Readout Card Preamp Chain

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_i(Z) = [1 + 2 x Z ^-1 + Z ^-2] / [1 + b_1i x Z ^-1 + b_2i x Z ^-2]

and the overall transfer function is:

H(Z) = k x H₁(Z) x H₂(Z) where k is the DC gain.

Rev B9 Bandwidth

Filter specification of the 4-pole Butterworth filter is 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 <math>\sqrt2</math> of its maximum value 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 / f_samp = 122.226Hz / 15151Hz

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 fitler type can be determined by reading back the MCE parameter called filter_type (rb rc1 filter_type). (To be implemented in firmware 5.0.a+)

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

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):

Here is the filter response for Scuba2 setting (f_samp=9.5kHz, f_readout=200Hz): Media: Filter fs10kHz 200Hzdecimated response.ps

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, then modified by Mandana on Jan. 14, 2009 for Scuba2 numbers.

Media: filter_pro.txt

Media: filter_pro2.txt

Media: low_pass_filter_model_pro.txt

Filter Coefficients

The filter coefficients are generated using fdatool (filter-design & Analysis tool) in Matlab/Simulink. 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/(k₁ * k₂)= 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 DC Gain (k)

Note that in firmware, instead of k₁ and k₂, an inter-biquad gain of k₃ (implemented as a binary shift) and k₄, representing number of bits dropped after the second biquad, are implemented.

Hence, the overall gain becomes (k₁ x k₂) / (k₃ x k₄).

(Note for firmware developers: see FILTER_GAIN_WIDTH and FILTER_SCALE_LSB in fsfb_calc_pack.vhd.)

Type 1

k3 is 2¹¹, the filter gain is estimated at 1184, but vhdl simulation results in a gain of 1216.

Type 2

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.