IN14 · A small peak beside a strong tone

FFT Spectrum Analyzer

Is a small spectral peak a real signal or leakage from a strong neighbour? A finite record spreads a tone into a main peak and sidelobes. Tapering its ends reduces distant leakage but widens the main peak. Compare windows and longer observations on saved samples: a weak neighbour may emerge, yet tones inside one unresolved lobe cannot be separated just by adding more plotted points.

Combine two signals and noise, capture ADC codes, then change the time gate and window on the same saved samples. Compare amplitude, spectral density and selected-band power.

Largest non-DC spectral bin $124\,\mathrm{Hz}$

Selected-band total RMS $8.01\,\mathrm{mV}$

Window noise bandwidth / RBW $3.00\,\mathrm{Hz}$

Window-weighted total RMS $0.400\,\mathrm{V}$

Hear the weak neighbour in the spectrum

Drag a physical knob or the time gate below. Window changes reprocess the same saved ADC samples.

Strong tone frequency

123.4

f_1/\mathrm{Hz}

Tone separation

22

\Delta f/\mathrm{Hz}

Second tone RMS voltage

0.008

V_2/\mathrm{V}

Time-gate samples

1024

N

Power averages

1

M

Gaussian pulse width

12

\sigma_t/\mathrm{ms}

Window on saved samples

A calibrated ADC record and a complete spectrum are saved.

Source, digitizer and processing details

Input sourceStrong tone RMS / pulse scale

0.4

V_1/\mathrm{V}

Input noise RMS

0.001

\sigma_V/\mathrm{V}

First gate sample

0

n_0

Integration lower edge

140

f_L/\mathrm{Hz}

Integration upper edge

152

f_H/\mathrm{Hz}

ADC channel gain

1.04

g

ADC channel offset

0.003

b/\mathrm{V}

Clock error

1250

\epsilon_f/\mathrm{ppm}

ADC positive range

2

V_{\max}/\mathrm{V}

Acquisition seed

187

s_0

Zero padding factorSegment overlapNominal ADC sample rateADC word lengthSubtract each segment’s unweighted mean

Calibration uses separate known voltage levels and a clock reference. Changing the ADC path invalidates old calibration. Overlapped averages share samples.

Model assumptions

This FFT instrument samples a flat voltage channel with gain, offset, a finite ADC range and a clock error. Independent known DC levels calibrate voltage; an ideal thirty-two-second reference and whole-tick counter calibrate the clock. The source and input are voltage signals with prescribed loading; no RF power in watts is inferred. No anti-alias filter is modeled, so samples alone cannot distinguish an alias. Rectangular or periodic Hann windows, zero padding and overlapped power averaging act only on saved ADC data. Line amplitudes use coherent gain and are accurate for isolated bin-centred sinusoids; PSD uses squared-window normalization. Band integration includes signal, leakage and noise; unresolved tones are not separated. The two-dimensional front projection shows physical ports and knobs; cable markers show slowed signal activity, not charge drift or actual clock speed. Spectra and time traces are outside the apparatus.

The saved voltage record

ADC record 1 · calibrated simulated voltage

$t/\mathrm{s},\qquad x/\mathrm{V}$

The overview preserves each pixel column’s sample minimum and maximum. Drag the gate’s centre to move it or its right handle to change the number of real samples.

A spectrum from those samples

Saved processing 1 · periodic Hann · single-sided PSD

$f/\mathrm{Hz},\qquad S_V/(\mathrm{V}^2\,\mathrm{Hz}^{-1})$

Calibration and selected ADC samples are valid.

Teal: processed ADC samples. Shading: the saved integration band. Its area includes signal, leakage and noise; overlapping tones are kept together.

Window, normalization, calibration and limits

$n,\qquad w_n$

$\sum_n w_n=512,\quad \sum_nw_n^2=384$

$\Delta f=2.00\,\mathrm{Hz},\quad B_{\mathrm{ENBW}}=3.00\,\mathrm{Hz}$

$\sum_kS_k\Delta f=0.160\,\mathrm{V}^2$

$n$	$c_n$	$t_n/\mathrm{s}$	$x_n/\mathrm{V}$

The ADC uses a flat gain and offset before 12- or 16-bit conversion, with inclusive end codes. Separate known minus 0.1, zero and plus 0.1 volt levels fit volts per code and offset. A whole-tick count over an ideal thirty-two-second reference estimates sample rate; its finite count resolution is retained. Calibration does not use the unknown input’s tone frequencies or amplitudes.

Periodic Hann and rectangular windows use their actual finite sums. PSD integrates to window-squared-weighted mean square, which can differ between windows on nonstationary data. Coherent line RMS applies to an isolated bin-centred sinusoid. DC and Nyquist are not doubled. Zero padding leaves that window’s energy and ENBW fixed. Overlapped power averages share samples, so no independent-frame uncertainty is claimed.

At DC and Nyquist, amplitude describes the sampled sequence. An arbitrary continuous sinusoid’s RMS or peak cannot be recovered at Nyquist without its phase.

This is an FFT analyzer, not a swept superheterodyne model. Its declared RBW is window ENBW; first-null main-lobe width and plotted grid spacing are reported separately. The sampler has no anti-alias filter: known out-of-band source settings are a teaching reference, and cannot be inferred uniquely from ADC samples. Voltage loading is prescribed; no watts or dBm are inferred.

Start with the essentials

Focus question: What changes when the same ADC samples are given a different window?
One-sentence intuition: The window trades sidelobe suppression for a wider main lobe; normalization must match the quantity being measured.

Core mathematical model

Window the saved voltages

F_k=\sum_{n=0}^{N-1}w_n x_n e^{-2\pi i kn/N_{\!F}}

Calibrated instantaneous voltages are multiplied by the stored weights. Extra zero samples change the Fourier grid without extending the observation.

Density and line amplitude use different sums

S_k=\frac{D_k|F_k|^2}{f_s\sum_n w_n^2},\qquad A_{k,\mathrm{RMS}}=\frac{\sqrt{D_k}|F_k|}{\sum_n w_n}

The one-sided factor is two at positive interior bins and one at DC and Nyquist. Density is in squared volts per hertz. The RMS line amplitude applies to an isolated bin-centred sinusoid; it is not total power in a bin.

Noise bandwidth differs from grid spacing

B_{\mathrm{ENBW}}=f_s\frac{\sum_n w_n^2}{(\sum_n w_n)^2},\qquad \Delta f_{\mathrm{grid}}=\frac{f_s}{N_F}

The periodic Hann window has a noise bandwidth of one and a half unpadded bins. Zero padding reduces plotted spacing but leaves the window’s noise bandwidth unchanged. This FFT instrument uses ENBW as its declared RBW; a swept superheterodyne analyzer commonly uses an IF filter’s three-decibel width.

Integrate density once

\sum_k S_k\Delta f_{\mathrm{grid}}=\frac{\sum_n w_n^2 x_n^2}{\sum_n w_n^2}

This is the window-weighted mean square, averaged across the chosen segments. It already includes density normalization; do not divide by noise bandwidth again. A selected band also includes leakage and noise.

Common difficulties

A band area is not a hidden tone label

Typical misconceptionIntegrating near a small peak reveals exactly the power of the second source.

Better mental modelThat band includes all signal, leakage and noise within its saved bins. When lobes overlap, this instrument reports their combined contribution.

Run the experiment

01
Keep the samples and change the window
Use Reveal a weak neighbour, then switch between Hann and rectangular windows.
What to observe: The distant sidelobes fall with Hann, but the main peak grows wider. Both views use the same ADC record.
02
Lengthen the real observation
Drag the analyzer time-gate knob or the right edge of the gate below the apparatus.
What to observe: More actual samples narrow both the bin interval and the window response. Increasing only zero padding changes the plotted grid.
03
Save a processing choice
Save the processing result, then change the window or average count and export both.
What to observe: The raw codes and earlier processing settings stay fixed. Overlapped segments share samples and are not independent repetitions.

FFT Spectrum Analyzer

The saved voltage record

A spectrum from those samples

A finite recording changes a spectral peak

Core mathematical model

Window the saved voltages

Density and line amplitude use different sums

Noise bandwidth differs from grid spacing

Integrate density once

Common difficulties

A band area is not a hidden tone label

Run the experiment

Keep the samples and change the window

Lengthen the real observation

Save a processing choice

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