Notes on Signals & Systems

These notes were created by Dr. Fred DePiero , et. al. of the CalPoly Electrical Engineering Dept and are used in EE 228, 302/342, 328/368, 419/459, 515, 525 and 528, at CalPoly.

General

Analog Signals and Transforms

- Concept of a Time-Frequency Transform
- Common Signals
- Finding System Response Directly via Differential Equation
- Applications of Concepts in Signals and Systems

Discrete Signals, Sampling, and Transforms

- Definition of Common Terms
- Properties of DT Systems
- Practical A/D's D/A's and the Sampling Process
- Frequency of Aliased Signals
- Useful Transforms and Relationships
- Sound I/O, Plotting Signals & Spectrums in SciLab
- Butterfly Diagram of 4-Point FFT
- Butterfly Diagram of 8-Point FFT
- Explanation of Musical Staff, with Analog Frequencies of Notes
- Hardware Structures for Digital Filters
- Formal Solution of Differential Equations
- Linear Phase FIR Filters

Analog Filters

- Grid Paper for Constructing Bode Plots
- Finding the Corner Frequencies of an Unknown System
- Capturing a Frequency Response Plot on the Scope

Digital Filters

- Filter Analysis
- Filter Design by Pole/Zero Placement
- Filter Design by Windowing
- Filter Design by Frequency Sampling
- Developing a Digital Filter to Process WAV Files
- Developing a Digital Filter on the TI-DSK Board
- Using the Daughter Card Interface on the TI DSK

- SciLab Version 3.0, with Additional Functions for Filter Analysis
Check out this image of |H(z)| for a notch filter, made with 'zplane3d_fwd()' by SciLab! The surface is clipped at the unit circle, revealing the contour of |H(z)|. The Filter Analysis notes explain the usage of SciLab with digital filters.

Advanced DSP

Analog Control Systems

- Simplifying Block Diagrams
- Finding a Transfer Function via Mason's Gain Rule
- Finding Stability via Routh-Hurwitz
- Determining Performance Measures
- Introduction to Root Locus Plots
- Finding a Root Locus: Step-By-Step
- Finding Bode Plots and Gain & Phase Margin
- Designing a Lead Compensator
- Compensator Example
- Nyquist Stability Criterion

Probability & Random Signals

- Basic Probability 1
- Basic Probability 2
- Basic Probability 3
- Multiple Random Variables
- Random Processes

Kalman Filter

Applied Math

Convolution DemoThis demo illustrates the process of evaluating a convolution integral to obtain y(t) = h(t)*x(t). Shows intermediate signals, including "flip & slide" version of x(), and its product with h(). User can select various signals for x() and h(), and vary the wave shape (duration, for example). The 'User' menu animates the calculation of the convolution integral, which can also be observed by moving the lower slider. User menu also provides a choice of adjusting either x(t) or h(t) using the sliders.

This demonstration illustrates a number of concepts in signals and systems including:

- Definition of common signals
- Time reversal and time shift of signals
- Product of signals
- Time invariance of systems
- Evaluation of convolution integral
- Time extent of convolution operation
Download the ZIP File and extract to any convenient location, such as the desktop. Open the extracted folder named 'SIPTool 1D Convolution', and run the 'SIPTool' executable.

Click on the 'x(l)', 'x(t-l)', and' h(l)x(t-l)' tabs to see intermediate signals. The lower slider adjusts 't', shifting the input and changing the overlapping functions. The product of 'h(l)x(t-l)' appears in blue. The 'y(t)' tab shows the output, as computed up to time 't'. See the 'User' menu to select which signal (x or h) the sliders affect. Left-click on the waveform names appearing in the 'x(t) Type' and 'h(t) Type' tabs to alter the type of signal.

Notch Filter DemoThis demo processes signals from the microphone in real-time. Input and output signals are displayed in both the time and frequency domains. The notch filter is described by its frequency response, a pole-zero plot, and the impulse response. The center frequency and pole radius of the filter are adjustable by sliders.

This demonstration illustrates a number of concepts associated with digital filters:

- Difference Equations
- Poles and Zeros
- Frequency Response
- Impulse Response
- Filter Structures
- Time and Frequency Domain Signals
Download the ZIP File and extract to any convenient location, such as the desktop. Open the extracted folder named 'Audio Notch', and run the 'Audio Notch SIPTool' executable.

To activate on-line processing, select on of the 'User' -> 'Process Mic' menu options. The signal that initially appears was from a whistle. Have Fun!

Short-Time Fourier Transform DemoThis demo uses WAV files for input and plots the Short-Time Fourier Transform, as well as a standard Fourier Transform and a time-domain plot. A plot similar to that of a spectrum analyzer is also shown.

This demonstration illustrates a number of concepts associated with digital signals:

- Time and frequency domain representations of a signal
- Magnitude and phase of a Fourier transform of a signal
- Time-varying frequency content of a signal
- Non-trivial frequency content of natural signals
- The Short-Time Fourier Transform of various signals, including music.
Download the ZIP File and extract to any convenient location, such as the desktop. Open the extracted folder named 'STFT for WAV', and run the 'SIPTool.exe' executable.

The 'STFT Magnitude' tab depicts the energy content of the signal as a function of time (horizontally) and frequency (vertically, lower frequencies at the top). Left-click in the tab windows and select 'Playback' to hear the WAV file played. You will also see a cursor sweep the STFT display (roughly) in time with the audio. Other left-click options are also available. Drag and drop a WAV file into the tab window to load a new file. Have Fun!

Root Locus DemoThis demonstration package addresses fundamental concepts in control systems, including contruction of root locus, transfer functions, construction of a Bode plot and deeterminiation of gain and phase margin. Nyquist plots are also illustrated. The step response is shown in a traditional format (output versus time) and via an X-Y plotter. In the X-Y display two independent positioners are animated. The horizontal positioner has dynamics and gain set by the input file and sliders. The vertical positioner has the same poles and zeros but half the gain value.

*All materials on this site are copyright by Professor DePiero, and others.*