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.

ΔΣ (1-bit) Modulator

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

Live graphs

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.

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