quiz.json

{
  "questions": [
    {
      "stage": "pre",
      "question": "What does the Fourier transform do to a signal?",
      "options": [
        "Compresses the signal to use less storage",
        "Decomposes the signal into sine waves of different frequencies, amplitudes, and phases",
        "Removes noise from the signal",
        "Converts the signal from analog to digital"
      ],
      "correct": 1,
      "explanation": "The Fourier transform converts a signal from the time domain to the frequency domain. Each frequency coefficient X[k] tells you the amplitude and phase of a sine wave at frequency k. The signal is re-expressed as a sum of these sine waves."
    },
    {
      "stage": "pre",
      "question": "What is the time complexity of the Fast Fourier Transform (FFT) compared to the direct DFT?",
      "options": [
        "Both are O(N^2)",
        "FFT is O(N log N), DFT is O(N^2)",
        "FFT is O(N), DFT is O(N log N)",
        "FFT is O(log N), DFT is O(N)"
      ],
      "correct": 1,
      "explanation": "The direct DFT computes N outputs, each summing over N inputs: O(N^2). The Cooley-Tukey FFT splits the signal into even/odd halves recursively, doing O(N) work at each of log2(N) levels, giving O(N log N). For N = 1 million, this is 20 million vs 1 trillion operations."
    },
    {
      "stage": "post",
      "question": "What does the convolution theorem state?",
      "options": [
        "Convolution in the time domain equals addition in the frequency domain",
        "Convolution in the time domain equals pointwise multiplication in the frequency domain",
        "Convolution always increases the length of a signal",
        "Convolution and correlation are identical operations"
      ],
      "correct": 1,
      "explanation": "The convolution theorem states that convolution in the time domain equals pointwise multiplication in the frequency domain: x * h = IFFT(FFT(x) . FFT(h)). This is why FFT-based convolution is O(N log N) instead of O(N*M) for large kernels."
    },
    {
      "stage": "post",
      "question": "Why does zero-padding a signal before FFT NOT increase the true frequency resolution?",
      "options": [
        "Zero-padding introduces noise that cancels the improvement",
        "Zero-padding interpolates between existing frequency bins but cannot reveal frequency detail absent from the original samples",
        "Zero-padding only works for power-of-2 signal lengths",
        "The FFT algorithm ignores zero-padded samples"
      ],
      "correct": 1,
      "explanation": "True frequency resolution depends on the observation time T = N/fs. Zero-padding adds more frequency bins (finer grid) but only interpolates the existing spectrum — it gives a smoother-looking result without resolving frequencies closer than 1/T Hz apart."
    },
    {
      "stage": "post",
      "question": "In the original Transformer's sinusoidal positional encodings, why are different dimension pairs assigned geometrically spaced frequencies?",
      "options": [
        "It reduces the computational cost of attention",
        "Each frequency provides a different resolution — high frequencies encode fine position, low frequencies encode coarse position, giving each position a unique fingerprint",
        "Geometric spacing is required by the FFT algorithm",
        "It ensures all encoding values are between 0 and 1"
      ],
      "correct": 1,
      "explanation": "High-frequency dimensions change rapidly with position (fine resolution), while low-frequency dimensions change slowly (coarse resolution). Together, the multi-frequency encoding gives every position a unique pattern — similar to how Fourier coefficients uniquely identify a signal."
    }
  ]
}