Examples are digital filters, Fourier transformers, modulators, and so on. In a multirate signal processing sys- tem, there are two new building blocks, called the M-fold decimator and the L-fold expander Fig. These will be defined and illustrated in Chapter 4.
For the purpose of the present discussion, the decimator is a device that reduces the sampling rate by an integer factor of M, whereas the expander is used to increase the rate by Z.
Multiplier Input Multiplication of 2 signals Figure 1. Some of the early references are Schafer and Rabiner [], Meyer and Burrus [], and Octkon, et al. The use of multiple sampling rates offers many advan- tages, such as reduced computational complexity for a given task, reduced transmission rate i.
M - fold decimator L - fold expander Figure 1. Broadly speaking, the idea is as follows. Suppose we wish to digitize an analog signal x, t. Ifthe signal has significant energy only up to a frequency fay.
The lowpass filter in this case has a sharp transition from passband to stopband. A second technique proceeds in two stages: a First use an antialias- ing filter with wider transition bandwidth, say by a factor of two.
Then oversample by a factor of two before digitizing. This two-stage process climinates the need for sharp-cutoff antialiasing analog filters, which not only are expensive, but also introduce severe phase distortion. Details of this technique will be con- sidered in Chap. A second application is in fractional sampling rate alteration, for ex- ample, converting a 48 kHz discrete-time signal to a Such requirements are common in the digital audio industry, where a nuinber of sampling rates coexist [Bloom, ].
For example, the sampling rate for studio work is 48 kllz, whereas that for CD production is The obvious way to perform the rate conversion would be to first convert the discrete-time signal into a continuous-time signal and then resample it at the lower rate. This method is expensive and involyes analog components, along with the associated inaccuracies. A direct digital nuultirate method is to perform the conversion directly in the discrete-Lime domain.
Such fractional decimation or interpolation is done by combining integer decimators, expanders and filters appropriately. This is more accurate as well as convenient. Details of this technique will be described in Chap. There are many more applications of multirate processing, and several of them are based on the so-called subband decomposition, to be described next, 2 Chap.
The ars tos reduce the sampling rate whenever possible is of course under standable, because it usually reduces the storage as well as the processi requirements. Now suppose 2 n is not bandlimited, but nevertheless has most of the energy in the low frequency region. Figure demonstrates the Fourier transform of such a signal.
Even though this cannot be decimated without aliasing, it seems only reasonable to expect that some kind of data rate re- duction is still feasible. This is indeed made possible by a technique called subband decomposition, implemented with the so-called quadrat ire mirror filter bank. In this technique, the average number of bits per sample is reduced, even though the average number of samples per unit time is un changed.
Here a discrete- lime signal z n is passed through a pair of digital filters M, z called anal- ysis filters, with frequency responses as demonstrated in the figure. The filtered signals x1 subband signals are thus approximately bandl ited lowpass and highpass, respectively.
They are then decimated by two, so that the number of samples per unit time [counting vo n as well as a n is the saine as that for x n. The decimated subband signals, vx n , are then quantized and transmitted. At the receiver end, these are recombined by us- sae oxpanders and synthesis filters 2. In this manner, an approximation 2 n of the signal 2 n is generated. This system will be studied in Chap. The above system can be regarded as a sophisticated quantizer, Thus, assume that we are allowed to trans: bbits per sample.
In the above filter-bank approach, we quantize the lower rate signals mg 1 and us 1 to Soc. This scheme is calicd subband coding [Croisicr, et al. More recently, the effectivencss of subband coding has been demon- strated for music signals. Using subband coding, it has been demonstrated that a major bit rate reduction can be obtained compared to the traditional 16 bit repesentation , with little compromise of quality [Veld- huis, et al.
This has been used in the digital compact casscte DCC. At the end of this section, more applications of subband splitting will be mentioned.
Reconstruction from subband signals. In many applications, the signals e, or, more properly, the quantized versions are recombined to obtain an approximation i n of the original signal z n. This recombina- tion is done by use of expanders which restore the sampling ratc followed by digital filters , z whose purpose will be explained in Sec. Such 4 Chap. Other will be discussed in due course. One of the major developments in multirate signal processing is the recognition of the fact that all of these errors except quantization ercor can be eliminated completely at finite cost.
The QMF bank, introduced in the mid seventies, has since been ex- tended to the case of more than two subbands. Figure 1. Two scts of typical frequency Tespanses are also sketched in the figure. One se has uniform filter bandwidths and spacing, while the other has rm octave spacing. For the moment, we note that a spectrum analyzer takes a signal z n and computes the Fourier transform of short blocks, af ter some preliminary processing such as windowing , Such a system can be interpreted as a filter bank Fig.
While the details will be presented only in Chap. While the Fourier transformer provides filters with overlapping responses, the generalized system can provide filters with arbitrarily sharp cutoff, better interband isolation and unequal bandwidths. Polyphase decomposition is useful in vir- tually every application of multirate signal processing, and often results in dramatic computational efficiency.
It is valuable in theoretical study, practi- cal design and actual implementation of filter banks. This will be introduced in Chap. As will be seen in Chap. It was shown in Croisicr. If this is not the case, then the alias-free system still suffers from residual distortion.
If the designer does not impose any specifications on the analysis filters such as large stopband attenuation. However, this is not very practical because, in order to utilize the benefits of subband coding, it is necessary to impose fairly stringent specifications on the attenuation characteristics of the filters.
For the two channel QMF bank a fundamental result was proved in- dependently by Smith and Barnwell [ and ], and Mintzer []. These papers showed that perfect reconstruction can be achicved even af- ter imposing such practical attenuation requirements. This involves careful design of the four filters, as will be seen in Chap. Smith and Barnwell as well as Ramstad independently showed how to formulate the porfcet reconstruction conditions in matrix form.
It was first recognized by Vetterli, and then independently by Vaidyanathan in the two references mentioned above that a polyphase component approach results in consid- erable simplification of the theory. It has since been shown that, by using a class of filter banks called paraunitary filter banks, perfect reconstruction can be achicved quite eas- ily. In these systems, the filter bank is constrained to have a paraunitary pelyphase matrix to be explained in Chap. These have the advantage that the cost of design as well as implementation is largely determined by the cost of one prototype filter, since all the other filters are derived from it The paraunitary property of filter banks offers many advantages, as elaborated in Chap.
Interestingly cnough, paraunitary matrices have their origin in classical electrical network theory see Sec. In the past, applications of these matrices have becn confined mostly within the nctwork theory and control theory communities. The use of parannitary matrices in digital signal processing, especially filter bank theory, is relatively recent.
Filter bank theory has been extended to the case of nonuniform band- widths and decimation ratios [Hoang and Vatdyanat han. Barnwell, and Stith, tal. The extensions to two dimensional signals has application in image compression and coding. A systematic study of anulti- dimensional filter hanks was first uadertaken by Vetterti []. Since then there has been major progress in multidimensional multirate systems [Ansari and Lau.
Reseasch results in multidimensional multirate systems are emerging at a rapid rate now. Research on these topics is still evolving; an excellent reference on the subject is provided by Shynk [].
Further applications in communications have been reported by some authors, for example, transmultiplexing [Vetterli. See Sec. This work has opened up considerable amount of research activity in both the signal processing, and mathematics communities.
It will be seen that wavelet analysis is closely related to the so called octave-band filter banks, introduced in the early seventies for analysis of sound signals. Research in wavelet transforms has grown very rapidly after the mid s and is still growing.
In particular, the detailed aspects of filter bank theory were developed largely during the last decade, subsequent to and im many cases triggered by the publication of Crochiere and Rabiner []. The theory of perfect reconstruction filter banks has now reached a state where such systems can be designed as well as implemented with ease.
The underlying theory is somewhat complicated, but as a reward it has immense potential for further research aud applications. For example, the theory can be applied directly to areas such as subband coding, voice privacy, image processing, uaultiresolution, and wavelct analysis.
The purpose of this text is to present an in-depth study of multirate sys- tems and filter banks. We have assumed that the reader has some exposure to signal processing eg.. Except for thi- Aequirement, the book is sclf-contamed. However, this background material ts reviewed in Chap. There are many exaniples, design methods, and tables which will aid the practicing professional as well. The chapters can 8 Chap. Chapters 2 to 4: Introductory Material A bricf review of linear system findamentals and digital filtenog, i provided in Chap.
Iu Chap. IMR elliptic filters, FER eigenfilters, and allpass filters have been treated in greater detail Decause of their special role in multirate systems. Chapter 4 is a detailed study of multirate building blocks, and their interconnections with other systems such as digital filters. Some of the ezrly sections overlap with the material covered im Croctecre and Rahincr [ AU the expense of this overlap, we have ensured Uhat the chapter is self- contained. A number of special types of digital filters, for example, Nyquist filters, power complementary filters and so on.
The polyphase decomposi- tion js introduced, along with special types of filter banks, for example, the form-DFT bank. Many applications of multirate processing are also described in Chap 4. This includes subband coding, digital audio, and transmmultiplexers, to name a few. Chapters 5 to 8: Maximally Decimated Filter Banks Chapter 5 is a study of the M-channel maximally decimated filter bank system shown in Fig.
Various distortions will be analyzed, foremost being aliasing caused by decimation. Conditions for alias cancelation znd perfect reconstruction will be established. The presentation will use some of the results ou paraunitary snatrices, which will be proved only in Chap.
We have chosen to defer the proofs to Chapter 14 which is devoted te paraunitary systems in order to ensure an easy and smooth flow. The results of Chap 14 will also be stated and used in some other chapters, e. Chapter 7 deals with Tincar-phase perfect reconstruction QM banks. In these systems the analysis filters have linear phase, which is a requirersent in some applications.
As a result, this system is very efficient both ftom the deswn aad implementation points of view. It tums out that one can eas Sec. L2 Scope and outtina 9.
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