Compute Fourier Transforms with step-by-step solutions. Analyze frequency content of signals, audio files, and images using Continuous FT, Discrete FT (DFT), and Fast Fourier Transform (FFT) algorithms. Professional-grade signal processing tools completely free.
Windowing Function ℹ️
Reduces spectral leakage in FFT analysis. Hamming window is good for general use, Hann for frequency resolution, Blackman for sidelobe suppression.
Hamming
Display Units
Hz/radians
Fourier Transform Results
Time Domain Signal
Frequency Spectrum (Magnitude)
Phase Spectrum
Power Spectral Density
Step-by-Step Solution
1
Original Function
f(t) = sin(2π·5t)
This is a continuous-time sinusoidal signal with frequency 5 Hz. The function is defined for all real values of t.
2
Fourier Transform Definition
F(ω) = ∫_{-∞}^{∞} f(t) e^{-jωt} dt
The Fourier Transform converts a time-domain function f(t) to its frequency-domain representation F(ω).
The integral computes the projection of f(t) onto complex exponentials e^{-jωt} at each frequency ω.
3
Apply Fourier Transform
F(ω) = ∫_{-∞}^{∞} sin(2π·5t) e^{-jωt} dt
Substitute the given function into the Fourier Transform integral. Use Euler's formula to express sin(2π·5t) in terms of complex exponentials.
4
Solve the Integral
F(ω) = (jπ)[δ(ω + 10π) - δ(ω - 10π)]
The integral evaluates to a pair of Dirac delta functions centered at ±10π rad/s (or ±5 Hz).
This shows the signal contains only two frequency components at ±5 Hz.
5
Final Result
F(ω) = jπ[δ(ω + 10π) - δ(ω - 10π)]
The Fourier Transform shows the signal's frequency content: pure tones at 5 Hz and -5 Hz with amplitude π and phase ±90° (from the j factor).
Understanding Fourier Transforms
The Fourier Transform is one of the most powerful tools in mathematics, engineering, and science. It decomposes a function of time (a signal) into the frequencies that make it up, similar to how a musical chord can be expressed as the frequencies (pitches) of its constituent notes.
🔍 What Does the Fourier Transform Do?
The Fourier Transform converts a signal from its original domain (often time or space) to a representation in the frequency domain. This reveals:
Which frequencies are present in the signal
How much of each frequency is present (amplitude)
Phase relationships between different frequency components
Hidden periodicities not obvious in the time domain
📊 Types of Fourier Transforms
Continuous Fourier Transform (FT)
For continuous, analog signals. Defined by an integral: F(ω) = ∫ f(t)e^{-jωt} dt
Discrete Fourier Transform (DFT)
For sampled, digital signals. Computed using summation: X[k] = Σ x[n]e^{-j2πkn/N}
Fast Fourier Transform (FFT)
Efficient algorithm for computing DFT. Reduces computation from O(N²) to O(N log N)
Communications: OFDM in 4G/5G, modem design, spectrum analysis
Medical Imaging: MRI reconstruction, ultrasound analysis
Seismology: Earthquake analysis, oil exploration
Finance: Time series analysis, option pricing
📈 Interpreting Fourier Transform Results
When you compute a Fourier Transform, you get:
Magnitude Spectrum: Shows amplitude of each frequency component
Phase Spectrum: Shows timing relationships between components
Power Spectral Density: Shows power distribution across frequencies
Frequency Bins: Discrete frequency values in DFT/FFT results
💡 Pro Tip: Understanding Frequency Resolution
The frequency resolution of your Fourier Transform depends on the sampling rate and number of points (N).
Frequency resolution = Sampling Rate / N. For example, with 44100 Hz sampling and 2048-point FFT,
each frequency bin represents 44100/2048 ≈ 21.53 Hz. Smaller bins give better frequency resolution.
Frequently Asked Questions
What is the difference between Fourier Transform and Fourier Series?
+
Fourier Series decomposes periodic functions into a sum of sines and cosines at discrete frequencies (harmonics). Fourier Transform extends this to non-periodic functions, representing them as integrals over a continuous range of frequencies. Fourier Series is for periodic signals; Fourier Transform is for both periodic and non-periodic signals.
Why do I get negative frequencies in Fourier Transform results?
+
Negative frequencies arise mathematically from using complex exponentials e^{-jωt}. Physically, a negative frequency component paired with its positive counterpart represents a real sinusoidal signal. For real-valued signals, the Fourier Transform has conjugate symmetry: F(-ω) = F*(ω). The magnitude spectrum is symmetric, while the phase spectrum is anti-symmetric.
What is spectral leakage and how can I reduce it?
+
Spectral leakage occurs when the signal frequency doesn't align exactly with FFT frequency bins, causing energy to "leak" into adjacent bins. To reduce leakage: 1) Use windowing functions (Hamming, Hann, Blackman), 2) Increase FFT size (more points), 3) Ensure integer number of signal cycles in the analysis window, 4) Use zero-padding for better frequency interpolation.
Can I analyze non-stationary signals with Fourier Transform?
+
Standard Fourier Transform assumes stationary signals (frequency content doesn't change over time). For non-stationary signals (like speech or music), use Short-Time Fourier Transform (STFT) or Wavelet Transform. STFT applies Fourier Transform to short, overlapping windows of the signal, creating a time-frequency representation called a spectrogram.
What is the Nyquist frequency and why is it important?
+
The Nyquist frequency is half the sampling rate (fs/2). According to the Nyquist-Shannon sampling theorem, to accurately represent a signal, you must sample at least twice the highest frequency present. Frequencies above Nyquist cause aliasing - higher frequencies masquerading as lower ones. Always apply an anti-aliasing filter before sampling.
How do I choose the right window function for my analysis?
+
Choose based on your priority: 1) Hamming - Good general purpose, 2) Hann - Better frequency resolution, 3) Blackman - Lower sidelobes (less leakage), 4) Rectangular - Best frequency resolution but worst leakage, 5) Flattop - Most accurate amplitude measurement. For unknown signals, start with Hamming window.
Related Calculators
Explore these other powerful mathematical tools on DerivativeCalculus.com: