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Digital Signal Processing Basics

To raise the sampling frequency by a factor of L, L − 1 zeros are inserted between successive samples (zero insertion). Which of the following correctly describes the processing needed next?

  1. AInserting zeros creates low-frequency components not present in the original signal, so a high-pass filter is applied to remove those low-frequency components
  2. BImages (copies of the spectrum) appear around integer multiples of the original sampling frequency, so a low-pass filter removes them and the gain is multiplied by L
  3. CInserting zeros creates no new components in the spectrum, so no filter is needed and the result can be used directly as a signal at the new sampling frequency
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Answer: B. Images (copies of the spectrum) appear around integer multiples of the original sampling frequency, so a low-pass filter removes them and the gain is multiplied by L

After zero insertion, copies (images) of the original spectrum appear on the new frequency axis around integer multiples of the original sampling frequency fs. For example, zero-inserting a 1 kHz sine wave sampled at fs = 8 kHz with L = 2 produces a component at 8 − 1 = 7 kHz as well as at 1 kHz. Removing them with a low-pass filter that cuts at the original Nyquist frequency fs/2 (an anti-imaging or interpolation filter) fills the zeros with smooth interpolated values. The inserted zeros reduce the amplitude to 1/L, so the filter gain is set to L to restore the original amplitude.

  • A. What must be removed are the images that appear at high frequencies, and the filter used is a low-pass filter.
  • B. Correct. A low-pass filter removes the images, and a gain of L restores the amplitude.
  • C. The images remain and the waveform stays interspersed with zeros, so a filter is needed.
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  1. 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)?
  2. 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?
  3. An FIR filter output is computed for every sample at a sampling frequency of 48 kHz on a processor clocked at 200 MHz that performs one multiply-accumulate per cycle. Ignoring the time for other processing, which of the following is closest to the maximum number of taps that can be processed in real time?
  4. In an FIR filter computation, products of 16-bit fixed-point data and coefficients are represented in 32 bits, and 256 of these products are added up. Which of the following is the minimum accumulator width that guarantees no overflow during the computation?
  5. A 6 kHz sine wave is sampled at a sampling frequency of 48 kHz. Which of the following is the normalized angular frequency ω (the phase advance per sample) of the resulting discrete-time signal?