IIR filters

 IIR – Feedback is infinity

Fir Filter are always stable, So we don’t have any poles & zeros

Biquad Filter is second order recursive filter with infinity feedback which has two poles & two zeros.

 

When Coefficients are normalized then a0 = 1

 

Why we use cascading in Biquad filters ?

Basically High-order infinite impulse response filters are highly sensitive to quantization of their coefficients and easily become unstable. But with first & second filters it is very less. So to make it stable, implement high order filters as serially cascaded Biquad sections which will make it stable by keeping all the poles inside the unit circle.

Direct Form 1:

From fig : 

𝑧−1Means delay by 1 sample

𝑧−2 Means delay by 2 samples

Y[n] = 1/a0 (b0x[n] + b1x[n-1] + b2x[n-2] – a1y[n-1] -a2y[n-2])

Y[n] = b0x[n] + b1x[n-1]+b2x[n-2]-a1y[n-1]-a2y[n-2]

Where b0, b1, b2 are current sample coefficients determine zeros

a1, a2 are previous output samples determines the poles

 

 

 

                              

Here path b0, b1, b2 is feed forward & a1, a2 is feed backward path

Direct Form II

 

W(n) = x(n) – a1 w(n-1) – a2 w(n-2)

Y(n) = b0w(n) + b1w(n-1) + b2(w-2)

 

 

                                

Transposition theorem rules

1. Reverse all signal paths: Change the direction of all arrows in the signal flow graph.
2. Interchange the input and output: The original input, 𝑥(𝑛), becomes the output, and the original output, 𝑦(𝑛), becomes the input.
3. Swap all summing and branching points.

                               

 

Direct Form 1

Transposed Direct Form1 1) In above picture . represents branch nodes which must replace with Summation   2) Swapped b2 & a2, a1 & b1, b0 & a0 (where a0=1)   3) Reverse arrows feedforward -> feed backward vice versa

 

                    

Direct Form II

Transposed Direct FormII

 

Y[n] = b0 x[n] + w1[n-1]

W1[n] = b1 x[n] – a1 Y[n] + w2[n-1]

W2[n] = b2 x[n] – a2 Y[n]

Direct Form I	Direct Form II
Requires 4 delay elements	Require 2 Delay elements
Requires more memory	Requires less memory
Direct form1 & transpose DF2 we are evaluating Numerator first & then denominator	Direct form2 & transpose DF1 we are evaluating denominator first & then Numerator

 

In Fixed point processor we prefer Direct Form1 over Direct Form2

If we except small values at numerator then we need to perform Numerator first. So, we will choose DF1 or TDF2

If we except small values at denominator then we need to perform denominator first. So, we will choose DF2 or TDF1

But choosing topology will depend on architecture and processor and filter coefficients

For SIMD processors TDF2 will be beneficial because of parallelism and easier to optimize

Cascade Form

In GTT we don’t have even number of biquads

 

For Magnitude & phase response: https://www.earlevel.com/main/2021/09/02/biquad-calculator-v3/

 

Parameter	Description	Tunable or Controllable	Range
Frequency	Filtering frequency to be applied	Control/Tunable	20Hz-20kHz
Gain	Filter gain	Control/Tunable	-18 to 18dB
Quality	Quality of the filtering coefficients	Control/Tunable	0.1-10
Type	Filter type	Tunable	Highpass,Lowpass,etc...
RampTime	Ramp time for filter coefficient to adapt to new coefficient	Tunable	0 to 500msec

 

As we have different biquads based on how you compute cofficients & how we are going to tune

1)       Parameter biquad

2)       Crossover biquad

3)       Tonecontrolled biquad

4)       Coefficient biquad

Parameter	Tone Control	Crossover	Coefficient
Used in equalizers where you want to boost or cut a narrow frequency band	Used in simpler EQ for treble/bass control Low-shelf - Boost or cuts bass region High shelf – Boosts or cuts treble region	Used in Multi audio systems (Subwoofers, mid, tweeters)	It is just representing or configuring the coefficients

 

What is sections in biquads

Basically sections is nothing but no of biquads using in a filter

Ex: 3 sections means using 3 biquads

1Biquad -> 2^nd Order

2Biquads -> 4^th Order

3Biquads -> 6^th Order

So 3 sections = three cascaded 2^nd order filters per channel

 

Example — 2 Channels × 3 Sections

Coefficients array (coeff)

Each section’s coefficients are stored contiguously, section by section:

Section	Coeff Index Range	Coefficients
0	0 – 4	b0, b1, b2, a1, a2
1	5 – 9	b0, b1, b2, a1, a2
2	10 – 14	b0, b1, b2, a1, a2

 

So :

float coeff[15] = {

   b00, b01, b02, a01, a02,   // section 0

   b10, b11, b12, a11, a12,   // section 1

   b20, b21, b22, a21, a22    // section 2

};

State array (state)

For 2 channels × 3 sections, we need:

2×3×2=12 state variables

Each section per channel keeps its own s1, s2. Where s1,s2 are state variables

Layout:

Channel	Section	State Indices	Contents
0 (Left)	0	0,1	s1L0, s2L0
0 (Left)	1	2,3	s1L1, s2L1
0 (Left)	2	4,5	s1L2, s2L2
1 (Right)	0	6,7	s1R0, s2R0
1 (Right)	1	8,9	s1R1, s2R1
1 (Right)	2	10,11	s1R2, s2R2

 

 

Ramping: Updating the biquad coefficients over time instead of changing them suddenly

In tuning perspective, When filter parameters like gain,cutoff changes we need to smoothly transition from old coefficients to new coefficients.

Why Ramping is needed ?

1.       In real time when coefficients jump suddenly cause audio pops or distortions

2.       It will cause sometimes move poles & zeros outside the unit circle which will cause filter un-stable