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Applications of DSP 1.Imaging 2.Medical Imaging 3.Bandwidth compression 4.sGraphic 5.Spectrum Analysis 6.Array Processors 7.Control and Guidance 8.Radar

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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

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

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

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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

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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

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Digital Filter Specification Digital Filter designed to pass signal components of certain frequencies without distortion. The frequency response should be equal to the signals frequencies to pass the signal. (passband) The frequency response should be equal to zero to block the signal. (stopband)

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

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Digital Filter Specification 4 Types

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

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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 δ p and δ s are peak ripple values, ω p are passband edge frequency and ω s are stopband edge frequency

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Digital filter specification are often given in terms of loss function, A (ω) = -20 log 10 |G(e jω )| Loss specification of a digital filter – Peak passband ripple, α p = -20 log 10 (1 – δ p ) dB – Minimum stopband attenuation, α s = -20 log 10 (δ s ) dB Digital Filter Specification Contd…

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

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14 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…

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

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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

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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 Digital Filter Type Contd…

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FIR FILTERS

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

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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!

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

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

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

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

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FIR Filter Design by Window function technique Contd… 5.Kaisers window 6.Blackman window

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5.Kaisers window 6.Blackman window FIR Filter Design by Window function technique Contd…

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Type of the windowTransition Bandwidth Minor Lobe attenuation in dB Rectangular4π/M-21 Triangu;ar8π/M-26 Hanning8π/M-44 Hamming8π/M-53 Blackmann12π/M-74 Kaiservariable Window Table

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351M Digital Signal Processing 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

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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

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

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

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Hamming Window Medium main lob –8/M Good side lobs –-53 dB Simpler than Blackman

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Blackman Window Large main lob –12/M Very good side lobs –-73 dB Complex equation

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Lowpass filter Desired frequency response Corresponding impulse response

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Highpass filter Corresponding impulse response

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Bandpass Filter The Impulse Response is

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Bandreject Filter

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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.

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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…

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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.

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Table: Frequency Response Ѡ Ø Attenuation 20log 0 0.2π 0.4π 0.6π 0.8π π

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42 Kaiser Window Filter Design Method Parameterized equation forming a set of windows – Parameter to change main-lob width and side-lob area trade-off – I 0 (.) represents zeroth-order modified Bessel function of 1 st kind

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Determining Kaiser Window Parameters Given filter specifications Kaiser developed empirical equations – Given the peak approximation error or in dB as A=-20log 10 – 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

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IIR FILTER

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

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Commonly used analog IIR filters Butterworth filter Chebyshev filters

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Butterworth filters It is governed by the magnitude squared response

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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 filters-Properties

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Butterworth Polynomial Order Butterworth polynomial 1S+1 2S 2 +2 S+1 3(S 2 +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

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Butterworth filter design Step1:Find the order of the filter n=log[(10 (k1/10)- 1)/ (10 (k2/10)- 1)]/2log(1/2) 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

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Analog Transformation Filter Type Normalise d Response Analog Transformati on Actual Response Backward Equation Low pass filterS=S/ C S = 2 / 1 High Pass FilterS= C /S S = 2 / 1

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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. Chebyshev filter design- Some Prelimnaries

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Analog Filter response

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n=log[(g+(g 2 -1) 1/2 ]/log(r/(r 2 -1) 1/2 Design Procedure for chebyshev filters Step1: Calculate the order of the filter \Where g-[(A 2 -1)/ 2 ] 1/2 r= 2 / 1 A=10 -K2/20 Step2:Obtained the normalised transfer function H n (s)=k/(S n +b n-I S n-1 +…+b 1 S+b0)

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Analog to Digital Conversion Impulse Invariance Transformation Bilinear Transformation

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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

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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 domainZ domain

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

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

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

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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.

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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. s domain

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

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Feb.2008DISP Lab65 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. Bilinear transformation method Contd…

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Filter Realization

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

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Copyright © 2005. Shi Ping CUC 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. z -1 a a

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Copyright © 2005. Shi Ping CUC a1a1 z -1 a2a2 b0b0 a1a1 a2a2 b0b0 1 2 3 4 5

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Introduction 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. The major factors that influence the choice of a specific structure

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IIR Filter Structures The characteristics of the IIR filter IIR filters have Infinite-duration Impulse Responses The system function H(z) has poles in The order of such an IIR filter is called N if a N0

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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

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Copyright © 2005. Shi Ping CUC Direct form I b1b1 b2b2 b0b0 z -1 b M-1 z -1 bMbM a1a1 a2a2 a N-1 z -1 aNaN

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Copyright © 2005. Shi Ping CUC Direct form II b1b1 b2b2 b0b0 z -1 b M-1 z -1 bMbM a1a1 a2a2 a N-1 aNaN For an LTI cascade system, we can change the order of the systems without changing the overall system response

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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

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Copyright © 2005. Shi Ping CUC Parallel form Structures Parallel form 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

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Copyright © 2005. Shi Ping CUC Example

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Copyright © 2005. Shi Ping CUC -12.9 z -1 7/8 -3/32 z -1 -14.75 26.82 z -1 1 -1/2 z -1 24. 5

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

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Feb.2008DISP Lab80 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

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Feb.2008DISP Lab81 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

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