Fourier Transform
Reference for the continuous Fourier Transform X(f) = ∫ x(t) e^(-i2πft) dt.
Converts time-domain signals to frequency spectra.
Foundation of DSP.
Continuous Fourier Transform
The Fourier transform converts a function of time x(t) into a function of frequency X(f). The output is in general complex-valued, with magnitude representing how much of each frequency is present in the signal and phase representing the timing offset of each frequency component.
Inverse Fourier Transform
The inverse transform reconstructs the original time-domain signal from its frequency spectrum. The forward and inverse transforms differ only in the sign of the exponent.
Discrete Fourier Transform (DFT)
For sampled signals of finite length N, the discrete Fourier transform produces N frequency components. The Fast Fourier Transform (FFT) is an O(N log N) algorithm for computing the DFT.
Variables
| Symbol | Meaning |
|---|---|
| x(t), x[n] | Time-domain signal (continuous, discrete) |
| X(f), X[k] | Frequency-domain spectrum |
| t | Time (seconds) |
| f | Frequency (Hz) |
| i | Imaginary unit (√−1) |
| N | Number of samples (DFT) |
| k | Discrete frequency bin index |
Example — Single Cosine
A signal x(t) = cos(2π × 100t). What is its Fourier transform?
cos(2π × 100t) = ½ (e^(i2π × 100t) + e^(−i2π × 100t))
Each complex exponential transforms to a Dirac delta in frequency
X(f) = ½ δ(f − 100) + ½ δ(f + 100)
Two spikes at f = +100 Hz and f = −100 Hz, each of magnitude ½
Real signals always produce frequency spectra that are symmetric about f = 0. The negative frequencies are mathematical artifacts of the complex exponential basis; physically, the signal contains only the positive frequency.
Example — Rectangular Pulse
A pulse x(t) = 1 for −T/2 ≤ t ≤ T/2, zero elsewhere. Find X(f).
X(f) = ∫₋T/2^T/2 1 × e^(−i2πft) dt
= [e^(−i2πft) / (−i2πf)] from −T/2 to T/2
= (e^(−iπfT) − e^(iπfT)) / (−i2πf)
X(f) = T × sin(πfT) / (πfT) = T × sinc(fT)
The sinc function appears whenever you transform a finite-duration pulse. Its main lobe width is inversely proportional to the pulse duration T — short pulses have wide frequency spectra. This is the basis of the uncertainty principle in signal processing.
Key Properties
| Property | Time domain | Frequency domain |
|---|---|---|
| Linearity | a × x(t) + b × y(t) | a × X(f) + b × Y(f) |
| Time shift | x(t − t₀) | X(f) × e^(−i2πft₀) |
| Modulation | x(t) × e^(i2πf₀t) | X(f − f₀) |
| Convolution | x(t) * h(t) | X(f) × H(f) |
| Multiplication | x(t) × y(t) | X(f) * Y(f) |
| Scaling | x(at) | (1/|a|) × X(f/a) |
| Parseval | ∫|x(t)|² dt | = ∫|X(f)|² df |
The convolution-multiplication duality is one of the most useful in practice: convolving signals in time corresponds to multiplying their spectra in frequency. This is why filters are designed in the frequency domain and applied in the time domain.
Common Transform Pairs
| Time domain | Frequency domain |
|---|---|
| δ(t) (impulse) | 1 (flat spectrum) |
| 1 (constant) | δ(f) (DC component) |
| cos(2πf₀t) | ½ [δ(f − f₀) + δ(f + f₀)] |
| Rectangular pulse of width T | T × sinc(fT) |
| Gaussian e^(−πt²) | Gaussian e^(−πf²) |
| Comb (Dirac comb) | Comb (Dirac comb) |
When to Use It
- Audio analysis — spectral display, pitch detection, EQ design
- Image processing — JPEG compression uses the DCT, a Fourier cousin
- Communications — modulation, demodulation, OFDM
- Vibration analysis — identify resonant frequencies in mechanical systems
- Medical imaging — MRI reconstruction is fundamentally a Fourier inversion
- Optics — diffraction patterns are Fourier transforms of apertures
- Solving differential equations — convert differentiation to multiplication by i2πf
The Uncertainty Principle
A signal cannot be arbitrarily concentrated in both time and frequency. If σ_t is the duration and σ_f is the bandwidth, then σ_t × σ_f ≥ 1 / (4π). A signal localized to a 1 ms pulse must span at least 80 Hz of frequency. This is mathematical fact, not a physical limitation — it follows from the structure of the Fourier transform itself.
FFT Practical Notes
The FFT requires N to be a power of 2 for the classical Cooley-Tukey algorithm. Modern FFT libraries (FFTW, Apple Accelerate) handle arbitrary N efficiently. The FFT outputs both positive and negative frequencies — for real signals, the negative-frequency half is the conjugate of the positive half and is often discarded.