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Digital Signal Processing Basics

To convert the sampling frequency from 44.1 kHz to 48 kHz, upsampling by L and downsampling by 1/M are combined. Which of the following is the smallest pair of integers (L, M)?

  1. AL = 147, M = 160
  2. BL = 480, M = 441
  3. CL = 160, M = 147
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Answer: C. L = 160, M = 147

The required ratio is 48/44.1 = 48000/44100 = 480/441, which reduces by the greatest common divisor 3 to 160/147 (160 = 2⁵ × 5 and 147 = 3 × 7², so it cannot be reduced further). Upsampling by 160 to 7.056 MHz, low-pass filtering and then downsampling by 1/147 gives 48 kHz. In practice, polyphase structures and the like compute only the outputs that are needed, avoiding computing everything at 7.056 MHz.

  • A. L and M are swapped, giving 44.1 kHz × 147/160 ≈ 40.5 kHz.
  • B. The ratio is right, but it can be reduced by 3, so it is not the smallest pair.
  • C. Correct. 44.1 kHz × 160/147 = 48 kHz.
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  1. A 256-tap FIR filter runs in real time at a sampling frequency of 44.1 kHz. Which of the following is the number of multiply-accumulate (MAC) operations needed per second?
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