Dijital işaret işleme ve FIR filtre tasarımı algoritmaları
Digital signal processing and FIR filter design algorithms
- Tez No: 22033
- Danışmanlar: DOÇ. DR. MEHMET BÜLENT ÖRENCİK
- Tez Türü: Yüksek Lisans
- Konular: Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve Kontrol, Computer Engineering and Computer Science and Control
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 257
Özet
ÖZET Dijital i gar ©t işleme entegre ve bilgisayar teknolojisindeki gelişmelerin artması sonucu oldukça önem kazanmıştır. Önceleri işlem hızı yüzünden gerçek zamanda yapılamayan işlemler hem bu gelişmeler hemde etkin algoritmaların bulunması sonucu gerçeklenme şansı bulmuşlardır. Bu algoritmalardan en önemlisi fast Fourier algoritması, kısaca FFT dir. Bu algoritma ile çok hızlı şekilde four i er dönüşümleri yapılabilir. Ayrıca direkt metod ve Goertzel algoritmaları FFT kadar hızlı olmamakla birlikte daha basit ve hatta sınırlı birkaç durumda FFT ye göre tercih edilmektedir. Bu çalışmada ilk önce birinci bölümde sürekli ve ayrık zaman işaretleri anlatılmıştır. İkinci bölümde ayrık zaman four i er dönüşümleri ve hesaplama yolları karşılaştırmalı olarak açıklanmış ve hangi durumlarda tercih edilmesi ger keti ği verilmiştir. Dördüncü bölümde FIR filtrelerin genel özellikleri verildikten sonra beşinci bölümde FIR filtre tasarım algoritmaları incelenmiştir. Frekans örnekleme metodu, pencere kullanımı, yakınsama kriterleri » hata kriterleri » Remez algoritması ve Parks-McCellan genel amaçlı FIR tasarım algoritması v. s. incelenmiştir. Bunlarla ilgili program ve çıktılar ek. 1 ve ek. 2 de yer almıştır. Sonuç olarak genel amaçlı FIR filtre tasarım ve Dijital İşaret İşleme algoritmalarını gerçekleştiren esnek bir program paketi ortaya çıkmıştır. iv
Özet (Çeviri)
SUMMARY DIGITAL SIGNAL PROCESSING AND FIR FILTER DESIGN ALGORITHMS In this study, digital signal processing, digital filter design methods and their algorithms has been given. Discrete Fourier transform and Fast Fourier Transform and other subroutines were written with MicrosftC. The main concept of this study is computing DFT, FFT and Finite Impulse Respose filter design by means of a user friendly software package. Digital signal processing is a field which has its roots 18th century mathematics, has become an important modern tool in a multitude of diverse fields of science and technology. Until recently, signal processing has typically been carried out using analog equipments. Some exeptions to this were evident in the 1930s particulary in area where sophisticated signal processing was required. Because of flexibility of digital computers it was often useful to simulate a signal processing system on a digital computer befor implementing it in analog hardware. The vocoder simulation which were carried out in Lincoln and Bell Laboratories is a good example of such simulations. Despite the fact that flexibility of digital computer in signal processing, The processing can always no be done in real time. In keeping with that style, early work on digital filtering was very much concerned with ways in which a filter could be programmed on a digital computer so that with analog to digital conversion of the signal, followed by the digital filtering, followed by the digital to analog conversion, the overall system approximated a good analog filter. The evolution of a new point of wiew toward digital signal processing was further acclerated by the disclosure in 196S of an efficent algorithm for computation of the Fast Fourier Transform or FFT. Many signal processing algorithms which had been developed on digital computers required processing times several orders of magnitude greater than real time. Often this was tied to the facts that spectrum was an important component of signal processing and that no efficent means had been known for implementing it.The fast four i er transform algorithm reduced computation time of the Fourier transform by orders of magbitude. This permitted the implementation of increasingly sophisticated signal processing algorithms with processing times that allowed interaction with the system. Futhermore, with the realization that the fast Fourier transform algorithm might, in fact, be i mpl ementabl e in special purpose digital hardware, many signal processing algorithms which previously had appeared to be impractical began to appear to have practical implementations with special purpose digital hardware. Another important implication of the fast Fourier transform algorithm was tied to the fact that it was an inherently discrete time concept. It was directed toward the computation of the Fourier transform of a discrete time signal or sequence and involved a sert of properties and mathematics that were exact in discrete time domain. It was not simply an approximation to a conti nous -time Fourier transform. The importance of this was that it had the effect of stimulating a reformultion of many signal processing concepts and algorithms in terms of discrete-time mathematics and these techniques then formed an exact set of relationship in discrete time domain. The Discrete Fourier Trasnforms plays an important role in the analysis, design and the implementation of digital signal processing algorithms and systems. The used notation in the sections is consistent with the standart setup by the IEEE. The Fourier Series : The fourier series may be viewed a way to represent a conti nous time signal sCtD as a superposition of harmonically related sine and cosine waves. Instead of sine and cosine representation exponantiel representation is often used, especially when there are many harmonics in the fourier expansion. The set of complex coefficents in fourier transforms are call d as spectrum of signal.Discrete Time Signals : A discrete time signal is obtined from a continuous time signal by sampling at equally spaced time signal. If a discrete time signal is denoted by sn then the sampling of sCtD every T second gives snCtD= sCnTD and n =.. -1,-2,0,.. Amiguity C Aliasing!) when Sampling a Cosine Wave : If the samples of a cosine wave are obtained at a rate of five samples per second shown as sn=l, -. 81,. 31,. 31. -. 81,.31. İt is reasonable to assume a sampling of a S Hz signal gives the same samples when sampling a 3 Hz cosine. The usual convention used to relate a unique signal to a set of samples is to assume that the signal is a lowpass band limited signal. As long as the sampling rate is higher than twice the highest frequeny present in the signal, there is a unique continuous time signal associated with a discrete time signal, viewed as a set of samples. The phenomenon of having other names for continuous time signals with the same samples is called aliasing. Sampling Theorem : A band limited signal,with highest frequency B Hz, can be uniquely recovered from its samples as long as the sampling rate is higher than 2B samples per second. Lower sampling rates may bring on aliasing. Comparison of continious time and discrete time transforms: Fourier Series-DFT : These are signal representations using discrete »harmonically related frequencies. The discretefrequeny representation results in periodic time functions, both continuous time and discrete time. The spectrum of discrete time signal is periodic while the spectrum of continuous time signal is not. Fourier Transforms.: The forier transform of a continuous time signal and the Fourier transform of a discrete time signal are continuous frequency represantations of the corresponding signals. The transforms are function of the real frequency variable w. The spectrum of the continuous time signal is not. Convolution and Linear Systems : While Fourier techniques certainly aid signal analysis, they also paly an important role in the study of linear time invariant systems. The time domain description of a linear time invariant system with an input xCtD, unit impulse response hCtD and output yCtD given by the convolution, -co yCt> = S hCt-u3 xCuZ) du. CO Discrete Fourier Transform CDFT3 The spectrum of signal : A discrete time signal can be viewed as a finite length sequence of numbers such as xCnD = 1,. 6,2, -1, -. 4... ni nk xCnD = cosC2nrj/ND and DFTCxCnDD = XClc> = £ xCn3 W Inverse DFT CIDFTD: xCn3 = IDFTCXCkDZ) = IDFT C DFTCxCnDDD Calculation of DFT from a signal has equal diffculties with calculation of IDFT.Cyclic Convolution Property : A fundamental operation in linear digital signal processing is convolution. Aperiodic convolution between two signals xCnD and hCnD is denoted by, yCnD = xCnD * hCnI> Cyclic convolution is denoted by, yCnD = V xCm D hC C n-nû D, the residu of n modulo N = CrO N DFT C yCnD D = DFT C xCnDD DFT C hCn3D yCnD = IDFTC DFT CxCnDD DFT ChCnDD The Periodic property of xCnD and XCk> : Strictly speaking, xCnD is defined only over the range of n from O to N-l. However, the I DFT of the DFT of xCnD can be calculated Shifting and modulation properties ; When the signal xCnD is shifted in time, the spectrum XCkD is multiplied by a linear phase shift. Sampling property : The DFT of a subset of the original signal values can be found in terms of the DFT of the signal if the subset is vi ewed as a set of sampl es of the or i gi nal si gnal. Goertzel algorithm » direct method and the fast fourier transforms are considered while calculating DFT. Direct Method : An obvious way to calculate the DFT of a signal xCnD is to implement the definition of the DFT directly. This is done by computing the desired values of XCkD. In the case xCnD is complex the real and the imaginary part of xCnD are denoted by XCnD and YCnD, respectively and the real and imaginary parts of the transform by ACkD and BCkD.Goertzel Algorithm for the DFT : First order filter with the input being the data sequence in reverse order and the value of the polynomial at z being the solution sampled at M=N. Applying this to the DFT gives the Goertzel Algorithm. Decimation in time Fast Fourier Transforms : To achieve a dramatic increase in efficency, it i a necessary to decompose the DFT computation into succesively smaller DFT computations. İn this process both the symmeetry and the periodicity of the complex exponantial is used. Algorithms in which the decomposition is based on decomposing the sequence xCnD» into succesively smaller subsequences are called decimation in time algorithms. Decimation in frequency Fast Fourier Algirithms : The decimation in time FFT algorithms were all based upon the decomposition of the DFT computation by forming smaller and smaller subsequences of input sequence xCnD. Alternatively divideing the output sequence XCkD samal 1 er and smal 1 er subsequences i n the same manner. This class of FFT algorithms based on this procedure is commonly referred as decimation in frequency. Properties of FIR filters : Digital filters with a finite duration impulse response CFIRD have characteristic that make them useful in many applications. Because of the method of implementation, The FIR filters is also called as nonrecursive or convolution filters. The duration or sequence length of the impulse response of these filters is by definition finite therefore, the output can be written as a finite convolution sum : N-i yCnD= £ hCmD xCn-nu by changing index variable, n-N+i yCnD= £ hCn-nû xCnu rn=o xCnD = input,yCrü) = output, hCrD = length-N impulse response. 1. FIR filters are always stable. 2. FIR filters can be exactly phase linear. 3. Coefficients can easily be set to 1 part in 65,000 and never change. C Analog filter response can be change because of temperature. D 4. Stopband floors of -50 or -60 dB are much more quickly reached using FIR filters than using analog filters. 5. FIR filters can be made to track simply by altering the clock rate. An example of where this is important would be in detecting CopticallyD the watermark on a moving paper web. As the paper speed moves faster or slower, the input signal will be shifted in frequency. if an angle encoder is installed on the pinch rollers, the pulses from it could be used to clock the filter. Thus the filter will automatically track to shift. 6. FIR filters can be designed directly. 7. Complex filters with many bands of different gains, which are virtually impossible to design with analog technics are just as easy as to design for FIR filters as is simple low-pass filter. There are three definitions used by FIR filter design. Trasition width, passband ripple and stopband attenuation. Furthermore calculation of filter length is another problem to be considered in designing FIR filters. FIR filters is defined in frequency domain. Thar e are four types of FIR filters. Two of them are according to symmetry C even, odd D. Two of them are according to filter length C even, odd D. Frequency sampling, Least squared error, using a transition region and windows, approximation criterias C Remez, Chebyshev, inteqral squared error, squared error D are given technics and are often used in designing FIR filters. The best window approximation function is given by Kaiser. İn addition, A transition region is used to minimize overshoot. Window is used for el emi na ting sidelobs in the frequency response. Remez exchange an Parks-McCellan algorithms are an effective way to design for FIR filters such as multiband, lowpass, highpass, differentiators and hi 11 bert transformers.
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