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Home/Math Visualization/ΔΣ (1-bit) Modulator

ΔΣ (1-bit) Modulator

First- and second-order discrete-time ΔΣ with ±1 quantizer: shaped quantization noise, sine test tone, boxcar reconstruction, and Hann-windowed error spectrum (last 1024 samples).

ΔΣ modulator

0.55
220Hz
48samples

Measured values

RMSE(x − g·y_LPF) tail0.31468
SNR (tail, crude)1.9dB
Best-fit gain g2.7477
Peak |integrator|4.037

Live graphs

About this model

Delta-sigma (ΔΣ) modulation represents a narrow-band analog signal using a high-rate stream of coarse quantization levels—here just ±1—so that quantization error is high-pass filtered by the loop and appears mostly out of band relative to the signal band of interest. The simulator implements textbook first- and second-order discrete-time loops: each stage integrates the difference between the input and the 1-bit DAC feedback, and a sign quantizer produces y[n] ∈ {−1, +1}. A boxcar (moving average) approximates the analog low-pass reconstruction after an ideal DAC. A final panel shows a Hann-windowed FFT of the weighted reconstruction error g·y_LPF − x on the last 1024 samples (relative dB), illustrating how second-order shaping pushes more error energy toward high frequencies than first order for comparable tone settings. The model is pedagogical: no DAC mismatch, clock jitter, multi-bit quantizers, or CIFF/CRFF coefficient tuning.

Who it's for: Introductory DSP, mixed-signal, or audio engineering students learning noise shaping, oversampling converters, and the intuition behind high-resolution ADCs/DACs from a coarse bitstream.

Key terms

  • Delta-sigma modulation
  • Noise shaping
  • Oversampling
  • Quantization noise
  • Sigma-delta ADC
  • Boxcar filter
  • NTF
  • Two-level quantizer

How it works

First- and second-order discrete ΔΣ modulators with a ±1 quantizer: the loop integrates the error between the input and the 1-bit feedback so that quantization noise is spectrally shaped—typically more high-frequency energy for higher order. A boxcar average approximates the DAC + analog low-pass reconstruction you would use after the bitstream. Compare 1st vs 2nd order at similar tone amplitude: the error spectrum (last graph) tilts upward more sharply when extra shaping is active. This toy model ignores DAC nonlinearity, excess loop delay, multi-bit stages, and stability analysis for aggressive inputs.

Key equations

1st: u[n] = u[n−1] + x[n] − y[n−1], y[n] = sign(u[n])
2nd: u₁ += x−y, u₂ += u₁−y, y = sign(u₂)

Frequently asked questions

Why can a ±1 bitstream approximate a smooth sine?
The feedback loop forces the local average of y[n] to track the input; rapid switching between −1 and +1 encodes amplitude through pulse density. A low-pass filter recovers the slow component—the sine—while attenuating the shaped high-frequency quantization noise.
What does “second-order” change compared to first order?
An extra integrator in the loop increases the order of the noise transfer function near DC, typically suppressing in-band quantization noise more strongly at the cost of a more aggressive high-frequency noise floor and tighter stability constraints for large inputs.
Is the boxcar the same as a real reconstruction filter?
No. Real converters use sharper analog or digital filters with controlled ripple and stopband rejection. The boxcar is a simple finite impulse response average that is easy to understand and tune with the “M” slider.
Why does the spectrum plot use a Hann window and a gain g?
The Hann window reduces FFT leakage when the sine frequency is not an exact bin; the scalar g matches the overall scale of the filtered bitstream to the input so the plotted error emphasizes spectral shape rather than a trivial amplitude mismatch.