A learning project by Huy · AI-assisted content · Updated
Signals14 min read
Fourier Transform
Understand how a waveform can be described as a combination of frequencies and how to read an FFT without confusing bins with physical components.
Mental model
A musical chord sounds like one waveform in time, but your ear separates it into notes. The Fourier transform performs a mathematical version of that separation.
The FFT changes the viewpoint: one complex time waveform becomes frequency components with magnitude and phase.UNDERSTAND THE BEHAVIOR
01
A signal does not physically split into separate drawings, but many waveforms can be represented as a weighted sum of sinusoids. The Fourier transform changes the description from amplitude versus time to magnitude and phase versus frequency.
02
A computer normally uses the discrete Fourier transform, calculated efficiently with an FFT algorithm. Because only N samples are observed, the spectrum has N discrete bins. Longer observation time creates narrower bins and therefore finer frequency resolution.
03
If the record ends midway through a cycle, the FFT must treat the sudden edge as part of a repeating signal. Energy then spreads into nearby bins. A window softens the record edges, reducing leakage while widening the measured peaks.
Why it matters
Fourier analysis reveals harmonics, switching noise, mechanical vibration, communication channels, and filter performance that may be difficult to recognize in a time trace.
Core ideas
The time domain shows when a signal changes; the frequency domain shows which repeating rates are present.
A finite measurement produces discrete frequency bins with spacing Δf = fs/N.
Windowing reduces spectral leakage when the captured record does not contain an exact number of cycles.
1Sample a sensor at fs = 8 kHz and collect N = 2,000 samples.
2The recording lasts N/fs = 0.25 s.
3Frequency-bin spacing is Δf = 8000/2000 = 4 Hz.
4A 1 kHz tone therefore appears near bin k = 1000/4 = 250.
Common mistake
A tall FFT bin is not automatically a real tone. Check units, window choice, sample rate, record length, DC offset, and aliasing before interpreting it.
RememberThe time domain shows when a signal changes; the frequency domain shows which repeating rates are present.
DEEPER EXPLANATION
What does a Fourier spectrum measure?
A complicated waveform can be described as a combination of sinusoidal components at different frequencies, amplitudes and phases. The Fourier transform expresses that change of viewpoint. For measured digital data, a discrete Fourier transform (DFT) compares a finite sequence with sine and cosine patterns at a set of frequency bins.
For N samples collected at fs samples per second, the record duration is T = N/fs and adjacent DFT bins are spaced Δf = fs/N = 1/T apart. Bin k corresponds to kΔf before interpreting negative-frequency bins. A fast Fourier transform (FFT) is an efficient algorithm for computing the same DFT; it does not add information that the samples did not contain.
A finite record acts like a window. If a sinusoid does not complete an integer number of cycles within that record, its energy spreads across bins (spectral leakage). Applying a window can control sidelobes but changes the shape and width of peaks. Zero padding draws a smoother-looking spectrum by interpolating bins; a longer observation is what improves the ability to separate close tones.
Walk through another example
Collect N = 1,000 samples at fs = 10,000 samples/s. Duration T = 0.1 s and Δf = 10 Hz.
A 200 Hz tone completes 20 cycles in the record and lines up with bin k = 20. A 205 Hz tone lies between the 200 and 210 Hz bins, so its energy generally spreads instead of appearing in a single bin.
To make bin spacing 5 Hz at the same sample rate, record N = 2,000 samples for 0.2 s. Adding zeros to the original 1,000 samples would not create that extra observation.
Check your intuition
Ask three separate questions: Is the input band limited enough before sampling? How long is the record? Which window was used? Each answers a different source of error.
Where this model can mislead you
Bin spacing is not automatically the exact ability to distinguish two nearby sinusoids; window shape, noise and the actual signal matter. The Nyquist sampling calculator helps with the first prerequisite only.
TEST YOUR UNDERSTANDING
Three quick checks
Choose one answer per question, then check your work. Each answer includes an explanation.
Practice and check your understanding
Try each question before opening the worked answer.
1. A DFT uses N = 1000 samples at fs = 10 kHz. What is the frequency-bin spacing?
Show answer and reasoning
Δf = fs/N = 10000/1000 = 10 Hz. Bin k = 20 represents 200 Hz in the positive-frequency range. The record lasts N/fs = 0.1 s; resolving nearby tones also depends on the window.
2. Does zero-padding add new measured information or truly resolve two previously unresolved tones?
Show answer and reasoning
No. It samples the existing spectrum more densely, making the display smoother. A longer measured record and a suitable window are needed to improve the ability to distinguish close tones.