SYSTEM REALIZATION
7.1 DIGITAL FILTER REALIZATION STRUCTURES
SYSTEM REALIZATION
This chapter discusses the steps and issues of system realization which involved synthesizing the filters used in the design, inserting the synthesized filter structures into the fixed-point system model and validating the system model with realized filter structures.
7.1 DIGITAL FILTER REALIZATION STRUCTURES
Realization structures are essentially the block or signal flow diagram representations of the different theoretically equivalent ways a digital filter transfer function can be arranged. In most of the cases, they consist of an interconnection of unit delay elements, multipliers, and adders.
Two types of realization methods have been proposed, namely, direct and indirect. In direct methods, the transfer function is put in some form that allows the identification of an interconnection of elemental digital filter subnetworks of low order. The most commonly used realization methods of this class are [13-17]:
1) Direct / Direct Form I 2) Transposed Direct Form I 3) Direct Canonic / Direct Form II 4) Transposed Direct Form II 5) Linear Phase
6) Transposed Linear Phase 7) Frequency Sampling 8) Fast Convolution
DESIGN OF AN INTEGRATED GFSK DEMODULATOR FOR A BLUETOOTH RECEIVER 64 9) Parallel
10) Series / Cascade 11) Lattice
12) Transposed Lattice 13) Ladder
14) Systolic 15) State-Space
In indirect methods, an analog filter network is converted into a topologically related digital filter network through the application of network theoretic concepts in conjunction with some simple transformations, e.g., wave structures.
Digital filter structures obtained by different methods can differ quite significantly with respect to complexity, number of elements, and their properties. One structure might require a large number of multipliers and yet be relatively insensitive to coefficient quantization errors, and a second structure might be economical in terms of elements but generate parasitic oscillations when signals are quantized, and so on.
Transversal / Direct Form I
The transversal / direct / tapped delay line structure is the most popular FIR structure. The input, x(k), and output of the FIR filter for the direct form structure are related simply by:
∑
=− −Figure 7.1 Direct Form I Filter Structure
The symbol z-1 represents a delay of one sample or unit of time. Thus x(n-1) is x(n) delayed by one sample. In digital implementations, the boxes labeled z-1 could represent shift registers. The output sample, y(n), is a weighted sum of the present input, x(n), and N-1 previous samples of the input, i.e., x(n-1) to x(n-N). For the transversal structure, the computation of each output sample, y(n), requires:
• N-1 memory locations to store the N-1 input samples,
• N memory locations to store the N coefficients,
• N multiplications
• N-1 additions
Alternative filter structures can be obtained by using the transposition theorem. In essence, a new filter structure is obtained by reversing the direction of all branches in a signal flow graph and changing inputs to outputs and vice versa. This new filter structure has the same transfer function as the original filter but the numerical properties are generally different.
Transposed Direct Form I
The transpose structure is similar to the direct structure, except that the partial sums feed into succeeding stages. This method is more susceptible to roundoff noise than the direct method.
Direct Canonic / Direct Form II
A digital network is said to be canonic if the number of unit delay elements employed is equal to the order of the transfer function.
Linear Phase Structure
A variation of the transversal structure is the linear phase structure which takes advantage of the symmetry or anti-symmetry in the impulse response coefficients for linear phase FIR filters to reduce the computational complexity of the filter implementation.
+
z-1
z-1 z-1
x(n)
h(0) h(1) h(2)
y(n)
z-1 z-1 z-1
+ +
+
h(3)
Figure 7.2 A Linear Phase Structure for an FIR Filter with 7 Coefficients
In a linear-phase filter, the coefficients are symmetrical, i.e., h(n)=±h(N-n-1). Thus the filter equation can be rewritten to take account of this symmetry with a consequent reduction in both the number of multiplications and additions. The number of multiplications is reduced from N to N/2 for N even and to (N-1)/2 for N odd. A major drawback is that the group delay for linear phase FIR filters is often too large to be useful in many applications.
Frequency Sampling
In the frequency sampling structure, the filters are characterized by the samples of the desired frequency response, H(k), instead of its impulse response coefficients. For
DESIGN OF AN INTEGRATED GFSK DEMODULATOR FOR A BLUETOOTH RECEIVER 66 narrowband filters, most of the frequency samples will be zero, and so the resulting frequency sampling filter will require a smaller number of coefficients and hence multiplications and additions than an equivalent transversal structure. However, the frequency sampling structures suffer from high coefficient sensitivity, low dynamic range, and severe stability problems.
Fast Convolution
The fast convolution method involves performing the convolution operation in the frequency domain. Convolution in the time domain is equivalent to multiplication in the frequency domain. Filtering is performed by first computing the DFTs of x(n) and h(n) (suitably zero padded), multiplying these together and then obtaining their inverse. In practice, techniques known as overlap-add and overlap-save are used in real-time filtering.
Lattice Form
FIR filters that are embedded in adaptive FIR filters are often realized by a lattice structure.
A drawback of such a structure is that the number of operations is high since there are two multiplications and two additions for each filter coefficient.
Series / Cascade
High-order IIR filters are often realized as a cascade of several low-order filters in order to reduce the sensitivity to coefficient errors. This approach, in principle, can also be used for FIR filters, but the benefits are offset by a decrease in dynamic signal range. In the cascade realization, the transfer function, H(z), is expressed as the product of second-order and first-order sections.
Parallel
Parallel filter structures, comprising first- and second-order filter sections, can be obtained by expanding the filter transfer function into partial fractions.