Case study -heart beat signal | Partial differential equations & Transforms | SNS Institutions

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Introduction
A heartbeat is a natural, repeating signal that shows the health of the heart. Doctors usually check this using an ECG (Electrocardiogram), which records the electrical activity of the heart. The ECG signal has a special shape with P, QRS, and T waves. If this signal is unclear due to noise, doctors may miss important details.

Problem
Real ECG signals are not always clean.

Noise can come from body movement, muscle contractions, or interference from electrical machines.

This noise hides the true shape of the ECG and makes it hard for doctors to read.

We need a method to remove noise and keep the real heartbeat signal clear.

Solution: Fourier Series
Fourier Series is a mathematical tool that can break a complicated repeating signal into a sum of many simple waves (sine and cosine functions).

Steps:

Take one complete ECG cycle as a periodic function f(t).

Write it as a Fourier Series:
f(t) = a₀ + Σ (aₙ cos(2πnt/T) + bₙ sin(2πnt/T))

Identify which terms (harmonics) carry the main ECG shape (low frequencies).

Remove the unwanted higher harmonics (noise).

Reconstruct the ECG with only the clean components.

This process gives a smooth ECG curve that is easy to read.

Result

Clearer P, QRS, and T waves

Accurate measurement of heart rate

Easier detection of irregular heartbeat

Applications

Used in hospital monitoring systems to show a clean ECG on screen

Helps in early diagnosis of heart problems

Reduces wrong readings and improves patient safety

Conclusion
By using Fourier Series, we can take a noisy heartbeat signal and make it clean. This makes it much easier for doctors to read and take the right decision quickly. Fourier analysis is a powerful example of how mathematics can save lives in the real world.

Видео Case study -heart beat signal | Partial differential equations & Transforms | SNS Institutions канала Chellapandi P.M.