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

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

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

Remember these circuits?

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.

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

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.

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

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

Instruction demo: Step 1: record a sound

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

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

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

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

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)

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′

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

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.

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.

An APP to illustrate basic concepts of signals

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

Link to Mathematica lecture on FFT numerical method