Crest Factor Optimization for Multisine Signals

1. Introduction

Crest factor optimization is a crucial technique in signal processing, particularly for multisine signals used in system identification and control. The crest factor is defined as the ratio of a signal's peak value to its root mean square (RMS) value:

Crest Factor = Peak Value / RMS Value

A lower crest factor is desirable because it allows for more efficient use of the available dynamic range in measurement systems and power amplifiers. This optimization process involves adjusting the phases of the individual sine components in a multisine signal to minimize the overall crest factor.

Key features of this optimization method include:

Combination of sigmoid transform and Gauss-Newton optimization
Frequency bin optimization for improved frequency resolution
Adaptive switching between optimization modes
Configurable parameters for fine-tuning the optimization process

In the following sections, we'll explore the steps of the optimization process, provide an interactive demo, and dive into the implementation details.

2. Steps of Crest Factor Optimization

Explore how different parameters affect the signal and optimization process using the interactive sliders below.

2.1. Signal Generation

Number of Frequencies: 10

Maximum Frequency (Hz): 100

2.2. Frequency Bin Optimization

Frequency Bin Optimization in Multisine Signals

Frequency bin optimization is a crucial technique in generating multisine signals with improved spectral properties and potentially lower crest factors. This method aligns the component frequencies of the multisine signal with the discrete frequency bins of the Fast Fourier Transform (FFT).

Key Concepts:

FFT Frequency Resolution: The frequency resolution of an FFT is given by:

Δf = f_s / N

Where f_s is the sampling frequency and N is the number of samples.

Frequency Alignment: Each component frequency is rounded to the nearest multiple of Δf.
Benefits:
- Reduces spectral leakage
- Improves frequency domain representation
- Can lead to lower crest factors

Implementation:


def optimize_frequencies(frequencies, fs, N):
    Δf = fs / N
    return [round(f / Δf) * Δf for f in frequencies]

This process ensures that each component of the multisine signal aligns precisely with an FFT bin, leading to a cleaner spectral representation and potentially improving the effectiveness of subsequent optimization steps.

Note: While frequency bin optimization can improve signal properties, it slightly alters the original frequency distribution. The trade-off between spectral clarity and frequency accuracy should be considered based on the specific application requirements.

Sampling Rate (Hz): 1000

Use Frequency Bin Optimization

2.3. Iterative Optimization

2.3.1. Sigmoid Transform Mode

Sigmoid Transform in Crest Factor Optimization

The sigmoid transform is a crucial step in our crest factor optimization process. It helps reshape the signal's amplitude distribution, effectively reducing peak values while maintaining the overall signal characteristics.

Key Points:

Definition: The sigmoid function is defined as:

f(x) = 1 / (1 + e^-kx) - 0.5

Effect: It compresses large amplitude values while leaving smaller values relatively unchanged.
Tunability: The 'k' factor allows for adjusting the strength of the transformation.
Preservation: It maintains the zero-crossing points of the original signal.
Reversibility: The transform is invertible, allowing for signal reconstruction if needed.

Process in Optimization:

Apply the sigmoid transform to the current signal.
Compute the Fourier transform of the transformed signal.
Extract new phase information from the Fourier transform.
Reconstruct the signal using the original frequencies and new phases.

This process helps to redistribute the signal energy, typically resulting in a lower crest factor. The sigmoid transform is particularly effective in the early stages of optimization, providing a good starting point for subsequent fine-tuning steps.

Sigmoid Factor: 0.5

Crest Factor:

2.3.2. Gauss-Newton Mode

The Role of p-norm in Gauss-Newton Optimization for Crest Factor Reduction

The p-norm plays a crucial role in the Gauss-Newton optimization process for crest factor reduction, particularly in approximating the maximum value of the signal. Here's why it's important:

Approximation of Maximum: The p-norm, especially with higher values, provides a smooth approximation of the maximum value of a signal. As p approaches infinity, the p-norm converges to the true maximum.
Numerical Stability: Using the p-norm instead of directly working with the maximum allows for a more numerically stable optimization process. It provides a differentiable objective function, which is essential for gradient-based optimization methods like Gauss-Newton.
Tunability: The choice of p allows for fine-tuning the optimization process. Higher p values place more emphasis on larger signal values, pushing the optimization towards minimizing the peak values more aggressively.
Gradual Transition: By using the p-norm, we create a gradual transition between optimizing for average power (p = 2) and peak power (p → ∞). This allows the algorithm to balance between reducing overall signal variance and specifically targeting peak values.
Mathematical Expression: The p-norm provides a mathematical expression that approximates the maximum value while still being amenable to optimization techniques. It's defined as:

||x||_p = (Σ|x_i|^p)^1/p

In the context of crest factor optimization:

As p increases, ||x||_p approaches max(|x_i|)
The optimization problem becomes: minimize ||x||_p / ||x||₂
This formulation allows us to use calculus-based optimization methods on a well-behaved function

By leveraging the p-norm in this way, we can effectively target the reduction of peak values (which define the crest factor) while maintaining a mathematically tractable optimization problem. This approach bridges the gap between theoretical signal processing concepts and practical numerical optimization techniques.

p-norm: 256

3. Interactive Demo

Click here for full screen

4. Details & Code

This section provides a simplified overview of the crest factor optimization algorithm. We'll examine key steps and their pseudo-code representations.

4.1 Signal Generation

We generate a multisine signal by combining multiple sine waves:

function generateMultisineSignal(frequencies, phases, time):
    signal = []
    for each t in time:
        value = 0
        for i = 0 to length(frequencies) - 1:
            value += cos(2π * frequencies[i] * t + phases[i])
        signal.append(value)
    return signal

4.2 Frequency Bin Optimization

We adjust frequencies to align with FFT bins for better spectral properties:

function optimizeFrequencies(frequencies, samplingRate):
    freqResolution = samplingRate / signalLength
    for i = 0 to length(frequencies) - 1:
        frequencies[i] = round(frequencies[i] / freqResolution) * freqResolution
    return frequencies

4.3 Sigmoid Transform

We apply a sigmoid transform to reshape the signal amplitude distribution:

function sigmoidTransform(signal, factor):
    for i = 0 to length(signal) - 1:
        signal[i] = 1 / (1 + exp(-factor * signal[i])) - 0.5
    return signal

4.4 Gauss-Newton Optimization

We use the Gauss-Newton method to optimize signal phases:

function gaussNewtonOptimization(signal, frequencies, phases):
    calculate Jacobian matrix J
    calculate residual vector r
    solve (J^T * J) * Δphases = -J^T * r
    update phases = phases + Δphases
    return updated phases

4.5 Crest Factor Calculation

We calculate the crest factor as the ratio of peak to RMS value:

function calculateCrestFactor(signal):
    peak = maximum absolute value in signal
    rms = square root of (average of squared values in signal)
    return peak / rms

Optimization Process

The overall optimization process alternates between sigmoid transform and Gauss-Newton steps:

while not converged and iterations < maxIterations:
    if current step is 1:
        apply sigmoid transform on multisine signal
        Do FFT on the transformed signal
        Inject the phases from excited frequencies into multisine
    else:
        apply Gauss-Newton optimization
    update signal with new phases
    calculate new crest factor
    check for convergence

Conclusion

This simplified implementation combines sigmoid transform and Gauss-Newton optimization to minimize the crest factor of multisine signals. The process iteratively adjusts signal phases to achieve a lower peak-to-RMS ratio, resulting in a more efficient use of the signal's dynamic range.

The interactive nature of the demo allows users to experiment with different parameters and observe their effects on the optimization process, providing valuable insights into the behavior of multisine signals and crest factor optimization techniques.