworth filters It is governed by the magnitude squared response

Butterworth filters-Properties The response is maximally flat at the origin Magnitude square is having a value of 0.5 at the cutoff frequency It is a monotonically decreasing function beyond the cutoff frequency.

Butterworth Polynomial Order Butterworth polynomial 1 S+1 2 S2+√2 S+1 3 (S2+S+1)(S+1) This Polynomial may be obtained by finding the roots for n is odd and even Then by considering the left half side poles the butterworth polynomial may be constructed

Butterworth 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

Analog Transformation Filter Type Normalised Response Analog Transformation Actual Response Backward Equation Low pass filter S=S/ΩC ΩS=Ω2/Ω1 High Pass Filter S=ΩC /S

Chebyshev filter design- Some Prelimnaries Relative linear scale The lowpass filter specifications on the magnitude-squared response are given by Where 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.

Analog Filter response

Design Procedure for chebyshev filters Step1: Calculate the order of the filter \Where g-[(A2-1)/ϵ2]1/2 Ωr= Ω2/Ω1 n=log[(g+(g2-1)1/2 ]/log(Ωr/(Ωr2-1)1/2 A=10-K2/20 Step2:Obtained the normalised transfer function Hn(s)=k/(Sn+bn-I Sn-1+…+b1S+b0)

Analog to Digital Conversion Impulse Invariance Transformation Bilinear Transformation

Impulse Invariance method The most straightforward of these is the impulse invariance transformation Let be the impulse response corresponding to , and define the continuous to discrete time transformation by setting We sample the continuous time impulse response to produce the discrete time filter

Impulse Invariance method contd… The impulse invariance transformation does map the -axis and the left-half s plane into the unit circle and its interior, respectively Re(Z) Im(Z) 1 S domain Z domain

Impulse invariance method contd… is expanded a partial fraction expansion to produce We have assumed that there are no multiple poles And thus The 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)

Impulse invariance method contd… Hence it is sufficient if we substitute

Impulse invariance method contd… Example: Expanding in a partial fraction expansion, it produce The impulse invariant transformation yields a discrete time design with the system function

Bilinear transformation method The most generally useful is the bilinear 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.

Bilinear transformation method Contd… We could first use a one-to-one transformation from to , which compresses the entire s plane into the strip Then could be transformed to z by with 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 domain s’ domain

Bilinear transformation method Contd… Hence by using this equation a digital transfer function may be obtained

Bilinear transformation method Contd… The discrete-time filter design is obtained from the continuous-time design by means of the bilinear transformation Unlike the impulse invariant transformation, the bilinear transformation is one-to-one, and invertible. Feb.2008 DISP Lab

Filter Realization

Filter Structures Direct form I Direct form II Cascaded form Parallel form

Basic elements of digital filter structures Adder has two inputs and one output. Multiplier (gain) has single-input, single-output. Delay element delays the signal passing through it by one sample. It is implemented by using a shift register. a Block diagram Signal-flow graph z-1 a z-1 Copyright © 2005. Shi Ping CUC

Copyright © 2005. Shi Ping CUC a1 z-1 a2 b0 a1 a2 b0 z-1 1 2 5 3 4 Copyright © 2005. Shi Ping CUC

Introduction The major factors that influence the choice of a specific structure Computational complexity refers to the number of arithmetic operations (multiplications, divisions, and additions) required to compute an output value y(n) for the system. Memory requirements refers 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 computations refers to the quantization effects that are inherent in any digital implementation of the system, either in hardware or in software.

IIR Filter Structures The characteristics of the IIR filter IIR filters have Infinite-duration Impulse Responses The system function H(z) has poles in 可参考教材P75～76 The order of such an IIR filter is called N if aN≠0

Direct form In 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

Copyright © 2005. Shi Ping CUC Direct form I b1 b2 b0 z-1 bM-1 bM a1 a2 z-1 aN-1 aN Copyright © 2005. Shi Ping CUC

Copyright © 2005. Shi Ping CUC Direct form II For an LTI cascade system, we can change the order of the systems without changing the overall system response b1 b2 b0 z-1 bM-1 bM a1 a2 aN-1 aN z-1 直接II型比直接I型节省存储单元（软件实现），或节省寄存器（硬件实现） 但不论是直接I型还是直接II型，其共同的缺点是系数对滤波器的性能控制作用不明显，这是因为它们与系统函数的零极点关系不明确因而调整困难。 另外，这两种结构极点对系数的变化过于敏感，从而使系统频率响应对系数的变化过于灵敏，也就是对有限精度（有限字长）运算过于灵敏，容易出现不稳定或产生较大误差。 Copyright © 2005. Shi Ping CUC

Copyright © 2005. Shi Ping CUC Cascade form In this form the system function H(z) is written as a product of second-order sections with real coefficients Copyright © 2005. Shi Ping CUC

Parallel form Structures In this form the system function H(z) is written as a sum of sections using partial fraction expansion. Each section is implemented in a direct form. The entire system function is implemented as a parallel of every section. Suppose M=N 当M<N，但M不等于N时，见教材P207（式5－7） Copyright © 2005. Shi Ping CUC

Copyright © 2005. Shi Ping CUC Example Copyright © 2005. Shi Ping CUC

Copyright © 2005. Shi Ping CUC -12.9 z-1 7/8 -3/32 -14.75 26.82 1 -1/2 24. 5 并联型可以用调整a1k和a2k的方法来单独调整一对极点的位置，但是不能单独调整零点的位置。 此外，并联型结构中，各并联基本节的误差相互没有影响，所以比级联型的误差一般来说要稍小一些，因此在要求准确的传输零点的场合下，宜采用级联型结构。 Copyright © 2005. Shi Ping CUC

Conclusions Discussed about the FIR filter design IIR Filter design Realization of structures

References [1]B. Jackson, Digital Filters and Signal Processing, Kluwer Academic Publishers 1986 [2]Dr. DePiero, Filter Design by Frequency Sampling, CalPoly State University [3]W.James MacLean, FIR Filter Design Using Frequency Sampling [5]Maurice G.Bellanger, Adaptive Digital Filters second edition, Marcel dekker 2001 Feb.2008 DISP Lab

References [6] Lawrence R. Rabiner, Linear Program Design of Finite Impulse Response Digital Filters, IEEE 1972 [7] Terrence J mc Creary, On Frequency Sampling Digital Filters, IEEE 1972 WWW.GOOGLE.COM Feb.2008 DISP Lab

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