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Alt-uzay dönüşüm yöntemi ile Fır süzgeç tasarımı

Finite-duration impulse response filter design using subspace transformations

  1. Tez No: 66658
  2. Yazar: MEHMET DEVRİM AZAK
  3. Danışmanlar: PROF. DR. ALİ NUR GÖNÜLEREN
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1997
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Elektronik-Haberleşme Eğitimi Ana Bilim Dalı
  12. Bilim Dalı: Devreler ve Sistemler Bilim Dalı
  13. Sayfa Sayısı: 87

Özet

ÖZET Sonlu darbe yanıtlı süzgeçlerin tasarım yöntemleri, kapalı ve iteratif yöntemler adı altında iki kısma ayrılabilir. Kapalı yöntemlerin tasarımdaki yetkinsizliği, araştırmacıları iteratif yöntemlere yöneltmiş ve bu yöntemler içinde en az işlem gerektiren ve tasarım başarımı en iyi olanlar elde edilmeye çalışılmıştır. Bu yolda ortaya konan en iyi yöntem Parks-McClellan algoritması olarak bilinmesine rağmen, sözkonusu yöntemin geniş geçiş bandları için çözüme yakınsaması güçtür ve işlemsel sorunlar ortaya çıkmaktadır. Bu çalışmada, FIR süzgeç tasarımı, çözümsüz çok-belirli bir doğrusal denklem sisteminin belirli kıstaslar altında yaklaşık çözümünün bulunması problemi olarak değerlendirilmiştir. Tasarımın ortaya koyduğu çözüm, bu denklem sisteminden, doğrusal bir dönüşüm aracılığı ile yeni dönüştürülmüş denklem sisteminin minimum kareler yaklaşımı altındaki yaklaşık çözümüdür. Temel doğrusal cebir kavramları yardımı ile ele alınan çözümsüz doğrusal denklem sistemlerinin minimum kareler yaklaşımı ile yaklaşık çözümleri incelenerek, bu çözümlere ilişkin hata vektörünün belirli bir alt-uzaya ait olabilmesi için yeter koşullar ortaya konabilir. Bu koşullar altında, hata vektörünün ait olduğu alt-uzayın önceden belirlenebilmesi, tasarım aşamasında, süzgecin farklı frekans bandlarındaki yerel maksimum hata bileşenlerinin oranının frekans ekseni boyunca denetlenebilmesine karşılık düşmektedir. Parks-McClellan algoritmasının dayandığı temel nokta alternasyon teoremidir. Alt-uzay dönüşüm yönteminin dayanak noktası ise; hata vektörünün ait olduğu alt-uzaylar arasında doğrusal dönüşümleri gerçekleyebilecek tersinir matrislerin varlığıdır. Tasarım için tanımlanmış doğrusal denklem sistemini çözmek yerine, doğrusal ve tersinir bir dönüşüm yolu ile dönüştürülmüş denklem sistemini çözerek, sözkonusu çözüm için asıl denklem sisteminde elde edilen hata vektörünü istenen bir alt-uzaya ait kılmak mümkündür. Bu durum, ancak denklem sayısı bilinmeyen sayısından bir fazla olduğunda mümkün olduğundan, alternasyon teoremi ile de çelişmemektedir. Bu yöntem ile elde edilen sonuçlar, her ne kadar teorik olarak Parks-McClellan algoritmasında elde edilenler ile teorik olarak özdeş ise de; işlemsel doğruluk açısından daha yetkindir. Bu anlamda; alt-uzay dönüşüm yöntemi ile elde edilen çözüm, Parks-McClellan algoritmasının ilk aşamasında elde edilen çözümün yerini alabilir. viii

Özet (Çeviri)

SUMMARY FINITE-DURATION IMPULSE RESPONSE FILTER DESIGN USING SUBSPACE TRANSFORMATIONS In this thesis, types of the discrete-time finite-duration impulse response filter design techniques are mentioned and a method is presented which can take place of the first step of Parks-McClellan algorithm [5]. The method mainly depends on determining the subspace which includes the error vector due to the least squares solutions to linear equation systems Digital signal processing is a field that has its roots 1 8th 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 equipment. Some exeptions to this were evident in the 1950s particularly 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 before implementing it in an analog hardware. Despite the fact that flexibility of digital computer in signal processing, the processing can always be done in real time. 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, and finally followed by the digital to analog conversion, the overall system approximated a good analog filter. The evolution of a new point of view toward digital signal processing was further accelerated by the disclosure in 1965 of an efficient algorithm for computation of the Discrete Fourier Transform; called Fast Fourier Transform or FFT. Many signal processing algorithms that had been developed on digital computers required much more processing rate rather than analog structures. A signal can be defined as a function that conveys information, generally about the state or behaviour of a physical system. Although signals can be represented in many ways, in all cases the information is contained in some pattern of variations. Signals are represented mathematically as functions of one or more independent variables. For example, a speech signal is represented as a function of time and a photographic image is represented as a brightness function of two spatial variables. The independent variable in the mathematical representation of a signal may be either continuous or discrete. Continuous-time signals are defined along a continuum of times and thus are represented by a continuous independent variable. Continuous- time signals are often referred to as analog signals. Discrete-time signals are defined at discrete times and thus the independent variable has discrete values, so discrete- IXtime signals are represented as sequences of numbers. Signals such as speech or images may have either a continuous or a discrete variable representation, and if certain conditions hold, these representations are entirely equivalent. Besides the independent variables being either continuous or discrete, the signal amplitude may be either continuous or discrete. Digital signals are those for which both time and amplitude are discrete [3], Discrete-time signals are represented mathematically as sequences of numbers. A sequence of numbers x, in which the n. number in the sequence is denoted as x[n], where n is an integer. Signal processing systems may be classified along the same lines as signals. So discrete-time systems are those for which both the input and the output are discrete- time signals.A discrete-time system is defined mathematically as a transformation or operator that maps an input sequence with values x[n] into an output sequence with values y[n].This can be denoted as, y[n] = T{x[n]}. Discrete-time systems are classified by detecting some constraints on the properties of the transformation T{-}:. A system is linear if and only if, T{x, [n] + ax2 [n]} = T{x, [n]} + aT{x2 [n]}. where a is an arbitrary constant.. A time-invariant system (shift-invariant system) is one for which a time shift or delay of the input sequence causes a corresponding shift in the output sequence. These properties which are defined above are properties of systems not of the inputs applied to the system. A paticularly important class of systems consists of those that are linear and time-invariant. These two properties in combination lead to especially convenient representations for these systems. Most important, this class of systems has significant signal processing applications. A linear time-invariant system (LTI) is completely characterized by its by its impulse response h[n] which is defined as, h[n] = T{8[n]}. As a direct result of linearity and time invariance, there exist a definition that explains the relation between output and input of the discrete-time systems, called the convolution sum; y[n]=£x[k]h[n-k]. Although the convolution sum expression is analogous to the convolution integral of continuous-time linear system theory, the convolution sum should not be thought ofas an approximation to the convolution integral. The convolution sum, in addition to its theoritical importance, often serves as an explicit realization of a discrete-time linear system. In addition,the properties of discrete-time systems discussed above can be detected by analyzing the impulse response of a LTI system. If the impulse response of a LTI system has finite number of nonzero samples, these systems are called finite-duration impulse response systems, shortly speaking, FIR systems.Otherwise, the system is called infinite-impulse response systems (HR). Filters are a particularly important class of linear time-invariant systems and, the term frequency selective filter suggests a system that passes certain frequency components and totally rejects all others, but in a broader context any system that modifies certain frequencies relative to others is also called a filter [4]. Generally the design of filters involves the following main stages; 1. The specification of the desired properties of the system. 2. The approximation of the specifications using a causal discrete-time system. 3. The realization of the system. The first stage is dependent on the application and the third dependent on the technology to be used for the implementation. The designed filter is often implemented with digital computation and used to filter a signal that is derivated from a continuous-time signal by means of periodic sampling followed by analog- digital conversion. For this reason, it has become common to refer to discrete-time filters as digital filters. The specifications for both the discrete-time filter and continuous-time filter are given in frequency domain. This is specially common for frequency-selective filters such as lowpass, bandpass, and highpass filters. The second stage has the most significant role in designing the filter, so that it is expected to determine the best approximation for some given specifications. As a class of discrete-time sequences, the finite-duration sequences possesses certain desirable properties from the point of view of filter design. For example, the question of stability and realizability never arise since FIR sequences are always stable and, with an appropriate finite shift, can always be made realizable. Hence filter design problems in which an arbitrary frequency response is desired can be tackled using FIR sequences. There are many reasons for studying how to design FIR filters. Among the advantages of FIR filters are:. FIR filters with exactly linear phase can be easily designed. This simplifies the approximation problem, in many cases, when one is only interested in designing a filter that approximates an arbitrary magnitude response. Linear phase filters are important for applications where frequency dispersion due to nonlinear phase is harmful (e.g., speech processing and data transmission). XI. Efficient realizations of FIR filters exist as both recursive and non- recursive structures.. FIR filters realized nonrecursively are always stable.. Roundoff noise, which is inherint in realizations with finite precision arithmetic, can easily be made small for nonrecursive realizations of FIR filters. Among the possible disadvantages of FIR filters are:. A large value of impulse response duration is required for better approximations to the desired filter characteristics.. The delay of linear phase FIR filters need not always be an integer number of samples.And this can lead to problems in some signal processing applications. [4] In a different point of view, FIR filter design techniques can classified into two groups: FIR Filter Design Methods Figure 2 - FIR Filter Design Methods Since the frequency response of a digital system is periodic, it can be expanded in a Fourier series. The coefficients of this Fourier series are the filter impulse response coefficients. Generally, there are infinite number of nonzero elements in impulse response. To obtain an FIR filter which approximates the original frequency response, the Fourier series must be truncated. Direct truncation of the series leads to Gibbs phenomenon because ot the discontinuity of the original frequency response. In order to control this situation, a weighting function is used to modify the Fourier coefficients. This time limited weighting function is called a window. Since the multiplication of Fourier coefficients by a window corresponds to convolving the original frequency response with the Fourier transform of the window. So the design criterion is to find a finite window whose Fourier transform has relatively small sidelobes. There are many types of windows used to truncate the impulse response of the filter. xuA second closed-form technique for approximating a filter with a given frequency response specifications is to sample the desired frequency response at equispaced frequencies, where the number of samples is the length of impulse response. By setting these frequency samples to be the DFT coefficients [7], [4] of the filter impulse response, one can derive an approximation to any desired continuous frequency response. The methods for design of FIR filters are based largely on closed-form solutions. As a result, they are easy to apply and entail a relatively insignificant amount of computation. But they usually lead suboptimal designs whereby the filter order required to satisfy a set of given specifications is not the lowest that can be achieved. So, the number of arithmetic operations required per output sample is not minimum, and the computational efficiency and speed of operation of the filter are not as high as could be. As a result; closed-form solutions are easier due to iterative methods. But for some specifications they are both not enough to establish the best design. In weigthed-Chebyshev method, an error function is formulated for the desired filter in terms of a linear combination of cosine functions and is then minimized by using a very efficient multivariable optimization algorithm known as Remez exchange algorithm. When convergence is achieved, the error function becomes equiripple and the the error in different frequency bands of interest is controlled by applying weigthing to the error function [4-3], The development of the weigthed-Chebyshev method began with a paper by Herrman published in 1970, which was followed soon after by a paper by Hofstetter, Oppenheim, and Siegel. These contributions were followed by a series of papers, during seventies, by Parks, McClellan, Rabiner, and Hermann. The least-squared error and the minimum Chebyshev error criteria are the two most commonly used in FIR filter design methods. Methods based on least-squares approximations are generally iterative like weigthed-Chebyshev method. In least- squares algorithms, there is generally a weighting function which differs at each iteration due to minimization of the maximum error. The main problem is to find the correct function to obtain the best approximation. The idea of using an iterative re-weigthed least-squared (IRLS) algorithm to achieve a Chebyshev or LM approximation was first developed by Lawson[10] and extended to Lp by Rice and Usow. Burrus [9], then developed a robust IRLS method to obtain different error criteria in pass and stopbands of a filter. And finally, Tseng [12] presented an efficient implementation of Lawson's algorithm which is very similar to the procedure in this thesis study. It was shown that the update terms in Lawson's algorithm can be efficiently achieved by computing a subspace projection, and equiripple designs can be obtained like weigthed-Chebyshev method. In this thesis, that re-weighting operation is viewed as a subspace transformation on. A linear transformation is defined due to the desired error function and is applied to a linear equation system which must be solved in least-square sense. So the inverse transform of error of transformed linear equation system is the desired error character. xmThe FIR filter design problem can be modelled as an linear equation like, Ch=d (1) which have one more equation than the unknowns. Let R(u) be the subspace which is desired to include the error vector due to the solution and t be the base of the N(C). Then the transformation matrix is defined as; K=(diag(u)diag(|t|)-1)"^, u = i/wcc»,) 1/W(0)2) Lı/w(û)L+2)J W((û) > 0. (2) where W(co) is a weighting function which is nonzero in the equation (2). The error vector (d-Ch) due to the least squares solution to the trasformed equation system; KCh=Kb (3) is an element of the subspace R(u). So there are the main steps of the design procedure: 1) Linear equation system in (1) which has one more equation than the unknowns (the coefficients of the filter) is defined due to the set of initial frequencies. The transformed linear equation system is then solved depending on the transformation matrix K in (2) which is a function of vectors u and t. 2) As in the second step of Parks-McClellan algorithm, a new set of frequencies are found due to data obtained from the first step. The structure of the thesis is as follows. In Chapter 2 some linear algebric background is given and solutions to linear equation systems are discussed. And then subspace transformation in solving linear equation systems is explained. Chapter 3 presents the FIR filter design problem and methods which were developed with the new method which will take the place of the first step of Parks-McClellan algorithm. In the last chapter, some results and comments are given about the design procedure. The following additional parts include a numerical example for the subspace transformation, Lagrange interpolation formulea and the explanation of the MATLAB® program which designs FIR digital filters using the new method. xiv

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