Sampling Rate Converison by a Fractional Factor

We have discussed so far the decimation and interpolation where the sampling rate conversion factor is an integer. However, the need for a non-integer sampling rate conversion appears when the two systems operating at different sampling rates have to be connected, or when there is a need to convert the sampling rate of the recorded data into another sampling rate for further processing or reproduction. Such applications are very common in telecommunications, digital audio, multimedia and others. In this chapter, we consider the sampling rate conversion by a rational factor, called sometimes a fractional sampling rate conversion. We use MATLAB functions from the Signal Processing and Filter Design Toolbox to demonstrate the fractional sampling rate conversion. We present the technique for constructing efficient fractional sampling rate converters based on FIR filters and the polyphase decomposition. In the sequel, we consider the sampling rate alteration with an arbitrary conversion factor. We present the polynomial-based approximation of the impulse response of a hybrid analog/digital model, and the implementation based on the Farrow structure. We also consider the fractional-delay filter problem. This chapter concludes with MATLAB exercises for individual study.

Filters in Multirate Systems

The role of filtering in sampling-rate conversion has been considered in Chapter II. The importance of filtering arises from the fact that the sampling theorem should be respected for all the sampling rates of the system at hand. Filters are required to bandlimit the spectrum of the signal to the prescribed bandwidth in accordance with the actual sampling rate. In sampling rate conversion systems, filters are used in decimation to suppress aliasing and in interpolation to remove imaging. Since an ideal frequency response cannot be achieved, the performance of the system for sampling rate conversion is mainly determined by filter characteristics. Obviously, an appropriate filter should enable the sampling rate conversion with minimal signal distortion. The main advantage of a multirate system is the computational efficiency, and therefore, a decimator (interpolator) that implements a high-order digital filter could not be tolerated. The specific role of a digital filter in sampling rate conversion demands high-performance filtering with the lowest possible complexity. To reach this goal one has to concentrate first on the choice of the appropriate design specifications in order to provide minimal signal distortion. Secondly, the multirate filter is to be designed in a manner to satisfy the prescribed characteristics and to provide a low-complexity implementation structure. In this chapter, we discus first the spectral characteristics of decimators and interpolators and introduce three commonly used types of filter specifications. In the sequel, we review the MATLAB functions that are appropriate for the design of FIR and IIR filters to satisfy the specifications. An approach to computation of aliasing characteristics of decimators is given and illustrated by examples. This chapter considers also the analysis of sampling rate conversion for band-pass signals. Chapter concludes with MATLAB exercises for individual study.

Basics to Multirate Systems

Linear time-invariant systems operate at a single sampling rate i.e. the sampling rate is the same at the input and at the output of the system, and at all the nodes inside the system. Thus, in an LTI system, the sampling rate doesn’t change in different stages of the system. Systems that use different sampling rates at different stages are called the multirate systems. The multirate techniques are used to convert the given sampling rate to the desired sampling rate, and to provide different sampling rates through the system without destroying the signal components of interest. In this chapter, we consider the sampling rate alterations when changing the sampling rate by an integer factor. We describe the basic sampling rate alteration operations, and the effects of those operations on the spectrum of the signal.

Single-Rate Discrete-Time Signals and Systems

This chapter is a concise review of time-domain and transform-domain representations of single-rate discrete-time signals and systems. We consider first the time-domain representation of discrete-time signals and systems. The representation in transform domain comprises the discrete-time Fourier transform (DTFT), the discrete Fourier transform (DFT), and the z-transform. The basic realization structures for FIR and IIR systems are briefly described. Finally, the relations between continuous and discrete signals are given.

Comb-Based Filters for Sampling Rate Conversion

Comb filters are developed from the structures based on the moving average (boxcar) filter. The combbased filter has unity-valued coefficients and, therefore, can be implemented without multipliers. This filter class can operate at high frequencies and is suitable for a single-chip VLSI implementation. The main applications are in communication systems such as software radio and satellite communications. In this chapter, we introduce first the concept of the basic comb filter and discuss its properties. Then, we present the structures of the comb-based decimators and interpolators, discuss the corresponding frequency responses, and demonstrate the overall two-stage decimator constructed as the cascade of a comb decimator and an FIR decimator. In the next section, we expose the application of the polyphase implementation structure, which is aimed to reduce the power dissipation. We consider techniques for sharpening the original comb filter magnitude response and emphasize an approach that modifies the filter transfer function in a manner to provide a sharpened filter operating at the lowest possible sampling rate. Finally, we give a brief presentation of the modified comb filter based on the zero-rotation approach. Chapter concludes with several MATLAB Exercises for the individual study. The reference list at the end of the chapter includes the topics of interest for further research.

Complementary Filter Pairs

Digital filters with complementary characteristics find many applications in practice. In this chapter, we concentrate on the properties and construction of complementary filters and filter pairs. An important application of complementary property is deriving a new transfer function from the existing one. A highpass filter can be obtained as a complement of the lowpass filter, and also a bandstop filter can be considered as a complement of the bandpass filter. Complementary lowpass/highpass and bandpass/bandstop filter pairs are popular because of very attractive implementations. Namely, the two complementary filters in the pair are implemented at the cost of a single one. Complementary filter pairs, usually lowpass/highpass filter pairs, are widely used whenever there is a need to split the signal into two adjacent subbands and reconstruct it after some processing performed in the subbands. They are used as basic building blocks in constructing analysis and synthesis multichannel filter banks. Moreover, the complementary filter pairs are used in constructing low sensitivity complex filtering structures. In some applications, such as signal analysis, the complementary filter pairs are used to separate a signal into two bands, and the filtered signals are processed without need to reconstruct it. Another application is digital audio where the signal is separated into two (three) bands resulting in the signals that are feed inside two (three) loudspeakers. In this chapter, at the beginning we introduce the basic definitions of the complementary properties that will be used through the chapter. We use then the complementary properties to construct FIR and IIR highpass filters from the existing lowpass filters. In the sequel, we consider the analysis and synthesis filter pairs. We present the design and efficient implementations of FIR and IIR complementary filter pairs. Chapter concludes with MATLAB Exercises for individual study.

Lth-Band Digital Filters

Digital Lth-band FIR and IIR filters are the special classes of digital filters, which are of particular interest both in single-rate and multirate signal processing. The common characteristic of Lth-band lowpass filters is that the 6 dB (or 3 dB) cutoff angular frequency is located at p/L, and the transition band is approximately symmetric around this frequency. In time domain, the impulse response of an Lth-band digital filter has zero valued samples at the multiples of L samples counted away from the central sample to the right and left directions. Actually, an Lth-band filter has the zero crossings at the regular distance of L samples thus satisfying the so-called zero intersymbol interference property. Sometimes the Lthband filters are called the Nyquist filters. The important benefit in applying Lth band FIR and IIR filters is the efficient implementation, particularly in the case L = 2 when every second coefficient in the transfer function is zero valued. Due to the zero intersymbol interference property, the Lth-band filters are very important for digital communication transmission systems. Another application is the construction of Hilbert transformers, which are used to generate the analytical signals. The Lth-band filters are also used as prototypes in constructing critically sampled multichannel filter banks. They are very popular in the sampling rate alteration systems as well, where they are used as decimation and interpolation filters in single-stage and multistage systems. This chapter starts with the linear-phase Lth-band FIR filters. We introduce the main definitions and present by means of examples the efficient polyphase implementation of the Lth-band FIR filters. We discuss the properties of the separable (factorizable) linear-phase FIR filter transfer function, and construct the minimum-phase and the maximum-phase FIR transfer functions. In sequel, we present the design and efficient implementation of the halfband FIR filters (L = 2). The class of IIR Lth-band and halfband filters is presented next. Particular attention is addressed to the design and implementation of IIR halfband filters. Chapter concludes with several MATLAB exercises for self study.

FIR Filters for Sampling Rate Conversion

The role of filters in sampling-rate conversion process has been discussed in Chapters II and III. Filters are used to suppress aliasing in decimators and to remove images in interpolators. The overall performance of a decimator or of an interpolator mainly depends on the characteristics of antialiasing and antiimaging filters. In Chapter III, we have considered the typical filter specifications and several methods for designing filter transfer functions that can meet the specifications. In this chapter, we are dealing with the implementation aspects of decimators and interpolators. The implementation problem arises from the unfavorable facts that filtering has to be performed on the side of the high-rate signal: in decimation filtering precedes the down-sampling, and in interpolation up-sampling precedes filtering. The goal is to construct a multirate implementation structure providing the arithmetic operations to be performed at the lower sampling rate. In this way, the overall workload in the sampling-rate conversion system can be decreased by the conversion factor M (L). The multirate filter implementation means that down-sampling or up-sampling operations are embedded into the filter structure. In this chapter, we are focused on the structures developed for finite impulse response (FIR) filters. The nonrecursive nature of FIR filters offers the opportunity to create implementation schemes that significantly improve the overall efficiency of FIR decimators and interpolators. This chapter concentrates on the direct implementation forms for decimators and interpolators and the implementation forms based on the polyphase decompositions. Memory saving solutions for polyphase decimators and interpolators are also presented. Finally, the efficiency of FIR polyphase decimators and interpolators is discussed. The chapter concludes with MATLAB exercises for the individual study.

Examples of Multirate Filter Banks

The purpose of this chapter is to illustrate by means of examples the construction of the analysis and synthesis filter banks with the use of FIR and IIR two-channel filter banks as the basic building blocks. In Chapter VIII, we have discussed the design and properties of several types of complementary filter pairs, and in Chapters IX and X we have shown how those filter pairs are used in the synthesis of digital filters with sharp spectral constraints. In this chapter, we demonstrate the application of the complementary filter pairs as two-channel filter banks used to decompose the original signal into two channel signals and to reconstruct the original signal from the channel signals. Signal decomposition is referred to as the signal analysis, whereas the signal reconstruction is referred to as the signal synthesis. Thereby, the filter bank used for the signal decomposition is called the analysis filter bank, and the bank used for signal reconstruction is called the synthesis filter bank. The two-channel filter bank is usually composed of a pair of lowpass and highpass halfband filters, which satisfy some complementary properties. The bandwidth that occupies each of two channel signals is a half of the original signal bandwidth. Hence, the channel signals can be processed with the sampling rate which is a half of the original signal sampling rate. At the output of the analysis bank, the channel signals are down-sampled-by-two and then processed at the lower sampling rate. For the signal reconstruction, each of two channel signals has to be up-sampled-by-two first, and then fed into the synthesis bank. The sampling rate alteration in the two-channel filter bank causes the unwanted effects: the downsampling produces aliasing, and the up-sampling produces imaging. The essential feature of the two-channel filter bank is that the aliasing produced in the analysis side can be compensated in the synthesis side. This is achieved by choosing the proper combination of filters in the analysis and synthesis banks. The elimination of aliasing opens the possibility of the perfect (and nearly perfect) reconstruction of the original signal. The perfect reconstruction means that the signal at the output of the cascade connection of the analysis and synthesis bank is a delayed replica of the original input signal. Constructing perfect reconstruction and nearly perfect reconstruction analysis/synthesis filter banks is an unbounded area of research. An important and widely used application of the two-channel filter banks is the construction of multichannel filter banks based on the tree-structures where the two-channel filter bank is used as a building block. In this way, a multilevel multichannel filter bank can be obtained with either uniform or nonuniform separation between the channels. The two-channel filter banks are particularly useful in generating octave filter banks. Depending on applications, the filter bank can be requested to provide frequency-selective separation between the channels, or to preserve the original waveform of the signal. The example applications of the frequency-selective filter banks are audio and telecommunication applications. The importance of preserving the original waveform is related with the images. In the case of the discrete-time wavelet banks, the frequency-selectivity is less important. The main goal is to preserve the waveform of the signal. The purpose of this chapter is to illustrate by means of MATLAB examples the signal analysis and synthesis based on the two-channel filter banks. We give first a brief review of the properties of the two-channel filter banks with the conditions for aliasing elimination. We discuss the perfect reconstruction and nearly perfect reconstruction properties and show the solutions based on FIR and IIR QMF banks and the orthogonal two-channel filter banks. In the sequel, the tree-structured multichannel filter banks are considered. The process of signal decomposition and reconstruction is illustrated by means of examples.

Frequency-Reponse Masking Techniques

The initial concept of the frequency-response masking technique was introduced by Neuvo, Cheng-Yu and Mitra (1984). It was shown that the complexity of a linear phase FIR filter can be considerably reduced by using the cascade connection of an interpolated FIR (IFIR) filter and a properly designed FIR filter. The IFIR filter transfer function is obtained by replacing the unit delay z-1 with the delay block z-M, where M is an integer. In this way, the frequency response of the IFIR filter is made periodic. The FIR filter in the cascade is used to eliminate (mask) the images from the IFIR filter frequency response. Two years later, Lim (1986) proposed a complete approach for the application of frequency-response masking technique in designing narrow-band and arbitrary-band linear phase FIR filters. It was shown that the approach given in (Lim, 1986) results in a linear phase FIR filter with a small fraction of nonzero coefficients, and thus is suitable for implementing sharp filters with arbitrary bandwidths. The arithmetic complexity is considerably smaller in comparison with the arithmetic complexity of an optimal FIR filter having the equivalent frequency response. This approach is applied later to IIR filters by Johansson and Wanhammar (1997, 2000). The overall filter is composed of an IIR periodic model filter and its complementary periodic filter, and FIR linearphase masking filters. In this way, the arbitrary-band filter can be designed. For a narrowband filter, the cascade of a periodic filter and masking filter can be used. The frequency-response masking approach is suitable for digital filters with sharp transition bands. Compared to the classical single-filter design, this technique offers the advantage of lower coefficients’ sensitivity, higher computation speed and lower power consumption. Recently, the application of frequency-response masking approach has been extended to filter banks to achieve a sharp band-separation with reduced computational complexity (Furtado, Diniz, Netto, and Saramäki, T. 2005; Rosenbaum, Lövenborg, and Johansson, 2007). In this chapter, we review the frequency-response masking techniques for narrow-band and arbitrary bandwidth IIR filters. We demonstrate through examples that very selective characteristics can be obtained using relatively low-order sub-filters. In this way, stable, low-sensitive filters are obtained.