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0.2.12-alpha.0 Apr 7, 2023
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0.1.42-alpha.0 Oct 27, 2021

#9 in #stereo


Used in 3 crates (2 directly)

GPL-3.0 license

1MB
13K SLoC

Stereo emphasis effect for the Surge synthesizer system


The surgefx-emphasize crate provides a stereo emphasis effect, designed to enhance the stereo image and spectral content of audio signals in the Surge synthesizer system.

The emphasize effect is achieved through a combination of mid/side processing, equalization, and harmonic enhancement techniques. The primary goal of this effect is to emphasize specific frequency ranges or spatial characteristics of the audio signal, resulting in a richer and more immersive listening experience.

Mathematical Ideas and Equations:

  1. Mid/Side Processing: This technique involves separating the audio signal into its mid (mono) and side (stereo) components, which can then be processed independently. The mid signal (M) is the sum of the left (L) and right (R) channels, while the side signal (S) is the difference between the two channels:

    M = L + R S = L - R

  2. Equalization: Emphasize effect can involve applying frequency-specific gain adjustments using equalizers, such as high-pass, low-pass, or band-pass filters. This allows the user to control the frequency content of the audio signal and emphasize desired frequency ranges. The gain adjustments can be applied in the frequency domain, using the transfer function of the filter:

    H(f) = K * (1 - e^(-j * 2 * π * f * τ))

    where f is the frequency, K is the gain, and τ is the time constant of the filter.

  3. Harmonic Enhancement: This technique involves generating harmonics of the input signal, which can be done using waveshaping or saturation algorithms. The harmonics can be used to enrich the sound and emphasize specific frequency ranges or spatial characteristics. The harmonic generation can be described by a nonlinear function:

    y(t) = f(x(t))

    where x(t) is the input signal and y(t) is the output signal after applying the nonlinear function f.

By processing and combining these techniques, the surgefx-emphasize crate offers a powerful effect that can be used to enhance the stereo image and spectral content of audio signals in the Surge synthesizer system.

30> could you show a detailed block diagram of the subcomponents?

Here is a detailed block diagram of the subcomponents for the surgefx-emphasize effect:

Input Signal (Stereo)
       |
       v
Mid/Side Encoder
       |
       +-----> Mid (Mono) Signal
       |
       v
Side (Stereo) Signal
       |
       v
Equalization (EQ)
       |
       v
Harmonic Enhancement
       |
       v
Side (Stereo) Signal (Processed)
       |
       v
Mid/Side Decoder
       |
       v
Output Signal (Stereo)
  1. Input Signal (Stereo): The stereo input signal enters the emphasize effect processor.

  2. Mid/Side Encoder: The input signal is split into mid (mono) and side (stereo) components using mid/side processing techniques.

  3. Mid (Mono) Signal: The mid signal bypasses the processing chain and goes directly to the Mid/Side Decoder.

  4. Side (Stereo) Signal: The side signal is processed through the following stages:

    a. Equalization (EQ): The side signal is equalized using high-pass, low-pass, or band-pass filters to emphasize specific frequency ranges.

    b. Harmonic Enhancement: The equalized side signal undergoes harmonic enhancement using waveshaping or saturation algorithms to enrich the sound and emphasize specific frequency ranges or spatial characteristics.

  5. Side (Stereo) Signal (Processed): The processed side signal is combined with the mid signal in the Mid/Side Decoder.

  6. Mid/Side Decoder: The mid and processed side signals are combined to create the stereo output signal.

  7. Output Signal (Stereo): The processed stereo output signal is the result of the emphasize effect.

This block diagram illustrates the main subcomponents and signal flow of the surgefx-emphasize effect. By processing the side signal independently and then combining it back with the mid signal, this effect can enhance the stereo image and spectral content of audio signals.

31> could you show me a dozen different techniques for using it? what parameters can we set? how do these parameter choices alter the sound?

Here are a dozen different techniques for using the surgefx-emphasize effect, including the parameters that can be set and how these choices alter the sound:

  1. Emphasize High Frequencies: Apply a high-pass filter to the side signal to emphasize high frequencies. This will make the audio sound brighter and more open. Adjust the filter cutoff frequency and resonance to control the amount of emphasis.

  2. Emphasize Low Frequencies: Apply a low-pass filter to the side signal to emphasize low frequencies. This will make the audio sound warmer and more full-bodied. Adjust the filter cutoff frequency and resonance to control the amount of emphasis.

  3. Emphasize Mid Frequencies: Apply a band-pass filter to the side signal to emphasize a specific frequency range in the mid-frequency spectrum. This can bring out certain elements in the mix or accentuate a specific tonal character. Adjust the filter center frequency, bandwidth, and resonance to control the amount of emphasis.

  4. Wide Stereo Image: Increase the level of the side signal relative to the mid signal to create a wider stereo image. This can make the audio sound more spacious and immersive.

  5. Narrow Stereo Image: Decrease the level of the side signal relative to the mid signal to create a narrower stereo image. This can help focus the audio and improve mono compatibility.

  6. Harmonic Saturation: Apply a subtle saturation algorithm to the side signal to introduce harmonic saturation. This can add warmth and character to the sound, especially in the mid and high-frequency ranges.

  7. Exciter Effect: Apply a more aggressive harmonic enhancement algorithm to the side signal to create an exciter effect. This can add sparkle and presence to the sound, making it more lively and engaging.

  8. Mid/Side EQ: Apply different equalization settings to the mid and side signals to shape the tonal balance and stereo image. For example, boost the low frequencies in the mid signal while cutting them in the side signal to create a focused low end with a wide stereo image.

  9. Dynamic Emphasis: Use a dynamic processor, such as a compressor or expander, to control the dynamics of the side signal. This can create a more controlled and balanced stereo image, allowing the emphasis effect to be more consistent across different audio material.

  10. Transient Shaping: Apply transient shaping to the side signal to emphasize or de-emphasize certain elements in the audio. This can help bring out specific instruments or create a more controlled stereo image.

  11. Emphasize Specific Instruments: If using the surgefx-emphasize effect on individual tracks or busses, apply it selectively to emphasize specific instruments or elements in the mix. For example, use the effect on a drum bus to bring out the stereo room sound or on a vocal track to add presence and depth.

  12. Creative Sound Design: Experiment with more extreme settings and combinations of the parameters to create unique and interesting sound design effects. For example, use a combination of aggressive harmonic enhancement and dynamic processing to create a distorted, lo-fi stereo image.

These techniques demonstrate various ways to use the surgefx-emphasize effect and how different parameter settings can alter the sound. By adjusting the filter settings, harmonic enhancement algorithms, and processing techniques, you can achieve a wide range of tonal and spatial enhancements to your audio.

32> how do we prevent unwanted artifacts when engineering this signal processor?

To prevent unwanted artifacts when engineering the surgefx-emphasize signal processor, consider the following best practices:

  1. Avoid Excessive Processing: Be cautious not to over-process the side signal, as this can lead to unnatural or harsh-sounding results. Use the effect subtly, and adjust the parameters gradually to achieve the desired sound without introducing artifacts.

  2. Smooth Transitions: When applying filters or other processing to the side signal, ensure that transitions between different settings are smooth to avoid abrupt changes or clicks in the audio. This can be achieved by using linear or exponential crossfades when changing parameter values or by incorporating smoothing algorithms for continuous control changes.

  3. Prevent Clipping: When using harmonic enhancement or other processing techniques that increase the level of the side signal, ensure that the output signal does not clip. This can be achieved by using a soft clipper or a limiter at the end of the processing chain to prevent digital clipping and distortion artifacts.

  4. Preserve Phase Coherence: When applying EQ or other processing to the side signal, ensure that the phase relationships between the mid and side signals are preserved to maintain a coherent stereo image. This can be achieved by using linear-phase EQ or other phase-coherent processing algorithms.

  5. Avoid Excessive Resonance: Be cautious with the resonance settings when using filters, as excessive resonance can introduce ringing artifacts and make the audio sound unnatural. Use moderate resonance settings, and adjust them carefully to achieve the desired emphasis without introducing unwanted artifacts.

  6. Manage Latency: When using filters or other processing techniques that introduce latency, ensure that the latency is properly compensated for in the processing chain to maintain accurate time alignment between the mid and side signals. This can be achieved by using delay compensation features in your DAW or by manually adjusting the latency of other processing elements in the chain.

  7. Optimize CPU Usage: When using complex algorithms for harmonic enhancement or other processing techniques, ensure that the processor is optimized for efficient CPU usage to avoid performance issues or audio dropouts. This can be achieved by using efficient algorithms, optimizing code, and enabling multi-threaded processing where appropriate.

  8. Test with Various Audio Material: Test the surgefx-emphasize processor with a wide range of audio material to ensure that it performs well under different conditions and does not introduce unwanted artifacts. This includes testing with various genres, instruments, and frequency ranges.

By following these best practices, you can help prevent unwanted artifacts when engineering the surgefx-emphasize signal processor, ensuring that it produces high-quality audio results while maintaining a coherent stereo image and optimal performance.

33> could you teach me how mid/side encoding/decoding work?

Mid/side (M/S) encoding and decoding are techniques used to manipulate stereo audio signals. They involve converting a stereo signal into a different representation that separates the information common to both channels (mid) from the information unique to each channel (side). This separation allows for more precise control over the stereo image and can be used for various audio processing tasks such as equalization, compression, and widening/narrowing the stereo image.

Mid/side encoding:

Mid/side encoding converts a stereo signal (left and right channels) into a mid signal and a side signal. The mid signal represents the sum of the left and right channels, while the side signal represents the difference between the left and right channels. The encoding process involves the following steps:

  1. Calculate the mid (M) signal by summing the left (L) and right (R) channels:

    M = L + R

  2. Calculate the side (S) signal by subtracting the right channel from the left channel:

    S = L - R

After encoding, the mid signal contains the information common to both channels (center-panned elements), and the side signal contains the unique information for each channel (stereo differences).

Mid/side decoding:

Mid/side decoding converts the mid and side signals back into a stereo signal (left and right channels). The decoding process involves the following steps:

  1. Calculate the left (L) channel by summing the mid (M) and side (S) signals:

    L = M + S

  2. Calculate the right (R) channel by subtracting the side signal from the mid signal:

    R = M - S

After decoding, the left and right channels are reconstructed from the mid and side signals, and the original stereo audio is restored.

Applications of mid/side processing:

Mid/side processing is useful in various audio engineering tasks, including:

  • Mid/side equalization: Apply different EQ settings to the mid and side signals to alter the tonal balance and stereo image.

  • Mid/side compression: Apply different compression settings to the mid and side signals to control the dynamics of the center and stereo elements separately.

  • Stereo widening/narrowing: Adjust the level of the side signal relative to the mid signal to create a wider or narrower stereo image.

In summary, mid/side encoding and decoding are techniques used to manipulate stereo audio signals by separating the center and stereo information. These techniques can be used for various audio processing tasks, providing more precise control over the stereo image and enabling more creative audio engineering possibilities.

34> what about equalization? what about harmonic enhancement?

Equalization and harmonic enhancement are two audio processing techniques used to shape the tonal balance and add harmonic content to audio signals, respectively. Let's dive into each of these techniques in more detail.

Equalization:

Equalization (EQ) is the process of adjusting the balance between different frequency components in an audio signal. By using EQ, you can boost, cut, or attenuate specific frequency ranges to shape the tonal characteristics of an audio signal, making it sound brighter, warmer, or more balanced. There are several types of equalizers:

  1. Graphic Equalizer: A graphic EQ divides the frequency spectrum into fixed bands, usually with sliders or knobs for adjusting the gain of each band. It provides a visual representation of the EQ curve, making it easy to see the adjustments being made.

  2. Parametric Equalizer: A parametric EQ provides more precise control over the frequency range, gain, and bandwidth (Q factor) of each band. It typically has fewer bands than a graphic EQ but offers greater flexibility in shaping the sound.

  3. Shelving Equalizer: A shelving EQ allows you to adjust the gain of all frequencies above or below a specific cutoff frequency. High-shelf filters boost or cut high frequencies, while low-shelf filters boost or cut low frequencies.

  4. High-pass/Low-pass/Band-pass Filters: These filters remove frequencies outside of a specific range. High-pass filters remove low frequencies, low-pass filters remove high frequencies, and band-pass filters remove frequencies outside a specified band.

Harmonic Enhancement:

Harmonic enhancement, also known as harmonic excitation or distortion, is a technique used to add harmonic content to an audio signal, giving it more warmth, presence, or character. Harmonic enhancement can be achieved using various methods:

  1. Tube/Valve Emulation: This technique emulates the characteristics of vacuum tube amplifiers, which are known for their warm, smooth distortion. Tube emulations add even and odd harmonics to the audio signal, enriching its harmonic content and creating a more pleasing sound.

  2. Tape Saturation: Tape saturation emulates the non-linear characteristics of analog tape machines, which can add a subtle form of distortion and compression. This type of harmonic enhancement can add warmth, presence, and character to an audio signal.

  3. Transistor/Console Emulation: Transistor and console emulations replicate the characteristics of analog mixing consoles and outboard gear that use solid-state electronics. These emulations add a subtle coloration to the audio signal, often described as "punchy" or "focused."

  4. Exciters: Exciters are signal processors that emphasize the high-frequency content of an audio signal by generating and adding harmonics. They can be used to brighten up dull recordings or add presence and air to vocals and instruments.

Both equalization and harmonic enhancement play crucial roles in shaping the sound of an audio signal. Equalization allows you to fine-tune the tonal balance of a recording, while harmonic enhancement adds warmth, presence, or character to the sound. These techniques are often used together in audio production to create a polished, professional sound.

35> could you please show me detailed block diagrams for each of these two components? then, could you describe how they are parametrized?

Here are the block diagrams and parameter explanations for equalization and harmonic enhancement.

Equalization:

                           ┌────────┐
Input Signal ─────────────►│ Filter ├────────────► Output Signal
                           └────────┘
                               │
                     ┌────────┴─────────┐
                     │ Frequency Control │
                     └────────┬─────────┘
                               │
                     ┌────────┴─────────┐
                     │ Gain Control     │
                     └────────┬─────────┘
                               │
                     ┌────────┴─────────┐
                     │ Q (Bandwidth)    │
                     └──────────────────┘

Equalization parameters:

  1. Frequency Control: Determines the center frequency of the filter, which is the frequency where the EQ has the most significant effect.

  2. Gain Control: Adjusts the amount of boost or cut applied to the specified frequency range. Positive values increase the amplitude, while negative values decrease it.

  3. Q (Bandwidth) Control: Sets the width of the frequency band affected by the filter. A higher Q value results in a narrower band, while a lower Q value affects a broader range of frequencies.

Harmonic Enhancement:

                           ┌──────────────┐
Input Signal ─────────────►│ Harmonic     ├────────────► Output Signal
                           │ Enhancer     │
                           └──────┬───────┘
                                  │
                     ┌───────────┴────────────┐
                     │ Harmonic Amount Control │
                     └───────────┬────────────┘
                                  │
                     ┌───────────┴────────────┐
                     │ Harmonic Type Control   │
                     └───────────┬────────────┘
                                  │
                     ┌───────────┴────────────┐
                     │ Input Drive Control     │
                     └───────────┬────────────┘
                                  │
                     ┌───────────┴────────────┐
                     │ Output Level Control    │
                     └────────────────────────┘

Harmonic Enhancement parameters:

  1. Harmonic Amount Control: Adjusts the amount of harmonic content added to the input signal. Higher values result in more noticeable harmonic enhancement.

  2. Harmonic Type Control: Selects the type of harmonic enhancement applied, such as tube/valve emulation, tape saturation, transistor/console emulation, or exciter.

  3. Input Drive Control: Adjusts the level of the input signal before harmonic enhancement is applied. Increasing the input drive can result in more pronounced harmonic distortion.

  4. Output Level Control: Sets the overall output level of the processed signal, allowing you to balance the enhanced signal with the original or other elements in the mix.

These block diagrams and parameters provide a high-level overview of equalization and harmonic enhancement. The actual implementation of these processes can vary depending on the specific algorithm, device, or software being used.

36> thanks -- what are the best algorithms for implementing harmonic enhancement?

There is no definitive answer to the question of the "best" algorithm for harmonic enhancement, as the choice often depends on the specific audio source and the desired sound. However, some well-regarded algorithms and techniques used for harmonic enhancement are as follows:

  1. Tube/Valve Emulation: This technique emulates the characteristics of vacuum tube amplifiers, which are known for their warm, smooth distortion. Tube emulations often use mathematical models of tube circuits or waveshaping functions that mimic the harmonic distortion characteristics of tubes.

  2. Tape Saturation: Tape saturation emulates the non-linear characteristics of analog tape machines, which can add a subtle form of distortion and compression. Tape saturation algorithms often model the hysteresis and saturation characteristics of magnetic tape, capturing the nuances of tape machine behavior.

  3. Transistor/Console Emulation: Transistor and console emulations replicate the characteristics of analog mixing consoles and outboard gear that use solid-state electronics. These emulations often model the non-linearities and frequency response characteristics of the hardware to add a subtle coloration to the audio signal.

  4. Exciters: Exciters are signal processors that emphasize the high-frequency content of an audio signal by generating and adding harmonics. This can be achieved using a variety of methods, including psychoacoustic processing, non-linear distortion, and multiband processing.

  5. Wave Shaping: Wave shaping is a non-linear distortion technique that maps an input signal to a new output signal based on a pre-defined transfer function or curve. This can be used to introduce harmonic content by altering the shape of the input waveform.

  6. Dynamic Convolution: Dynamic convolution uses impulse responses of real hardware devices, such as tube amplifiers, tape machines, or other analog processors, to emulate their characteristics. The convolution process involves analyzing the impulse responses of the devices and applying their characteristics to the input signal.

Each of these algorithms and techniques has its own strengths and weaknesses, and the choice depends on the audio material and desired sound. The best approach is often to experiment with different harmonic enhancement methods and choose the one that best suits your needs and aesthetic preferences.

37> let's talk about tube/value emulation in more mathematical depth -- how is it implemented?

Tube/valve emulation algorithms aim to recreate the distinctive characteristics of vacuum tube amplifiers, which are known for their warm, smooth distortion. There are various approaches to achieving this, but most techniques involve modeling the non-linear behavior of tubes using mathematical functions or models.

Here are two common methods for implementing tube/valve emulation:

  1. Waveshaping:

Waveshaping is a method of applying a non-linear transfer function to the input signal. In the case of tube emulation, the transfer function is chosen to resemble the response curve of a vacuum tube. The function often takes the form of a polynomial, such as:

y(x) = a0 + a1 * x + a2 * x^2 + a3 * x^3 + ... + an * x^n

Where x is the input signal and y(x) is the output signal. The coefficients a0, a1, ..., an determine the shape of the transfer function and are chosen to match the behavior of a specific tube type.

The non-linear transfer function can also be represented as a lookup table, where the input signal level is used to find the corresponding output level in the table.

  1. Circuit Modeling:

Circuit modeling involves creating a mathematical model of the tube amplifier circuit, including the tube itself and other components such as resistors, capacitors, and transformers. There are several techniques for modeling tube circuits, including:

  • Spice-like modeling: This approach involves analyzing the tube amplifier circuit using techniques from electrical engineering, such as nodal analysis, Kirchhoff's laws, and Laplace/Fourier transforms. The resulting system of equations can be solved numerically to find the output signal given an input signal.

  • Volterra Series: The Volterra series is a mathematical framework that can be used to model the behavior of non-linear systems, such as tube amplifiers. The output signal is represented as a series of terms that involve the input signal and its derivatives, multiplied by kernel functions that describe the non-linear behavior of the system.

  • State-space modeling: State-space models represent the behavior of a system using a set of state variables and a system of differential equations. The tube amplifier circuit can be represented in state-space form, and the output signal can be found by numerically integrating the differential equations.

Each of these methods has its own trade-offs in terms of computational complexity and accuracy. Some tube emulation algorithms combine different modeling techniques or use approximations to achieve a balance between computational efficiency and sound quality.

38> let's talk about tape saturation in more mathematical depth -- how is it implemented?

Tape saturation is the process of emulating the non-linear characteristics of analog tape machines, which can add a subtle form of distortion and compression to audio signals. There are several approaches to implementing tape saturation, but most techniques involve modeling the hysteresis and saturation characteristics of magnetic tape. Here are two common methods for implementing tape saturation:

  1. Hysteresis modeling:

Hysteresis is the phenomenon in which the magnetization of a magnetic material (like tape) depends on its previous magnetization state. In audio processing, hysteresis causes non-linear behavior in the input-output relationship, resulting in a unique form of distortion.

A common approach to modeling hysteresis is to use the Preisach model or the Jiles-Atherton model. These models describe the relationship between the input signal (audio) and the output signal (magnetization) using non-linear differential equations. The models typically involve parameters like coercivity, saturation magnetization, and anhysteretic magnetization, which can be adjusted to match the characteristics of a specific tape machine.

  1. Saturation modeling:

Saturation is the phenomenon where the magnetic material reaches its maximum magnetization and cannot be further magnetized. In audio processing, saturation results in a soft-clipping effect, where the amplitude of the output signal is limited, and the waveform is smoothly rounded off.

One common approach to modeling saturation is to use a smooth, non-linear transfer function, such as a sigmoid or hyperbolic tangent function:

y(x) = tanh(ax)

Where x is the input signal, y(x) is the output signal, and a is a parameter that controls the amount of saturation. This function smoothly maps the input signal to a limited range, emulating the soft-clipping behavior of tape saturation.

Another approach is to use a piecewise function that combines linear and non-linear regions to model the saturation curve, such as:

y(x) = x, if abs(x) <= T
y(x) = sgn(x) * (T + (x - sgn(x) * T) / (1 + ((x - sgn(x) * T) / (1 - T))^2)), if abs(x) > T

Where x is the input signal, y(x) is the output signal, T is the saturation threshold, and sgn(x) is the sign function. This function provides a more flexible way to model the saturation curve and can be adjusted to match the characteristics of different tape machines.

These hysteresis and saturation models can be combined with other elements, such as frequency response and noise modeling, to create a more comprehensive tape saturation algorithm. The specific implementation details and parameter choices depend on the desired sound and the characteristics of the tape machine being emulated.

39> let's talk about Transistor/Console Emulation in more mathematical depth -- how is it implemented?

Transistor/console emulation aims to recreate the non-linear characteristics and frequency response of analog mixing consoles and their components, such as transistors, operational amplifiers, and transformers. Here are two key aspects of implementing transistor/console emulation:

  1. Non-linear behavior modeling:

The non-linear behavior of transistors and other components in a console can be modeled using mathematical functions, such as polynomials or sigmoid functions. These functions can be used to represent the input-output relationship of the components and emulate the unique distortion they produce. One approach is to use a polynomial function like:

y(x) = a0 + a1 * x + a2 * x^2 + a3 * x^3 + ... + an * x^n

Where x is the input signal, y(x) is the output signal, and the coefficients a0, a1, ..., an determine the shape of the transfer function. The coefficients can be chosen based on measurements of the specific console or component being emulated.

Another approach is to use a sigmoid function, such as a hyperbolic tangent function, to model the non-linear behavior:

y(x) = tanh(ax)

Where x is the input signal, y(x) is the output signal, and a is a parameter that controls the amount of non-linearity. This function can be adjusted to match the characteristics of a specific console or component.

  1. Frequency response modeling:

Analog consoles and their components have unique frequency response characteristics, which can be modeled using digital filters. The frequency response can be described as a transfer function in the frequency domain, and then a corresponding digital filter can be designed to replicate this behavior.

For example, if the frequency response of a console or component is known, an impulse response can be calculated, which represents the time-domain response of the system. A digital filter, such as a finite impulse response (FIR) or infinite impulse response (IIR) filter, can then be designed to match the impulse response.

In some cases, it might be more convenient to design a digital filter directly in the frequency domain using techniques like filter approximation (e.g., Butterworth, Chebyshev, or elliptic filters) or frequency sampling.

The non-linear behavior and frequency response models can be combined to create a comprehensive transistor/console emulation algorithm. The specific implementation details and parameter choices depend on the desired sound and the characteristics of the console or component being emulated. Various optimizations and approximations can be applied to balance sound quality and computational efficiency.

40> let's talk about Exciters in more mathematical depth -- how are they implemented?

Exciters are audio processing tools that enhance the high-frequency content of an audio signal, adding brightness, presence, and clarity to the sound. They can be implemented using various techniques, including harmonic generation, dynamic equalization, and psychoacoustic processing. Here are three common methods for implementing exciters:

  1. Harmonic generation:

Harmonic generation involves adding new harmonics to the audio signal based on its existing frequency content. One way to achieve this is by using a non-linear transfer function, such as a waveshaping function, that introduces distortion and creates harmonics. For example, a simple polynomial waveshaper can be used:

y(x) = a0 + a1 * x + a2 * x^2 + ... + an * x^n

Where x is the input signal, y(x) is the output signal, and a0, a1, ..., an are coefficients that determine the shape of the transfer function. By adjusting the coefficients, different types of harmonic distortion can be created.

  1. Dynamic equalization:

Dynamic equalization involves applying time-varying gain to specific frequency bands in the audio signal, emphasizing or de-emphasizing certain frequencies based on the signal's amplitude. One way to implement a dynamic equalizer is to use a combination of band-pass filters and envelope followers:

  • First, the input signal is split into multiple frequency bands using band-pass filters.

  • An envelope follower is applied to each band, tracking the amplitude of the signal within the band.

  • The amplitude information from the envelope followers is used to control the gain applied to each band, emphasizing or de-emphasizing the frequencies based on the input signal's amplitude and a user-defined threshold.

  • The processed bands are then summed to create the output signal.

  1. Psychoacoustic processing:

Psychoacoustic processing techniques exploit the characteristics of human hearing to create the perception of enhanced high-frequency content. For example, one approach is to use spectral masking, where higher-frequency content is boosted in the presence of lower-frequency content. This can be achieved using a combination of filters and dynamic processing:

  • First, the input signal is split into low-frequency and high-frequency bands using a crossover filter.

  • An envelope follower is applied to the low-frequency band, tracking the amplitude of the signal in that band.

  • The amplitude information from the envelope follower is used to control the gain applied to the high-frequency band, emphasizing the high frequencies based on the amplitude of the low-frequency content.

  • The processed bands are then summed to create the output signal.

Each of these methods can be combined or adjusted to create a unique exciter algorithm, with specific implementation details and parameter choices depending on the desired sound and application.

41> let's talk about Wave Shaping and Dynamic Convolution in more mathematical depth -- how are they implemented?

Wave shaping and dynamic convolution are both audio processing techniques that can be used for distortion, harmonic enhancement, or modeling of analog devices. Let's dive deeper into the mathematical aspects of their implementation.

Wave Shaping:

Wave shaping is a non-linear distortion technique that applies a transfer function to the input signal to create an output signal. The transfer function defines the relationship between input and output levels, and it can be chosen to create various types of distortion or to emphasize specific harmonics. Mathematically, the output signal y(x) is obtained by applying the transfer function f(x) to the input signal x:

y(x) = f(x)

One common approach is to use a polynomial function as the transfer function:

f(x) = a0 + a1 * x + a2 * x^2 + ... + an * x^n

Where a0, a1, ..., an are coefficients that determine the shape of the transfer function. By adjusting these coefficients, different types of distortion or harmonic enhancement can be created. Other transfer functions, such as piecewise-linear, sigmoid, or exponential functions, can also be used for wave shaping.

Dynamic Convolution:

Dynamic convolution is an advanced technique used to model the behavior of analog devices or complex digital systems. Unlike standard convolution, which uses a fixed impulse response, dynamic convolution uses an impulse response that varies depending on the input signal's amplitude, frequency content, or other factors.

Dynamic convolution can be described as a time-varying linear system with an impulse response h(t, x), where t is time and x is the input signal. The output signal y(t) is obtained by convolving the input signal x(t) with the time-varying impulse response h(t, x):

y(t) =x(τ) * h(t - τ, x)

Dynamic convolution can be implemented in the time domain or the frequency domain, depending on the specific application and computational requirements. In the time domain, the implementation typically involves segmenting the input signal into overlapping blocks, estimating the time-varying impulse response for each block, and convolving the input signal with the impulse responses using techniques like overlap-add or overlap-save.

In the frequency domain, dynamic convolution can be implemented using techniques like partitioned convolution or real-time partitioned convolution, which involve dividing the input signal and impulse response into frequency-domain segments and performing the convolution using fast Fourier transforms (FFTs).

Dynamic convolution can be used for various applications, such as modeling the behavior of analog devices (e.g., amplifiers, compressors, or reverberation units) or simulating complex acoustic environments (e.g., rooms, halls, or stadiums). The specific implementation details and parameter choices depend on the desired sound and the characteristics of the system being modeled.

42> what is Volterra series? could you describe this idea in depth and how it applies to the concepts we have been discussing? what about the spectral theory? how does it apply

Volterra Series:

The Volterra series is a generalization of the Taylor series that can be used to describe the behavior of nonlinear systems with memory. The Volterra series can be thought of as a multi-dimensional polynomial expansion that expresses the output of a system as a function of its input and its past inputs. It is a powerful tool for modeling and analyzing nonlinear systems, including analog devices, audio effects, and many other applications.

The nth-order Volterra series can be represented as:

y(t) =h_n(x(t), x(t-τ_1), ..., x(t-τ_n)) * x(t-τ_1) * ... * x(t-τ_n) dτ_1 ... dτ_n

Here, y(t) is the output signal, x(t) is the input signal, h_n are the Volterra kernels (which are the generalized impulse responses), and τ_i are time delays.

In the context of audio processing and the concepts we've been discussing, the Volterra series can be used to model nonlinear systems, such as distortion effects, harmonic enhancement, or analog device emulation. The advantage of using the Volterra series is that it can capture the complex interactions between the input signal and the system's nonlinearities, providing a more accurate model than simpler methods like static nonlinearities or memoryless transfer functions.

Spectral Theory:

Spectral theory is a branch of mathematics that deals with the properties and behavior of linear operators on vector spaces, especially those associated with spectral decompositions, such as eigenvalues, eigenvectors, and singular values. In the context of signal processing and audio effects, spectral theory can be applied to the analysis and manipulation of signals in the frequency domain.

Spectral theory is particularly relevant to audio processing when working with linear time-invariant (LTI) systems, as these systems can be represented by their impulse response or their transfer function, which are related through the Fourier transform. The spectral properties of LTI systems, such as their frequency response and phase response, provide valuable insights into their behavior and can help guide the design of audio effects and signal processing algorithms.

In the context of the concepts we've been discussing, spectral theory can be applied to analyze and design linear components of audio effects, such as filters, equalizers, or reverberation algorithms. For example, equalization can be thought of as modifying the frequency response of a signal by applying a linear filter with a specific transfer function, which can be designed and analyzed using spectral theory.

When dealing with nonlinear systems or effects, like distortion or harmonic enhancement, spectral theory becomes more challenging to apply directly. However, it can still provide valuable insights into the behavior of the system by analyzing the spectral content of the input and output signals or by examining the linear components of the system, such as filters or other linear processing stages.

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~296K SLoC