ECE 445S Real-Time Digital Signal Processing Laboratory - Lecture 3

Lecture by Prof. Brian L. Evans

Lecture

Lecture 3 slides on Discrete-Time Signals and Systems in PowerPoint format.
Part 1
- Board: Periodicity for CT and DT Sinusoids
- Board: Homogeneity for Integrator Under Observation (initial condition must be zero)
Part 2
- Board: Initial conditions for ideal delay
- Tuneup: In-Lecture Assignment #1 on time-frequency representations
Part 3
How a generated sound is heard -- it's complicated
Demo: filtering audio signals. Noise and sweep signals filtered by lowpass, bandpass and highpass filters.
Demo: "Sandstorm", Darude, 1999, requested by Fall 2016 student. Synthesia Animation shows a time-frequency visualization of the song with frequency (piano keys) on the horizontal axis and time on the vertical axis.
Spring 2014 lecture on video: Part 1 and Part 2

Supplemental Material

Spring 2026
- Spring 2026 Part 1
  - Board: Review of Three Algorithms to Compute Cosine Signals
  - Board: Quantization
- Spring 2026 Part 2
  - Board: Linearity for an integrator with initial conditions
  - Board: Squaring block output frequencies when input is a sinusoidal signal
  - Tuneup: In-Lecture Assignment #1 on time-frequency representations
- Spring 2026 Part 3
  - Demo: "Sandstorm", Darude, 1999, requested by Fall 2016 student. Synthesia Animation shows a time-frequency visualization of the song with frequency (piano keys) on the horizontal axis and time on the vertical axis.
  - Audio signal processing
    - Mathematical Modeling of a Repeated Drum Beat (impulse train in the time domain which is an impulse train in the frequency domain; i.e., a repeating drum beat has low, medium, and high frequencies in it)
    - Frequency response of laptop/tablet speakers (very low response below 100 Hz and above 15 kHz)
    - Wavelength as propagation speed over frequency to explain why audible frequencies below 100 Hz need a larger speaker than can fit into a laptop or tablet: wavelength = 343m/s / 100 Hz = 3.43m
    - Normal human hearing can hear from 20 Hz to an upper frequency that reduces with age beyond 20: 19 kHz at 20, 17 kHz at 24, 16 kHz at 30, 15 kHz at 40, 12 kHz at 50, etc. How Old Are Your Ears?
    - Single-frequency threshold test vs. age from 500 Hz to 6000 Hz (see Fig. 4) that shows that 1 kHz is a sweet spot for human hearing across all age groups and the frequency responses curves get weaker with increasing age group
    - "A-weighting" approximation of the human auditory system frequency response over the audible range of 20 Hz to 20 kHz
  - Ideal delay has linearity and time invariance properties
    - always when observing for all time
    - if initial conditions are zero when observing for t ≥ 0
  - Handouts: Time-invariance for an integrator and delay under observation
- Spring 2026 Part 4
  - Weighted signal-to-noise ratio (WSNR) to bring application information into the SNR measure
  - Connection between discrete-time and continuous-time frequencies (e.g., pi rad/sample maps to fs/2)
  - Finite impulse response (FIR) filter initial conditions
  - Definition of the sinc function which is sinc(x) = sin(pi x) / (pi x) for this course and in Matlab (another defintion is also common)
- Demo: filtering audio signals. Noise and sweep signals filtered by lowpass, bandpass and highpass filters.
Fall 2025
- Board: Review of sinusoidal generation
- Board: Signum function
- Board: Frequency resolution in spectrograms (Spring 2024)
- Board: Aliasing Analyzed in Frequency Domain
- Board: Reconstruction of a Discrete-Time Impulse
- Board: Review of Dirac deltas
- Board: Linearity for an integrator with initial conditions
- Board: Averaging filter time response
Spring 2025
- Board: Convolution with delayed Dirac delta
- Board: Review of Sinusoidal Generation
Fall 2024
- Averaging and First-Order Difference Filters
Spring 2024 notes
- Spring 2024 Part 1. Board: Rectangular Pulse
- Spring 2024 Part 2
  - Board: Discrete-time unit step in Matlab
  - Board: Linearity of an integrator when observing for t ≥ 0
  - Board: Time invariance of a delay when observing for t ≥ 0
  - Board: Amplitude modulation is time-varying (after lecture)
  - Board: All pointwise operations are time-invariant (after lecture)
- Spring 2024 Part 3
  - Board: Effects of finite observation in time
  - Board: Continuous-time frequencies captured by sampling
  - Board: Sampling to obtain, and reconstruction of, a discrete-time impulse
Fall 2023 notes
- Fall 2023 Part 1
  - Board: Review of modulation and demodulation
  - Demo: filtering audio signals. Noise and sweep signals filtered by lowpass, bandpass and highpass filters.
- Fall 2023 Part 2
  - Demo: "Sandstorm", Darude, 1999, requested by Fall 2016 student. Synthesia Animation to show a time-frequency representation of the song.
  - Board: Review of System Properties and Linearity of integration under observation
  - Board: Scalar gain has linearity property
  - Handouts: Time-invariance for an integrator and delay under observation
  - Tuneup: In-Lecture Assignment #1
- Fall 2023 Part 3
  - Board:. Review of chirps and spectrograms from in-lecture assignment #1
  - Board:. Spectrogram windowing, and time and frequency resolution for a deeper dive into in-lecture assignment #1
  - Board: Reconstruction of Discrete-Time Impulse
  - Board: Sampling 30 kHz sinusoidal at 48 kHz
  - Demo: Human hearing vs. age: How Old Are Your Ears? and large-scale study (Fig. 4)
- Designing averaging filters
Spring 2023 notes
- Part 1
  - Dirac delta
  - Filtering demo: lowpass filter and chirp signal (sweep)
  - GPS
  - Lecture Notes by TA Dan Jacobellis
- Part 2
  - System properties
  - Lecture Notes by TA Alice Liu
- Part 3
Fall 2022 marker board notes
- Part 1
  - Digital vs. Analog and the Dirac delta
  - Fourier transform of the Dirac delta
  - Cellular Communications: basestations, 4G and 5G frequency bands, and companies
- Part 2
  - Takeaways from Lecture 3 Part 1
  - Determining system linearity property
  - Homogeneity property for integrator under observation during class and after class
  - In-lecture #1 assignment chirp signal parameters and spectrogram time and frequency resolution
  - Handout on Time Invariance for an Integrator
- Part 3
  - Handout on time-invariance of an ideal delay under observation
  - Continuous-time LTI properties for integrator, ideal delay and sinuosidal amplitude modulation
  - Continuous-time finite and infinite impulse response systems
  - After-lecture discussion on linearity for an ideal delay
- Part 4
Spring 2022 marker board notes
- Part 1: Continuous-Time Signals
- Part 2: Continuous-Time Systems
  - takeaways from part 1
  - pointwise systems
  - initial conditions for an integrator and tapped delay line
  - Handout on Time-Invariance of a System Under Observation
  - linearity for system described by linear constant coefficient differential equation
- Part 3: Discrete-Time Systems
Fall 2021 marker board notes
- time-invariance for an ideal delay
- aliasing analyzed in the time and frequency domains
- LTI property after implementation
Spring 2021 marker board notes
- Part 1: Z and Fourier transforms, signal properties and Dirac delta properties
- Part 2: continuous-time and discrete-time convolution, and ideal delay
- Part 3: sampling and aliasing
Fall 2020 marker board notes

Last updated 09/17/26. Send comments to (Mailbox) bevans@ece.utexas.edu