ECE4782: Biosystems Analysis

[Virtual Presenter] ECE4782: Biosystems Analysis Lecture 2: Refresher on Linear Systems Professor Omer T. Inan Spring, 2026.

[Audio] ECE4781: ECE4782: Bioinstrumentation Biosystems Analysis Healthcare Provider Sensor(s) PreProcessing Feature Extraction Diagnosis / Knowledge Amplifier, Conditioning, and/or A/D Sensor Feedback Patient Feedback Ongoing Research!.

[Audio] Application of Linear Systems to Biosignals: Noise Reduction Selective noise and interference removal (sometimes). SWWC must be considered when "de-noising". Some good reasons: Improved feature extraction Improved classification An Important Concept for Biosystems Analysis: Patients don't read papers; so always seek clinical relevance! Source: mathworks.com.

[Audio] Application of Frequency Domain Analysis: Deciphering Brain Waves Frequency analysis uncovers features that may be less apparent in the time domain. Alpha rhythms in an electroencephalogram (EEG) recording are an example. EEGO1 Source: Inan, et al. IEEE TBioCAS 2010 +.

[Audio] Overview Brief Refresher on Signals and Systems Time and Frequency Domain Representation An excellent resource for refreshing your signals / systems fundamentals, which is the basis for many of these slides, is: www.stanford.edu/~boyd/ee102/.

[Audio] Signals T(t) T(t) is a continuous time signal, a representation of the temperature, T, as a function of time, t. t H[k] H[k] is a discrete time signal, a representation of when I woke up, H, as a function of days, k. k.

[Audio] Examples of Biosignals Aortic Pressure LV Pressure Pressure Tracings Electrocardiogram Lung Volume Heel Strike Force.

[Audio] Measuring the Size of a Signal The size of a signal, u, is measured in many ways, assuming u(t) is defined for t ≥ 0. For example: ∞ 𝑢𝑢 𝑡𝑡 2𝑑𝑑𝑡𝑡 Integral square (or total energy): � 0 Square root of total energy Peak or maximum absolute value: 𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡≥0 𝑢𝑢 𝑡𝑡 𝑇𝑇 𝑢𝑢 𝑡𝑡 2𝑑𝑑𝑡𝑡 1/2 Root-mean-square (RMS) value: lim 𝑇𝑇→∞ 1 𝑇𝑇 � 0 For some signals, these are infinite or undefined..

[Audio] Application: Electromyography Source: C. Vinyard, et al., Integ Compar Biol, 2008..

[Audio] Systems (SISO) S u y Input, u, gets transformed to output, y System, S, maps input signals into output signals S is an example of a single-input, single-output (SISO) system Examples: amplification, attenuation, differentiation, integration Notation: y = S(u) or y = Su.

[Audio] Systems (MISO) u1 S y u2 Multiple inputs, u1 and u2, get transformed to a single output, y System, S, maps input signals into output signal S is an example of a multiple-input, single-output (MISO) system Examples: summing, difference, multiplication, comparator.

[Audio] Examples of MISO With inputs u1, u2, and output y 𝑦𝑦 1. Summing system: + 𝑢𝑢1 𝑦𝑦 𝑡𝑡 = 𝑢𝑢1 𝑡𝑡 + 𝑢𝑢2 𝑡𝑡 𝑢𝑢2 2. Difference system: 𝑦𝑦 + + 𝑢𝑢1 𝑦𝑦 𝑡𝑡 = 𝑢𝑢1 𝑡𝑡 − 𝑢𝑢2 𝑡𝑡 𝑢𝑢2 𝑦𝑦 3. Multiplier system: X 𝑢𝑢1 𝑦𝑦 𝑡𝑡 = 𝑢𝑢1 𝑡𝑡 𝑢𝑢2 𝑡𝑡 𝑢𝑢2 4. Comparator system: + 𝑢𝑢1 𝑦𝑦 𝑡𝑡 = � 1, 𝑢𝑢1 𝑡𝑡 ≥ 𝑢𝑢2 𝑡𝑡 −1, 𝑢𝑢1 𝑡𝑡 < 𝑢𝑢2 𝑡𝑡 𝑦𝑦 𝑢𝑢2.

[Audio] Examples of Biosystems Biological Neuron Baroreceptor Widrow's electrical "neuron" concept (more on this later!).

[Audio] Linearity A system, S, is linear if the following two properties hold: 1. Homogeneity: if u is a signal and a is any scalar, S(au) = aS(u) 2. Superposition: if u and v are any two signals, S(u + v) = S(u) + S(v) In words: Scaling before or after the system is the same. Summing before or after the system is the same..

@Randy Glasbergen. www.glasbergen.com ACCOUNTING DEPT. REORGANIZATION, PUN B 2+2=22 3 + 3 33 "For years, we've been playing by old rules and the results have been dismal. It's time for a bold new direction!".

[Audio] TimeInvarianceAsystem,S,istime-invariantifthefollowingpropertyholds: If: S(u(t))=v(t) Then: S(u(t+t0))=v(t+t0) Inwords: Thesystemisnotchangingwithtime(althoughthesignalscanbe). Iftheanswerwasxyesterday,it'sstillxtoday(notlikeDCpolitics…)..

[Audio] Interconnections of Systems We can interconnect systems to form new systems: Cascade (or series): y = S(R(u)) S u R y Sum (or parallel): y = S(u) + R(u) S y u + R.