DSP 101
M2 · L4
Module 2 — Sampling & Quantization
Reconstruction
& DAC

Sampling and quantization convert the world to numbers. Reconstruction converts them back — turning a stream of discrete values into the smooth, continuous analog signal the real world requires.

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DSP 101
M2 · L4
The Goal
Back to Continuous

A DAC must produce the unique bandlimited signal consistent with the stored samples. When the Nyquist criterion was met during sampling, that signal is exactly the original. Reconstruction is the theorem’s other half.

Key Insight
If sampling was done correctly, reconstruction can be perfect in theory — the samples contain all the information.
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DSP 101
M2 · L4
The Math
Sinc Interpolation

Ideal reconstruction sums infinitely many sinc functions, each centered on a sample and scaled by its value. In frequency, this equals perfect low-pass filtering below fs/2.

Whittaker–Shannon Formula
x(t) = \sum_{n} x[n]\, \operatorname{sinc}\!\left(\tfrac{t - nT_s}{T_s}\right)

Non-causal: requires “future” samples — only an idealization.

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DSP 101
M2 · L4
Real DACs
Zero-Order Hold & Images

Real DACs output a staircase waveform: each sample held constant for one period. This zero-order hold (ZOH) creates spectral images — unwanted copies of the signal at multiples of fs.

  • Images appear at fs, 2fs, 3fs, …
  • ZOH also attenuates high frequencies within the baseband (sinc roll-off)
  • First-order hold (linear interpolation) is smoother — a sinc² response — but adds a sample of delay; only the infinite sinc is ideal
  • A reconstruction filter must remove images before the signal reaches the output
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DSP 101
M2 · L4
The Filter
Anti-Imaging Filter

The reconstruction filter low-passes the DAC output: suppress images, correct sinc roll-off, maintain flat passband. It mirrors the anti-aliasing filter at the input.

  • Stopband: −80 dB or better for audio applications
  • Passband: flat magnitude, linear phase to avoid coloration
  • Sinc correction boosts high-frequency content to compensate ZOH attenuation
  • Analog implementation — must be realized with real components
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DSP 101
M2 · L4
The Modern Solution
Oversampling

Run the DAC at 8×, 64×, or 256× the audio rate. Images are pushed far up in frequency — the analog filter needs only a gentle, low-order design. Hard filtering moves to the digital domain, where it’s cheap and precise.

8–64×
Typical audio
256×+
Sigma-delta
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DSP 101
M2 · L4
The Clock Problem
Jitter & Timing Noise

If samples arrive at slightly wrong times, the waveform is distorted even with perfect amplitude values. Jitter converts timing uncertainty into amplitude noise, degrading SNR proportionally to signal frequency.

Jitter-Limited SNR
\text{SNR}_{\text{jitter}} \approx -20\log_{10}(2\pi f_0\,\sigma_t)

1 ns rms jitter limits a 20 kHz signal to ~98 dB SNR — just 16-bit performance.

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DSP 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
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DSP 101
M2 · L4
Key Takeaways
What You Learned
  • Ideal reconstruction = sinc interpolation = ideal low-pass filtering below fs/2
  • Real DACs produce a staircase (ZOH) with spectral images at multiples of fs
  • Reconstruction filter removes images and corrects ZOH sinc roll-off
  • Oversampling pushes images up in frequency, simplifying the analog filter
  • Sigma-delta DACs use 256×+ oversampling and noise shaping for high dynamic range
  • Jitter converts timing uncertainty into amplitude noise, proportional to frequency
Module 2 Complete
You now understand the full ADC–DAC chain. Next: Discrete-Time Signals & Systems.
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