NxSignal

https://hexdocs.pm/nx_signal
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NxSignal is an Nx library extension providing digital signal processing functions including STFT,...
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### Install Nx.Signal and related libraries

Source: https://hexdocs.pm/nx_signal/spectrogram.html

Install the necessary libraries for signal processing and visualization. This is typically the first step in using these tools.

```elixir
Mix.install([
  {:nx_signal, "~> 0.3"},
  {:vega_lite, "~> 0.1"},
  {:kino_vega_lite, "~> 0.1"}
])
```

--------------------------------

### Find Local Minima Excluding Global Minimum

Source: https://hexdocs.pm/nx_signal/NxSignal.PeakFinding.html

This example demonstrates finding local minima using `argrelextrema` with a comparator that excludes the global minimum. It ensures that only local minima, not the absolute smallest value in the tensor, are identified.

```elixir
iex> x = Nx.tensor([0, 1, 0, 2, 1, 3, 0, 1])
iex> global_minimum = Nx.reduce_min(x)
iex> comparator = fn x, y ->
...>   x_not_global = Nx.not_equal(x, global_minimum)
...>   y_not_global = Nx.not_equal(y, global_minimum)
...>   both_not_global = Nx.logical_and(x_not_global, y_not_global)
...>   Nx.logical_and(Nx.less(x, y), both_not_global)
...> end
iex> result = NxSignal.PeakFinding.argrelextrema(x, comparator)
iex> result.valid_indices
#Nx.Tensor<
  u32
  1
>
iex> result.indices[0..0]
#Nx.Tensor<
  s32[1][1]
  [
    [4]
  ]
>
```

--------------------------------

### Generate a sawtooth waveform

Source: https://hexdocs.pm/nx_signal/NxSignal.Waveforms.html

Example of defining a 5Hz waveform sampled at 500Hz for 1 second.

```elixir
t = Nx.linspace(0, 1, n: 500)
n = Nx.multiply(2 * :math.pi() * 5, t)
wave = NxSignal.Waveforms.sawtooth(n)
```

--------------------------------

### Find Relative Extrema with Custom Comparator

Source: https://hexdocs.pm/nx_signal/NxSignal.PeakFinding.html

Use `argrelextrema` with a custom comparator function to find elements that meet specific criteria relative to their neighbors. This example defines a maximum as a value greater than or equal to double its neighbors.

```elixir
iex> comparator = fn x, y -> Nx.greater_equal(x, Nx.multiply(y, 2)) end
iex> x = Nx.tensor([0, 1, 3, 2, 0, 1, 0, 0, 0, 2, 1])
iex> result = NxSignal.PeakFinding.argrelextrema(x, comparator)
iex> result.valid_indices
#Nx.Tensor<
  u32
  3
>
iex> result.indices[0..2]
#Nx.Tensor<
  s32[3][1]
  [
    [5],
    [7],
    [9]
  ]
>
```

--------------------------------

### Generate polynomial sweeps with NxSignal

Source: https://hexdocs.pm/nx_signal/NxSignal.Waveforms.html

Demonstrates generating polynomial sweeps using different coefficients and phase offsets.

```elixir
iex> t = Nx.linspace(0, 10, n: 5)
iex> NxSignal.Waveforms.polynomial_sweep(t, Nx.tensor([2, 0, 1]))
#Nx.Tensor<
  f32[5]
  [1.0, 0.8660273, -0.5000064, 1.7942519e-5, -0.4999892]
>
iex> NxSignal.Waveforms.polynomial_sweep(t, Nx.tensor([2, 0, 1]), phi: :math.pi() / 2)
#Nx.Tensor<
  f32[5]
  [-4.371139e-8, 0.49999946, -0.8660195, 1.0, 0.86603385]
>
iex> NxSignal.Waveforms.polynomial_sweep(t, Nx.tensor([1, 0]))
#Nx.Tensor<
  f32[5]
  [1.0, 0.70710653, -1.0, 0.7071085, 1.0]
>
iex> NxSignal.Waveforms.polynomial_sweep(t, Nx.tensor([1, 0]), phi: 180, phi_unit: :degrees)
#Nx.Tensor<
  f32[5]
  [-1.0, -0.70710695, 1.0, -0.70711297, -1.0]
>
```

--------------------------------

### NxSignal Module Overview

Source: https://hexdocs.pm/nx_signal/api-reference.html

Overview of the available modules within the NxSignal library for digital signal processing.

```APIDOC
## NxSignal Modules

### Description
The NxSignal library provides several modules to handle digital signal processing tasks using Nx tensors.

### Modules
- **NxSignal.Convolution**: Provides functions for convolution operations.
- **NxSignal.Filters**: Contains common filter functions for signal processing.
- **NxSignal.PeakFinding**: Implements algorithms for identifying peaks in signals.
- **NxSignal.Waveforms**: Includes functions to calculate waveforms based on time tensors.
- **NxSignal.Windows**: Provides common window functions used in signal analysis.
```

--------------------------------

### Windowing Utility

Source: https://hexdocs.pm/nx_signal/NxSignal.html

Returns a tensor of K windows of length N from an input tensor.

```APIDOC
## AS_WINDOWED as_windowed(tensor, opts)

### Description
Returns a tensor of K windows of length N.

### Parameters
#### Options
- **:window_length** (number) - Required - The number of samples in a window.
- **:stride** (number) - Optional - The number of samples to skip between windows. Defaults to 1.
- **:padding** (atom/string) - Optional - Padding mode: :reflect, :valid, :same, or :zeros. Defaults to :valid.
```

--------------------------------

### Prepare signal data

Source: https://hexdocs.pm/nx_signal/filtering.html

Generate synthetic signal data and prepare it for visualization.

```elixir
fs = 16.0e3
window_duration_seconds = 100.0e-3
window_length = 2 ** ceil(:math.log2(fs * window_duration_seconds))
signal_duration_seconds = 3
signal_length = ceil(signal_duration_seconds * fs)

sin = fn freq, n ->
  Nx.sin(Nx.multiply(2 * :math.pi() * freq / fs, n))
end

half_n = Nx.iota({div(signal_length, 2)})

sin220 = sin.(220, half_n)
sin440 = sin.(440, half_n)
sin1000 = sin.(440 * 5 / 2, half_n)
sin3000 = sin.(220 * 4 / 3 * 4, half_n)

n = Nx.iota({signal_length})
t = Nx.divide(n, fs)
data = Nx.concatenate([Nx.add(sin440, sin1000), Nx.add(sin220, sin3000)])

# Data for plotting

slice = (signal_length - 3000)..(signal_length - 2750)
plot_data = %{y: Nx.to_flat_list(data[[slice]]), x: Nx.to_flat_list(t[[slice]])}
```

```elixir
VegaLite.new(width: 600, height: 400, title: "Signal sample")
|> VegaLite.data_from_values(plot_data, only: ["x", "y"])
|> VegaLite.mark(:line)
|> VegaLite.encode_field(:x, "x", type: :quantitative)
|> VegaLite.encode_field(:y, "y", type: :quantitative)
```

--------------------------------

### Plot Audio Sample Data

Source: https://hexdocs.pm/nx_signal/spectrogram.html

Visualizes a segment of the generated audio data as a line plot using Vega-Lite. Ensure the data `d` is available in the scope.

```elixir
VegaLite.new(width: 600, height: 600, title: "Audio Sample")
|> VegaLite.data_from_values(d, only: ["n", "data"])
|> VegaLite.mark(:line)
|> VegaLite.encode_field(:x, "n", type: :ordinal)
|> VegaLite.encode_field(:y, "data", type: :quantitative)
```

--------------------------------

### Design and visualize FIR filter

Source: https://hexdocs.pm/nx_signal/filtering.html

Create a low-pass FIR filter and visualize its time-domain coefficients and frequency response.

```elixir
# Cutoff frequency in Hz
fc = 600

# Design the FIR filter coefficients using the window method
h = NxSignal.Filters.firwin(window_length, fc, sampling_rate: fs, window: :hann)

# Separate periodic Hann window for STFT analysis
stft_window = NxSignal.Windows.hann(window_length)

hfft =
  h
  |> Nx.fft(length: window_length)
  |> Nx.abs()
  |> Nx.add(1.0e-10)

hfft_power =
  hfft
  |> Nx.log()
  |> Nx.divide(Nx.log(10))
  |> Nx.multiply(20)

f_idx = Nx.iota({window_length}) |> Nx.subtract(div(window_length, 2))

f = Nx.multiply(f_idx, fs / window_length)

plot_data = %{
  n: Enum.to_list(0..(window_length - 1)),
  h: Nx.to_flat_list(h),
  hfft:
    Nx.to_flat_list(
      Nx.take_along_axis(
        hfft_power,
        Nx.select(Nx.less(f_idx, 0), Nx.add(f_idx, window_length), f_idx)
      )
    ),
  f: Nx.to_flat_list(f)
}
```

```elixir
VegaLite.new(
  width: 600,
  height: 400,
  title: "600 Hz Low-Pass filter (Hann window); L = 2048"
)
|> VegaLite.data_from_values(plot_data, only: ["n", "h"])
|> VegaLite.mark(:line)
|> VegaLite.encode_field(:x, "n", type: :quantitative)
|> VegaLite.encode_field(:y, "h", type: :quantitative)
```

```elixir
VegaLite.new(
  width: 600,
  height: 400,
  title: "FFT - 600 Hz Low-Pass filter (Hann window); L = 2048"
)
|> VegaLite.data_from_values(plot_data, only: ["f", "hfft"])
|> VegaLite.mark(:line)
|> VegaLite.encode_field(:x, "f", type: :quantitative)
|> VegaLite.encode_field(:y, "hfft", type: :quantitative)
```

--------------------------------

### Windowing Utilities

Source: https://hexdocs.pm/nx_signal

Functions for manipulating signal tensors into windowed formats.

```APIDOC
## as_windowed

### Description
Returns a tensor of K windows of length N from the input tensor.

### Method
FUNCTION

### Endpoint
NxSignal.as_windowed(tensor, opts)

### Parameters
#### Arguments
- **tensor** (Tensor) - Required - The input signal tensor.

#### Options
- **window_length** (number) - Optional - The number of samples in a window.
- **stride** (number) - Optional - The number of samples to skip between windows. Defaults to 1.
- **padding** (atom/string) - Optional - Padding mode: :reflect, :valid, :zeros, or custom padding. Defaults to :valid.
```

--------------------------------

### Design an FIR filter with firwin

Source: https://hexdocs.pm/nx_signal/NxSignal.Filters.html

Computes FIR filter coefficients using the window method.

```elixir
iex> coeffs = NxSignal.Filters.firwin(5, [0.3], window: :hamming, sampling_rate: 2.0)
iex> Nx.shape(coeffs)
{5}
```

--------------------------------

### Create windowed views of tensors with NxSignal.as_windowed

Source: https://hexdocs.pm/nx_signal/NxSignal.html

Generates sliding window views of a tensor. Supports custom window lengths, strides, and padding strategies like reflection.

```elixir
iex> NxSignal.as_windowed(Nx.tensor([0, 1, 2, 3, 4, 10, 11, 12]), window_length: 4)
#Nx.Tensor<
  s32[5][4]
  [
    [0, 1, 2, 3],
    [1, 2, 3, 4],
    [2, 3, 4, 10],
    [3, 4, 10, 11],
    [4, 10, 11, 12]
  ]
>
```

```elixir
iex> NxSignal.as_windowed(Nx.tensor([0, 1, 2, 3, 4, 10, 11, 12]), window_length: 3)
#Nx.Tensor<
  s32[6][3]
  [
    [0, 1, 2],
    [1, 2, 3],
    [2, 3, 4],
    [3, 4, 10],
    [4, 10, 11],
    [10, 11, 12]
  ]
>
```

```elixir
iex> NxSignal.as_windowed(Nx.tensor([0, 1, 2, 3, 4, 10, 11]), window_length: 2, stride: 2, padding: [{0, 3}])
#Nx.Tensor<
  s32[5][2]
  [
    [0, 1],
    [2, 3],
    [4, 10],
    [11, 0],
    [0, 0]
  ]
>
```

```elixir
iex> t = Nx.iota({7});
iex> NxSignal.as_windowed(t, window_length: 6, padding: :reflect, stride: 1)
#Nx.Tensor<
  s32[8][6]
  [
    [3, 2, 1, 0, 1, 2],
    [2, 1, 0, 1, 2, 3],
    [1, 0, 1, 2, 3, 4],
    [0, 1, 2, 3, 4, 5],
    [1, 2, 3, 4, 5, 6],
    [2, 3, 4, 5, 6, 5],
    [3, 4, 5, 6, 5, 4],
    [4, 5, 6, 5, 4, 3]
  ]
>
```

```elixir
iex> NxSignal.as_windowed(Nx.iota({10}), window_length: 6, padding: :reflect, stride: 2)
#Nx.Tensor<
  s32[6][6]
  [
    [3, 2, 1, 0, 1, 2],
    [1, 0, 1, 2, 3, 4],
    [1, 2, 3, 4, 5, 6],
    [3, 4, 5, 6, 7, 8],
    [5, 6, 7, 8, 9, 8],
    [7, 8, 9, 8, 7, 6]
  ]
>
```

--------------------------------

### Generate periodic square waves

Source: https://hexdocs.pm/nx_signal/NxSignal.Waveforms.html

Demonstrates square wave generation with constant and time-varying duty cycles.

```elixir
iex> t = Nx.iota({10}) |> Nx.multiply(:math.pi() * 2 / 10)
iex> NxSignal.Waveforms.square(t, duty: 0.1)
#Nx.Tensor<
  s32[10]
  [1, -1, -1, -1, -1, -1, -1, -1, -1, -1]
>
iex> NxSignal.Waveforms.square(t, duty: 0.5)
#Nx.Tensor<
  s32[10]
  [1, 1, 1, 1, 1, -1, -1, -1, -1, -1]
>
iex> NxSignal.Waveforms.square(t, duty: 1)
#Nx.Tensor<
  s32[10]
  [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
>

iex> t = Nx.iota({10}) |> Nx.multiply(:math.pi() * 2 / 10)
iex> duty = Nx.tensor([0.1, 0, 0.3, 0, 0.5, 0, 0.7, 0, 0.9, 0])
iex> NxSignal.Waveforms.square(t, duty: duty)
#Nx.Tensor<
  s32[10]
  [1, -1, 1, -1, 1, -1, 1, -1, 1, -1]
>
```

--------------------------------

### Generate Triangular Window

Source: https://hexdocs.pm/nx_signal/NxSignal.Windows.html

Creates a triangular window. Options include window length, output type, and axis name.

```elixir
iex> NxSignal.Windows.triangular(3)
#Nx.Tensor<  f32[3]
  [0.5, 1.0, 0.5]
>
```

--------------------------------

### NxSignal.Windows Functions

Source: https://hexdocs.pm/nx_signal/NxSignal.Windows.html

Common window functions for signal processing.

```APIDOC
## bartlett(n, opts)

### Description
Bartlett triangular window.

### Parameters
#### Path Parameters
- **n** (integer) - Required - The window length.

#### Options
- **:type** (tuple) - Optional - The output type for the window. Defaults to {:f, 32}.
- **:name** (atom) - Optional - The axis name. Defaults to nil.

## blackman(n, opts)

### Description
Blackman window.

### Parameters
#### Path Parameters
- **n** (integer) - Required - The window length.

#### Options
- **:is_periodic** (boolean) - Optional - If true, produces a periodic window, otherwise produces a symmetric window. Defaults to true.
- **:type** (tuple) - Optional - The output type for the window. Defaults to {:f, 32}.
- **:name** (atom) - Optional - The axis name. Defaults to nil.

## hamming(n, opts)

### Description
Hamming window.

### Parameters
#### Path Parameters
- **n** (integer) - Required - The window length.

#### Options
- **:is_periodic** (boolean) - Optional - If true, produces a periodic window, otherwise produces a symmetric window. Defaults to true.
- **:type** (tuple) - Optional - The output type for the window. Defaults to {:f, 32}.
- **:name** (atom) - Optional - The axis name. Defaults to nil.

## hann(n, opts)

### Description
Hann window.

### Parameters
#### Path Parameters
- **n** (integer) - Required - The window length.

#### Options
- **:is_periodic** (boolean) - Optional - If true, produces a periodic window, otherwise produces a symmetric window. Defaults to true.
- **:type** (tuple) - Optional - The output type for the window. Defaults to {:f, 32}.
- **:name** (atom) - Optional - The axis name. Defaults to nil.

## kaiser(n, opts)

### Description
Creates a Kaiser window of size window_length using a Bessel function.

### Parameters
#### Path Parameters
- **n** (integer) - Required - The window length.

#### Options
- **:is_periodic** (boolean) - Optional - If true, produces a periodic window, otherwise produces a symmetric window. Defaults to true.
- **:type** (tuple) - Optional - The output type for the window. Defaults to {:f, 32}.
- **:beta** (float) - Optional - Shape parameter. Defaults to 12.0.
- **:eps** (float) - Optional - Epsilon value to avoid division by zero. Defaults to 1.0e-7.
- **:axis_name** (atom) - Optional - The axis name. Defaults to nil.

## rectangular(n, opts)

### Description
Rectangular window, useful when no window function should be applied.

### Parameters
#### Path Parameters
- **n** (integer) - Required - The window length.

#### Options
- **:type** (atom) - Optional - The output type. Defaults to s64.

## triangular(n, opts)

### Description
Triangular window.

### Parameters
#### Path Parameters
- **n** (integer) - Required - The window length.

#### Options
- **:type** (tuple) - Optional - The output type for the window. Defaults to {:f, 32}.
- **:name** (atom) - Optional - The axis name. Defaults to nil.
```

--------------------------------

### Find relative minima with order

Source: https://hexdocs.pm/nx_signal/NxSignal.PeakFinding.html

Use the order option to define the neighborhood size for relative minima detection.

```elixir
iex> x = Nx.tensor([2, 1, 2, 3, 2, 0, 1, 0])
iex> %{indices: indices, valid_indices: valid_indices} = NxSignal.PeakFinding.argrelmin(x, order: 3)
iex> valid_indices
#Nx.Tensor<
  u32
  1
>
iex> indices
#Nx.Tensor<
  s32[8][1]
  [
    [1],
    [-1],
    [-1],
    [-1],
    [-1],
    [-1],
    [-1],
    [-1]
  ]
>
iex> Nx.slice_along_axis(indices, 0, Nx.to_number(valid_indices), axis: 0)
#Nx.Tensor<
  s32[1][1]
  [
    [1]
  ]
>
```

--------------------------------

### Plot Spectrogram with Default Window

Source: https://hexdocs.pm/nx_signal/spectrogram.html

Calculates and plots the spectrogram of the audio data using a default window duration. This provides a baseline visualization of the signal's frequency content over time.

```elixir
Spectrogram.calculate_stft_and_plot_spectrogram(data, fs, 50.0e-3)
```

--------------------------------

### Visualize reconstructed signal

Source: https://hexdocs.pm/nx_signal/filtering.html

Plots the reconstructed time-domain signal using VegaLite.

```elixir
VegaLite.new(width: 600, height: 400, title: "Reconstructed Signal")
|> VegaLite.data_from_values(plot_data, only: ["x", "y"])
|> VegaLite.mark(:line)
|> VegaLite.encode_field(:x, "x", type: :quantitative)
|> VegaLite.encode_field(:y, "y", type: :quantitative)
```

--------------------------------

### Prepare spectrogram data for plotting

Source: https://hexdocs.pm/nx_signal/filtering.html

Iterates over time and frequency indices to format spectrogram data into a map suitable for visualization.

```elixir
plot_data =
  for t_idx <- 0..(Nx.size(t) - 1),
      f_idx <- 0..max_f,
      Nx.to_number(f[[f_idx]]) <= 4000,
      reduce: %{"t" => [], "f" => [], "s" => [], "filtered_s" => []} do
    %{"t" => t_acc, "f" => f_acc, "s" => s_acc, "filtered_s" => filtered_s_acc} ->
      %{
        "t" => [Nx.to_number(t[[t_idx]]) | t_acc],
        "f" => [Float.round(Nx.to_number(f[[f_idx]]), 3) | f_acc],
        "s" => [Nx.to_number(spectrogram[[t_idx, f_idx]]) | s_acc],
        "filtered_s" => [Nx.to_number(filtered_spectrogram[[t_idx, f_idx]]) | filtered_s_acc]
      }
  end
```

--------------------------------

### Perform overlap-and-add reconstruction with NxSignal.overlap_and_add

Source: https://hexdocs.pm/nx_signal/NxSignal.html

Reconstructs a signal from windowed segments by overlapping and adding them. Requires specifying the overlap length.

```elixir
iex> NxSignal.overlap_and_add(Nx.iota({3, 4}), overlap_length: 0)
#Nx.Tensor<
  s32[12]
  [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
>
```

```elixir
iex> NxSignal.overlap_and_add(Nx.iota({3, 4}), overlap_length: 3)
#Nx.Tensor<
  s32[6]
  [0, 5, 15, 18, 17, 11]
>
```

```elixir
iex> t = Nx.tensor([[[[0, 1, 2, 3], [4, 5, 6, 7]]], [[[10, 11, 12, 13], [14, 15, 16, 17]]]]) |> Nx.vectorize(x: 2, y: 1)
iex> NxSignal.overlap_and_add(t, overlap_length: 3)
#Nx.Tensor<
  vectorized[x: 2][y: 1]
  s32[5]
  [
    [
      [0, 5, 7, 9, 7]
    ],
    [
      [10, 25, 27, 29, 17]
    ]
  ]
>
```

--------------------------------

### Generate Bartlett Triangular Window

Source: https://hexdocs.pm/nx_signal/NxSignal.Windows.html

Creates a Bartlett triangular window of a specified size. Options include output type and axis name.

```elixir
iex> NxSignal.Windows.bartlett(3)
#Nx.Tensor<
  f32[3]
  [0.0, 0.6666667, 0.6666666]
>
```

--------------------------------

### Define Spectrogram plotting module

Source: https://hexdocs.pm/nx_signal/filtering.html

Creates a reusable module for generating VegaLite spectrogram visualizations.

```elixir
defmodule Spectrogram do
  alias VegaLite, as: Vl

  def plot(title, dataset) do
    Vl.new(title: title, width: 500, height: 500)
    |> Vl.mark(:rect)
    |> Vl.data_from_values(dataset)
    |> Vl.encode_field(:x, "t",
      type: :quantitative,
      title: "Time (seconds)",
      axis: [tick_min_step: 0.1],
      grid: false
    )
    |> Vl.encode_field(:y, "f",
      type: :quantitative,
      sort: "-x",
      title: "Frequency (Hz)",
      axis: [tick_count: 25],
      grid: false
    )
    |> Vl.encode_field(:color, "s",
      aggregate: :max,
      type: :quantitative,
      scale: [scheme: "viridis"],
      legend: [title: "dBFS"]
    )
    |> Vl.config(view: [stroke: nil])
  end
end
```

--------------------------------

### Spectrogram Calculation and Plotting Module

Source: https://hexdocs.pm/nx_signal/spectrogram.html

Defines a module for calculating and plotting spectrograms from audio input. It includes functions for STFT, data transformation, and visualization.

```elixir
defmodule Spectrogram do
  alias VegaLite, as: Vl
  import Nx.Defn

  def calculate_stft_and_plot_spectrogram(
        input,
        fs,
        window_duration_ms,
        plot_cutoff_frequency \ 4000
      ) do
    n_window = ceil(fs * window_duration_ms)
    {spectrogram, f, t, max_f} = stft(input, fs: fs, n_window: n_window)

    max_f = Nx.to_number(max_f)
    spectrogram = Nx.slice(spectrogram, [0, 0], [Nx.size(t), max_f])
    f = Nx.slice(f, [0], [max_f])

    spectrogram
    |> to_plot_data(f, t, plot_cutoff_frequency)
    |> plot()
  end

  defn stft(input, opts) do
    fs = opts[:fs]
    n_window = opts[:n_window]

    # ms to samples
    window = NxSignal.Windows.hann(n: n_window, is_periodic: true)

    # use the default overlap of 50%
    {s, t, f} = NxSignal.stft(input, window, sampling_rate: fs, fft_length: 1024)

    max_f =
      Nx.select(f >= fs / 2, Nx.iota(f.shape), Nx.size(f) + 1)
      |> Nx.argmin()

    spectrogram = Nx.abs(s)
    # to dBFS
    spectrogram = 20 * Nx.log(spectrogram / Nx.reduce_max(spectrogram)) / Nx.log(10)

    {spectrogram, f, t, max_f}
  end

  defp to_plot_data(s, f, t, plot_cutoff_frequency) do
    for t_idx <- 0..(Nx.size(t) - 1),
        f_idx <- 0..(Nx.size(f) - 1),
        Nx.to_number(f[[f_idx]]) <= plot_cutoff_frequency,
        reduce: %{"t" => [], "f" => [], "s" => []} do
      %{"t" => t_acc, "f" => f_acc, "s" => s_acc} ->
        %{"t" => [Nx.to_number(t[[t_idx]]) | t_acc], "f" => [Float.round(Nx.to_number(f[[f_idx]]), 3) | f_acc], "s" => [Nx.to_number(s[[t_idx, f_idx]]) | s_acc]}
    end
  end

  defp plot(dataset) do
    Vl.new(title: "Spectrogram", width: 500, height: 500)
    |> Vl.mark(:rect)
    |> Vl.data_from_values(dataset)
    |> Vl.encode_field(:x, "t",
      type: :quantitative,
      title: "Time (seconds)",
      axis: [tick_min_step: 0.1],
      grid: false
    )
    |> Vl.encode_field(:y, "f",
      type: :quantitative,
      sort: "-x",
      title: "Frequency (Hz)",
      axis: [tick_count: 25],
      grid: false
    )
    |> Vl.encode_field(:color, "s",
      aggregate: :max,
      type: :quantitative,
      scale: [scheme: "viridis"],
      legend: [title: "dBFS"]
    )
    |> Vl.config(view: [stroke: nil])
  end
end
```

--------------------------------

### Windowing Functions

Source: https://hexdocs.pm/nx_signal

Functions for creating and manipulating windowed signals.

```APIDOC
## as_windowed(tensor, opts \\ [])

### Description
Returns a tensor of K windows of length N.

### Method
N/A (Function Call)

### Endpoint
N/A

### Parameters
#### Arguments
- **tensor** (Nx.Tensor) - Required - The input tensor.

#### Options
- **:window_size** (integer) - Required - The size of each window.
- **:step** (integer) - Optional - The step size between windows. Defaults to `window_size`.
- **:padding` (integer) - Optional - The amount of zero-padding to apply to the ends of the tensor. Defaults to `0`.

### Request Example
```elixir
input_tensor = Nx.tensor(1..10)
NxSignal.as_windowed(input_tensor, window_size: 3, step: 2)
```

### Response
#### Success Response (200)
- **Tensor** (Nx.Tensor) - A tensor where each row represents a windowed segment of the input.

#### Response Example
```json
{
  "example": "#Nx.Tensor<\n  i32[4, 3]\n  [[1, 2, 3], [3, 4, 5], [5, 6, 7], [7, 8, 9]]\n>"
}
```
```

```APIDOC
## overlap_and_add(tensor, opts \\ [])

### Description
Performs the overlap-and-add algorithm over an {..., M, N}-shaped tensor, where M is the number of windows and N is the window size.

### Method
N/A (Function Call)

### Endpoint
N/A

### Parameters
#### Arguments
- **tensor** (Nx.Tensor) - Required - The input tensor containing windowed segments.

#### Options
- **:step** (integer) - Required - The step size between windows (must be less than `window_size`).
- **:window_size** (integer) - Optional - The size of each window. Defaults to the last dimension of the input tensor.

### Request Example
```elixir
windowed_tensor = Nx.tensor([[1, 2, 3], [3, 4, 5], [5, 6, 7]])
NxSignal.overlap_and_add(windowed_tensor, step: 2)
```

### Response
#### Success Response (200)
- **Tensor** (Nx.Tensor) - The reconstructed signal after overlap-and-add.

#### Response Example
```json
{
  "example": "#Nx.Tensor<\n  i32[7]\n  [1, 5, 12, 15, 18, 13, 7]\n>"
}
```
```

--------------------------------

### Prepare signal data for line plot

Source: https://hexdocs.pm/nx_signal/filtering.html

Converts reconstructed signal slices into a flat list format for line chart visualization.

```elixir
plot_data = %{
  y: Nx.to_flat_list(data_out[[slice]]),
  x: Nx.to_flat_list(Nx.divide(n, fs)[[slice]])
}
```

--------------------------------

### Generate Kaiser Window

Source: https://hexdocs.pm/nx_signal/NxSignal.Windows.html

Creates a Kaiser window using a Bessel function. Customizable with shape parameter 'beta', periodicity, output type, and epsilon.

```elixir
iex> NxSignal.Windows.kaiser(4, beta: 12.0, is_periodic: true)
#Nx.Tensor<  f32[4]
  [5.277619e-5, 0.21566667, 1.0, 0.21566667]
>
```

```elixir
iex> NxSignal.Windows.kaiser(5, beta: 12.0, is_periodic: true)
#Nx.Tensor<  f32[5]
  [5.277619e-5, 0.10171464, 0.792937, 0.792937, 0.10171464]
>
```

```elixir
iex> NxSignal.Windows.kaiser(4, beta: 12.0, is_periodic: false)
#Nx.Tensor<  f32[4]
  [5.277619e-5, 0.5188395, 0.51883906, 5.277619e-5]
>
```

--------------------------------

### Plot Spectrogram with Shorter Window

Source: https://hexdocs.pm/nx_signal/spectrogram.html

Generates a spectrogram using a shorter window duration (25ms). Shorter windows improve time resolution but decrease frequency resolution.

```elixir
Spectrogram.calculate_stft_and_plot_spectrogram(data, fs, 25.0e-3)
```

--------------------------------

### Generate Hamming Window

Source: https://hexdocs.pm/nx_signal/NxSignal.Windows.html

Creates a Hamming window. Can be periodic or symmetric, with options for output type and axis name.

```elixir
iex> NxSignal.Windows.hamming(5, is_periodic: true)
#Nx.Tensor<  f32[5]
  [0.08000001, 0.3978522, 0.9121479, 0.9121478, 0.39785212]
>
```

```elixir
iex> NxSignal.Windows.hamming(5, is_periodic: false)
#Nx.Tensor<  f32[5]
  [0.08000001, 0.54, 1.0, 0.54, 0.08000001]
>
```

--------------------------------

### Generate Chirp Waveform with Logarithmic Interpolation

Source: https://hexdocs.pm/nx_signal/NxSignal.Waveforms.html

Generates a chirp waveform using logarithmic frequency interpolation. `f0` and `f1` must be non-zero and have the same sign.

```elixir
t = Nx.linspace(0, 10, n: 5)
NxSignal.Waveforms.chirp(t, 10, 10, 1, method: :logarithmic)
```

--------------------------------

### Plot spectrograms

Source: https://hexdocs.pm/nx_signal/filtering.html

Generates visualizations for both raw and filtered spectrogram data.

```elixir
Spectrogram.plot("Spectrogram", Map.take(plot_data, ["t", "f", "s"]))
```

```elixir
Spectrogram.plot(
  "Filtered Spectrogram",
  Map.take(plot_data, ["t", "f"]) |> Map.put("s", plot_data["filtered_s"])
)
```

--------------------------------

### Generate Blackman Window

Source: https://hexdocs.pm/nx_signal/NxSignal.Windows.html

Generates a Blackman window. Supports periodic and symmetric window types, with options for output type and axis name.

```elixir
iex> NxSignal.Windows.blackman(5, is_periodic: false)
#Nx.Tensor<
  f32[5]
  [-1.4901161e-8, 0.34000003, 0.99999994, 0.34000003, -1.4901161e-8]
>
```

```elixir
iex> NxSignal.Windows.blackman(5, is_periodic: true)
#Nx.Tensor<
  f32[5]
  [-1.4901161e-8, 0.20077012, 0.84922993, 0.84922993, 0.20077012]
>
```

```elixir
iex> NxSignal.Windows.blackman(6, is_periodic: true, type: {:f, 32})
#Nx.Tensor<
  f32[6]
  [-1.4901161e-8, 0.13, 0.63, 0.99999994, 0.63, 0.13]
>
```

--------------------------------

### Generate Rectangular Window

Source: https://hexdocs.pm/nx_signal/NxSignal.Windows.html

Produces a rectangular window, useful when no window function is needed. Supports different output types.

```elixir
iex> NxSignal.Windows.rectangular(5)
#Nx.Tensor<  s64[5]
  [1, 1, 1, 1, 1]
>
```

```elixir
iex> NxSignal.Windows.rectangular(5, type: :f32)
#Nx.Tensor<  f32[5]
  [1.0, 1.0, 1.0, 1.0, 1.0]
>
```

--------------------------------

### Find relative maxima with order

Source: https://hexdocs.pm/nx_signal/NxSignal.PeakFinding.html

Use the order option to define the neighborhood size for relative maxima detection.

```elixir
iex> x = Nx.tensor([2, 1, 2, 3, 2, 0, 1, 0])
iex> %{indices: indices, valid_indices: valid_indices} = NxSignal.PeakFinding.argrelmax(x, order: 3)
iex> valid_indices
#Nx.Tensor<
  u32
  1
>
iex> indices
#Nx.Tensor<
  s32[8][1]
  [
    [3],
    [-1],
    [-1],
    [-1],
    [-1],
    [-1],
    [-1],
    [-1]
  ]
>
iex> Nx.slice_along_axis(indices, 0, Nx.to_number(valid_indices), axis: 0)
#Nx.Tensor<
  s32[1][1]
  [
    [3]
  ]
>
```

--------------------------------

### Generate Audio Data with Sine Waves

Source: https://hexdocs.pm/nx_signal/spectrogram.html

Generates a digital audio signal composed of multiple sine waves at specified frequencies. This code sets up the sample rate and duration, then creates and concatenates sine waves.

```elixir
# You can load an audio file here.  For this example,
# we're producing 3 seconds of 220Hz, 440Hz, 1kHz and 3kHz sine waves

fs = 44.1e3
t_max = 3

full_n = ceil(fs * t_max)
half_n = div(full_n, 2)

# samples/sec * sec = samples
n = Nx.iota({half_n})

sin = fn freq, n ->
  Nx.sin(Nx.multiply(2 * :math.pi() * freq / fs, n))
end

sin220 = sin.(220, n)
sin440 = sin.(440, n)
sin1000 = sin.(1000, n)
sin3000 = sin.(3000, n)

data = Nx.concatenate([Nx.add(sin440, sin1000), Nx.add(sin220, sin3000)])
n = Nx.iota({full_n})

d = %{data: Nx.to_flat_list(data[[1000..1250]]), n: Nx.to_flat_list(n[[1000..1250]])}
```

--------------------------------

### Generate Chirp Waveform with Quadratic Interpolation

Source: https://hexdocs.pm/nx_signal/NxSignal.Waveforms.html

Generates a chirp waveform using quadratic frequency interpolation. The `vertex_zero` option controls the vertex position.

```elixir
t = Nx.linspace(0, 10, n: 5)
NxSignal.Waveforms.chirp(t, 10, 10, 1, method: :quadratic)
```

```elixir
t = Nx.linspace(0, 10, n: 5)
NxSignal.Waveforms.chirp(t, 10, 10, 1, method: :quadratic, vertex_zero: false)
```

--------------------------------

### Generate Hann Window

Source: https://hexdocs.pm/nx_signal/NxSignal.Windows.html

Generates a Hann window, which can be periodic or symmetric. Options include output type and axis name.

```elixir
iex> NxSignal.Windows.hann(5, is_periodic: false)
#Nx.Tensor<  f32[5]
  [0.0, 0.5, 1.0, 0.5, 0.0]
>
```

```elixir
iex> NxSignal.Windows.hann(5, is_periodic: true)
#Nx.Tensor<  f32[5]
  [0.0, 0.34549153, 0.90450853, 0.9045085, 0.34549144]
>
```

--------------------------------

### fft_frequencies

Source: https://hexdocs.pm/nx_signal/NxSignal.html

Computes the frequency bins for a FFT with given options.

```APIDOC
## fft_frequencies(sampling_rate, opts)

### Description
Computes the frequency bins for a FFT with given options.

### Parameters
#### Arguments
- **sampling_rate** (float) - Required - Sampling frequency in Hz.

#### Options
- **:fft_length** (integer) - Optional - Number of FFT frequency bins.
- **:type** (tuple) - Optional - Optional output type. Defaults to {:f, 32}.
- **:name** (atom) - Optional - Optional axis name for the tensor. Defaults to :frequencies
```

--------------------------------

### Generate Chirp Waveform with Hyperbolic Interpolation

Source: https://hexdocs.pm/nx_signal/NxSignal.Waveforms.html

Generates a chirp waveform using hyperbolic frequency interpolation. Ensure time tensor `t` is defined.

```elixir
t = Nx.linspace(0, 10, n: 5)
NxSignal.Waveforms.chirp(t, 10, 10, 1, method: :hyperbolic)
```

--------------------------------

### Plot Spectrogram with Medium Window

Source: https://hexdocs.pm/nx_signal/spectrogram.html

Generates a spectrogram using a medium window duration (100ms). This offers a balance between time and frequency resolution.

```elixir
Spectrogram.calculate_stft_and_plot_spectrogram(data, fs, 100.0e-3)
```

--------------------------------

### Compute Inverse Short-Time Fourier Transform (ISTFT)

Source: https://hexdocs.pm/nx_signal/NxSignal.html

Reconstructs a time-domain signal from its Short-Time Fourier Transform (STFT) representation. Ensure the window and options match those used for the STFT. Note that edge distortions may occur due to windowing.

```elixir
iex> t = Nx.tensor([10, 10, 1, 0, 10, 10, 2, 20])
iex> w = NxSignal.Windows.hann(4)
iex> opts = [sampling_rate: 1, fft_length: 4]
iex> {z, _time, _freqs} = NxSignal.stft(t, w, opts)
iex> result = NxSignal.istft(z, w, opts)
iex> Nx.as_type(result, Nx.type(t))
#Nx.Tensor<
  s32[8]
  [0, 10, 1, 0, 10, 10, 2, 20]
>
```

```elixir
iex> t = Nx.tensor([10, 10, 1, 0, 10, 10, 2, 20])
iex> w = NxSignal.Windows.hann(4)
iex> opts = [scaling: :spectrum, sampling_rate: 1, fft_length: 4]
iex> {z, _time, _freqs} = NxSignal.stft(t, w, opts)
iex> result = NxSignal.istft(z, w, opts)
iex> Nx.as_type(result, Nx.type(t))
#Nx.Tensor<
  s32[8]
  [0, 10, 1, 0, 10, 10, 2, 20]
>
```

```elixir
iex> t = Nx.tensor([10, 10, 1, 0, 10, 10, 2, 20], type: :f32)
iex> w = NxSignal.Windows.hann(4)
iex> opts = [scaling: :psd, sampling_rate: 1, fft_length: 4]
iex> {z, _time, _freqs} = NxSignal.stft(t, w, opts)
iex> result = NxSignal.istft(z, w, opts)
iex> Nx.as_type(result, Nx.type(t))
#Nx.Tensor<
  f32[8]
  [0.0, 10.0, 0.99999994, -2.1900146e-7, 10.0, 10.0, 2.0000002, 20.0]
>
```

--------------------------------

### Generate Chirp Waveform with Linear Interpolation

Source: https://hexdocs.pm/nx_signal/NxSignal.Waveforms.html

Generates a chirp waveform using linear frequency interpolation. Ensure time tensor `t` is defined.

```elixir
t = Nx.linspace(0, 10, n: 5)
NxSignal.Waveforms.chirp(t, 10, 10, 1, method: :linear)
```
Tokens:
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