CHANNEL PROCESSING TWIN DETECTORS CHANNEL PROCESSING TWIN DETECTORS CHANNEL PROCESSING TWIN DETECTORS CHANNEL PROCESSING TWIN DETECTORS

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Joan Hancock
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1 BIQUAD FILERS IN DIGIAL HYBRID INFORMAION NOE INRODUCION he DIGIAL offers great flexibility in adjusting input/output characteristics as well as shaping of the system's frequency response. he input/output characteristics of the system are adjusted by up to 4 channels of independent signal processing, AGC-O, peak clipping and low level expansion (squelch) blocks. he system's frequency response can be controlled by adjusting individual channel's gain and the crossover frequencies between channels. Also, the system's frequency response can be controlled using DIGIAL biquad filters. x b 1 b b 0 Σ a 0 = 1 -a 1 -a y he DIGIAL hybrid has four biquad filters available. he first two of them, Pre1 and Pre are located before the Band Split Filter in the signal path. hose filters are cascaded and it is important to note that if the Pre1 filter is disabled, the hardware assumes that the Pre filter is also disabled. he other two filters, PostA and PostB are located before the Notch Filter and the Peak Clipping blocks respectively. Filters PostA and PostB can be used completely independently. Please refer to the GB310-S0 Pal Configuration Block Diagram for locations of the biquad filters in the signal path. BIQUAD FILER SRUCURE Each of the DIGIAL biquad filters is an IIR (Infinite Impulse Response) digital filter implemented as Direct Form 1 structure. he block diagram of the Direct Form 1 structure is shown in Fig. Fig. Direct Form 1 IIR Structure Block Diagram he transfer function of each of the biquad filter can be expressed as follows: b 0 + b 1 + b z - H(z) = 1 + a 1 + a z - By multiplying the numerator and the denominator by z the transfer function can also be expressed as: b 0 z + b 1 z 1 + b H(z) = z + a 1 z 1 + a Note that the coefficient a 0 is hard-wired to always be a 1. he coefficients are each 16 bits in length and include one sign bit, one bit to the left of the decimal point, and 14 bits FMIC RMIC nF 70nF 60nF A/D FRONWAVE A/D FRONWAVE PRE 1 & BIQUAD FILERS WIN DEECOR BAND SPLI FILER WIN DEECORS WIN DEECORS WIN DEECORS WIN DEECORS Σ POS A BIQUAD FILER NOCH FILER VC SQUELCH ONE GENERAOR AGC-O GB310-S0 PEAK CLIPPING POS B BIQUAD FILER GB310 D/A HBRIDGE 14 1 OU+ OU- Fig. 1 GB310-SO Pal Configuration Block Diagram Doc.No [Rev. November 001] 1

2 All multiplications and additions are done with a single multiplier-accumulator (MAC) and therefore the following rules should be considered in order for the biquad filter to function properly: 1. All coefficients {b0, b1, b, a1, a} must be in the range [-;+].. he calculated filter's output signal level should not exceed the full scale representation in the filter. For example, if the input signal to the biquad is -10dBFS (10dB below Full Scale), therefore the biquad has only 10dB of headroom before it starts overflowing. Please take note that the underlying code in the product components automatically checks all of the filters in the system for stability (that is, the poles have to be within the unit circle) before updating the graphs on the screen or programming the coefficients into the hybrid. If the Interactive Data Sheet receives an exception from the underlying stability checking code, it will automatically disable the biquad being modified and display a warning message. When the filter is made stable again, it can be reenabled. DIGIAL SYSEM LIMIAIONS When designing and implementing an IIR filter, it is important to consider the limitations of any DSP system. he limitations described here might cause the IIR filter to become noisy, distorting or even unstable. In that case, it may be necessary to adjust some of the filter's parameters so it will perform as expected. When the signal is converted from analog to digital form, the precision is limited by the number of bits available. he signals are not only subject to limited precision, filter coefficients follow the same rule - they are also subject to limited precision of digital systems. he limited precision of all digital systems leads to quantization errors. hose errors due to limited precision are nonlinear and signal dependent. he nonlinearity can eventually lead to instability, particularly with IIR filters. here is no method that can eliminate quantization errors. However those errors can be minimized by keeping values large so that the maximum number of bits is used to represent them. But, there is a limit to how large the numbers can be. here is also another potential source of error in DSP systems, particularly in IIR filters. It is possible for a result of IIR filter processing to exceed the maximum number size. When that happens, the hardware may saturate or overflow. o avoid the overflow or saturation, the input signal and/or filter coefficients should be scaled down. DIGIAL IIR FILER DESIGN he following examples illustrate one of the IIR filter design algorithms. Each filter is first designed as an analog filter, the filter's characteristics (order, corner frequency, passband gain) are translated into an analog transfer function. Poles and zeros of the analog transfer function are then matched to the poles and zeros of the digital filter's transfer function using bilinear transformation. he digital filter's coefficients (from the digital filter's transfer function) can then be entered into the Biquad Section of the Interactive Data Sheet to implement that particular filter. hose examples require some knowledge of complex math and "s" and "z" operators used in analog and digital transfer functions. DIGIAL FILER DESIGN EXAMPLES: Example 1: 1st order low pass filter with 4kHz corner frequency and 0dB gain he transfer function of the 1st order analog low pass filter ω c H(s) = where ω c = πƒ c s + ω c ƒ c is -3dB corner frequency in Hz, π = he transfer function has pole s p1 at -ω c and zero s z1 at he bilinear transformation is used to convert the analog transfer function to a digital transfer function. he pole and the zero of the digital transfer function are matched to the pole and zero of the analog transfer function according to the following formula: 1 + s z = 1 - s where is the sampling period, = 1/ƒ s, and ƒ s is the sampling frequency. he corner frequency of the analog filter ƒ c = 4000Hz therefore ω c = 513. he transfer function of this analog low pass filter is: 513 H(s) =, the pole s p1 = -513, the zero s z1 = s

3 Using the bilinear transformation, sampling frequency ƒ s = 3000Hz: z p1 = z p1 = z z1 = he transfer function of the first order digital filter is: z - z z1 H(z) = G z - z p1 Gain parameter G has to be calculated so the digital filter has unity gain at DC: 1- z z1 1 - z p1 H(z) = G = 1 thus G = /z=1 1 - z p1 G = he transfer function of the digital filter: z z H(z) = = z z he filter's coefficients have to be converted to 14-bit integer numbers. he conversion is done by multiplying the fractional coefficients of the transfer function by 14 and rounding the results to the nearest integer number. he rounded integers can then be entered into one of the biquad sections in the Interactive Data Sheet and programmed into the DIGIAL hybrid. COEFFICIEN 14-BI INEGER b b b 0 0 Example : 1st order high pass filter with 500Hz corner frequency and 0dB gain he transfer function of the 1st order analog high pass filter s H(s) = where ω c = πƒ c s + ω c ƒ c is -3dB corner frequency in Hz, π = he transfer function has pole s p1 at -ω c and zero s z1 at 0. he bilinear transformation is used to convert the analog transfer function to a digital transfer function. he pole and the zero of the digital transfer function are matched to the pole and zero of the analog transfer function according to the following formula: 1 + s z = 1 - s where is sampling period, = 1/ƒ s, and ƒ s is the sampling frequency. he corner frequency of the analog high pass filter ƒ c = 500Hz therefore ω c = he transfer function of this analog low pass filter is: s H(s) = s p1 = -3141, s z1 = 0 s Using the bilinear transformation, sampling frequency ƒ s = 3000Hz: z p1 = z p1 = z z1 = he transfer function of the first order digital filter: a a 0 0 z - z z1 H(z) = G z - z p

4 Gain parameter G has to be calculated so the digital filter has unity gain at infinite frequency: -1 - z z1 1 + z p1 H(z) = G = 1 thus G = /z= z p1 G = z z H(z) = = z z he filter's coefficients have to be converted to 14-bit integer numbers. he conversion is done by multiplying the fractional coefficients of the transfer function by 14 and rounding the results to the nearest integer number. he rounded integers can then be entered into one of the biquad sections in the Interactive Data Sheet and programmed into the DIGIAL hybrid: COEFFICIEN 14-BI INEGER b b he corner frequency of the nd order analog filter ƒ c = 3000Hz therefore ω c = For the nd order filter to have -3dB at the corner frequency the damping ratio should be set to: ζ = = he transfer function of the analog low pass filter: H(s) = s s the poles s p1, p = ± j 1339, the zeros s z1 = s z = Using the bilinear transformation, sampling frequency ƒ s = 3000Hz: ( j 1339) 1 + z p1 = ( j 1339) 1 - b 0 0 a a 0 0 Example 3: nd order low pass filter with 3000Hz corner frequency and 0dB gain he transfer function of a nd order analog low pass filter ω c H(s) = s + ζ ω c s + ω c ζ = damping ratio (0 to 1) ω c = πƒ c, ƒ c = corner frequency he transfer function has poles located at s p1, p = -ζ ω c ± j ω c 1- ζ and zeros s z1 = s z at z p1 = j ( j 1339) 1 + z p = ( j 1339) 1 - z z1 = z z = -1 z p = j he second order digital filter transfer function: (z - z z1 ) (z - z z ) H(z) = G (z - z p1 ) (z - z p ) z + z +1 H(z) = G z z

5 Gain parameter G has to be calculated so the digital filter has unity gain at DC: (1 - z z1 ) (1 - z z ) H(z) = G = 1 /z=1 (1 - z p1 ) (1 - z p ) For the nd order filter to have -3dB at the corner frequency the damping ratio should be set to: ζ = = (1 - z p1 ) (1 - z p ) thus G = 4 G = he transfer function of the digital filter: z z H(z) = z z he filter's coefficients have to be converted to 14-bit integer numbers. he conversion is done by multiplying the fractional coefficients of the transfer function by 14 and rounding the results to the nearest integer number. he rounded integers can then be entered into one of the biquad sections in the Interactive Data Sheet and programmed into the DIGIAL hybrid: he transfer function of the analog high pass filter: H(s) = s s the poles s p1, p = ± j 4443, the zeros s z1 = s z = 0 Using the bilinear transformation, sampling frequency ƒ s = 3000Hz: ( j 4443) 1 + z p1 = ( j 4443) 1 - s COEFFICIEN 14-BI INEGER z p1 = j b b b a a Example 4: nd order high pass filter with 1000Hz corner frequency and 0dB gain he transfer function of a nd order analog high pass filter s H(s) = s + ζ ω c s + ω c ζ = damping ratio (0 to 1) ω c = πƒ c, ƒ c = corner frequency he transfer function has poles located at s p1, p = -ζ ω c ± j ω c 1- ζ and zeros s z1 = s z at 0 z z1 = z z = 1 ( j 4443) 1 + z p = ( j 4443) 1 - z p = j he second order digital filter transfer function: (z - z z1 ) (z - z z ) H(z) = G (z - z p1 ) (z - z p ) z - z +1 H(z) = G z z he corner frequency of the nd order analog filter ƒ c = 1000Hz therefore ω c =

6 Gain parameter G has to be calculated so the digital filter has unity gain at infinity: (-1 - z z1 ) (-1 - z z ) H(z)= G = 1 /z= -1 (-1 - z p1 ) (-1 - z p ) (-1 - z p1 ) (-1 - z p ) thus G = 4 G = he transfer function of the digital filter: z z H(z) = z z he filter's coefficients have to be converted to 14-bit integer numbers. he conversion is done by multiplying the fractional coefficients of the transfer function by 14 and rounding the results to the nearest integer number. he rounded integers can then be entered into one of the biquad sections in the Interactive Data Sheet and programmed into the DIGIAL hybrid: COEFFICIEN 14-BI INEGER b b b a a CAUION ELECROSAIC SENSIIVE DEVICES DO NO OPEN PACKAGES OR HANDLE EXCEP A A SAIC-FREE WORKSAION DOCUMEN IDENIFICAION INFORMAION NOE he product is in a development phase and specifications are subject to change without notice. Gennum reserves the right to remove the product at any time. Listing the product does not constitute an offer for sale. REVISION NOES: Corrections to errors in filter design examples. GENNUM CORPORAION MAILING ADDRESS: P.O. Box 489, Stn A, Burlington Ontario, Canada L7R 3Y3 el. +1 (905) fax: +1 (905) GENNUM JAPAN CORPORAION C-101, Miyamae Village, Miyamae, Suginami-ku, okyo , Japan el. +81 (3) Fax: +81 (3) SHIPPING ADDRESS: 970 Fraser Drive, Burlington, Ontario, Canada L7L 5P5 Gennum Corporation assumes no responsibility for the use of any circuits described herein and makes no representations that they are free from patent infringement. Copyright October 001 Gennum Corporation. All rights reserved. Printed in Canada

Chapter 7: IIR Filter Design Techniques

IUST-EE Chapter 7: IIR Filter Design Techniques Contents Performance Specifications Pole-Zero Placement Method Impulse Invariant Method Bilinear Transformation Classical Analog Filters DSP-Shokouhi Advantages