Posted on by Valentin Schmidtke

Filter Objects General Information

About Biquad Topology

  • Direct Form I (DFI)
  • Direct Form II (DFII)
  • Transposed Direct Form II (TDFII)

Direct Form I (DFI)

Structure: The DFI structure implements the zeros of the transfer function first, followed by the poles. It uses two separate delay lines for the input and output signals.

Advantages Disadvantages
  • Straightforward Implementation: It directly implements the biquad difference equation, making it conceptually simple.
  • Single Summation Point: This can be advantageous in fixed-point processors, as it reduces the risk of overflow at intermediate summation stages if a double-width accumulator is used.
  • Less Susceptible to Overflow: The zeros are implemented first, which can attenuate the signal before it enters the recursive (pole) section, potentially reducing the internal signal levels and the risk of overflow.
  • Better for Parameter Modulation: Some sources suggest DFI handles parameter modulation more gracefully than DFII, with fewer artifacts.
  • Higher Memory Requirement: It requires twice the number of delay elements compared to Direct Form II, increasing memory usage.
  • Potentially Higher Quantization Noise: Without a double-width accumulator, the quantization noise introduced after each multiplication in the feedback path can be amplified by the poles.
  • Longer Critical Path: The signal passes through all the feedforward and feedback coefficients sequentially, which can limit the maximum achievable clock rate in hardware implementations.

Direct Form II (DFII)

Structure: The DFII structure rearranges the DFI structure by using a single delay line for both the input and output signals. It implements the poles first, followed by the zeros.

Advantages Disadvantages
  • Memory Efficient (Canonical Form): It is a canonical form, meaning it uses the minimum number of delay elements required for a second-order filter.
  • Potentially Faster Code Execution: Some implementations might lead to slightly faster code compared to DFI.
  • Higher Risk of Overflow: The poles are implemented first. If the filter has a high Q-factor or gain near the resonant frequency, the internal signal levels in the delay line can become very large, leading to overflow, especially in fixed-point implementations.
  • More Sensitive to Quantization Noise: Quantization errors introduced within the feedback loop are directly fed back and shaped by the poles, potentially leading to more noticeable noise.
  • Poor Parameter Modulation Behaviour: DFII is known to produce artifacts like clicks and pops when filter parameters are changed rapidly due to the high gain that can occur in the state variables.

Transposed Direct Form II (TDFII)

Structure: The transposed form of DFII is obtained by reversing the signal flow graph. Branch points become summers, and summers become branch points. The transfer function remains the same. It also uses a single delay line.

Advantages Disadvantages
  • Memory Efficient (Canonical Form): Like DFII, it is also a canonical form with minimal delay elements.
  • Lower Quantization Noise: Generally considered to have better numerical properties and lower quantization noise compared to the direct forms, especially in floating-point implementations where each multiplication and potentially each addition is rounded.
  • Better Robustness: It often exhibits robustness similar to DFI in terms of handling internal signal levels compared to the direct DFII. The zeros are effectively implemented before the feedback path influences the output.
  • Preferred for Floating-Point DSP: Due to its better noise characteristics, it is often preferred in floating-point digital signal processing.
  • Longer Critical Path: Similar to DFI, the transposed structure can have a longer critical path through the feedback loop, potentially limiting the maximum clock rate in hardware.
  • Can be Less Intuitive: The transposed structure might be less directly related to the standard difference equation, making the implementation and analysis slightly less intuitive.
  • Potential for Limit Cycles: In fixed-point implementations, transposed forms can sometimes be more prone to limit cycles (self-sustained oscillations at low levels) under certain conditions.

Summary

Feature Direct Form I (DFI) Direct Form II (DFII) Direct Form II Transposed
Memory Usage High Low (Canonical) Low (Canonical)
Overflow Risk Lower Higher Moderate
Quantization Noise Potentially Higher Higher Lower
Computational Cost Moderate Moderate Moderate
Critical Path Longer Shorter Longer
Parameter Modulation Better Poor Good
Fixed-Point DSP Good (with double acc.) Can be problematic Generally Good
Floating-Point DSP Acceptable Acceptable Preferred