Quantized Filter Analysis

CHAPTER 7 Quantized Filter Analysis 7.1 INTRODUCTION The analysis and design of discrete-time systems, digital filters, and their realiza- tions, computation of DFT-IDFT, and so on discussed in the previous chapters of this book were carried out by using mostly the functions in the Signal Pro- cessing Toolbox working in the MATLAB environment, and the computations were carried out with double precision. This means that all the data representing the values of the input signal, coefficients of the filters, or the values of the unit impulse response, and so forth were represented with 64 bits; therefore, these numbers have a range approximately between 10 −308 and 10 308 and a precision of ∼2 −52 = 2.22 × 10 −6 . Obviously this range is so large and the precision with which the numbers are expressed is so small that the numbers can be assumed to have almost “infinite precision.” Once these digital filters and DFT-IDFT have been obtained by the procedures described so far, they can be further analyzed by mainframe computers, workstations, and PCs under “infinite precision.” But when the algorithms describing the digital filters and FFT computations have to be implemented as hardware in the form of special-purpose microprocessors or application-specific integrated circuits (ASICs) or the digital signal processor (DSP) chip, many practical considerations and constraints come into play. The registers used in these hardware systems, to store the numbers have finite length, and the memory capacity required for processing the data is determined by the number of bits—also called the wordlength —chosen for storing the data. More memory means more power consumption and hence the need to minimize the wordlength. In microprocessors and DSP chips and even in workstations and PCs, we would like to use registers with as few bits as possible and yet obtain high computational speed, low power, and low cost. But such portable devices such as cell phones and personal digital assistants (PDAs) have a limited amount of mem- ory, containing batteries with low voltage and short duration of power supply. These constraints become more severe in other devices such as digital hearing aids and biomedical probes embedded in capsules to be swallowed. So there is a Introduction to Digital Signal Processing and Filter Design, by B. A. Shenoi Copyright © 2006 John Wiley & Sons, Inc. 354 FILTER DESIGN –ANALYSIS TOOL 355 great demand for designing digital filters and systems in which they are embed- ded, with the lowest possible number of bits to represent the data or to store the data in their registers. When the filters are built with registers of finite length and the analog-to-digital converters (ADCs) are designed to operate at increasingly high sampling rates, thereby reducing the number of bits with which the samples of the input signal are represented, the frequency response of the filters and the results of DFT-IDFT computations via the FFT are expected to differ from those designed with “infinite precision.” This process of representing the data with a finite number of bits is known as quantization, which occurs at several points in the structure chosen to realize the filter or the steps in the FFT computation of the DFT-IDFT. As pointed out in the previous chapter, a vast number of structures are available to realize a given transfer function, when we assume infi- nite precision. But when we design the hardware with registers of finite length to implement their corresponding difference equation, the effect of finite wordlength is highly dependent on the structure. Therefore we find it necessary to analyze this effect for a large number of structures. This analysis is further compounded by the fact that quantization can be carried out in several ways and the arithmetic operations of addition and multiplication of numbers with finite precision yield results that are influenced by the way that these numbers are quantized. In this chapter, we discuss a new MATLAB toolbox called FDA Tool avail- able 1 for analyzing and designing the filters with a finite number of bits for the wordlength. The different form of representing binary numbers and the results of adding and multiplying such numbers will be explained in a later section of this chapter. The third factor that influences the deviation of filter performance from the ideal case is the choice of FIR or IIR filter. The type of approximation chosen for obtaining the desired frequency response is another factor that also influences the effect of finite wordlength. We discuss the effects of all these factors in this chapter, illustrating their influence by means of a design example. 7.2 FILTER DESIGN –ANALYSIS TOOL An enormous amount of research has been carried out to address these problems, but analyzing the effects of quantization on the performance of digital filters and systems is not well illustrated by specific examples. Although there is no analytical method available at present to design or analyze a filter with finite precision, some useful insight can be obtained from the research work, which serves as a guideline in making preliminary decisions on the choice of suitable structures and quantization forms. Any student interested in this research work should read the material on finite wordlength effects found in other textbooks [1,2,4]. In this chapter, we discuss the software for filter design and analysis that has been developed by The MathWorks to address the abovementioned 1 MATLAB and its Signal Processing Toolbox are found in computer systems of many schools and universities but the FDA Tool may not be available in all of them. 356 QUANTIZED FILTER ANALYSIS problem 2 . This FDA Tool finite design–analysis (FDA) tool, found in the Filter Design Toolbox, works in conjunction with the Signal Processing (SP) Toolbox. Unlike the SP Toolbox, the FDA Tool has been developed by making extensive use of the object-oriented programming capability of MATLAB, and the syntax for the functions available in the FDA Tool is different from the syntax for the functions we find in MATLAB and the SP Toolbox. When we log on to MATLAB and type fdatool , we get two screens on display. On one screen, we type the fdatool functions as command lines to design and analyze quantized filters, whereas the other screen is a graphical user interface (GUI) to serve the same purpose. The GUI window shown in Figure 7.1a displays a dialog box with an immense array of design options as explained below. First we design a filter with double precision on the GUI window using the FDA Tool or on the command window using the Signal Processing Toolbox and then import it into the GUI window. In the dialog box for the FDA Tool, we can choose the following options under the Filter Type panel: 1. Lowpass 2. Highpass 3. Bandpass 4. Bandstop 5. Differentiator. By clicking the arrow on the tab for this feature, we get the following additional options. 6. Hilbert transformer 7. Multiband 8. Arbitrary magnitude 9. Raised cosine 10. Arbitrary group delay 11. Half-band lowpass 12. Half-band highpass 13. Nyquist Below the Filter Type panel is the panel for the design method. When the button for IIR filter is clicked, the dropdown list gives us the following options specifying the type of frequency response: • Butterworth • Chebyshev I • Chebyshev II • Elliptic • Least-pth norm • Constrained least-pth norm 2 The author acknowledges that the material on the FDA Tool described in this chapter is based on the Help Manual for Filter Design Toolbox found in MATLAB version 6.5. FILTER DESIGN –ANALYSIS TOOL 357 (a) (b) Figure 7.1 Screen capture of fdatool window: (a) window for filter design; (b) window for quantization analysis. the following options are available for the FIR filter: • Equiripple • Least squares • Window • Maximally flat • Least-pth norm • Constrained equiripple 358 QUANTIZED FILTER ANALYSIS To the right of the panel for design method is the one for filter order. We can either specify the order of the filter or let the program compute the minimum order (by use of SP Tool functions Chebord, Buttord , etc.). Remember to choose an odd order for the lowpass filter when it is to be designed as a parallel connection of two allpass filters, if an even number is given as the minimum order. Below this panel is the panel for other options, which are available depending on the abovementioned inputs. For example, if we choose a FIR filter with the window option, this panel displays an option for the windows that we can choose. By clicking the button for the windows, we get a dropdown list of more than 10 windows. To the right of this panel are two panels that we use to specify the frequency specifications, that is, to specify the sampling frequency, cutoff fre- quencies for the passband and stopband, the magnitude in the passband(s) and stopband(s), and so on depending on the type of filter and the design method chosen. These can be expressed in hertz, kilohertz, megahertz, gigahertz, or nor- malized frequency. The magnitude can be expressed in decibels, with magnitude squared or actual magnitude as displayed when we click Analysis in the main menu bar and then click the option Frequency Specifications in the drop- down list. The frequency specifications are displayed in the Analysis panel, which is above the panel for frequency specifications, when we start with the filter design. The options available under any of these categories are dependent on the other options chosen. All the FDA Tool functions, which are also the functions of the SP Tool, are called overloaded functions. After all the design options are chosen, we click the Design Filter button at the bottom of the dialog box. The program designs the filter and displays the magnitude response of the filter in the Analysis area. But it is only a default choice, and by clicking the appropriate icons shown above this area, the Analysis area displays one of the following features: • Magnitude response • Phase response • Magnitude and phase response • Group delay response • Impulse response • Step response • Pole–zero plot • Filter coefficients This information can also be displayed by clicking the Analysis button in the main menu bar, and choosing the information we wish to display in the Anal- ysis area. We can also choose some additional information, for example, by clicking the Analysis Parameters . At the bottom of this dropdown list is the option Full View Analysis . When this is chosen, whatever is displayed in the Analysis area is shown in a new panel of larger dimensions with features that FILTER DESIGN –ANALYSIS TOOL 359 are available in a figure displayed under the SP Tool. For example, by clicking the Edit button and then selecting either Figure Properties , Axis Properties , or Current Object Properties ,the Property Editor becomes active and properties of these three objects can be modified. Finally, we look at the first panel titled Current Filter Information . This lists the structure, order, and number of sections of the filter that we have designed. Below this information, it indicates whether the filter is stable and points out whether the source is the designed filter (i.e., reference filter designed with double precision) or the quantized filter with a finite wordlength. The default structure for the IIR reference filter is a cascade connection of second-order sections, and for the FIR filter, it is the direct form. When we have completed the design of the reference filter with double precision, we verify whether it meets the desired specification, and if we wish, we can convert the structure of the reference filter to any one of the other types listed below. We click the Edit button on the main menu and then the Convert Structure button. A dropdown list shows the structures to which we can convert from the default structure or the one that we have already converted. For IIR filters, the structures are 1. Direct form I 2. Direct form II 3. Direct form I transposed 4. Direct form II transposed 5. Lattice ARMA 6. Lattice-coupled allpass 7. Lattice-coupled allpass—power complementary 8. State space Items 6 and 7 in this list refer to structures of the two allpass networks in parallel as described in Chapter 6, with transfer functions G(z) = 1 2 [A 1 (z) + A 2 (z)]andH(z) = 1 2 [A 1 (z) − A 2 (z)], respectively. The allpass filters A 1 (z) and A 2 (z) are realized in the form of lattice allpass structures like the one shown in Figure 6.19b. The MA and AR structures are considered special cases of the lattice ARMA structure, which are also discussed in Chapter 6. For FIR filters, the options for the structures are • Direct-form FIR • Direct-form FIR transposed • Direct-form symmetric FIR When we have converted to a new structure, the information that can be displayed in the Analysis area, like the coefficients of the filter, changes. We also like to point out that any one of the lowpass, highpass, bandpass, and bandstop filters that we have designed can be converted to any other type, by clicking 360 QUANTIZED FILTER ANALYSIS the first icon on the left-hand bar in the dialog box and adding the frequency specifications for the new filter. 7.3 QUANTIZED FILTER ANALYSIS When we have finished the analysis of the reference filter, we can move to construct the quantized filter as an object, by clicking the last icon on the bar above the Analysis area and the second icon on the left-hand bar, which sets the quantization parameters. The panel below the Analysis area now changes as shown in Figure 7.1b. We can construct three objects inside the FDA Tool: qfilt , qfft ,and quantizer . Each of them has several properties, and these properties have values, which may be strings or numerical values. Currently we use the objects qfilt and quantizer to analyze the performance of the reference filter when it is quantized. When we click the Turn Quantization On button and the Set Quantization Parameters icon, we can choose the quantization parameters for the coefficients of the filter. Quantization of the filter coefficients alone are sufficient for finding the finite wordlength effect on the magnitude response, phase response, and group delay response of the quantized filter, which for comparison with the response of the reference filter displayed in the Analysis area. Quantization of the other data listed below are necessary when we have to filter an input signal: • The input signal • The output signal • The multiplicand: the value of the signal that is multiplied by the multiplier. • The product of the multiplicand and the multiplier constant • The output signal The object quantizer is used to convert each of these data, and this object has four properties: Mode, Round Mode, Overflow mode ,and Format .Inorderto understand the values of these properties, it is necessary to review and understand the binary representation of numbers and the different results of adding them and multiplying them. These will be discussed next. 7.4 BINARY NUMBERS AND ARITHMETIC Numbers representing the values of the signal, the coefficients of both the filter and the difference equation or the recursive algorithm and other properties cor- responding to the structure for the filter are represented in binary form. They are based on the radix of 2 and therefore consist of only two binary digits, 0 and 1, which are more commonly known as bits, just as the decimal numbers based on a radix of 10 have 10 decimal numbers from 0 to 9. Placement of the bits in a string determines the binary number as illustrated by the example x 2 = 1001  1010, BINARY NUMBERS AND ARITHMETIC 361 which is equivalent to x 10 = 1 × 2 0 + 1 × 2 3 + 2 −1 + 2 −3 = 9.625. In this dis- cussion of binary number representation, we have used the symbol  to separate the integer part and the fractional part and the subscripts 2 and 10 to denote the binary number and the decimal number. Another example given by x 2 = b 2 b 1 b 0 b −1 b −2 b −3 b −4 (7.1) has a decimal value computed as x 10 = b 2 2 + b 1 1 + b 0 0 + b −1 −1 + b −2 −2 + b −3 −3 + b 4 −4 (7.2) where the bits b 2 ,b 1 ,b 0 ,b −1 ,b −2 ,b −3 ,b −4 are either 1 or 0. In general, when x 2 is represented as x 2 = b I −1 b I −2 ···b 1 b 0 b −1 b −2 ···b −F (7.3) the decimal number has a value given by x 10 = I −1  i=−F b i 2 i (7.4) In the binary representation (7.3), the integer part contains I bits and the bit b I −1 at the leftmost position is called the most significant bit (MSB); the fractional part contains F bits, and the bit b −F at the rightmost position is called the least significant bit (LSB). This can only represent the magnitude of positive numbers and is known as the unsigned fixed-point binary number. In order to represent positive as well as negative numbers, one more bit called the sign bit is added to the left of the MSB. The sign bit, represented by the symbol s in (7.5), assigns a negative sign when this bit is 1 and a positive sign when it is 0. So it becomes a signed magnitude fixed-point binary number. Therefore a signed magnitude number x 2 = 11001  1010 is x 10 =−9.625. In general, the signed magnitude fixed-point number is given by x 10 = (−1) s I −1  i=−F b i 2 i (7.5) and the total number of bits is called the wordlength w = 1 + I + F .When two signed magnitude numbers with widely different values for the integer part and/or the fractional part have to be added, it is not easy to program the adders in the digital hardware to implement this operation. So it is common practice to choose I = 0, keeping the sign bit and the bits for the fractional part only so that F = w − 1 in the signed magnitude fixed-point representation. But when two numbers larger than 0.5 in decimal value are added, their sum is larger than 1, and this cannot be represented by the format shown above, where I = 0. 362 QUANTIZED FILTER ANALYSIS So two other form of representing the numbers are more commonly used: the one’s-complement and two’s-complement forms (also termed one-complementary and two-complementary forms) for representing the signed magnitude fixed-point numbers. In the one’s-complement form, the bits of the fractional part are replaced by their complement, that is, the ones are replaced by zeros and vice versa. By adding a one as the least significant bit to the one’s-complement form, we get the two’s-complement form of binary representation; the sign bit is retained in both forms. But it must be observed that when the binary number is positive, the signed magnitude form, one’s-complement form, and two’s-complement form are the same. Example 7.1 Given: x 2 = 0  1100 is the 5-bit, signed magnitude fixed-point number equal to x 10 =+2 −1 + 2 −2 = 0.75 and v 2 = 1  1100 is equal to v 10 =−0.75. The one’s complement of v 2 = 1  1100 is 1  0011, whereas the two’s complement of v 2 is 1  0011 +  0001 = 1  0100. The values that can be represented by the signed magnitude fixed-point repre- sentation range from −2 w−F −1 to 2 w−F −1 − 2 −F . In order to increase the range of numbers that can be represented, two more formats are available: the floating- point and block floating-point representations. The floating-point representation of a binary number is of the form X 10 = (1) s M(2 E ) (7.6) where M is the mantissa, which is usually represented by a signed magnitude, fixed-point binary number, and E is a positive- or negative-valued integer with E bits and is called the exponent. To get both positive and negative exponents, the bias is provided by an integer, usually the bias is chosen as e 7 − 1 = 127 when the exponent E is 8 bits or e 10 − 1 = 1023 when E is 11 bits. Without the bias, an 8-bit integer number varies from 0 to 255, but with a bias of 127, the exponent varies from −127 to 127. Also the magnitude of the fractional part F is limited to 0 ≤ M<1. In order to increase the range of the mantissa, one more bit is added to the most significant bit of F so that it is represented as (1.F ). Now it is assumed to be normalized, but this bit is not counted in the total wordlength. The IEEE 754-1985 standard for representing floating-point numbers is the most common standard used in DSP processors. It uses a single-precision format with 32 bits and a double-precision format with 64 bits. The single-precision floating point number is given by X 10 = (−1) s (1.F )2 E−127 (7.7) According to this standard, the (32-bit) single-precision, floating-point number uses one sign bit, 8 bits for the exponent, and 23 bits for the fractional part BINARY NUMBERS AND ARITHMETIC 363 (b) b 11 s b 10 b 0 E (11 bits) F (52 bits) b − 1 b −52 (a) b 8 s b 7 b 0 E (8 bits) F (23 bits) b − 1 b −23 Figure 7.2 IEEE format of bits for the 32- and 64-bit floating-point numbers. F (and one bit to normalize it). A representation of this format is shown in Figure 7.2a. But this formula is implemented according to the following rules in order to satisfy conditions other than the first one listed below: 1. When 0 <E<255, then X 10 = (−1) s (1  F)2 E−127 . 2. When E = 0andM = 0, then X 10 = (−1) s (0  F)(2 −126 ). 3. When E = 255 and M = 0, then X 10 is not a number and is denoted as NaN. 4. When E = 255 and M = 0, then X 10 = (−1) s ∞. 5. When E = 0andM = 0, then X 10 = (−1) s (0). Here, (1  F)is the normalized mantissa with one integer bit and 23 fractional bits, whereas (0  F) is only the fractional part with 23 bits. Most of the commercial DSP chips use this 32-bit, single-precision, floating-point binary representation, although 64-bit processors are becoming available. Note that there is no provision for storing the binary point (  ) in these chips; their registers simply store the bits and implement the rules listed above. The binary point is used only as a notation for our discussion of the binary number representation and is not counted in the total number of bits. The IEEE 754-1985 standard for the (64-bit), double-precision, floating-point number is expressed by X 10 = (−1) s (1.F )2 E−1023 (7.8) It uses one sign bit, 11 bits for the exponent E, and 52 bits for F (one bit is added to normalize it but is not counted). The representation for this format is shown in Figure 7.2b. Example 7.2 Consider the 16-bit floating-point number with 8 bits for the unbiased exponent and 4 bits for the denormalized fractional part, namely, E = 8andF = 4. The [...]... response 1.4 Filter #1: Reference Filter #1: Quantized 1.2 Magnitude 1 0.8 0.6 Lattice ARMA Lowpass: Elliptic, IIR Filter Format [8 7] for Filter Coefficients 0.4 0.2 0 0 5 10 15 Frequency (kHz) 20 Figure 7.8 Magnitude responses of reference filter and quantized filter with format [8 7] and lattice ARMA structure 372 QUANTIZED FILTER ANALYSIS Magnitude response in dB Filter #1: Reference Filter #1: Quantized. .. Format [9 8] for Filter Coefficients Filter #1: Reference magnitude Filter #1: Quantized magnitude 0.28 Phase (degrees) Magnitude 1.12 −320 Filter #1: Reference phase Filter #1: Quantized phase 0 0 5 10 15 Frequency (kHz) 20 −400 Figure 7.5 Magnitude response of reference filter and quantized filter with format [9 8] in cascade connection of second-order sections 370 QUANTIZED FILTER ANALYSIS Magnitude... left column are shown the corresponding coefficients of the quantized filter with a 8 bit wordlength in the fixed-point, signed magnitude format [8 7] QUANTIZED FILTER ANALYSIS Magnitude and phase responses 0 1.2 Magnitude 0.96 −80 0.72 −160 0.48 Filter #1: Reference magnitude Filter #1: Quantized magnitude Filter #1: Reference phase Filter #1: Quantized phase 0.24 0 −320 Magnitude and Phase Response of... 0.84 −360 0.56 −540 0.28 0 −720 Filter #1: Reference magnitude Filter #1: Quantized magnitude Filter #1: Reference phase Filter #1: Quantized phase 0 5 10 15 Frequency (kHz) Phase (degrees) Magnitude 1.12 20 −900 Figure 7.16 Magnitude and phase responses of reference FIR filter and quantized filter with format [7 6] for filter coefficients QUANTIZATION ANALYSIS OF FIR FILTERS Magnitude and phase responses... Response of Lowpass, Equiripple FIR Filter Format [8 7] for Filter Coefficients −180 0.84 −360 0.56 −540 0.28 0 −720 Filter #1: Reference magnitude Filter #1: Quantized magnitude Filter #1: Reference phase Filter #1: Quantized phase 0 5 10 15 Frequency (kHz) Phase (degrees) Magnitude 1.12 −900 20 Figure 7.17 Magnitude and phase responses of FIR reference filter and quantized filter with format [8 7] for... dB Filter #1: Reference Filter #1: Quantized 1.5 Magnitude (dB) 1 0.5 0 −0.5 −1 Magnified Magnitude in dB in the passband of the Lowpass, Equiripple FIR Filter Format [8 7] for Filter Coefficients −1.5 1 2 3 4 5 6 Frequency (kHz) 7 8 9 Figure 7.18 Magnified magnitude responses (in decibels) of reference FIR filter and quantized filter with format [8 7] for the filter coefficients 378 QUANTIZED FILTER ANALYSIS. .. the quantized filter with the format [9 8], the same as the direct form QUANTIZATION ANALYSIS OF IIR FILTERS 371 Magnitude response 1.4 Filter #1: Reference Filter #1: Quantized 1.2 Magnitude 1 0.8 0.6 0.4 Magnified Response of Direct Form II, Second Order Sections in Cascade Format [8 7] for Filter Coefficients 0.2 0 0 5 15 10 Frequency (kHz) 20 Figure 7.7 Magnitude responses of reference filter and quantized. .. of a lattice-coupled allpass structure, quantized with a format [9 8] QUANTIZATION ANALYSIS OF IIR FILTERS 373 Magnitude response 1.2 Filter #1: Reference Filter #1: Quantized 1 Magnitude 0.8 0.6 0.4 Lattice Coupled Allpass Structure Format [7 6] for Filter Coefficients 0.2 0 0 5 10 15 Frequency (kHz) 20 Figure 7.11 Magnitude responses of reference filter and quantized filter with format [7 6], in lattice-coupled... wordlength 376 QUANTIZED FILTER ANALYSIS Magnitude response 1.4 Filter #1 1.2 Magnitude 1 0.8 0.6 0.4 Magnitude Response of Lowpass, Equiripple, Reference Filter Order 16 0.2 0 0 5 10 15 Frequency (kHz) 20 Figure 7.15 Magnitude response of a lowpass equiripple FIR (reference filter) filter Magnitude and phase responses 1.4 0 Magnitude and Phase Response of Lowpass, Equiripple FIR Filter Format [7 6] for Filter. .. Elliptic Filter Format [8 7] for Filter Coefficients −1.5 1 2 3 4 5 6 Frequency (kHz) 7 8 9 Figure 7.9 Magnified plot of the magnitude responses (in decibels) of reference filter and quantized filter with format [8 7] in lattice ARMA structure Magnitude response 1.4 Filter #1: Reference Filter #1: Quantized 1.2 Magnitude 1 0.8 0.6 0.4 0.2 0 Lattice Coupled Allpass Structure Format [9 8] for filter Coefficients . 10 20 15 Frequency (kHz) Filter #1: Reference magnitude Filter #1: Quantized magnitude Filter #1: Reference phase Filter #1: Quantized phase Magnitude. cascade connection of second-order sections. 370 QUANTIZED FILTER ANALYSIS Filter #1: Reference Filter #1: Quantized 2.5 2 1.5 0.5 0 −0.5 −1 −1.5 −2 1 Magnitude