Documentation

Overview

This program generates a synthetic signal composed of a sine wave with added noise, computes its Fast Fourier Transform (FFT) to analyze its frequency components, detects peaks in the spectrum, and displays the results.

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Included Libraries

• <iostream>: for input/output operations.
• <vector>: to handle dynamic arrays.
• <complex>: for complex number operations (though not directly used here).
• <fftw3.h>: FFTW library for performing FFTs.
• <cmath>: mathematical functions like sin() and sqrt().
• <cstdlib>, <ctime>, <random>: for random number generation.
• <algorithm>: for functions like max_element().

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Constants and Data Structures

• PI: a constant for π.
• struct Peak: to store information about detected peaks (their index and magnitude).

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Functions

1. generateSignal()

• Purpose: Creates a synthetic signal of length N samples containing a sine wave of a specified frequency (freq) plus random noise.
• Parameters:
  • N: number of samples.
  • freq: sine wave frequency.
  • sampleRate: sampling rate in Hz.
  • generator: random engine for noise.
  • noiseAmplitude: amplitude of added noise (default 0.5).
• Process:
  • For each sample n, compute sin(2π * freq * n / sampleRate).
  • Add random noise uniformly distributed between -0.5 and 0.5, scaled by noiseAmplitude.
• Returns: a vector containing the noisy sine wave.

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2. performFFT()

• Purpose: Computes the FFT of the input signal and returns the magnitude spectrum.
• Process:
  • Allocates memory for FFT input (in) and output (out) arrays using FFTW.
  • Copies the real input signal into the complex input array (in), setting imaginary parts to zero.
  • Creates an FFT plan and executes it.
  • Calculates the magnitude of the first half of the FFT output (since the FFT of real signals is symmetric).
  • Frees FFTW resources.
• Returns: a vector of magnitudes representing the spectrum.

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3. detectPeaks()

• Purpose: Finds local maxima in the spectrum that are above a specified threshold.
• Parameters:
  • spectrum: magnitude spectrum.
  • threshold: minimum magnitude to consider a peak.
  • minDistance: minimum separation between peaks (not fully implemented here).
• Process:
  • Iterates through the spectrum (excluding first and last points).
  • Checks if a point is greater than its neighbors and above the threshold.
  • Collects such points as peaks.
• Returns: a vector of Peak objects.

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4. displaySpectrum()

• Purpose: Prints a summarized view of the spectrum, limiting the number of printed points for readability.
• Parameters:
  • spectrum: the magnitude spectrum.
  • maxPoints: maximum number of points to display (default 50).

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5. displayPeaks()

• Purpose: Prints information about the detected peaks.

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Main Function Workflow

1. Initialize Random Generator:

   • Seeds with current time for noise randomness.

2. Parameters:

   • sampleRate = 1024 Hz
   • N = 1024 samples
   • freq = 50 Hz (signal frequency)

3. Generate Signal:

   • Calls generateSignal() to create a noisy sine wave.

4. Compute FFT:

   • Calls performFFT() to get the spectrum.

5. Display Spectrum:

   • Calls displaySpectrum() to print a summarized spectrum.

6. Peak Detection:

   • Finds the maximum magnitude in the spectrum.
   • Sets a threshold at 30% of this maximum.
   • Calls detectPeaks() to find peaks above this threshold.

7. Display Detected Peaks:

   • Calls displayPeaks().

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Summary

This code performs a basic spectral analysis:

• Generates a noisy sine wave signal.
• Uses FFT to analyze its frequency content.
• Detects prominent peaks in the spectrum, which correspond to significant frequency components.
• Outputs the spectrum and the peaks for inspection.

This approach is common in signal processing applications for identifying dominant frequencies in noisy signals.

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Note: To run this code, you need to have FFTW installed on your system and compile with the appropriate flags, e.g., -lfftw3.