Digital 4-pole Butterworth Low-pass filter

A digital 4-pole Butterworth low-pass filter is implemented as 2 cascaded biquads (2-pole topology) in the Read-Out card firmware of the MCE. Each biquad has the following functional form:

H(Z) = [1 + 2*Z^{-1} + Z^{-2}] / [1+b1*Z^{-1} + b2*Z^{-2}]

Filter Specification

Filter specification are determined by the filter coefficients b11, b12, b21, b22. These coefficients are currently hard-coded in the Readout-card firmware and depending on the firmware revision, the filter specification may vary.

Assume:

f_3dB is the frequency at which the signal is sqrt(2) of its maximum and

f_samp is the sampling frequency calculated as 50MHz/(num_rows * row_len). For example 50MHz(100*33)=15151.5Hz.

(Note that the filter f_3dB scales if f_samp changes.)

Type 1

• (supported in all rc except 5.0.7)
  
• DC amplification=1217.9148
  
• F_3dB=122.226 Hz
  

Gain @ 200Hz=0.14189148 (wrt the DC gain)

Type 2

• (supported in rc version 5.0.7)
  
• DC amplification=2044
  
• F_3dB/F_samp=75Hz/30000Hz
  

Type 256

Work in Progress: Filter coefficients b11, b12, b21, b22 are parametrized and programmable by software.

The following plot shows the magnitude and the phase of the filter type 1.

Here is Elia's original plot Media:BW_filter.ps. Then we see the same plot with the impulse-response also added (by Joe, Nov 29, 2007):

IDL Code

In this little IDL program you can find the functional form as a function of the frequency of the filter. First, Elia's original program, then as modified by Joe on November 29, 2007.

Media: filter_pro.txt

Media: filter_pro2.txt

Filter Coefficients

The filter coefficients are generated using fdatool (filter-design & Analysis tool) in Matlab. Once you launch the fdatool, choose the following settings:

• Response Type: Low Pass

• Design Method: Butterworth

• Filter Order: 4

• Frequency Specifications:
  • For Type 1: F_s = 12195 = (50000/100*41), F_c=100
  • For Type 2: F_s = 30000 = (50000/100*41), F_c=75
  • (The attenuation at F_c is fixed at 3dB (half the passband power))

Then click on Design Filter and you will get the following coefficients:

Type 1

• Section 1:
  • Numerator: 1 2 1
  • Denominator: 1 -1.9587428340882587 0.96134553442399129
  • Gain = 1/k1 = 0.00065067508393319923 (not implemented)
• Section 2:
  • Numerator: 1 2 1
  • Denominator: 1 -1.9066292518523014 0.90916270571237567
  • Gain = 1/k2= 0.00063336346501859835 (not implemented)

Type 2

• Section 1:
  • Numerator: 1 2 1
  • Denominator: 1 -1.9711486088510415 0.97139181456687917
• Section 2:
  • Numerator: 1 2 1
  • Denominator: 1 -1.9878047097960421 0.98804997058724808
  • Gain = 1/k= 0.0000000037280516432624239 (not implemented)

The following ONLY concerns the MCE firmware developers:
Now to include these floating numbers (coefficients) in MCE firmware, you need to convert the coefficients to signed binary fractional (SBF) 1.14 format where the right-most bit is the sign and the rest of the bits are the magnitude. For example, to convert b11=-1.9587428340882587, you multiply it by -2¹⁴ and convert it to hex, it becomes 0x7D5C. Then the binary is 111 1101 0101 1100.

These values are then plugged in fsfb_calc_pack.vhd (FILTER_B11_COEF, FILTER_B12_COEF, FILTER_B21_COEFF, FILTER_B22_COEFF).

Calculating the Filter Gain

Note that k1 and k2 are not implemented and instead an inter-biquad gain of k3 is implemented as a binary shift in fsfb_calc_pack.vhd (FILTER_GAIN_WIDTH). Also a FILTER_SCALE_LSB or k4 is the number of bits dropped after the second biquad.

Then the overall gain becomes k1*k2/k3*k4.

In the current implementation: For Type 1: where k3 is 2¹¹, the filter gain is estimated at 1184, but vhdl simulation results in a gain of 1216. For Type 2: where k3 is 2¹⁴ and k4 is 2³, the filter gain is estimated at 2046, but vhdl simulation results in a gain of ????. The gain difference can be attributed to the coefficient quantization effects.