FFT: From Signal to Spectrum

How an analyser computes a spectrum from N samples: frequency resolution, leakage, window functions and aliasing to explore.

Explanation Tasks

What is happening?

What an analyser really does

The spectrum analyser in your DAW looks as if it showed the spectrum “of the signal”. In fact it always takes just a short frame of N samples (“FFT length N”) and decomposes it into sine waves with the FFT (Fast Fourier Transform). The result is N/2 + 1 frequency points, the bins, at a fixed spacing Δf = fs/N (fs is the sample rate, i.e. the number of samples per second) from 0 Hz up to the Nyquist frequency fs/2. Bin number k lies at k · Δf. Like a ruler with fixed tick marks: whatever lies between two ticks has to be shared out among the ticks.

The module shows both sides. At the top, “Analysis frame (time domain)”: the frame x[n] in blue (n counts the samples in the frame, from 0 to N−1), the window w[n] dashed and the windowed signal x[n]·w[n] in orange – only this goes into the FFT. Below, “Spectrum (result of the FFT)” with levels in dB. The dashed markers f₁ and f₂ show where the frequencies of tone 1 and tone 2 really lie.

Worked example with the default settings: fs = 8 kHz, N = 256 → Δf = 8000/256 = 31.25 Hz, the frame lasts N/fs = 32 ms, and there are 129 bins. A 1000 Hz tone lies exactly on bin 1000/31.25 = 32. These values appear on the right under “Frequency resolution Δf = fs/N”, “Frame length N/fs” and “Tone 1 lies at bin”.

Frequency resolution costs time

Since Δf = fs/N and the frame length is T = N/fs, simply Δf = 1/T. For 1 Hz resolution you need one second of signal – whatever the sample rate. Typical in the studio: fs = 48 kHz, N = 4096 gives Δf ≈ 11.7 Hz and a frame of 85 ms.

Common misconception: “A higher sample rate gives finer frequency resolution.” At the same N it is the other way round: if you change fs from 8 to 48 kHz, the bins become six times wider, because the frame becomes six times shorter. Resolution only gets finer with more time.

Try it: switch on “Second tone (sine)”, set tone 1 to 1000 Hz and tone 2 to 1040 Hz at 0 dB, window Hann, frequency axis “zoom ±16 bins”. At N = 128 (Δf = 62.5 Hz) you see a single peak. From N = 512 (15.6 Hz, 64 ms) two peaks appear with a shallow dip between them; at N = 1024 (7.8 Hz, 128 ms) they are clearly separated. The price: a long frame averages over its whole duration and smears fast changes. That is why long FFTs are used for the bass range and short ones for transients.

Leakage

The FFT treats the frame as if it repeated endlessly. Switch on “Show periodic continuation”: the time diagram then shows the frame three times side by side, with the seams marked by dashed lines. If a whole number of periods fits in, the curve continues seamlessly, the tone lies exactly on a bin and appears as a single line at 0 dB.

Otherwise the signal jumps at the seam, and the marker turns orange. This jump does not exist in the real signal, but the FFT sees it and needs many additional frequencies to represent it. The energy spreads over many bins: leakage. At the same time the displayed level drops. Example: at fs = 8 kHz and N = 256, 1015.625 Hz lies exactly at bin 32.5. With the rectangular window the highest bin shows only −3.9 dB, and 58 bins lie above −40 dB.

Important: the tone has not changed – leakage is a property of the measurement, not of the signal. Try it: click “Let the frequency glide”. Tone 1 moves up half a bin per second, and you see the line become sharp and smear out again in step with the bins.

Windows: narrow lobe or deep skirts

A window fades the frame in and out gently at its edges. Both ends go to zero, and the jump at the seam disappears from the windowed signal. In the spectrum, each tone then becomes a main lobe with side lobes to its left and right. The “Window properties” block shows the figures: rectangular 2 bins main lobe, highest side lobe −13.3 dB; Hann 4 bins, −31.5 dB; Hamming 4 bins, −42.7 dB; Blackman 6 bins, −58.1 dB.

This is a real trade-off: low side lobes make quiet tones next to loud ones visible; the wider main lobe makes closely spaced tones blur more easily. Try it: tone 1 at 1015.625 Hz, tone 2 at 1500 Hz with −60 dB, N = 256, frequency axis linear. With the rectangular window the leakage of tone 1 at 1500 Hz is still about −35 dB and hides tone 2. With Hamming it is about −52 dB – still too much, because although its first side lobe is the lowest, the distant ones hardly fall off. With Hann and Blackman tone 2 appears clearly at −60 dB.

Rule of thumb: Hann is a good all-rounder, Blackman is used for large dynamic range. The rectangular window only suits tones that lie exactly on bins.

Bins are only samples

Set the frequency axis to “zoom ±16 bins”. The grey curve is the continuous spectrum of the windowed frame, the orange stems are the bins. The bins are only samples of this curve. Where they hit the curve depends on where the tone lies between the bins: exactly on a bin, one stem hits the top of the main lobe; at half a bin, two stems lie to the left and right of it.

The levels are scaled so that a full-scale sine exactly on a bin shows 0 dB (dBFS), with every window. If it lies in between, “Measured level tone 1” reads too low. The largest error, exactly between two bins, is what the module calls “Loss between two bins”: 3.9 dB for rectangular, 1.4 dB for Hann, 1.8 dB for Hamming, 1.1 dB for Blackman.

A practical example: a 1 kHz test tone at 0 dBFS lies at bin 1000/46.875 ≈ 21.33 at 48 kHz and N = 1024, i.e. between two bins. With Hann the analyser then reads about 0.6 dB too low. The generator is not broken – for accurate level measurements use a flat-top window or a level meter.

Aliasing: mirroring at fs/2

After sampling, frequencies above fs/2 can no longer be distinguished from lower ones: they appear mirrored at fs/2. Try it: at fs = 8 kHz set tone 1 to 5000 Hz. The marker shows “f₁ 5 kHz → appears at 3 kHz”, because 8000 − 5000 = 3000 Hz. The readout “Tone 1 lies at bin” also gives the frequency at which it appears.

A sampled square wave contains infinitely many overtones, most of them above fs/2. Example: 700 Hz square wave, fs = 8 kHz, N = 4096, Blackman, frequency axis linear. Below 4 kHz only 700, 2100 and 3500 Hz belong to the harmonic series. All other lines are mirrored higher overtones: 4900 Hz appears at 3100 Hz, 6300 Hz at 1700 Hz, 7700 Hz at 300 Hz and so on.

Common misconception: “Whatever shows up below fs/2 in the spectrum belongs to the signal.” After sampling, alias lines can no longer be told apart from real tones and cannot be removed. That is why every AD converter has an analogue anti-aliasing filter in front of it, and distortion or synthesizer plug-ins often work internally with oversampling.

In the studio and limits of the model

Many analysers let you set FFT length and window; use long FFTs for bass and fine pitch detail, short ones for fast events. Keep the constant bin width in mind, too: pink noise has equal energy in every octave, but an octave higher spreads it over twice as many bins, so in an FFT analyser it falls by 3 dB per octave. White noise appears flat.

The model analyses a single frame of ideally computed signals: no noise, no overlapping or averaging of successive frames. Square, sawtooth and triangle are sampled without band limiting, which is why their alias lines appear. The impulse is a single sample per period; tone 2 is always a sine. The display reaches down to −120 dB.

Tasks

Practice questions

Questions from studio, stage and mastering – to think about, discuss and check with the module.

1. In your DAW's analyser (48 kHz) you see one broad hump at FFT length 1024, although bass and kick sit at 41 and 44 Hz. With 32,768 they separate, but the display reacts sluggishly. Why?

Show answer

At N = 1024 the resolution is Δf = 48,000/1024 ≈ 47 Hz – two tones 3 Hz apart land in the same bin. At 32,768 it is about 1.5 Hz, but each frame covers 683 ms of signal: fast changes are averaged over that time. For the bass range one uses long FFTs, for transients short ones; some analysers combine both.
2. You measure a 1 kHz test tone at 0 dBFS, but the analyser shows −0.6 dB. With another setting it is −1.4 dB. Is the generator broken?

Show answer

Probably not. At 48 kHz and N = 1024, 1 kHz lies at bin 21.33, i.e. between two bins. With a Hann window the analyser then reads too low, up to 1.4 dB exactly between two bins, with the rectangular window even up to 3.9 dB. For accurate level measurements use a flat-top window (error below 0.1 dB) or a level meter instead of the analyser.
3. Pink noise sounds balanced, but in the FFT analyser it falls by 3 dB per octave towards the top. White noise, on the other hand, looks flat. Why?

Show answer

Every FFT bin has the same width (Δf). Pink noise has equal energy in every octave; since an octave higher contains twice as many bins, each bin gets half as much – 3 dB less per octave. White noise has equal energy per hertz and therefore looks flat in the FFT analyser. Many analysers offer a slope (e.g. +3 dB/octave) or a third-octave display so that pink noise appears flat – matching hearing, which works roughly in relative bandwidths.

Tasks

Predict first, then try it out, then open the solution. “Load starting point” sets the matching initial values.

1. Separate two tones

Two equally loud tones lie at 1000 and 1040 Hz. At N = 128 you see only one peak. From which FFT length on do you see two separate peaks? How long is the frame then?

Load starting point

Show solution

From N = 512 (Δf ≈ 15.6 Hz, frame 64 ms) two peaks appear, though with a shallow dip between them. At N = 1024 (Δf ≈ 7.8 Hz, 128 ms) they are clearly separated. The tones are then about five bins apart, more than the width of the Hann main lobe (4 bins). Rule of thumb: the spacing must be larger than the main lobe – and it only gets narrower with a longer frame.
2. Leakage

The 1000 Hz sine lies exactly on bin 32. What does the spectrum look like? Then let the frequency glide or set about 1016 Hz (bin 32.5). What happens to the line and the measured level?

Load starting point

Show solution

On the bin there is exactly one line at 0 dB; all other bins are practically empty. At bin 32.5 the energy spreads over very many bins: with the rectangular window 58 bins lie above −40 dB and the highest shows only −3.9 dB. The tone has not changed – only a whole number of periods no longer fits into the frame. With Hann only 6 bins remain and the level error is at most 1.4 dB.
3. A quiet tone next to a loud one

Next to the loud tone 1 there is a tone 2 at 1500 Hz with −60 dB. With the rectangular window it is invisible. Try all windows: with which does it appear?

Load starting point

Show solution

With Hann and Blackman it appears as a clear peak at −60 dB. With the rectangular window the leakage of tone 1 at this point is still about −35 dB, with Hamming about −52 dB – both hide it. Hamming has the lowest first side lobe, but its more distant side lobes hardly fall off. For large dynamic range Hann or Blackman are better.
4. Lines that do not belong

A 700 Hz square wave is sampled at 8 kHz. Which lines belong to the harmonic series (multiples of 700 Hz)? Where do the others come from?

Load starting point

Show solution

Below 4 kHz only 700, 2100 and 3500 Hz belong to it. All other lines are aliases of higher overtones, mirrored at fs/2: 4900 Hz appears at 3100 Hz, 6300 Hz at 1700 Hz, 7700 Hz at 300 Hz and so on. They are quieter than the real partials, but the strongest (at 3100 Hz) is only just under 3 dB below the partial at 3500 Hz – and they lie inharmonically in between. After sampling they cannot be removed – that is why every AD converter needs an analogue anti-aliasing filter in front of it.
5. The seam

The periodic continuation is switched on. At 1000 Hz exactly eight periods fit into the frame. Then set a frequency between two bins, for example 1060 Hz. What happens at the seam? Then switch to the Hann window.

Load starting point

Show solution

At 1000 Hz the signal continues seamlessly and the spectrum shows a single line. At 1060 Hz the signal jumps at the seam: this jump does not exist in the real signal, but the FFT sees it and needs many additional frequencies to represent it – that is leakage. The Hann window pulls both ends to zero, the jump disappears from the windowed signal and the leakage drops sharply.

Settings

Sound

Level calibration (dB SPL)

MIDI keyboard

FFT: From Signal to Spectrum