Implementation Structures for Recursive Digital Filters
This chapter introduces the four direct-form filter implementations, and discusses implementation of filters as parallel or series combinations of smaller filter sections. A careful study of filter forms can be important when numerical issues arise, such as when implementing a digital filter in a fixed-point processor. (The least expensive ``DSP chips'' use fixed-point numerical processing, sometimes with only a 16-bit word length in low-cost audio applications.) For implementations in floating-point arithmetic, especially at 32-bit word-lengths or greater, the choice of filter implementation structure is usually not critical. In matlab software, for example, one rarely uses anything but the filter function, which is implemented in double-precision floating point (typically 64 bits at the time of this writing, and more bits internally in the floating-point unit).The Four Direct Forms
Direct-Form I
As mentioned in §5.5, the difference equation![]() |
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(10.1) |
specifies the Direct-Form I (DF-I) implementation of a digital filter [60]. The DF-I signal flow graph for the second-order case is shown in Fig.9.1. The DF-I structure has the following properties:
- It can be regarded as a two-zero filter section followed in series by a two-pole filter section.
- In most fixed-point arithmetic schemes (such as two's complement,
the most commonly used
[84]10.1)
there is no possibility of internal filter overflow. That is,
since there is fundamentally only one summation point in the filter,
and since fixed-point overflow naturally ``wraps around'' from the
largest positive to the largest negative number and vice versa, then
as long as the final result
is ``in range'', overflow is avoided, even when there is overflow of intermediate results in the sum (see below for an example). This is an important, valuable, and unusual property of the DF-I filter structure.
- There are twice as many delays as are necessary. As a result,
the DF-I structure is not canonical with respect to delay. In
general, it is always possible to implement an
th-order filter using only
delay elements.
- As is the case with all direct-form filter structures (those which have coefficients given by the transfer-function coefficients), the filter poles and zeros can be very sensitive to round-off errors in the filter coefficients. This is usually not a problem for a simple second-order section, such as in Fig.9.1, but it can become a problem for higher order direct-form filters. This is the same numerical sensitivity that polynomial roots have with respect to polynomial-coefficient round-off. As is well known, the sensitivity tends to be larger when the roots are clustered closely together, as opposed to being well spread out in the complex plane [18, p. 246]. To minimize this sensitivity, it is common to factor filter transfer functions into series and/or parallel second-order sections, as discussed in §9.2 below.
Two's Complement Wrap-Around
In this section, we give an example showing how temporary overflow in two's complement fixed-point causes no ill effects. In 3-bit signed fixed-point arithmetic, the available numbers are as shown in Table 9.1.
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Let's perform the sum




![$ [-4,3]$](http://www.dsprelated.com/josimages_new/filters/img1129.png)



Direct Form II
The signal flow graph for the Direct-Form-II (DF-II) realization of the second-order IIR filter section is shown in Fig.9.2. The difference equation for the second-order DF-II structure can be written as
- It can be regarded as a two-pole filter section followed by a two-zero filter section.
- It is canonical with respect to delay. This happens because delay elements associated with the two-pole and two-zero sections are shared.
- In fixed-point arithmetic, overflow can occur at the delay-line input (output of the leftmost summer in Fig.9.2), unlike in the DF-I implementation.
- As with all direct-form filter structures, the poles
and zeros are sensitive to round-off errors in the coefficients
and
, especially for high transfer-function orders. Lower sensitivity is obtained using series low-order sections (e.g., second order), or by using ladder or lattice filter structures [86].
More about Potential Internal Overflow of DF-II
Since the poles come first in the DF-II realization of an IIR filter, the signal entering the state delay-line (see Fig.9.2) typically requires a larger dynamic range than the output signal
Transposed Direct-Forms
The remaining two direct forms are obtained by formally transposing direct-forms I and II [60, p. 155]. Filter transposition may also be called flow graph reversal, and transposing a Single-Input, Single-Output (SISO) filter does not alter its transfer function. This fact can be derived as a consequence of Mason's gain formula for signal flow graphs [49,50] or Tellegen's theorem (which implies that an LTI signal flow graph is interreciprocal with its transpose) [60, pp. 176-177]. Transposition of filters in state-space form is discussed in §G.5. The transpose of a SISO digital filter is quite straightforward to find: Reverse the direction of all signal paths, and make obviously necessary accommodations. ``Obviously necessary accommodations'' include changing signal branch-points to summers, and summers to branch-points. Also, after this operation, the input signal, normally drawn on the left of the signal flow graph, will be on the right, and the output on the left. To renormalize the layout, the whole diagram is usually left-right flipped. Figure 9.3 shows the Transposed-Direct-Form-I (TDF-I) structure for the general second-order IIR digital filter, and Fig.9.4 shows the Transposed-Direct-Form-II (TDF-II) structure. To facilitate comparison of the transposed with the original, the inputs and output signals remain ``switched'', so that signals generally flow right-to-left instead of the usual left-to-right. (Exercise: Derive forms TDF-I/II by transposing the DF-I/II structures shown in Figures 9.1 and 9.2.)![]() |
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Numerical Robustness of TDF-II
An advantage of the transposed direct-form II structure (depicted in Fig.9.4) is that the zeros effectively precede the poles in series order. As mentioned above, in many digital filters design, the poles by themselves give a large gain at some frequencies, and the zeros often provide compensating attenuation. This is especially true of filters with sharp transitions in their frequency response, such as the elliptic-function-filter example on page![[*]](../icons/crossref.png)

Series and Parallel Filter Sections
In many situations it is better to implement a digital filter
Series Second-Order Sections
For many filter types, such as lowpass, highpass, and bandpass filters, a good choice of implementation structure is often series second-order sections. In fixed-point applications, the ordering of the sections can be important. The matlab function tf2sos10.5 converts from ``transfer function form'',
BAMatrix = tf2sos(B,A);converts the real filter specified by polynomial vectors B and A to a series of second-order sections (biquads) specified by the rows of BAMatrix. Each row of BAMatrix is of the form
![$ [b_0,b_1,b_2,\,1,a_1,a_2]$](http://www.dsprelated.com/josimages_new/filters/img1145.png)
function [sos,g] = tf2sos(B,A) [z,p,g]=tf2zp(B(:)',A(:)'); % Direct form to (zeros,poles,gain) sos=zp2sos(z,p,g); % (z,p,g) to series second-order sections
Matlab Example
The following matlab example expands the filter
B=[1 0 0 0 0 1]; A=[1 0 0 0 0 .9]; [sos,g] = tf2sos(B,A) sos = 1.00000 0.61803 1.00000 1.00000 0.60515 0.95873 1.00000 -1.61803 1.00000 1.00000 -1.58430 0.95873 1.00000 1.00000 -0.00000 1.00000 0.97915 -0.00000 g = 1The g parameter is an input (or output) scale factor; for this filter, it was not needed. Thus, in this example we obtained the following filter factorization:

% Numerically challenging "clustered roots" example: [B,A] = zp2tf(ones(10,1),0.9*ones(10,1),1); [sos,g] = tf2sos(B,A); [Bh,Ah] = sos2tf(sos,g); format long; disp(sprintf('Relative L2 numerator error: %g',... norm(Bh-B)/norm(B))); % Relative L2 numerator error: 1.26558e-15 disp(sprintf('Relative L2 denominator error: %g',... norm(Ah-A)/norm(A))); % Relative L2 denominator error: 1.65594e-15Thus, in this test, the original direct-form filter is compared with one created from the second-order sections. Such checking should be done for high-order filters, or filters having many poles and/or zeros close together, because the polynomial factorization used to find the poles and zeros can fail numerically. Moreover, the stability of the factors should be checked individually.
Parallel First and/or Second-Order Sections
Instead of breaking up a filter into a series of second-order sections, as discussed in the previous section, we can break the filter up into a parallel sum of first and/or second-order sections. Parallel sections are based directly on the partial fraction expansion (PFE) of the filter transfer function discussed in §6.8. As discussed in §6.8.3, there is additionally an FIR part when the order of the transfer-function denominator does not exceed that of the numerator (i.e., when the transfer function is not strictly proper). The most general case of a PFE, valid for any finite-order transfer function, was given by Eq.
where





First-Order Complex Resonators
For distinct poles, the recursive terms in the complete partial fraction expansion of Eq.

Real Second-Order Sections
In practice, however, signals are typically real-valued functions of time. As a result, for real filters (§5.1), it is typically more efficient computationally to combine complex-conjugate one-pole sections together to form real second-order sections (two poles and one zero each, in general). This process was discussed in §6.8.1, and the resulting transfer function of each second-order section becomeswhere










![[*]](../icons/crossref.png)
Implementation of Repeated Poles
Fig.9.5 illustrates an efficient implementation of terms due to a repeated pole with multiplicity three, contributing the additive terms
Formant Filtering Example
In speech synthesis [27,39], digital filters are often used to simulate formant filtering by the vocal tract. It is well known [23] that the different vowel sounds of speech can be simulated by passing a ``buzz source'' through a only two or three formant filters. As a result, speech is fully intelligible through the telephone bandwidth (nominally only 200-3200 Hz). A formant is a resonance in the voice spectrum. A single formant may thus be modeled using one biquad (second-order filter section). For example, in the vowel![$ [a]$](http://www.dsprelated.com/josimages_new/filters/img1161.png)
![$ [a]$](http://www.dsprelated.com/josimages_new/filters/img1161.png)
F = [700, 1220, 2600]; % Formant frequencies (Hz) BW = [130, 70, 160]; % Formant bandwidths (Hz) fs = 8192; % Sampling rate (Hz) nsecs = length(F); R = exp(-pi*BW/fs); % Pole radii theta = 2*pi*F/fs; % Pole angles poles = R .* exp(j*theta); % Complex poles B = 1; A = real(poly([poles,conj(poles)])); % freqz(B,A); % View frequency response: % Convert to parallel complex one-poles (PFE): [r,p,f] = residuez(B,A); As = zeros(nsecs,3); Bs = zeros(nsecs,3); % complex-conjugate pairs are adjacent in r and p: for i=1:2:2*nsecs k = 1+(i-1)/2; Bs(k,:) = [r(i)+r(i+1), -(r(i)*p(i+1)+r(i+1)*p(i)), 0]; As(k,:) = [1, -(p(i)+p(i+1)), p(i)*p(i+1)]; end sos = [Bs,As]; % standard second-order-section form iperr = norm(imag(sos))/norm(sos); % make sure sos is ~real disp(sprintf('||imag(sos)||/||sos|| = %g',iperr)); % 1.6e-16 sos = real(sos) % and make it exactly real % Reconstruct original numerator and denominator as a check: [Bh,Ah] = psos2tf(sos); % parallel sos to transfer function % psos2tf appears in the matlab-utilities appendix disp(sprintf('||A-Ah|| = %g',norm(A-Ah))); % 5.77423e-15 % Bh has trailing epsilons, so we'll zero-pad B: disp(sprintf('||B-Bh|| = %g',... norm([B,zeros(1,length(Bh)-length(B))] - Bh))); % 1.25116e-15 % Plot overlay and sum of all three % resonator amplitude responses: nfft=512; H = zeros(nsecs+1,nfft); for i=1:nsecs [Hiw,w] = freqz(Bs(i,:),As(i,:)); H(1+i,:) = Hiw(:).'; end H(1,:) = sum(H(2:nsecs+1,:)); ttl = 'Amplitude Response'; xlab = 'Frequency (Hz)'; ylab = 'Magnitude (dB)'; sym = ''; lgnd = {'sum','sec 1','sec 2', 'sec 3'}; np=nfft/2; % Only plot for positive frequencies wp = w(1:np); Hp=H(:,1:np); figure(1); clf; myplot(wp,20*log10(abs(Hp)),sym,ttl,xlab,ylab,1,lgnd); disp('PAUSING'); pause; saveplot('../eps/lpcexovl.eps'); % Now synthesize the vowel [a]: nsamps = 256; f0 = 200; % Pitch in Hz w0T = 2*pi*f0/fs; % radians per sample nharm = floor((fs/2)/f0); % number of harmonics sig = zeros(1,nsamps); n = 0:(nsamps-1); % Synthesize bandlimited impulse train for i=1:nharm, sig = sig + cos(i*w0T*n); end; sig = sig/max(sig); speech = filter(1,A,sig); soundsc([sig,speech]); % hear buzz, then 'ah'Notes:
- The sampling rate was chosen to
be
Hz because that is the default Matlab sampling rate, and because that is a typical value used for ``telephone quality'' speech synthesis.
- The psos2tf utility is listed in §J.7.
- The overlay of the amplitude responses are shown in Fig.9.6.
Butterworth Lowpass Filter Example
This example illustrates the design of a 5th-order Butterworth lowpass filter, implementing it using second-order sections. Since all three sections contribute to the same passband and stopband, it is numerically advisable to choose a series second-order-section implementation, so that their passbands and stopbands will multiply together instead of add.fc = 1000; % Cut-off frequency (Hz) fs = 8192; % Sampling rate (Hz) order = 5; % Filter order [B,A] = butter(order,2*fc/fs); % [0:pi] maps to [0:1] here [sos,g] = tf2sos(B,A) % sos = % 1.00000 2.00080 1.00080 1.00000 -0.92223 0.28087 % 1.00000 1.99791 0.99791 1.00000 -1.18573 0.64684 % 1.00000 1.00129 -0.00000 1.00000 -0.42504 0.00000 % % g = 0.0029714 % % Compute and display the amplitude response Bs = sos(:,1:3); % Section numerator polynomials As = sos(:,4:6); % Section denominator polynomials [nsec,temp] = size(sos); nsamps = 256; % Number of impulse-response samples % Note use of input scale-factor g here: x = g*[1,zeros(1,nsamps-1)]; % SCALED impulse signal for i=1:nsec x = filter(Bs(i,:),As(i,:),x); % Series sections end % %plot(x); % Plot impulse response to make sure % it has decayed to zero (numerically) % % Plot amplitude response % (in Octave - Matlab slightly different): figure(2); X=fft(x); % sampled frequency response f = [0:nsamps-1]*fs/nsamps; grid('on'); axis([0 fs/2 -100 5]); legend('off'); plot(f(1:nsamps/2),20*log10(X(1:nsamps/2)));The final plot appears in Fig.9.7. A Matlab function for frequency response plots is given in §J.4. (Of course, one can also use freqz in either Matlab or Octave, but that function uses subplots which are not easily printable in Octave.) Note that the Matlab Signal Processing Toolbox has a function called sosfilt so that ``y=sosfilt(sos,x)'' will implement an array of second-order sections without having to unpack them first as in the example above.
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Summary of Series/Parallel Filter Sections
In summary, we noted above the following general guidelines regarding series vs. parallel elementary-section implementations:- Series sections are preferred when all sections contribute to the same passband, such as in a lowpass, highpass, bandpass, or bandstop filter.
- Parallel sections are usually preferred when the sections have disjoint passbands, such as a formant filter bank used in voice models. Another example would be the phase vocoder filter bank [21].
Pole-Zero Analysis Problems
See http://ccrma.stanford.edu/~jos/filtersp/Pole_Zero_Analysis_Problems.html.Next Section:
Filters Preserving Phase
Previous Section:
Pole-Zero Analysis