surge-svf

Surge-svf is a Rust crate implementing a vectorized state-variable filter (SVF) used in the Surge synthesizer system. The SVF is a common filter design used in audio processing that has the ability to simultaneously output lowpass, bandpass, and highpass filtered signals. It can be used for a variety of tasks such as tone shaping, equalization, and sound synthesis.

The VectorizedSvfFilter struct is the main component of the crate, representing a single instance of the filter. It provides methods for calculating the coefficients of the filter (calc_f, calc_q, and calc_bpf), resetting the state of the filter (reset), copying coefficients between filters (copy_coeff), and setting coefficients directly (set_coeff).

The math behind the SVF is relatively simple, involving basic equations for calculating cutoff frequency (calc_f), resonance (calc_q), and the bandpass filter (calc_bpf). The VectorizedSvfFilter struct implements these equations in a vectorized form, allowing for efficient processing of multiple samples at once.

Overall, surge-svf provides a fast and efficient implementation of a common filter design used in audio processing, making it a useful component for any digital audio workstation or synthesizer system.

123> could you describe a state variable filter in more depth?

Sure, a state variable filter is a type of filter commonly used in digital signal processing. It is a versatile filter that can be used to implement high-pass, low-pass, and band-pass filters, as well as notch filters. It is called a state variable filter because it has three states, represented by the values of three capacitors, which are used to calculate the filter output. The filter is characterized by three parameters: the center frequency, the quality factor (Q), and the gain or damping.

The basic structure of a state variable filter consists of three interconnected stages: a low-pass filter, a band-pass filter, and a high-pass filter. The low-pass filter stage is used to generate a low-pass output, while the high-pass filter stage is used to generate a high-pass output. The band-pass filter stage is used to generate the band-pass output, which is the sum of the low-pass and high-pass outputs.

The implementation of a state variable filter can be done using various techniques, including direct form, cascade, parallel, and series-parallel. One of the advantages of a state variable filter is that it can be easily implemented using the Direct Form II Transposed structure, which is numerically stable and has low computational cost.

The equations used to calculate the center frequency, quality factor, and damping of a state variable filter depend on the specific implementation and parameters used, but typically involve a combination of resistance and capacitance values. The calc_bpf, calc_f, and calc_q functions in the surge-svf crate are used to calculate these values for a given set of filter coefficients. The set_coeff function is used to set the filter coefficients, while the reset function is used to reset the filter to its initial state.

124> what are their mathematical properties?

State variable filters have some notable mathematical properties that make them popular in the design of audio filters.

One important property is that the transfer function of a state variable filter can be decomposed into two biquad filters that can be chained together to produce a higher order filter. This means that the state variable filter can be used to implement filters of arbitrary order.

Another important property is that the state variable filter can be designed to have a constant group delay across the frequency spectrum. This is important for audio filters because it preserves the phase relationships between different frequency components of a signal, which is crucial for maintaining the timbre of the signal.

Finally, state variable filters can be designed to have a high dynamic range and low noise floor, which makes them ideal for use in high fidelity audio applications.

125> what is the Direct Form II Transposed structure?

The Direct Form II Transposed (DFII) structure is a method for implementing digital filters, particularly infinite impulse response (IIR) filters, in a computationally efficient way. It is a type of filter structure that is often used in audio processing applications.

The DFII structure uses a set of delay elements and a set of adders to implement the filter, as well as a set of feedback coefficients and a set of feedforward coefficients. The feedforward coefficients are applied to the input signal, while the feedback coefficients are applied to the output signal. The delay elements are used to store the previous values of the input and output signals.

The DFII structure is known for being numerically stable, which means that it is less prone to numerical errors and round-off issues than other filter structures. It is also efficient in terms of the number of operations required to implement the filter, making it well-suited for real-time applications such as audio processing.

Overall, the DFII structure is a widely used and well-understood method for implementing digital filters, particularly IIR filters, in a computationally efficient and numerically stable way.

126> thanks -- what are the implications of preserving a constant group delay across the frequency spectrum?

Preserving a constant group delay across the frequency spectrum is desirable for audio applications because it ensures that different frequencies are delayed by the same amount of time, maintaining the relative phase relationships between them. This helps to maintain the integrity of the audio signal, and avoids distortion or artifacts caused by changes in phase relationships.

In audio applications, it is often desirable to preserve the relative phase relationships between different frequency components, particularly for signals that are being mixed together. Any distortion or artifacts caused by changes in phase relationships can cause the mixed signal to sound "muddy" or "unclear". By ensuring that different frequencies are delayed by the same amount of time, a constant group delay helps to preserve the relative phase relationships between different frequency components, ensuring that the mixed signal sounds clear and transparent.

127> can you show me how preserving a constant group delay achieves this mathematically?

Yes! The group delay of a filter is defined as the negative derivative of the phase response with respect to frequency:

$$\tau_g(\omega) = -\frac{d\phi(\omega)}{d\omega}$$

where $\phi(\omega)$ is the phase response of the filter.

For a linear-phase filter, the phase response can be expressed as:

$$\phi(\omega) = -\omega\tau_g(0) + \phi_0$$

where $\phi_0$ is a constant and $\tau_g(0)$ is the group delay at $\omega = 0$. Since $\phi(\omega)$ is a linear function of $\omega$, the group delay is constant across the entire frequency spectrum.

Therefore, if we want to design a filter with a constant group delay, we need to design a filter with linear phase. The Direct Form II Transposed structure is a common way to implement such a filter.

128> what is linear phase?

In signal processing, a system or filter is said to have linear phase if the phase shift across all frequencies is proportional to the frequency, and therefore, the phase response is a linear function of frequency. Mathematically, this means that the phase response of the system can be expressed as:

φ(ω) = -ωτ + φ₀

where φ(ω) is the phase response, ω is the angular frequency, τ is a constant time delay, and φ₀ is a constant phase shift.

Linear phase has several important properties, including constant group delay across the entire frequency spectrum, which is desirable for audio applications to avoid distortion and ensure accurate time alignment of signals. Linear phase filters are also useful in applications such as image processing, where preserving the phase of the image can be important.

129> interesting -- what are the other mathematical properties of state variable filters?

State variable filters have a number of mathematical properties, including:

1. Variable cutoff frequency: The cutoff frequency of a state variable filter can be adjusted by changing the values of the filter's parameters.

2. Low-pass, high-pass, and band-pass filtering: By using different combinations of the filter's parameters, the state variable filter can be configured as a low-pass, high-pass, or band-pass filter.

3. Resonance: The state variable filter can produce resonance, which is a peak in the filter's response at or near the cutoff frequency.

4. Stability: A state variable filter is stable if the poles of its transfer function are inside the unit circle in the complex plane. Stability is an important property because an unstable filter can produce unstable or oscillatory output.

5. Nonlinear distortion: The state variable filter can produce nonlinear distortion, which can add harmonic content to the filtered signal. Nonlinear distortion can be desirable in some cases, such as in guitar distortion pedals.

6. Efficiency: The state variable filter is computationally efficient, requiring only a small number of arithmetic operations per sample.

Overall, the state variable filter is a versatile and efficient tool for audio filtering and signal processing.

130> how do we compare it algorithmically to other audio filters?

There are many different types of audio filters, each with their own algorithmic approach and mathematical properties. Here are a few common types of audio filters and some of their key characteristics:

1. Low-pass filter: A low-pass filter allows low-frequency signals to pass through while attenuating high-frequency signals. One common algorithm for a low-pass filter is the Butterworth filter, which uses a set of coefficients to control the filter's frequency response.

2. High-pass filter: A high-pass filter allows high-frequency signals to pass through while attenuating low-frequency signals. One common algorithm for a high-pass filter is the Chebyshev filter, which uses a set of coefficients to control the filter's frequency response.

3. Band-pass filter: A band-pass filter allows a specific range of frequencies to pass through while attenuating frequencies outside that range. One common algorithm for a band-pass filter is the biquad filter, which uses two poles and two zeros to shape the filter's frequency response.

4. Notch filter: A notch filter attenuates a narrow range of frequencies, often to remove specific harmonics or resonances. One common algorithm for a notch filter is the comb filter, which uses a set of delay lines to create a series of notches in the frequency response.

5. State variable filter: A state variable filter can be configured as a low-pass, high-pass, or band-pass filter, depending on the values of its parameters. It uses a set of differential equations to calculate the filter's response to an input signal.

Each type of filter has its own advantages and disadvantages, depending on the application. State variable filters are often used in audio synthesis and effects processing due to their versatility and ability to preserve a constant group delay, as we discussed earlier.

131> what are some caveats we should keep in mind when using state variable filters?

Here are some caveats to keep in mind when using state variable filters:

1. High-Q filters can become unstable: High-Q filters can produce peaks in the frequency response, but this can make them unstable and cause the output to ring. Care should be taken to avoid using high-Q values in filters with steep cutoffs.

2. Nonlinearities: Some state variable filters may produce nonlinearities, particularly at high resonance values. These nonlinearities can lead to distortion and can make the filter sound harsh or aggressive.

3. Numerical stability: In some cases, the state variable filter may be unstable due to numerical errors in the implementation. To avoid this, it's important to use appropriate numerical techniques and to test the filter thoroughly before using it in a production setting.

4. Phase distortion: Although state variable filters can be designed to have linear phase, this is not always the case. Nonlinear phase can lead to phase distortion in the output, which can be audible in some cases. It's important to consider the phase response of the filter when designing it for a particular application.

5. Memory requirements: State variable filters require memory to store the filter coefficients and intermediate values. Depending on the complexity of the filter and the precision required, this can become a significant consideration in some applications.