Library / Signal processing and digital filters / DSP-16
Signal processing and digital filters
Peak sine amplitude from an interior DFT line
Peak amplitude of a real sinusoid that fits the N-sample record exactly (coherent), from the magnitude of its unwindowed DFT coefficient at a strictly positive interior bin.
- Ref
- DSP-16
- Rev
- v1
- Field
- Signal processing and digital filters
- Result
- Apeak [V]
A real sinusoid of peak amplitude A that fits the record exactly puts A·N/2 into bin k and A·N/2 into bin N − k, so A = 2|X[k]|/N. The factor 2 recovers the energy of the negative-frequency bin. At DC or at the Nyquist bin of an even-length record there is no mirror bin, so the factor is 1. A window changes the scaling by its coherent gain.
Peak amplitude of a real sinusoid that fits the N-sample record exactly (coherent), from the magnitude of its unwindowed DFT coefficient at a strictly positive interior bin. The factor 2 recovers the energy in the negative-frequency bin; for DC or an even-length Nyquist bin the factor is one.
Calculator
- Magnitude of the coefficient, |X|16 V
- RMS amplitude, A/√20.7071 V
- Power in the tone, A²/20.5 V²
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A coefficient of magnitude 16 V in a 32-sample record means a tone of 1 V peak, 0.7071 V RMS.
- The formula is exact for a coherent tone; otherwise the tone’s energy leaks into neighbouring bins, and the line reads low by up to 3.9 dB with a rectangular window.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Im X Imaginary part of the coefficient | −10,000 to 10,000 V | 0 | 625 | 625 | +400 V | +2,300 % |
| Re X Real part of the coefficient | −10,000 to 10,000 V | 1 | 625 | 624 | +400 V | +2,400 % |
| N Record length | 2 to 65,540 samples | 4.883e-4 | 16 | 16 | +1 samples | −3.03 % |
Sweep of Record length
| Record length (samples) | Peak amplitude (V) |
|---|---|
| 2 | 16 |
| 7.336 | 4.571 |
| 26.91 | 1.185 |
| 98.7 | 0.3232 |
| 362 | 0.0884 |
| 1,328 | 0.0241 |
| 4,871 | 0.006569 |
| 17,870 | 0.001791 |
| 65,540 | 4.883e-4 |
Sweep one variable
Variables and how each one contributes
Peak amplitude
What the equation gives you. Everything else feeds it.
Real part of the coefficient
The real part of the DFT coefficient, in V. Raises the magnitude and so the amplitude. Raising it by 10 % (to 0) moves the result by +0 %, from 1 to 1 V.
Imaginary part of the coefficient
The imaginary part of the DFT coefficient, in V. Raises the magnitude and so the amplitude. Raising it by 10 % (to −17.6) moves the result by +10 %, from 1 to 1.1 V.
Record length
The number of samples in the record. A longer record spreads the same amplitude over a larger coefficient, so the amplitude falls in inverse proportion for a fixed X. Raising it by 10 % (to 35.2) moves the result by −8.57 %, from 1 to 0.9143 V.
Worked example with the values above
- Magnitude: |X| = √(0² + −16²) = 16 V.
- Double it: 2|X| = 32 V.
- A = 2|X|/N = 32 / 32 = 1 V.
Assumptions and limits
- A stationary signal with a tone of constant amplitude and frequency, and a record that holds a whole number of cycles for the coherent formulas.
- A single, well-isolated spectral peak for interpolation, and a bandwidth that is stated.
Common mistakes
- Using these coherent formulas on a record with a non-integer number of cycles.
- Adding noise or a DC offset into the harmonic sum.
- Using a peak that is not the largest of its three neighbouring bins in the parabolic interpolation.
Statistics and uncertainty
Tone amplitude and phase from a single FFT line are accurate to a fraction of a per cent only for a coherent record; for a non-coherent tone, scalloping loss of up to 3.9 dB (rectangular window) or 1.4 dB (Hann) adds error unless it is corrected.
Where it is used
- Measuring amplitude, phase and distortion of tones in an FFT record.
- Refining a frequency estimate without a longer record.
Maths you need
- Coherent sampling: a whole number of signal cycles in the record
- Spectral peak: a local maximum in the spectrum
- Parseval’s theorem: power in the time domain equals power in the frequency domain