Why LLMs Hallucinate: The Signal Processing Explanation

The Real Cause of LLM Hallucination

Every week, a new headline says LLMs "make things up." Most people blame "stochastic parrots" or gaps in training data. But I found a cleaner answer from signal processing: hallucination is aliasing caused by undersampled prompts.

When you send a short, vague prompt to an LLM, you give it only one data point about what you want. The Nyquist-Shannon sampling theorem explains what comes next. The model fills in the rest, but it fills in its guess, not your intent. That filled-in guess is aliasing. That aliasing is hallucination.

What the Nyquist-Shannon Theorem Says

The theorem is simple: to reconstruct a signal with bandwidth B, you need at least 2B samples. Use fewer samples and the output gains extra frequencies that were never there. They look real, but they are invented.

For LLM prompts, the "signal" is your specification, meaning what you actually want the model to do. My research on 275 production prompts across 11 agents found 6 specification bands that every good prompt must cover:

1. PERSONA: who should answer
2. CONTEXT: background facts about the situation
3. DATA: the specific inputs the model needs
4. CONSTRAINTS: rules and limits (42.7% of output quality)
5. FORMAT: how the output should look (26.3% of output quality)
6. TASK: the goal

Aliasing in Practice: Real Examples

Consider this prompt: "Write me a marketing email." That is 1 sample of a 6-band signal, a 6:1 undersampling ratio. The model must guess your persona, context, data, constraints, format, and half the task. Every guess is a chance to hallucinate.

Now consider the same request decomposed into 6 bands:

PERSONA: Senior B2B SaaS copywriter
CONTEXT: Series A fintech, 50 employees, launching new API product
DATA: Product name "PayFlow", pricing $99/mo, target audience: CFOs
CONSTRAINTS: Max 200 words, no jargon, include one CTA, compliance-safe
FORMAT: Subject line + 3 paragraphs + CTA button text
TASK: Write a cold outreach email for the product launch

Same request. Six bands instead of one. Now the model has enough to work from. It does not need to guess. Hallucination drops because there is nothing left to invent.

Empirical Evidence: 275 Observations

My sinc-LLM paper studied 275 real prompt-response pairs across 11 agents. The results are clear:

The CONSTRAINTS band by itself drives 42.7% of output quality. When a prompt skips constraints, the model invents its own. Those invented constraints are hallucinations by definition.

How to Fix Hallucination Today

The fix is a process, not a trick. For any prompt:

1. Find the 6 bands your prompt needs to cover
2. Write clear content for each band, especially CONSTRAINTS
3. Put roughly 50% of your prompt tokens into CONSTRAINTS and FORMAT
4. Use the free sinc-LLM transformer to break down any raw prompt automatically

I made the sinc-LLM framework open source. It applies these principles for you, turning any raw prompt into a 6-band Nyquist-compliant specification.

{
  "formula": "x(t) = Σ x(nT) · sinc((t - nT) / T)",
  "T": "specification-axis",
  "fragments": [
    {
      "n": 0,
      "t": "PERSONA",
      "x": "You are an AI systems researcher specializing in LLM failure modes, hallucination classification, and output reliability analysis. You diagnose root causes, not symptoms."
    },
    {
      "n": 1,
      "t": "CONTEXT",
      "x": "A production chatbot is generating confident but factually wrong responses 23% of the time. The model is Claude Sonnet, the system prompt is 47 tokens long, and there are no constraints or format specifications."
    },
    {
      "n": 2,
      "t": "DATA",
      "x": "Hallucination rate: 23%. System prompt: 47 tokens. CONSTRAINTS band: 0 tokens. FORMAT band: 0 tokens. Model: Claude Sonnet. Use case: customer support for a SaaS product."
    },
    {
      "n": 3,
      "t": "CONSTRAINTS",
      "x": "State facts directly. Never hedge with 'I think' or 'probably'. Cite the specific specification band that is missing for each hallucination type. Every claim must reference a concrete token count or percentage. Do not suggest 'more training data' as a fix. The fix must be at the prompt level."
    },
    {
      "n": 4,
      "t": "FORMAT",
      "x": "Return: (1) Hallucination Classification Table with columns: Type, Frequency, Missing Band, Fix. (2) Root Cause Analysis in one paragraph with exact numbers. (3) Before/After prompt comparison showing the fix."
    },
    {
      "n": 5,
      "t": "TASK",
      "x": "Diagnose why this chatbot hallucinates 23% of the time and provide the exact prompt-level fix using sinc band analysis."
    }
  ]
}