6.2 Signals of the Muscles and Motions

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bolic exhaustion⁸ of this motor unit has already set in, is shifted to lower frequencies

compared to the spectrum at the beginning of the contraction. A suitable parameter

for observing the frequency shift is the frequency centroid of the power density spec-

trum. The power density spectrum Φ(ω) of a function s(t) is calculated from the Four-

ier transform of the autocorrelation of s(t) according to

Φ(ω) = ∫φ(τ)e^−jωτdτ ,

(6.1)

with the autocorrelation function

φ(τ) = ∫s(t) s(t + τ)dt .

(6.2)

The integration in Equation 6.1 and Equation 6.2 runs over all times τ and t, respect-

ively. Φ(ω) describes the power contained per infinitesimal frequency interval. In the

case of the sequence in Figure 6.7 (left), the result for the power density spectrum is

as shown in Figure 6.7 (right).

The spectral centroid fmean of the power density spectrum is calculated according

to

fmean =

∫f Φ(f)df

∫Φ(f)df

.

(6.3)

The integration limits in Equation 6.3 correspond to the first Nyquist range, i.e. from

0 to half the sample frequency. The numerator in Equation 6.3 corresponds to the first

moment of the power density spectrum, the denominator describes the power con-

tained in the signal. The implementation of the described algorithm in Matlab from

Listing 6.2.1 is given below.

Listing 6.2.1: Analysis of an EMG signal in relation to the median frequency.

%% reads the electromyogram s1(t) from wav file, the power density

%% spectrum is formed and displayed graphically as both a

%% logarithmic as well as linear plot,

%% finally, the centre frequency of the spectrum is determined.

% Reading in and graphical representation of the signal

[s1,F1s] = audioread('emg.wav');

L1 = length(s1);

% sample number

p1 = 1:L1;

t1s = p1/F1s;

str_Fs = ['Sampling frequency F1s = ', num2str(F1s), ' 1/s.']

str_L = ['Signal length L1 = ', num2str(L1)]

8 Metabolic exhaustion occurs when the nutrient reserve within the cell has been within the cell has

been depleted, causing the cell’s action potential to be triggered less frequently.