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Signal Processing

Fourier Transform

Reading a waveform in terms of its frequencies and amplitudes

1. Adding Waves and Separating Them

Even a complicated sound or vibration can often be understood as a combination of sine and cosine waves at different frequencies. Building a periodic waveform from such components introduces the idea of a Fourier series. A Fourier transform asks the reverse question: which frequencies are present in the original signal?

Let us add just two waves and compare their time-domain waveform with the frequency-domain picture.

2. Example: Add a 5 Hz and a 12 Hz Wave

Construct a one-second signal from a 5 Hz sine wave of amplitude 1 and a 12 Hz cosine wave of amplitude 0.5.

s(t) = sin(2π · 5t) + 0.5 cos(2π · 12t)

The first component oscillates five times per second; the second oscillates twelve times. At t = 0, their sum is 0.5. At t = 0.25 s, both reach a positive crest, giving 1.5. The red dots mark these values.

A 5 Hz sine wave, a 12 Hz cosine wave with amplitude 0.5, and their combined waveform. Red dots mark values at zero and a quarter second.
Figure 1: The two components and their sum over the same one-second interval. The 5 Hz and 12 Hz components are not obvious from the shape of the combined waveform alone.

3. Look at the Frequency Spectrum

Sample this one-second signal 64 times per second to obtain 64 values s[n]. For a finite set of samples, we use the discrete Fourier transform (DFT).

X[k] = ∑n=063 s[n] e−i2πkn/64

Because the observation lasts one second, adjacent frequency bins are 1 Hz apart. Plotting the positive-frequency amplitude 2|X[k]|/64 for 1 ≤ k ≤ 31 reveals peaks at exactly 5 Hz and 12 Hz.

One-sided DFT amplitude spectrum with a peak of 1.0 at 5 Hz and a peak of 0.5 at 12 Hz; all other frequencies are nearly zero.
Figure 2: One-sided amplitude spectrum from 64 samples. The two components appear at 5 Hz with amplitude 1.0 and at 12 Hz with amplitude 0.5.
ComponentFrequencyAmplitudePhase
Sine wave5 Hz1.0−90°
Cosine wave12 Hz0.50°

The amplitude spectrum tells us how strongly each frequency is present, but it does not tell us where each wave starts. Reconstructing the original waveform also requires the phase values in the table.

4. Fourier Transform, DFT, and FFT

The Fourier transform describes the frequency content of a continuous signal. The DFT applies the idea to a finite set of samples. The fast Fourier transform (FFT) is an efficient algorithm for computing the same DFT result.

The two peaks are especially clean here because exactly five and twelve cycles fit into the one-second observation window. In real data, the observation length and where the window is cut can spread energy into neighboring bins. This is called spectral leakage.

The sample rate here is 64 Hz, more than twice the highest signal frequency of 12 Hz. If sampling is too slow, a high frequency can appear as a lower one; this is aliasing.

Adding components produces a time-domain waveform; the DFT takes us back to frequency components. That round trip underlies the analysis of sound, vibration, and images.