English
Menu

Digital Signal Processing Basics

Learn with multiple-choice questions about normalized frequency and the period of discrete-time sinusoids, the unit impulse and unit step, energy and power, linearity, time invariance, causality and BIBO stability, impulse response and convolution, difference equations, DFT frequency resolution and bins, FFT workload, zero padding, spectral leakage and window functions, Parseval's theorem, the z-transform with poles and zeros, FIR and IIR filters, linear phase and group delay, moving averages, the bilinear transform, coefficient quantization, decimation and interpolation, and multiply-accumulate counts and real-time processing.

Practice 10 questionsRandom questions from this topic · about 5 minutes Practice this topicAll questions in this topic, in random order Weak questionsRetry only the questions you last got wrong

Digital Signal Processing Basics: key points

Once an AD converter has turned a signal into a sequence of numbers, processors and FPGAs filter it, analyze its frequency content and change its sampling rate. Digital signal processing runs inside almost every electronic product, from audio and communications to sensors, motor control and measuring instruments.

This topic starts with the normalized frequency and period of discrete-time signals, the unit impulse and step, and energy and power, and then deals with the properties of linear time-invariant systems (linearity, time invariance, causality and BIBO stability), impulse response and convolution, and computing with difference equations. It goes on to check DFT bin spacing and bin indices, the symmetry of the spectra of real signals, FFT workload, zero padding, spectral leakage and window functions, Parseval's theorem, transfer functions with poles and zeros via the z-transform, FIR and IIR filters, linear phase and group delay, moving averages and exponential smoothing, the bilinear transform, coefficient quantization, decimation and interpolation, and estimating multiply-accumulate counts and real-time processing budgets. The values in the calculation questions, including those of the wrong choices, have been checked by computing them in Python (NumPy). The sampling theorem, aliasing, quantization noise, oversampling and delta-sigma ADCs are covered in "A/D and D/A conversion", analog filters and op-amps in "Analog and electronic circuits", OFDM in "Wireless Communication Basics", convolution in CNNs in "Machine Learning Basics", and fixed-point representation in "Computer Arithmetic".

All Digital Signal Processing Basics questions (40)
  1. 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?
  2. Which of the following is the fundamental period (the smallest positive integer N such that x[n + N] = x[n] for every integer n) of the discrete-time signal x[n] = cos(0.3πn)?
  3. Which of the following signals takes the same value as the discrete-time signal x[n] = cos(1.75πn) for every integer n?
  4. Which of the following correctly relates the unit impulse δ[n] (1 at n = 0, 0 elsewhere) and the unit step u[n] (1 for n ≥ 0, 0 for n < 0)?
  5. Which of the following is the energy E = Σ|x[n]|² of x[n] = (0.5)ⁿu[n] (0.5 to the power n for n ≥ 0, 0 for n < 0)?
  6. Which of the following is the average power (the mean of |x[n]|² over one period) of x[n] = 2cos(πn/4)?
  7. Which of the following systems with input x[n] and output y[n] is both linear and time-invariant (LTI)?
  8. Which of the following correctly describes the system y[n] = n·x[n]?
  9. Which of the following systems with input x[n] and output y[n] is causal (its output depends only on inputs up to the present time)?
  10. Let u[n] be the unit step. Which of the following LTI systems, given by their impulse responses, is BIBO stable (every bounded input gives a bounded output)?
  11. The sequence x[n] = {1, 2, 3} (n = 0, 1, 2) is applied to an LTI system with impulse response h[n] = {1, −1} (n = 0, 1). Which of the following are the outputs y[0] to y[3]?
  12. A unit step (x[n] = 1 for n ≥ 0) is applied from n = 0 to the system described by the difference equation y[n] = x[n] + 0.5y[n − 1]. With y[−1] = 0, which of the following is y[2]?
  13. A signal sampled at 48 kHz is analyzed with a 1024-point discrete Fourier transform (DFT). Which of the following is the spacing of the frequency bins?
  14. Let k = 0 be the DC bin. When a 1 kHz sine wave sampled at 8 kHz is analyzed with a 256-point DFT, which of the following is the index k of the bin where the peak appears on the positive-frequency side?
  15. A 64-point DFT of a real signal x[n] gives X[10] = 3 + 4j. Which of the following is X[54]?
  16. The DFT of the 8-point real signal x[n] = {3, 1, 3, 1, 3, 1, 3, 1} is defined as X[k] = Σx[n]e^(−j2πkn/8). Which of the following is the value of X[4], the bin at the Nyquist frequency?
  17. Suppose an N-point DFT takes about N² operations when computed from its definition and about N log₂N operations with the FFT (fast Fourier transform). For N = 1024, which of the following is closest to the reduction in the number of operations achieved by the FFT?
  18. A 256-point signal is followed by 768 zeros (zero padding) and a 1024-point DFT is computed. Compared with computing the DFT of the 256 points as they are, which of the following is correct?
  19. Which of the following is the main cause of spectral leakage in a DFT?
  20. Which of the following correctly compares the rectangular window (no window applied) and the Hann (Hanning) window applied before a DFT?
  21. Assume the number of points N is large enough. In a DFT with a rectangular window, a sine wave lies exactly halfway between two adjacent bins. Which of the following is closest to the drop in the amplitude read at the nearest bin, compared with a sine wave exactly at a bin center?
  22. The DFT of the 4-point signal x[n] = {1, −1, 2, 0} is defined as X[k] = Σx[n]e^(−j2πkn/4). Which of the following is Σ|X[k]|² (k = 0 to 3)?
  23. When the frequency response of a discrete-time LTI system is obtained from its transfer function H(z), along which of the following is z moved?
  24. Which of the following is the transfer function H(z) = Y(z)/X(z) of the system described by the difference equation y[n] = x[n] + 0.5x[n − 1] + 0.25y[n − 1]?
  25. Which of the following correctly gives the poles and the stability of the causal system with transfer function H(z) = 1/(1 − 1.3z⁻¹ + 0.4z⁻²)?
  26. An FIR filter with transfer function H(z) = 1 + z⁻² (y[n] = x[n] + x[n − 2]) runs at a sampling frequency of 8 kHz. Which of the following is the frequency of a sine-wave input for which the output is 0?
  27. Which of the following gives the gain magnitudes at DC and at the Nyquist frequency of the filter y[n] = x[n] − x[n − 1] (the difference from the previous input)?
  28. Which of the following is a characteristic of an FIR (finite impulse response) filter?
  29. Which of the following 5-tap FIR filters has linear phase?
  30. Which of the following is the group delay of a 101-tap linear-phase FIR filter (with symmetric coefficients)?
  31. At a sampling frequency of 8 kHz, a moving-average filter y[n] = (x[n] + x[n − 1] + … + x[n − 7])/8 averages the latest 8 samples. Which of the following is the lowest frequency at which its output is 0?
  32. Which of the following gives the gains at DC and at the Nyquist frequency of the first-order IIR low-pass filter given by the exponential smoothing y[n] = 0.2x[n] + 0.8y[n − 1]?
  33. Which of the following correctly describes the bilinear transform used to design a digital filter from the transfer function H(s) of an analog filter?
  34. Which of the following is a correct precaution when implementing a high-order IIR filter in fixed point with coefficients of limited bit width?
  35. A signal sampled at 48 kHz is reduced to 12 kHz by keeping one sample out of every four (decimation by 4). Which of the following frequency components should the low-pass filter placed before the downsampling remove?
  36. 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?
  37. 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)?
  38. 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?
  39. 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?
  40. 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?

Books on Digital Signal Processing Basics Ad

As an Amazon Associate I earn from qualifying purchases.

Other topics in Fundamentals