Comparison of Finite-Precision IIR Bandpass Filter Structures for Software Radio Systems

L. M. Naji, K. Choi

Abstract—In software radio systems, a digital IIR filter

offers good roll-off and transition bandwidth

characteristics using a low amount of arithmetic

components, thus simplifying hardware implementation.

However, IIR filter structures are very sensitive to finite

precision compared to FIR filter structures. In order to

design a high quality, low component count, digital

narrowband bandpass filter onto an FPGA or directly on

VLSI, structures of 2nd

order bandpass filters are

compared according to their component count, component

latency, band narrowness, ease of tuning, and sensitivity to

low finite precision. A Direct form structure is shown to

be the best method for hardware implementation when

designing a digital 2nd

order tunable bandpass filter

structure.

Keywords—IIR filter structures, finite-precision filters, tunable bandpass filters.

I. INTRODUCTION

IGITAL filters provide many versatility advantages overanalog filters. A tunable bandpass filter is typically used

for radio receiver systems in order to tune a specific band from the entire spectrum of the received signal. Analog filtersusually consist of electrical components, such as resistors,capacitors, inductors, and semi-conductor devices, operatingon a continuous-time domain with infinite precision. Analogfilters can cheaply be implemented in order to achieve highfiltering performance using low-cost components. However, when an analog filter needs to be updated, the entire filtering hardware usually needs to be modified, providing little versatility.

Digital filters are usually implemented using software,which is executed by a digital microprocessor. The source signal is first pre-filtered to eliminate aliasing, then it issampled and quantized as a stream of binary data, which ispassed through a digital filtering system executed by the microprocessor used. If a digital filter requires an update, its corresponding software can be updated and reloaded onto itscorresponding hardware device. This process yields a lowercost per update compared to analog filters.

A radio transmitter/receiver system which consists of adigital tuner, digital modem (modulator/demodulator), and

digital processor is known as a software radio system. Inorder to reduce the cost of digital hardware, the digitalcomponents implemented on-chip must be of relatively lowsize and low count. Therefore, the bandpass filter used fortuning must be a high-performance filter of low order.Because digital IIR band-pass filters can achieve narrowbandwidths under 2nd order structures, the scope of thissubmission is limited to 2nd order IIR digital bandpass filter structures. Each structure holds different performancecharacteristics under finite-precision and latency-dependentprocessing. Therefore, these structures are compared based upon their finite-precision sensitivity, processing latency, andease of tuning. Table 1 provides descriptions of commonlyused symbols throughout this paper.

II. STANDARD 2ND ORDER IIR DIGITAL FILTER

A standard 2nd order IIR digital filter is a function based ontwo parameters: a center frequency (tuning) parameter and a bandwidth adjusting parameter. A filter’s frequency response is described by a z-transform function, where z is e2 Fc/Fs. The roots of the numerator of this function are known as “zeros”. The roots of the denominator of this function are known as“poles”. Zeros and poles are usually of complex-numbervalues. The angles of the values of zeros determine the frequencies where notches, or dips, appear in the frequencyresponse magnitude plot. The angles of the values of polesdetermine the frequencies where peaks appear in the frequency response magnitude plot. The closer themagnitudes of zeros and poles are to 1, the more effectivethese zeros and poles are (i.e. resulting in sharper dips andpeaks). The z-transform of a standard 2nd order IIR bandpass filter is:

TABLE ICOMMONLY USED UNITS FOR THIS PAPER

Symbol Quantity Unit

Discrete-time frequency Radians/sample

Fc Center Frequency Hz

Fs Sampling Frequency Samples/second

BW 3dB Bandwidth Hz

ADLAT Adder Latency Seconds

MULAT Multiplier Latency seconds

D

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21

2

)cos(11

1

2

1

zz

zzH

c

(1)

In a standard narrow-band 2nd order filter, a peak is requiredat a frequency of FC, therefore, poles must be located atfrequencies (or angles) of =±2 FC/FS. In order to ensure stability, poles must have a magnitude less than 1. is themagnitude-square of the pole locations, which relates to the3dB bandwidth by:

2

1

1

2cos

2SF

BW (2)

The closer the pole magnitude is to 1, the narrower thebandwidth. This results in a common z-transformdenominator format for 2nd-order narrowband IIR bandpass filters. Because frequency notches in bandpass filters may notbe necessary, the numerator of the transfer function above isnot very critical. The numerator in (1) places notches at =0and = (thus, notches are located at frequencies of DC and FS/2). Depending on the application the filter is used for, thismay be removed, reducing the numerator to 1.

The transfer functions of all the filter structures compared inthis paper have the same denominator, the critical and mosteffective component in bandpass filtering, but may vary in numerator expressions, which may be discarded.

III. FILTER STRUCTURES

A. DIRECT FORM

A Direct form structure is a direct implementation of thefilter’s transfer function. Direct form structures usuallyconsist of two sub-structures, a structure for the numerator(zeros), and a structure for the denominator (poles) of thetransfer function. Because of finite precision error, and because of importance of pole locations in IIR bandpassfilters, the denominator structure is cascaded in front of thenumerator structure. Also, because adders and multipliersrequire time for processing (latency), delay free paths, consisting of adders and multipliers in series, accumulate atotal amount of latency. Therefore, the sampling interval mustnot be shorter than the worst-case latency in the filter structure. Path latency can be reduced by placing sampledelays (1/z) between filter and between numerator anddenominator structures. A Direct form implementation of (1)is shown in Fig. 1.

Fig. 1: Direct Form structure

In this structure, a1 is the denominator coefficientcorresponding to z-1, a2 is the denominator coefficientcorresponding to z-2, and G is the gain component of (1- )/2.In order to maintain a high SNR, a high signal magnitudemust be processed by the filter. Therefore, since G is usuallyan attenuation coefficient instead of a gain coefficient, it is included in the post-filtering stage.

This Direct form structure requires the use of 3 multipliersand 3 adders. The longest delay-free path latency is2×(ADLAT) + 1×( MULAT). This structure consists of 2 BW

dependent coefficients and 1 BW and FC dependentcoefficient.

B. ALL-PASS FORM.

An All-pass filter is a filter with constant magnitudethroughout the entire frequency spectrum, but may have afrequency-varying phase response. A standard 2nd order IIR All-pass filter [1] is described by:

21

21

)1(1

)1(

zzb

zzbzA (3)

In (3), poles and zeros are balanced in order to maintain a constant frequency response magnitude. The above transferfunction can be used to create equivalent bandpass and bandstop filters, where a bandpass filter transfer function is described by:

)](1[*5.)( zAzH BP (4)

In (3), b=cos( c). Applying (4) to (3) results in the standard form 2nd order bandpass filter described in (1).Applying a bandpass transformation to the structure of (3) results in the structure in Fig. 2.

Fig. 2: All-pass structure of 2nd order bandpass IIR filter

In Fig. 2, B is cos( c), a center frequency adjustingparameter, and A is , a bandwidth adjusting parameterdescribed in (2). The All-pass structure only requires twocoefficients to be adjusted, one for tuning, and one for bandwidth adjustment, making this structure simple for tuningand bandwidth adjustment. However, the All-pass structurerequires a total of 7 adders and 2 multipliers, and the longest delay-free path latency is 4×(ADLAT) + 1×( MULAT). Due to the many adders in series of several paths, this structure has poor immunity to quantization error under floating-pointprecision.

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C. LATTICE FORM.

The Lattice structure is known to perform reasonably wellunder finite-precision coefficients and operations [1]. The Lattice structure is an all-pole structure, therefore any zeros of a transfer function used for structure implementation requireselimination of the numerator term. If the numerator term iscritical for the filtering process, then either a Direct form all-zero filter must be used in cascade of the structure, or a Grey-Markel Lattice-Ladder structure must be used. Because thenumerator term has little significance in our filtering process,an all-pole band-pass transfer function is used as described by:

22

111

1

zazaGzH (5)

The denominator term of (5) is the same as the denominatorterm of (1). The gain term (G) in (5) is (1- )/2. The coefficients are then converted to Lattice-form reflection coefficients described as:

22

2

21211 ,1 akaaaak (6)

The resulting Lattice-form filter structure is shown in Fig. 3.This structure requires the use of 3 adders and 4 multipliers.A total of 3 coefficients are required, 1 coefficient is FC and BW dependent, and 2 coefficents are BW dependent only. Thelongest delay-free path latency is 3×(ADLAT) + 2×( MULAT).

Fig. 3: Lattice structure of 2nd order bandpass IIR filter

D. COUPLED FORM

In standard implementations of IIR bandpass filter structures, poles are placed in a semi-polar format under finite precision as shown in Fig. 4a. A Coupled form structure isbased on an implementation directly from a state-space transformation, which causes poles to be placed in a Cartesian-grid format as shown in Fig. 4b.

(a)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

(b)Fig, 4. Finite precision pole placement. (a) Pole placement due to Direct

Form structure. (b) Pole placement due to Coupled Form structure.

For a 2nd order IIR bandpass filter, the Coupled form state space system is described by:

)sin()cos(

00

1

1

0

2/12/1cc

T

vh

DCBhv

vhA (7)

is the bandwidth adjustment parameter described in (2).The implementation of this state-space system is shown in Fig. 5. This system requires 3 adders and 4 multipliers. A total of2 coefficients are used in this system, both are dependent on both FC and BW. The longest delay-free path latency is2×(ADLAT) + 1×( MULAT).

Fig. 5. Coupled form implementation of IIR bandpass filter

The Coupled form structure allows good tuning accuracy atspectrum ends. However, unlike a Direct form structure, this structure does not provide constant bandwidth at differenttuning frequencies under finite precision.

IV. SIMULATION AND COMPARISON

The four different bandpass filter structures are simulated

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 -0.8 -0.6 -0.4 -0

Im(z)

Re(z)

.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Im(z)

Re(z)

h

h

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under a floating point precision of 1 sign bit, 10 mantissa bits,and 8 exponent bits. For all comparisons, FS of 500MSa/S isused, Fc is set to 120MHz, and is set to .999, which providesthe sharpest possible BW setting under the 10-bit mantissasystem. All structures are simulated using a white-noise inputof 0 mean and unit variance. All filter response magnitudesare normalized to a 0dB peak for comparison purposes. Spectral averaging is used in order to obtain smooth frequencyresponse results. Fig. 6 and Fig. 7 show the frequencyresponse magnitude of the structures used in this simulation.Fig. 6 shows the results for 1 filter stage only, and Fig. 7shows the result of 2 filters cascaded in series.

As shown in Fig. 6, the All-pass structure has poor noisefloor attenuation, and the Coupled-form structure has poor BW

sharpness compared to other structures. The Lattice-form and Direct-Form structures have similar frequency responsemagnitude characteristics. When cascading 2 filters in series,all structures have similar BW characteristics. The All-pass structure, however, shows harmonics.

90 100 110 120 130 140 150-40

-35

-30

-25

-20

-15

-10

-5

0

Freq (MHz)

dB

All-pass form

Coupled Form

Direct Form, Lattice Form

Fig. 6: Comparion of structures using single filter

90 100 110 120 130 140 150-40

-35

-30

-25

-20

-15

-10

-5

0

MHz

dB

All structures

All-pass structure harmonics

Fig. 7. Comparison of structures using double filter.

V. CONCLUSION

An All-pass structure requires many adders in a single path,and is therefore sensitive to finite precision errors. Undersimulation, All-pass structure shows error due to poor noiseattenuation and harmonics.

Due to its pole-placement under finite precision, a Coupled Form structure experiences varying bandwidth when tuning.Although this structure provides better pole precision atspectrum ends than the Direct-form structure, tuning error in Direct-Form structure can be compensated by implementing a feedback system which fine-tunes the sampling rate. Thus,the desired tuning frequency can be matched exactly to a result pole location under finite-precision by sampling rate

adjustment. Also, error at spectrum ends can be eliminated by limiting the tuning range to exclude spectral areas with highpole placement error.

Although a Lattice structure is theoretically a better choice in finite precision than a Direct-Form structure, they have a very similar response under finite precision for a 2nd order system. However, a Direct-Form structure requires lessarithmetic components and has a lower worst-case pathcomputation latency than the Lattice structure.

Because minimal hardware is a requirement, coefficient computation for desired Fc and BW specifications requires anon-chip hardware arithmetic system. This arithmetic systemcan use a significant amount of on-chip area, which defeatsthe purpose of using low finite precision and a low amount of filter components. Also, an arithmetic system requiresprocessing latency, resulting in slower frequency tuning.Qui ck frequency tuning is required for features such asfrequency-hopping or signal searching. Therefore, a ROM-based look-up table is used in conjunction with the filterstructure. The look-up table contains coefficient values forevery Fc and every BW used by the system. When a new Fc

or a new BW is specified, the corresponding filter coefficients are accessed from the look-up table, and stored in multiplierregisters in the filter structure.

A Modern optimization technique can be applied to theDirect form structure by replacing essential delays with aninverse detla operator of -1= z-1/(1-z-1) [4]. Coefficient is calculated so that the filter achieves optimal performanceunder finite precision. This modification results in a morerobust structure under finite precision. However, this methodadds 4 additional adders and 4 additional multipliers to thesystem, with a longest delay-free path latency of 2×(ADLAT)

+ 1×( MULAT). Such addition of components and increase inlatency results in a more expensive structure and a slowersampling rate limit.

In conclusion, the Direct form structure, although simple, isan excellent structure for a finite-precision 2nd order bandpass IIR filter when bandwidth sharpness, arithmetic latency,component count, and accuracy are a concern.

REFERENCES

[1] S. K. Mitra. Digital Signal Processing, A Computer-Based Approach.McGraw-Hill, New York, NY, 2nd edition, 2001.

[2] J. G. Proakis and D.G. Manolakis. Digital Signal Processing,

Principles, Algorithms, and Applications. Prentice-Hall, 3rd edition, 1996.

[3] T. Bose. Digital Signal and Image Processing. John Wiley & Sons,Inc., Hoboken, NJ, 2004

[4] H. H. Dam, S. Norbedo, and L. Svensson, “Approximation of Classical IIR Filters with Additional Specifications,” IEEE Trans. Circuits and

Systems, vol. 47, pp. 1533 – 1536, Dec. 2000.[5] J. Kauraniemi, T. I. Laakso, I. Hartimo, and S.J. Ovaska, “Delta operator

realizations of direct-form IIR filters,” IEEE Trans. Circuits Syst. II,vol. 45, pp. 41-52, Jan 1998.

[6] M. Laddomada, F. Daneshgaran, M. Mondin, R.M. Hickling, “A PC-based software receiver using a novel front-end technology,” IEEE Communications Magazine, vol. 39, issue 8, pp. 136 – 145, Aug. 2001.

[7] B. Lincoln. (1991, May 10). Implementation of Discrete Time

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