SIGNAL PROCESSING - Biomechanics and Motor Control of Human Movement - page 21

Stimulus S1 S2

t

d

V = d/t

R S1S2 ( τ )

0 τ 1 = t

Figure 2.5 Cross-correlation of a neural or muscular signal recorded at two sites, S1
and S2, separated by a distance, d . R xy (τ ) reaches a peak when S2 record is shifted
τ 1 = t sec. Thus, the velocity of the transmission V = d /t .

cycle will sum to zero. However, shifting the cosine wave forward 90 ◦ will
bring the two signals into phase such that all the cross products are + ve and

1. Shifting the cosine wave backward 90 ◦ will bring the two signals
180 ◦ out of phase so that all the “cross products” are

R xy (τ ) =

1.
A physiological example is the measurement of transmission delays (neural
or muscular) to determine the conduction velocity of the signal. Consider
Figure 2.5, where the signal is stimulated and is recorded at sites S1 and S2;
the distance between the sites is d . The time delay between the S1 and S2 is
t as determined from R S1S2 (τ ) , the cross-correlation of S1 and S2. Figure 2.5
shows a peak at τ 1 =

− ve and R xy (τ ) =−

t when S2 is shifted so that it is in phase with S1.

Property #3. The Fourier transform of the cross-correlation function is the
cross spectral density function, which is used to calculate the coherence func-
tion, which is a measure of the common frequencies present in the two
signals. This is a valuable tool in determining the transfer function of a
system in which you cannot control the frequency content of the input signal.
For example, in determining the transfer function of a muscle with EMG as
an input and force an output, we cannot control the input frequencies (Bobet
and Norman, 1990).

2.1.5 Importance in Removing the Mean Bias from the Signal
A caution that must be heeded when cross correlating two signals is that the
mean (dc bias) in both signals must be removed prior calculating R xy (τ ) .Most