How to calculate a Fourier series in Numpy?

How to calculate a Fourier series in Numpy?

Calculating a Fourier series in NumPy involves summing a series of sine and cosine functions to approximate a given function. The Fourier series represents a periodic function as an infinite sum of trigonometric functions. Here's how you can calculate a Fourier series using NumPy:

  • Import NumPy and other necessary libraries:
import numpy as np import matplotlib.pyplot as plt
  • Define the function you want to approximate using its Fourier series. In this example, we'll approximate a square wave function with a given period and amplitude:
def square_wave(x, period, amplitude): return amplitude * np.sign(np.sin(2 * np.pi * x / period))
  • Define the Fourier series formula and calculate the coefficients for the series. The coefficients are calculated using integration or summation, but in this example, we'll use a simplified approach for a square wave:
def fourier_series(x, period, amplitude, num_terms): result = 0.5 * amplitude for n in range(1, num_terms + 1): result += (amplitude / np.pi) * (1 / n) * np.sin(2 * np.pi * n * x / period) return result
  • Define the x-values (time domain) over which you want to calculate the Fourier series and the number of terms in the series:
num_points = 500 # Number of points x_values = np.linspace(0, 2, num_points) # Time domain num_terms = 10 # Number of Fourier series terms to use
  • Calculate the Fourier series values for the given x-values:
period = 2 # Period of the square wave amplitude = 1 # Amplitude of the square wave fourier_series_values = [fourier_series(x, period, amplitude, num_terms) for x in x_values]
  • Plot the original function and the Fourier series approximation:
original_values = [square_wave(x, period, amplitude) for x in x_values] plt.figure(figsize=(10, 6)) plt.plot(x_values, original_values, label="Original Function") plt.plot(x_values, fourier_series_values, label="Fourier Series Approximation") plt.legend() plt.title("Fourier Series Approximation") plt.xlabel("x") plt.ylabel("y") plt.grid(True) plt.show()

This code will approximate the square wave function using a Fourier series with a specified number of terms and plot both the original function and the approximation.

You can adjust the num_terms variable to control the number of terms in the Fourier series, which will affect the accuracy of the approximation.

Examples

  1. How to calculate a Fourier series using Numpy in Python?

    • Description: This query seeks information on utilizing Numpy, a popular numerical computing library in Python, to compute a Fourier series.
    import numpy as np def fourier_series(x, n_terms): series = np.zeros_like(x) for n in range(1, n_terms + 1): series += (4 / (np.pi * n)) * np.sin(n * x) return series x = np.linspace(-np.pi, np.pi, 1000) n_terms = 10 result = fourier_series(x, n_terms)

  2. How to compute Fourier series coefficients using Numpy in Python?

    • Description: This query looks for instructions on using Numpy to compute the coefficients of a Fourier series in Python.
    import numpy as np def fourier_coefficients(func, n_terms): coefficients = [] for n in range(n_terms): a_n = 2 * np.trapz(func * np.cos(n * x), x) / np.pi b_n = 2 * np.trapz(func * np.sin(n * x), x) / np.pi coefficients.append((a_n, b_n)) return coefficients x = np.linspace(-np.pi, np.pi, 1000) func = np.sin(x) n_terms = 10 coefficients = fourier_coefficients(func, n_terms)

  3. Calculating Fourier series expansion in Python using Numpy?

    • Description: This query seeks guidance on calculating the expansion of a Fourier series in Python using the Numpy library.
    import numpy as np def fourier_series_expansion(x, n_terms): series = np.zeros_like(x) for n in range(1, n_terms + 1): series += (4 / (np.pi * n)) * np.sin(n * x) return series x = np.linspace(-np.pi, np.pi, 1000) n_terms = 10 series_expansion = fourier_series_expansion(x, n_terms)

  4. How to implement Fourier series approximation in Python using Numpy?

    • Description: This query looks for an implementation of Fourier series approximation in Python, leveraging the capabilities of the Numpy library.
    import numpy as np def fourier_series_approximation(x, n_terms): series = np.zeros_like(x) for n in range(1, n_terms + 1): series += (4 / (np.pi * n)) * np.sin(n * x) return series x = np.linspace(-np.pi, np.pi, 1000) n_terms = 10 series_approximation = fourier_series_approximation(x, n_terms)

  5. How to compute a Fourier series using Numpy and visualize the result?

    • Description: This query aims to compute a Fourier series in Python using Numpy and visualize the resulting series.
    import numpy as np import matplotlib.pyplot as plt def fourier_series(x, n_terms): series = np.zeros_like(x) for n in range(1, n_terms + 1): series += (4 / (np.pi * n)) * np.sin(n * x) return series x = np.linspace(-np.pi, np.pi, 1000) n_terms = 10 result = fourier_series(x, n_terms) plt.plot(x, result) plt.xlabel('x') plt.ylabel('Fourier Series') plt.title('Fourier Series Approximation') plt.grid(True) plt.show()

  6. How to find Fourier series coefficients using Numpy in Python?

    • Description: This query seeks a method for finding Fourier series coefficients in Python using the Numpy library.
    import numpy as np def fourier_coefficients(func, n_terms): coefficients = [] for n in range(n_terms): a_n = 2 * np.trapz(func * np.cos(n * x), x) / np.pi b_n = 2 * np.trapz(func * np.sin(n * x), x) / np.pi coefficients.append((a_n, b_n)) return coefficients x = np.linspace(-np.pi, np.pi, 1000) func = np.sin(x) n_terms = 10 coefficients = fourier_coefficients(func, n_terms)

  7. Calculating Fourier series coefficients and plotting the result using Numpy in Python?

    • Description: This query aims to calculate Fourier series coefficients and plot the result using Numpy in Python.
    import numpy as np import matplotlib.pyplot as plt def fourier_coefficients(func, n_terms): coefficients = [] for n in range(n_terms): a_n = 2 * np.trapz(func * np.cos(n * x), x) / np.pi b_n = 2 * np.trapz(func * np.sin(n * x), x) / np.pi coefficients.append((a_n, b_n)) return coefficients x = np.linspace(-np.pi, np.pi, 1000) func = np.sin(x) n_terms = 10 coefficients = fourier_coefficients(func, n_terms) plt.plot(range(1, n_terms + 1), [coef[0] for coef in coefficients], label='a_n') plt.plot(range(1, n_terms + 1), [coef[1] for coef in coefficients], label='b_n') plt.xlabel('n') plt.ylabel('Coefficient Value') plt.title('Fourier Series Coefficients') plt.legend() plt.grid(True) plt.show()

  8. How to calculate a Fourier series with a custom function in Python using Numpy?

    • Description: This query seeks guidance on calculating a Fourier series with a custom function in Python, utilizing the Numpy library.
    import numpy as np def custom_fourier_series(x, n_terms, custom_function): series = np.zeros_like(x) for n in range(1, n_terms + 1): series += custom_function(n, x) return series def custom_function(n, x): return np.sin(n * x) / n x = np.linspace(-np.pi, np.pi, 1000) n_terms = 10 result = custom_fourier_series(x, n_terms, custom_function)

  9. Calculating Fourier series expansion with specific functions in Python using Numpy?

    • Description: This query aims to calculate a Fourier series expansion with specific functions in Python using the Numpy library.
    import numpy as np def fourier_series_expansion(x, n_terms): series = np.zeros_like(x) for n in range(1, n_terms + 1): series += (4 / (np.pi * n)) * np.sin(n * x) return series x = np.linspace(-np.pi, np.pi, 1000) n_terms = 10 series_expansion = fourier_series_expansion(x, n_terms)

  10. How to implement Fourier series approximation with a custom function in Python using Numpy?

    • Description: This query looks for an implementation of Fourier series approximation with a custom function in Python, utilizing the Numpy library.
    import numpy as np def custom_fourier_series_approximation(x, n_terms, custom_function): series = np.zeros_like(x) for n in range(1, n_terms + 1): series += custom_function(n, x) return series def custom_function(n, x): return np.sin(n * x) / n x = np.linspace(-np.pi, np.pi, 1000) n_terms = 10 result = custom_fourier_series_approximation(x, n_terms, custom_function)

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