Biosignal Processing Fundamentals and Recent Applications with MATLAB (Stefan Bernhard, Andreas Brensing etc.)

2 Fundamentals of Information, Signal and System Theory

electrical power from the effective values of the respective AC voltages Ueff and currents

Ieff. With the associated Ohm’s law Ieff = Ueff/R, the following relationship holds:

P = ¹

∫

R ^u²^d^t⁼^U^eff^⋅^I^eff⁼⁽^U^eff⁾²

(2.5)

Converting leads to an expression for the root mean square of an AC voltage, which is

often referred to as RMS value⁵:

Ueff = ^√¹

t0+T

∫

u²dt = ^√u².

(2.6)

Corresponding equations apply to the RMS value of the current intensity and gener-

alised to any other periodic or stochastic signal. The signal-noise-distance therefore

corresponds exactly to the ratio of the squared RMS values of the effective voltages:

SNR = ^P^signal

Pnoise

U²

effsignal

U²

effnoise

(2.7)

Due to the large possible range of numbers, the signal-to-noise ratio is often given in

logarithmic scale with the unit decibel as follows:

SNR = 10 log(SNR)dB = 10 log (

U²

effSignal

U²

effRauschen

) dB = 20 log ( ^U^eff^Signal

UeffRauschen

) dB ≥0 , (2.8)

where in the second line of the equation the squares of the RMS voltages have been

pulled out of the logarithm for simplicity. According to the equation, the signal-noise-

ratio SNR ≥0, and a SNR = 20 dB, for example, is equivalent to the ratio of the root

mean square values of the amplitudes of the signal and the noise signal being 100 to

1, corresponding to a SNR of 100. Regardless of the available bandwidth B, this results

in practice in data transmission rates of about a factor of 6.7 compared to the data

transmission rate for SNR = 1, i.e. Cmax = B ⋅log2(101) ≈B ⋅6.7.

Bandwidth and Modulation

The bandwidth of a signal describes the frequency interval (e.g. of a transmission

channel of a radio link) in which the dominant frequency components of the signal to

be transmitted or stored. It is characterised by a lower and an upper cut-off frequency

fu, fo of a signal. If the lower cut-off frequency is zero, one speaks of a baseband posi-

tion, otherwise of a bandpass position. The difference in magnitude of the two cut-off

frequency values is called the bandwidth: B = |fo −fu|. Table 2.2 shows common (bio-)

signals and their approximate bandwidths in comparison.

5 The RMS value is the abbreviation for Root Mean Square