TAS5825M: Calculating filter coefficients on the fly

Richard Walborn1

Part Number: TAS5825M

I have an application , that I need to be able to adjust EQ parameters on the fly through software. .This is something I have done many times before with the SigmaStudio processors Anyhow, I captured the I2S output from PurePath for a 48Khz peaking, filter Freq=1000, Q=.71, Gain = 12dB.which generates a I2S address, sub address and a series of 5 32 bit numbers in hex.. However, nowhere in the documentation is there an indication the coefficient load order and the actual coefficients don't seem to make sense. I understand that your implementation of Direct Form 1 has a 2x gain for B! and A1.

Do you have any additional information of time?

over 5 years ago

0 Andy Liu (BMS) over 5 years ago

TI__Guru 52115 points

Hi Richard,

Let me contact our software team. I think we should be provide something to help you.

Andy

0 Richard Walborn1 over 5 years ago in reply to Andy Liu (BMS)

Intellectual 580 points

Thanks

0 Andy Liu (BMS) over 5 years ago in reply to Richard Walborn1

TI__Guru 52115 points

Hi Richard,

See the two files.

You can find the equations in the Audio-EQ-Cookbook.txt.

The excel file shows how to convert a coefficient to a 32-bit data in 5.27 format.

EQ format calculation tool.xlsx

Fullscreen Audio-EQ-Cookbook.txt Download

         Cookbook formulae for audio EQ biquad filter coefficients
----------------------------------------------------------------------------
           by Robert Bristow-Johnson  <rbj@audioimagination.com>


All filter transfer functions were derived from analog prototypes (that
are shown below for each EQ filter type) and had been digitized using the
Bilinear Transform.  BLT frequency warping has been taken into account for
both significant frequency relocation (this is the normal "prewarping" that
is necessary when using the BLT) and for bandwidth readjustment (since the
bandwidth is compressed when mapped from analog to digital using the BLT).

First, given a biquad transfer function defined as:

            b0 + b1*z^-1 + b2*z^-2
    H(z) = ------------------------                                  (Eq 1)
            a0 + a1*z^-1 + a2*z^-2

This shows 6 coefficients instead of 5 so, depending on your architechture,
you will likely normalize a0 to be 1 and perhaps also b0 to 1 (and collect
that into an overall gain coefficient).  Then your transfer function would
look like:

            (b0/a0) + (b1/a0)*z^-1 + (b2/a0)*z^-2
    H(z) = ---------------------------------------                   (Eq 2)
               1 + (a1/a0)*z^-1 + (a2/a0)*z^-2

or

                      1 + (b1/b0)*z^-1 + (b2/b0)*z^-2
    H(z) = (b0/a0) * ---------------------------------               (Eq 3)
                      1 + (a1/a0)*z^-1 + (a2/a0)*z^-2


The most straight forward implementation would be the "Direct Form 1"
(Eq 2):

    y[n] = (b0/a0)*x[n] + (b1/a0)*x[n-1] + (b2/a0)*x[n-2]
                        - (a1/a0)*y[n-1] - (a2/a0)*y[n-2]            (Eq 4)

This is probably both the best and the easiest method to implement in the
56K and other fixed-point or floating-point architechtures with a double
wide accumulator.



Begin with these user defined parameters:

    Fs (the sampling frequency)

    f0 ("wherever it's happenin', man."  Center Frequency or
        Corner Frequency, or shelf midpoint frequency, depending
        on which filter type.  The "significant frequency".)

    dBgain (used only for peaking and shelving filters)

    Q (the EE kind of definition, except for peakingEQ in which A*Q is
        the classic EE Q.  That adjustment in definition was made so that
        a boost of N dB followed by a cut of N dB for identical Q and
        f0/Fs results in a precisely flat unity gain filter or "wire".)

     _or_ BW, the bandwidth in octaves (between -3 dB frequencies for BPF
        and notch or between midpoint (dBgain/2) gain frequencies for
        peaking EQ)

     _or_ S, a "shelf slope" parameter (for shelving EQ only).  When S = 1,
        the shelf slope is as steep as it can be and remain monotonically
        increasing or decreasing gain with frequency.  The shelf slope, in
        dB/octave, remains proportional to S for all other values for a
        fixed f0/Fs and dBgain.



Then compute a few intermediate variables:

    A  = sqrt( 10^(dBgain/20) )
       =       10^(dBgain/40)     (for peaking and shelving EQ filters only)

    w0 = 2*pi*f0/Fs

    cos(w0)
    sin(w0)

    alpha = sin(w0)/(2*Q)                                       (case: Q)
          = sin(w0)*sinh( ln(2)/2 * BW * w0/sin(w0) )           (case: BW)
          = sin(w0)/2 * sqrt( (A + 1/A)*(1/S - 1) + 2 )         (case: S)

        FYI: The relationship between bandwidth and Q is
             1/Q = 2*sinh(ln(2)/2*BW*w0/sin(w0))     (digital filter w BLT)
        or   1/Q = 2*sinh(ln(2)/2*BW)             (analog filter prototype)

        The relationship between shelf slope and Q is
             1/Q = sqrt((A + 1/A)*(1/S - 1) + 2)

    2*sqrt(A)*alpha  =  sin(w0) * sqrt( (A^2 + 1)*(1/S - 1) + 2*A )
        is a handy intermediate variable for shelving EQ filters.


Finally, compute the coefficients for whichever filter type you want:
   (The analog prototypes, H(s), are shown for each filter
        type for normalized frequency.)


LPF:        H(s) = 1 / (s^2 + s/Q + 1)

            b0 =  (1 - cos(w0))/2
            b1 =   1 - cos(w0)
            b2 =  (1 - cos(w0))/2
            a0 =   1 + alpha
            a1 =  -2*cos(w0)
            a2 =   1 - alpha



HPF:        H(s) = s^2 / (s^2 + s/Q + 1)

            b0 =  (1 + cos(w0))/2
            b1 = -(1 + cos(w0))
            b2 =  (1 + cos(w0))/2
            a0 =   1 + alpha
            a1 =  -2*cos(w0)
            a2 =   1 - alpha



BPF:        H(s) = s / (s^2 + s/Q + 1)  (constant skirt gain, peak gain = Q)

            b0 =   sin(w0)/2  =   Q*alpha
            b1 =   0
            b2 =  -sin(w0)/2  =  -Q*alpha
            a0 =   1 + alpha
            a1 =  -2*cos(w0)
            a2 =   1 - alpha


BPF:        H(s) = (s/Q) / (s^2 + s/Q + 1)      (constant 0 dB peak gain)

            b0 =   alpha
            b1 =   0
            b2 =  -alpha
            a0 =   1 + alpha
            a1 =  -2*cos(w0)
            a2 =   1 - alpha



notch:      H(s) = (s^2 + 1) / (s^2 + s/Q + 1)

            b0 =   1
            b1 =  -2*cos(w0)
            b2 =   1
            a0 =   1 + alpha
            a1 =  -2*cos(w0)
            a2 =   1 - alpha



APF:        H(s) = (s^2 - s/Q + 1) / (s^2 + s/Q + 1)

            b0 =   1 - alpha
            b1 =  -2*cos(w0)
            b2 =   1 + alpha
            a0 =   1 + alpha
            a1 =  -2*cos(w0)
            a2 =   1 - alpha



peakingEQ:  H(s) = (s^2 + s*(A/Q) + 1) / (s^2 + s/(A*Q) + 1)

            b0 =   1 + alpha*A
            b1 =  -2*cos(w0)
            b2 =   1 - alpha*A
            a0 =   1 + alpha/A
            a1 =  -2*cos(w0)
            a2 =   1 - alpha/A



lowShelf: H(s) = A * (s^2 + (sqrt(A)/Q)*s + A)/(A*s^2 + (sqrt(A)/Q)*s + 1)

            b0 =    A*( (A+1) - (A-1)*cos(w0) + 2*sqrt(A)*alpha )
            b1 =  2*A*( (A-1) - (A+1)*cos(w0)                   )
            b2 =    A*( (A+1) - (A-1)*cos(w0) - 2*sqrt(A)*alpha )
            a0 =        (A+1) + (A-1)*cos(w0) + 2*sqrt(A)*alpha
            a1 =   -2*( (A-1) + (A+1)*cos(w0)                   )
            a2 =        (A+1) + (A-1)*cos(w0) - 2*sqrt(A)*alpha



highShelf: H(s) = A * (A*s^2 + (sqrt(A)/Q)*s + 1)/(s^2 + (sqrt(A)/Q)*s + A)

            b0 =    A*( (A+1) + (A-1)*cos(w0) + 2*sqrt(A)*alpha )
            b1 = -2*A*( (A-1) + (A+1)*cos(w0)                   )
            b2 =    A*( (A+1) + (A-1)*cos(w0) - 2*sqrt(A)*alpha )
            a0 =        (A+1) - (A-1)*cos(w0) + 2*sqrt(A)*alpha
            a1 =    2*( (A-1) - (A+1)*cos(w0)                   )
            a2 =        (A+1) - (A-1)*cos(w0) - 2*sqrt(A)*alpha





FYI:   The bilinear transform (with compensation for frequency warping)
substitutes:

                                  1         1 - z^-1
      (normalized)   s  <--  ----------- * ----------
                              tan(w0/2)     1 + z^-1

   and makes use of these trig identities:

                     sin(w0)                               1 - cos(w0)
      tan(w0/2) = -------------           (tan(w0/2))^2 = -------------
                   1 + cos(w0)                             1 + cos(w0)


   resulting in these substitutions:


                 1 + cos(w0)     1 + 2*z^-1 + z^-2
      1    <--  ------------- * -------------------
                 1 + cos(w0)     1 + 2*z^-1 + z^-2


                 1 + cos(w0)     1 - z^-1
      s    <--  ------------- * ----------
                   sin(w0)       1 + z^-1

                                      1 + cos(w0)     1         -  z^-2
                                  =  ------------- * -------------------
                                        sin(w0)       1 + 2*z^-1 + z^-2


                 1 + cos(w0)     1 - 2*z^-1 + z^-2
      s^2  <--  ------------- * -------------------
                 1 - cos(w0)     1 + 2*z^-1 + z^-2


   The factor:

                    1 + cos(w0)
                -------------------
                 1 + 2*z^-1 + z^-2

   is common to all terms in both numerator and denominator, can be factored
   out, and thus be left out in the substitutions above resulting in:


                 1 + 2*z^-1 + z^-2
      1    <--  -------------------
                    1 + cos(w0)


                 1         -  z^-2
      s    <--  -------------------
                      sin(w0)


                 1 - 2*z^-1 + z^-2
      s^2  <--  -------------------
                    1 - cos(w0)


   In addition, all terms, numerator and denominator, can be multiplied by a
   common (sin(w0))^2 factor, finally resulting in these substitutions:


      1         <--   (1 + 2*z^-1 + z^-2) * (1 - cos(w0))

      s         <--   (1         -  z^-2) * sin(w0)

      s^2       <--   (1 - 2*z^-1 + z^-2) * (1 + cos(w0))

      1 + s^2   <--   2 * (1 - 2*cos(w0)*z^-1 + z^-2)


   The biquad coefficient formulae above come out after a little
   simplification.

Andy

0 Richard Walborn1 over 5 years ago in reply to Andy Liu (BMS)

Intellectual 580 points

Andy,

I am very familiar with the Audio Eq Cookbook .

Here is my problem.

The calculated values of the normalized coefficients for a a peaking filter 1000hz, Q=.71, Boost=12 dB. at a 48khz sample

Freq.	1000	Hz
Q	0.707
Gain	12	dB

Fs	48000	Hz

Digital coefficients: supply to the
MiniDSP plugin - advanced biquad
a0	1
a1	1.895208759112340
a2	-0.911562440785883

b0	1.131819352739890
b1	-1.895208759112340
b2	0.779743088045993

Status	Stable

A	1.995262315
w0	0.130899694
alpha	0.092309895
a0	1.046264541

Here are the 5.27 values for above

BQ Coeff	[5.27] Format
1.8952087591	0F296336	a1
-0.9115624408	F8B51EC0	a2
1.1318193527	090DF74E	b0
-1.8952087591	F0D69CCA	b1
0.779743088	063CE9F2	b2

Here is the I2s dump output of the same filter from Purepath 3 TAS5825 EVal board

w 98 30 0a 03 4b ef f1 79 cb ec 04 a2 fd b5 0e 86 34 14 f9 59 b6 5b

You can easily see that the the hex values do not match .

0 Richard Walborn1 over 5 years ago in reply to Richard Walborn1

Intellectual 580 points

I took the I2c output from the eval board and converted to the floating point values .

Calculated
a0	1		Eval output	1
a1	1.895208759112340	0F296336	0E863414	1.815529019
a2	-0.911562440785883	F8B51EC0	F959B65B	-0.831195153

b0	1.131819352739890	090DF74E	0A034BEF	1.251609676
b1	-1.895208759112340	F0D69CCA	F179CBEC	-1.815529019
b2	0.779743088045993	063CE9F2	04A2FDB5	0.57958547

0 Andy Liu (BMS) over 5 years ago in reply to Richard Walborn1

TI__Guru 52115 points

Hi Richard,

I will take a look and get back to you tomorrow.

Andy

0 Andy Liu (BMS) over 5 years ago in reply to Andy Liu (BMS)

TI__Guru 52115 points

Hi Richard,

Actually PPC3 can tell you the calculated coefficients. See the screenshot below.

If you convert these coefficients to their corresponding values in 5.27 format, you will get these values below.

B0 = 0A054C43
B1 = F17B20B3
B2 = 049FA5A9
A1 = 0E84DF4D
A2 = F95B0E14

Here is the i2c dump output I see in the PPC3 I2C Monitor.

w 98 30 0a 05 4c 43 f1 7b 20 b3 04 9f a5 a9 0e 84 df 4d f9 5b 0e 14

Andy

0 Richard Walborn1 over 5 years ago in reply to Andy Liu (BMS)

Intellectual 580 points

Andy,

Maybe I haven't been clear on the problem. The coefficients calculated by PurePath do not match the values calculated by the formulas in the Audio EQ Cook Book.. The questions is how our these values calculated?

0 Andy Liu (BMS) over 5 years ago in reply to Richard Walborn1

TI__Guru 52115 points

Hi Richard,

Here is some Matlab/Octave code to show how to calculate coefficients shown in the screenshot above.

https://e2e.ti.com/cfs-file/__key/communityserver-discussions-components-files/6/bq_5F00_coeff_5F00_example.m

Andy

0 Richard Walborn1 over 5 years ago in reply to Andy Liu (BMS)

Intellectual 580 points

Andy,

Perfect ! do you have the formulas for the rest of the filters

High Pass Variable Q

Low Pas Variable Q

Bass Shelf

Treble shelf.

If use the Audio EQ Cook Book formulas , just let me know .

0 Andy Liu (BMS) over 5 years ago in reply to Richard Walborn1

TI__Guru 52115 points

Hi Richard,

Probably. I would need some time to prepare the sample code.

Andy

0 Andy Liu (BMS) over 5 years ago in reply to Richard Walborn1

TI__Guru 52115 points

See two more examples below.
a) Bass Shelf

https://e2e.ti.com/cfs-file/__key/communityserver-discussions-components-files/6/Bass-Shelf.m

b) Treble Shelf

https://e2e.ti.com/cfs-file/__key/communityserver-discussions-components-files/6/Treble_5F00_Shelf.m

Andy

0 Richard Walborn1 over 5 years ago in reply to Richard Walborn1

Intellectual 580 points

Andy,

Did you get a chance to look at the formulas for the variable Q 2nd order low pass and high pass filters?

Rich.

0 Richard Walborn1 over 5 years ago in reply to Richard Walborn1

Intellectual 580 points

I look at the coefficients generated for the variable Q low and High and they are the classical calculations so you don't need to provide any additional information

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TAS5825M: Calculating filter coefficients on the fly