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# Circuit elements: Analyzing electrocardiogram voltage signals

## Problem

An electrocardiogram (ECG) measures the heart’s electrical activity using a series of probes placed on the skin. In addition to detecting rhythmic electrical activity due to the heart’s beating, an ECG can detect other processes, like spasms, that occur with different frequencies than the heartbeat. The ECG is able to convert data indicating the magnitude of the electrical activity as a function of time into information about specific processes that occur in the heart over different timescales using a mathematical operation known as a Fourier transform.
In a Fourier transform, a time-varying signal is converted into a histogram showing the various frequencies that constitute the signal. As a result, an ECG voltage versus time graph can be replotted as an amplitude versus frequency graph, which indicates the relative presence of various frequencies in the original voltage reading. For example, The Fourier transform of a time series showing an undulating sine wave with frequency $1$ hz would be a histogram with peaks at $±1$ hz (Figure 1).
For a more complex signal consisting of many different sine waves superimposed, the Fourier transform shows the relative amounts of various frequencies present in the signal. For example, in the Figure 2, the Fourier transform of the voltage signal $V\left(t\right)=\left(1/3\right)\mathrm{sin}\left(2\pi \phantom{\rule{0.167em}{0ex}}3t\right)+1\mathrm{sin}\left(2\pi \phantom{\rule{0.167em}{0ex}}t\right)$ (Figure 2A) produces a bar chart with a higher peak at $1$ hz than at $3$ hz (Figure 2B). A mathematical theorem guarantees that a valid Fourier transform can be generated for any periodic signal, which equivalently suggest that any periodic ECG signal can be represented as a sum of sine waves with various frequencies and amplitudes.
The raw output of an ECG is a periodic signal indicating the electrical activity arising from a patient’s heart as a function of time, but the shape of this signal is generally NOT purely sinusoidal-instead, it consists of a superposition of many different periodic processes in the body which generally ARE sinusoidal. As a result, the Fourier transform of an ECG signal can be used to indicate the presence or absence of various processes that occur with different characteristic frequencies based on their presence or absence in the Fourier histogram.
Figure 1: A) A sinusoidal output from an ECG and B) its Fourier transform.
Figure 2: A) An ECG signal consisting of two superimposed sinusoids and B) its Fourier transform.
Figure 3: The output of an ECG test for a healthy patient contains many different frequencies and, as a result, is generally not a single sine wave.
Suppose a rare genetic mutation doubles the diameters of the electrically conducting cells that transmit electrical activity throughout the heart and body. Which of the following describes the most direct effect this would have on observed ECG voltage fluctuations if all other aspects of the patient (height, weight, composition) remain the same?