Discrete-Time Signals#

Once an analog signal has been sampled, it becomes a sequence of numbers — discrete in time and (after quantization) discrete in amplitude. Working with these sequences requires a slightly different vocabulary and a few concepts that don’t exist in the continuous world: sample indexing, block processing, windowing, and the subtle errors that come from treating finite chunks of data as if they represent infinite signals.

Samples and Sequences#

A discrete-time signal x[n] is a sequence of values indexed by an integer n. Each value corresponds to a sample taken at time t = n/f_s, where f_s is the sample rate. The square-bracket notation x[n] (vs parentheses x(t) for continuous signals) is a deliberate convention that flags “this is discrete.”

Sequence operations are analogous to continuous-time operations:

• Delay: x[n - k] shifts the sequence by k samples. One sample of delay at 48 kHz is about 20.8 µs
• Scaling: a × x[n] multiplies every sample by a constant
• Addition: x[n] + y[n] adds corresponding samples (same rate assumed)
• Convolution: y[n] = Σ h[k] × x[n - k] — the fundamental operation of digital filtering

The sample rate is metadata — the sequence itself is just numbers. This means the same sequence of values could represent a 1 kHz signal at 48 kHz sample rate, or a 10 kHz signal at 480 kHz sample rate. All frequency information is relative to the sample rate.

Frames and Blocks#

Real-time digital signal processing doesn’t operate on individual samples one at a time (usually). Instead, samples are collected into blocks (frames) of N samples, and the DSP algorithm processes entire blocks:

• Block size (N): Typical values range from 32 to 4096 samples. Larger blocks are more computationally efficient (setup overhead is amortized) and provide better frequency resolution for spectral analysis
• Block latency: Processing can’t begin until a full block is collected, adding at minimum N/f_s seconds of latency. A 256-sample block at 48 kHz adds 5.3 ms minimum

The latency-efficiency tradeoff:

For real-time audio, block sizes above ~512 samples at 48 kHz (>10 ms) start to become perceptible as latency. Musical performance monitoring needs ≤5 ms — see Latency & Throughput.

Windowing#

When analyzing a finite block of samples, the block is implicitly multiplied by a rectangular window — ones inside the block, zeros outside. This abrupt truncation creates artifacts in the frequency domain (spectral leakage): energy from a single frequency smears across many frequency bins.

Window functions taper the edges of the block to reduce leakage, trading off frequency resolution for reduced sidelobes:

The tradeoff: Wider main lobe means worse ability to distinguish nearby frequencies. Lower sidelobes mean less leakage from strong signals into weak-signal bins. There’s no window that wins on both — this is another manifestation of the time-frequency uncertainty principle.

Overlap processing: To avoid losing information at the tapered block edges, consecutive blocks typically overlap by 50-75%. Each sample appears in multiple blocks, weighted by the window function at its position. The overlap-add or overlap-save method reconstructs a continuous output from overlapping windowed blocks.

Circular vs Linear Convolution#

A subtlety that causes real bugs: the FFT inherently computes circular (periodic) convolution, not the linear convolution needed for filtering. A block of N samples convolved with a filter of length M using a straightforward FFT approach produces wrap-around artifacts unless the FFT length is at least N + M - 1.

Practical solution: Zero-pad both the signal block and the filter to length N + M - 1 before computing the FFT. The overlap-add and overlap-save methods systematize this for continuous processing of sequential blocks.

Tips#

• Choose block size based on latency requirements first, then optimize for efficiency
• Use Hann or Blackman windows for general spectral analysis — rectangular windows cause excessive leakage
• When using FFT-based convolution, always zero-pad to avoid circular convolution artifacts

Caveats#

• Off-by-one errors are everywhere — A block of N samples spans indices 0 to N-1. An N-point FFT produces N/2 + 1 unique frequency bins (the rest are conjugate mirrors for real signals). The highest frequency bin represents f_s/2, not f_s. These fencepost errors compound in multi-stage processing
• Sample rate must be tracked explicitly — The sequence x[n] has no inherent notion of time or frequency. If a sequence is processed without knowing its sample rate, the results can’t be interpreted in physical units. Every processing chain should carry the sample rate as metadata
• Block boundaries create discontinuities — If a signal is processed in blocks without proper overlap and windowing, the block boundaries can introduce clicks, pops, or spectral artifacts. This is especially problematic for time-varying operations (filters with changing coefficients)
• Zero-padding does not create resolution — Padding a block with zeros and taking a larger FFT interpolates the spectrum (smoother appearance) but does not increase the fundamental frequency resolution. True resolution depends on the signal duration, not the FFT size
• Integer indexing hides timing precision — A sample at index n represents a moment in time, but the actual sampling instant has jitter. For high-resolution systems, the assumption of perfectly uniform spacing breaks down — see Clocking & Jitter

In Practice#

• Clicks or pops at regular intervals in processed audio indicate block boundary discontinuities — verify overlap and windowing
• Spectral analysis showing smeared peaks rather than clean lines indicates spectral leakage — use an appropriate window function
• Unexpected frequency components in FFT output may be wrap-around artifacts from insufficient zero-padding
• A processing chain that works at one sample rate but fails at another often has hardcoded values that should be sample-rate-dependent