1. ECE3340 Review of Numerical Methods for Fourier and Laplace Transform Applications – Part 1 Fourier
Spring 2016
Prof. Han Q. Le
Note: PPT file is the main outline of the chapter topic – associated Mathematica file(s) contain details and assignments

2. Sound wave discussion
What is the frequency of your sound? (whistling, singing, talking, etc…?)
How to compute it?

3. Remember this? Discussion:
Sound is periodic in time, but it also travels. Hence, it must be a wave! In fact, sound is an air pressure wave.
Voltage signal
Air pressure wave from whistling
Microphone diaphragm

4. Remember these circuits?

5. When design a circuit, we want to know how it works, what it does for the intended application. We want numerical simulation results such as this:
input
output
This is what “numerical methods” is about. This is your objective: learn how to use computers to apply to ECE problems.

6. What do those things discussed above have in common regarding numerical computation methods?
Numerical application of Fourier transform

7. Overview (or the “big” perspective)
We have learned Fourier transform, Laplace transform, and their applications in scientific/engineering problem – specifically for ECE, circuit and signal analysis. (Math 3321)
Fourier transform is most applicable for harmonic, steady state phenomena (or when a starting time is not important).
Laplace transform is most applicable when the initial condition is essential.
In computation and digital applications, we use numerical methods such as FFT (or Z-transform especially for digital control of physical analog systems).
Learning objectives: review and practice numerical computation of these methods with specific examples of circuits and signal processing.

8. Checklist of reviewing/learning topics
Harmonic signal analysis: Fourier transform
Power spectral density
Frequency response of a system (circuit)
Transfer function
Bode plot (Nyquist plot if time permits)
Filters
Analog
Digital
whether you have learned all these things and only need a refresher or you have not learned much, it’s good to go through the complete list of topics, do exercises and be proficient at numerical methods

9. An example
We have a recorded sound signal. But there are some background noises. Can we reduce the noises so that we can listen to our signal?
Principle: filter out the noises from the signal.
How to implement it with numerical methods?
IOW, how to do it on a computer like the APP:
the ability to apply the principle we have learned into algorithm and software code to solve the problem

10. Instruction demo: Step 1: record a sound

11. Instruction demo: Step 2: we can select portion of the sound for analysis

12. Instruction demo: Step 3
This is the heart of the matter:
Analyze the signal: power spectral density
Can we tell the signal we want from the background noises or interfering signals that we don’t want?
Key concept: what is exactly “noises”?
randomness (which means unpredictable because we don’t know. A secret code can be made noise-like: pseudo-random noise code for encryption or in CDM (code division multiplexing) for example.
Something with distinctive features in harmonic analysis but not the signal we want: this is the same as interfering signals.
How do we reduce noises from our signal? digital filtering

13. Objective: write computer code to perform these steps like the APP
Instruction demo: Step 3 (continued)
Turn on various square filter bands
Observe the effect on the output signal: this is the simplest filtering we can apply
There is a whole field about designing various types of digital filters for numerous different applications:
sounds
images
sensor signals: radars, sonars, lidars, optical signals, virtually all types of physical sensors: e. g. temperature, pressure, strains, … all signals require some form of filtering.
Objective: write computer code to perform these steps like the APP

14. Note: checklist of review and learning topics:
Link to review of FT and numerical FFT file
Note: checklist of review and learning topics:
concept of harmonic spectrum
practical code to calculate power spectral density (do a few examples)
multi-dimension: 2 D FFT example in image processing

15. We have seen post-signal acquisition analysis
Can we do something pre-signal acquisition? Can we predict and/or design the output before it happens?
(we may learn Kalman filter near the end if time permit)

17. Linear Time-Invariant (LTI) System Time domain
Link to Mathematica file
Control
Input
Output
System
Impulse response function
Impulse function
ℎ 𝑡
𝛿 𝑡
Arbitrary input
LTI
𝑥 𝑖 𝑡 = 𝑥 𝑖 𝑡′ 𝛿 𝑡−𝑡′ 𝑑𝑡′
𝑥 𝑜 𝑡 = 𝑥 𝑖 𝑡′ ℎ 𝑡−𝑡′ 𝑑𝑡′

18. Linear Time-Invariant (LTI) System Frequency domain
Control
Input
Output
System
Harmonic transfer function
Harmonic function
𝐻 𝜔 𝑒 𝑖𝜔𝑡
𝑒 𝑖𝜔𝑡
Arbitrary input
LTI
𝑥 𝑖 𝑡 = 𝐴 𝑖 𝜔 𝑒 𝑖𝜔𝑡 𝑑𝜔
𝑥 𝑜 𝑡 = 𝐴 𝑖 𝜔 𝐻 𝜔 𝑒 𝑖𝜔𝑡 𝑑𝜔

19. Concept Review: Signal Processing
All electronics around us involve signal processing.
Signal represents information. That information can be something we generate. The APP is an example of sound signal. Other common types: texts, images, all sensors (as discussed previously)
Electronics deal with signals: preprocessing, post-acquisition, analog, digital.

20. Concept Review: Signal Processing (cont.)
Signal processing is a general concept, not a single specific operation. It includes:
signal synthesis or signal acquisition
signal conditioning (transforming): shaping, filtering, amplifying
signal transmitting (telecomm) or applying for control (robotics)
signal receiving and analysis: transforming the signal, converting into information, for implementing certain controls.
Signal processing is mathematical operation; electronics are simply tools.
Some computation are high-level signal processing: dealing directly with encode or embedded information rather than raw signal.

21. An APP to illustrate basic concepts of signals

22. Concept review: numerical methods
Signal and AC circuit problems
RLC or any time-varying linear circuits. Applicable to linear portion of circuits that include nonlinear elements
Signal processing
signal analysis (spectral decomposition)
filtering, conditioning (inc amplification)
synthesizing
Fourier transform
Harmonic function
Complex number &analysis
Phasors

23. Link to Mathematica lecture on FFT numerical method