How to implement FFT using JavaScript

In the field of signal processing, Fast Fourier Transform (FFT) is a widely used algorithm for converting time domain signals into frequency domain signals. The efficiency and accuracy of FFT make it widely used in fields such as audio, video, speech, image, and electricity. As a highly portable and flexible scripting language, JavaScript is widely used in web development, so it is very necessary to implement the JavaScript version of FFT.

This article will introduce how to use JavaScript to implement FFT.

Algorithm Introduction

FFT algorithm is based on the Fast Fourier Transform algorithm, which can convert a discrete time domain signal into a discrete frequency domain signal. In the field of computers, there are two types of FFT algorithms: discrete Fourier transform (DFT) and fast Fourier transform (FFT). The discrete Fourier transform is the basis of FFT.

The formula of discrete Fourier transform is:

$$X_k=\sum_{n=0}^{N-1}x_ne^{-i2\pi kn/N}, k=0,1,2,\cdots,N-1$$

Among them, $x_n$ represents the value of the $n$th sampling point in the time domain signal $x$, and $X_k$ represents the frequency The value of the $k$th frequency component in the domain signal $X$. Its computational complexity is $O(N^2)$ and its time complexity is high.

The fast Fourier transform is an algorithm based on the divide-and-conquer strategy, which can optimize the computational complexity of the discrete Fourier transform to $O(N\log N)$, significantly improving the computational complexity. speed.

JavaScript implements FFT

Next, we will introduce how to use JavaScript to implement the FFT algorithm.

First, we need to clarify the input and output of the FFT algorithm. The input of the FFT algorithm is a set of time domain signals, and the output is the component of the signal in the frequency domain. In JavaScript, we can use an array to represent a set of discrete time domain signals, where the value of each element represents the sampled value of the signal at that moment.

When implementing the FFT algorithm, we need the following steps:

1. Calculate the input signal to obtain the time domain sampling points.
2. Rearrange the obtained sampling points according to the Bit-Reversal algorithm to reduce cache misses in calculations and improve calculation efficiency.
3. Use recursive calculation of FFT algorithm. The recursive process divides and conquers the signal. In each recursive level, the signal is divided into two subsets of even points and odd points, and then the two subsets are recursively calculated and then joined.
4. Calculate the amplitude and phase of the frequency domain signal. According to the formula $|X_k|=\sqrt{Re(X_k)^2 Im(X_k)^2}$ and $\angle X_k=\tan^{-1}\left(\frac{Im(X_k)}{Re (X_k)}\right)$ to calculate frequency amplitude and phase.

The following is a sample code for implementing the FFT algorithm in JavaScript:

function fft(signal) {
  const N = signal.length;
  const X = new Array(N);

  if (N === 1) {
    X[0] = signal[0];
    return X;
  }

  const even = new Array(N / 2);
  const odd = new Array(N / 2);

  for (let i = 0; i < N / 2; i++) {
    even[i] = signal[2 * i];
    odd[i] = signal[2 * i + 1];
  }

  const E = fft(even);
  const O = fft(odd);

  for (let i = 0; i < N / 2; i++) {
    const w = Math.exp((-2 * Math.PI * i) / N);
    const b = w * O[i];

    X[i] = E[i] + b;
    X[i + N / 2] = E[i] - b;
  }

  return X;
}

function amplitudeAndPhase(X) {
  const N = X.length;
  const amplitude = new Array(N);
  const phase = new Array(N);

  for (let i = 0; i < N; i++) {
    const Re = X[i].real;
    const Im = X[i].imaginary;

    amplitude[i] = Math.sqrt(Re * Re + Im * Im);
    phase[i] = Math.atan2(Im, Re);
  }

  return { amplitude, phase };
}

function bitReversal(signal) {
  const N = signal.length;
  const X = new Array(N);

  for (let i = 0; i < N; i++) {
    X[reverseBits(i, Math.log2(N))] = signal[i];
  }

  return X;
}

function reverseBits(num, bits) {
  let reversed = 0;

  for (let i = 0; i < bits; i++) {
    reversed = (reversed << 1) | (num & 1);
    num >>= 1;
  }

  return reversed;
}

In this sample code, we define several auxiliary functions, including calculating amplitude and phase, Bit-Reversal Algorithms etc. The most important is the fft function, which accepts an array as an input signal and computes the FFT algorithm using recursion.

Conclusion

FFT algorithm is a commonly used signal processing algorithm, widely used in audio, video, speech, image and other fields. This article describes how to implement the FFT algorithm using JavaScript. In specific implementation, we need to adopt some optimization methods, such as Bit-Reversal algorithm and recursive method. By implementing and using the FFT algorithm, we can make signal processing more convenient and help with work in web development and other fields.

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