{
  "$schema": "https://siliconverify.com/schemas/tool-v1.json",
  "slug": "comm-modulation",
  "title": "Modulation Workbench — AM, FM, PM in the Frequency Domain — Communication",
  "category": "communication",
  "category_label": "Communication",
  "urls": {
    "interactive": "https://siliconverify.com/comm-modulation.html",
    "static_sheet": "https://siliconverify.com/ref-comm-modulation.html",
    "api_doc": "https://siliconverify.com/api/comm-modulation.json",
    "blog_index": "https://siliconverify.com/llms-communication.txt"
  },
  "description": "The Analog Modulation workbench visualizes AM (DSB-FC, DSB-SC, SSB), FM (with Bessel expansion), and PM in the frequency domain. The continuous-time spectrum (carrier + sidebands) is shown as impulses, then the same impulses replicated at every n·F_samp to show the bandpass-sampled spectrum. When F_samp is too low, the replicas overlap and a red \"ALIASING\" warning appears.",
  "keywords": [
    "communication",
    "modulation",
    "AM",
    "FM",
    "PM",
    "DSB-SC",
    "DSB-FC",
    "SSB",
    "sideband",
    "carrier",
    "Bessel",
    "Carson's rule",
    "modulation index",
    "aliasing",
    "sampling",
    "frequency domain",
    "GATE ECE"
  ],
  "pricing": {
    "class": "workbench-tool",
    "protocol": "x402",
    "network": "base",
    "currency": "USDC",
    "price_atomic": 1000,
    "price_usd": 0.001,
    "payTo": "0x97DAA5649Fd7Dfe3e46fCd9f75516e36E750eed7",
    "static_sheet_price_atomic": 1000,
    "export_price_atomic": 5000
  },
  "content": {
    "formulas": [
      "AM: x_AM(t) = A_c·[1 + m·cos(2πf_m·t)]·cos(2πf_c·t) = A_c·cos(ω_c·t) + (m·A_c/2)·[cos((ω_c − ω_m)·t) + cos((ω_c + ω_m)·t)]",
      "AM modulation index: m = A_m / A_c (must be < 1 for no overmodulation)",
      "FM: x_FM(t) = A_c·cos(2π·f_c·t + β·sin(2π·f_m·t))",
      "Carson bandwidth: BW ≈ 2·(Δf + f_m) = 2·f_m·(β + 1)",
      "Bessel expansion: cos(β·sin θ) = J₀(β) + 2·Σ J_n(β)·cos(2n·θ) (n = 1, 2, ...)"
    ],
    "citations": [
      "Proakis, J. G., \"Digital Communications,\" 5th ed., McGraw-Hill, 2008, Ch. 2-3.",
      "Carlson, A. B., \"Communication Systems,\" 4th ed., McGraw-Hill, 2002."
    ],
    "faq": [
      {
        "q": "Why does my AM signal distort at m > 1?",
        "a": "Overmodulation. The envelope of x_AM(t) goes through zero and flips phase, creating harmonics of f_m that distort the demodulated audio."
      },
      {
        "q": "How wide is an FM signal?",
        "a": "Carson rule: BW ≈ 2·(Δf + f_m) = 2·f_m·(β + 1). For β = 5, f_m = 1 kHz: BW ≈ 12 kHz."
      }
    ]
  },
  "accessibility": {
    "crawlable": true,
    "javascript_required": true,
    "static_alternative": "/ref-comm-modulation.html",
    "structured_data": [
      "TechArticle",
      "SoftwareApplication",
      "FAQPage",
      "BreadcrumbList"
    ]
  },
  "x402_challenge_example": {
    "request": "curl -A 'GPTBot/1.0' -i https://siliconverify.com/comm-modulation.html",
    "response_headers": [
      "HTTP/1.1 402 Payment Required",
      "X-Siliconverify-Resource-Class: workbench-tool",
      "X-Siliconverify-Price-Atomic: 1000",
      "X-Siliconverify-Price-USD: 0.001"
    ],
    "response_body": "{\n  \"x402Version\": 1,\n  \"accepts\": [\n    {\n      \"scheme\": \"exact\",\n      \"network\": \"base\",\n      \"maxAmountRequired\": \"1000\",\n      \"resource\": \"https://siliconverify.com/comm-modulation.html\",\n      \"description\": \"SiliconVerify workbench tool — interactive browser-based engineering calculator. Client-side computation, no server cost.\",\n      \"mimeType\": \"text/html\",\n      \"payTo\": \"0x97DAA5649Fd7Dfe3e46fCd9f75516e36E750eed7\",\n      \"validAfter\": 1783274656,\n      \"validBefore\": 1783274956\n    }\n  ]\n}",
    "paid_request": "curl -A 'GPTBot/1.0' -H 'X-PAYMENT: 0x<txhash>' -i https://siliconverify.com/comm-modulation.html"
  },
  "last_updated": "2026-07-05"
}