From an analytical perspective, a signal consists of multiple components.

Analyte signal: the signal of interest

Matrix signal: e.g., interference, suppression, or enhancement

System signal: e.g., noise, drift, or baseline

Goal of signal processing: Extract the analyte signal.

What is Convolution?

Convolution is a moving process that uses a kernel (analysis function) to transform a signal. \[ (f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau \] with \(f\) as the signal and \(g\) as the kernel

We can use convolution to smooth a signal.

What are Common Kernels for Smoothing?

Boxcar: all weights are equal
e.g. [0.25, 0.25, 0.25, 0.25]
or [0.33 0.33 0.33] \[ k(t) = \frac{1}{n} \]

This is used for simple moving averages.

Span: 4

Gaussian: weights follow a Gaussian distribution
e.g. [0.05, 0.25, 0.4, 0.25, 0.05] \[ k(t) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{t^2}{2\sigma^2}\right) \]

This is used for weighted moving averages.

Span: 4

Savitzky-Golay: weights are polynomial coefficients
e.g. [-3, 12, 17, 12, -3] for a quadratic polynomial
\[ y(t) = \sum_{i=-n}^{n} c_i \cdot f(t + i) \] Coefficients \( c_i \) are derived to fit a polynomial of order \( p \) in a moving window.

This method is ideal for smoothing while preserving peak shapes in signals.

Window Size: 5
Quadratic (2) Cubic (3) Quartic (4) Polynomial Order

What is the problem with Smoothing?

Smoothing reduces noise, but it also alters the signal.

Stronger smoothing:

Removes more noise

But reduces peak intensity

And widens the signal

Weaker smoothing:

Preserves signal intensity and shape

But leaves more noise in the data

The choice of smoothing parameters (kernel size, type, etc.) is a trade-off between noise reduction and signal preservation.

The elbow criterion is a method to determine the optimal smoothing parameter.

By plotting the error vs. the smoothing span, we can find a point where:

Increasing the span no longer significantly reduces the error.

This point, resembling an elbow in the plot, indicates a balance between noise reduction and signal preservation.

Steps to apply:

1. Choose a range of smoothing spans.

2. Compute the error (e.g., root mean squared error) for each span.

3. Plot the error vs. the span and find the elbow point.

The 1st derivative provides both smoothing and baseline correction by highlighting changes in the signal while reducing noise.

How it works: Savitzky-Golay computes derivatives directly from a polynomial fit within a moving window.

Applications:

Removing baseline drifts by zeroing the derivative of constant or linear trends.

Smoothing noisy data while retaining peak features and enhancing changes.

Example Polynomial Fit: \[ y'(t) = \frac{d}{dt} \sum_{i=-n}^{n} c_i \cdot f(t + i) \]

How are Savitzky-Golay Coefficients generated?

The coefficients are derived by fitting a polynomial over a moving window using a Vandermonde matrix and its pseudo-inverse.

Vandermonde Matrix:
(e.g., for 5 points & 2 orders) \[ \mathbf{V} = \begin{bmatrix} 1 & t_{-2} & t_{-2}^2 \\ 1 & t_{-1} & t_{-1}^2 \\ 1 & t_{0} & t_{0}^2 \\ 1 & t_{1} & t_{1}^2 \\ 1 & t_{2} & t_{2}^2 \end{bmatrix} = \begin{bmatrix} 1 & -2 & 4 \\ 1 & -1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{bmatrix} \] where \( t_i \in \{-2, -1, 0, 1, 2\} \) for a window of 5 points.

Pseudo-Inverse: \[ \mathbf{V}^+ = (\mathbf{V}^T \mathbf{V})^{-1} \mathbf{V}^T \] The rows of \( \mathbf{V}^+ \) are the coefficients for smoothing and derivatives.

What is Fourier Transformation and how does it work?

Fourier Transformation transfers axial frequencies to radial frequencies and analyzes interferences.

Axial Frequency:
Periodic signal along an axis. \[ \cos(x \cdot f) \] Frequency: 10

Radial:
Number of loops around a circle. \[ \exp(i \cdot x \cdot f) \]

The radial frequency is the number of loops around the circle per unit of time.

Fourier Transformation combines axial and radial frequencies to identify patterns: \[ \hat{y}_k = \sum_{j=0}^{N-1} \left(e^{-2\pi i \cdot \frac{jk}{N}} \cdot y_j\right) \] Where \( y \) is the dataset with \( N \) points.

When frequencies differ, the integral approaches zero. For matching frequencies, the sum increases.

Fourier Transformation systematically varies radial frequencies to identify all frequencies in the dataset.

This process works in both directions and is reversible:

High-frequency noise can be filtered by removing frequencies above a threshold:

\[ I(f) = \begin{cases} I(f), & \text{if } f < \text{threshold} \\ 0, & \text{otherwise.} \end{cases} \]

The result is a denoised dataset.

When the signal contains sharp peaks, the Fourier Transformation may not be suitable for denoising.

Peaks contain not a single frequency but a range of frequencies.

The sharper the peak, the broader the frequency range.

When the signal contains broad peaks, the Fourier Transformation may be suitable for denoising.

Fourier Transformation extracts frequencies from a signal: \[ \hat{y}_k = \sum_{j=0}^{N-1} \left(e^{-2\pi i \cdot \frac{jk}{N}} \cdot y_j\right) \] Output: Frequency spectrum showing which frequencies are present in the signal.

Frequencies can be filtered (e.g., high-frequency noise removal).

The signal can be transformed back to the time domain after filtering.

Discrete Wavelet Transformation (DWT):
A method that decomposes a signal into components using wavelets.

Unlike FFT, DWT provides both frequency and location information of the signal's features.

\[ DWT(x) = \sum_{j,k} x(t) \cdot \psi_{j,k}(t) \] Where \(\psi_{j,k}(t)\) are scaled and shifted versions of a mother wavelet.

Wavelets are localized in both time and frequency.

In DWT, we use two functions to analyze signals.

The Haar wavelet operates on two functions:

1. Scaling Function (\(\phi(t)\)):
Used to calculate averages (low-frequency components).
\[ \phi(t) = \begin{cases} 1, & 0 \leq t < 1 \\ 0, & \text{otherwise.} \end{cases} \]

2. Wavelet Function (\(\psi(t)\)):
Used to calculate differences (high-frequency components).
\[ \psi(t) = \begin{cases} 1, & 0 \leq t < 0.5 \\ -1, & 0.5 \leq t < 1 \\ 0, & \text{otherwise.} \end{cases} \]

These functions decompose a signal into approximations and details.

The Haar Wavelet Transformation processes data by:

1. Splitting the data \( y \) into vertical packages: (e.g., scale of 4) \[ y = \begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ y_4 \\ \vdots \\ y_n \end{bmatrix} \rightarrow P = \begin{bmatrix} y_1 & y_{5} & y_{9} \\ y_2 & y_{6} & y_{10} \\ y_3 & y_{7} & y_{11} \\ y_4 & y_{8} & y_{12} \end{bmatrix} \]

2. Applying the Scaling Function: \[ \phi = \frac{1}{4} \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix} \] Averages: \[ A = \phi \cdot P \]

The DWT decomposes a signal into approximations and details.

On the left side, we see the original signal.

On the left side, we see the original and reconstructed signal.

Depending on the Wavelet, the signal may look step-like (e.g., Haar Wavelet).

Depending on the Wavelet, the signal may look step-like (e.g., Haar Wavelet).

On the left side, we see the original and reconstructed signal.

The Discrete Wavelet Transformation (DWT) provides a way to analyze signals in both:

Frequency domain (like FFT).

Time domain (unlike FFT).

By using Wavelets:

Approximations: Low-frequency components for trends (smooth signal).

Details: High-frequency components for transient features (sharp changes).