5.3 Methods for Analysis and Processing of Discrete Biosignals

|

183

5.3.3 Signal Analysis in the Time-Frequency Domain

f(t) =

∞

∫

−∞

F(f)e^j2πftdf ,

F(f) =

∞

∫

−∞

f(t)e^−j2πftdt .

(5.63)

This integral quantity does not obtain a statement about when a particular event oc-

curred in the signal waveform, but only a spectral average. E.g. one would have to

record a phonocardiogram (PCG) from beginning to end and would not know after-

wards when a certain heart sound occurred. This is of course very unfavourable for

heart sound recognition, and ideally the doctor would like to know not only the exact

frequency but also the time of occurrence. But this is not possible for the following

reasons:

In order to examine a certain time interval around a certain time t0, a signal should

only be present in this range. This can be achieved by multiplying the signal by a win-

dow function w(t −t0) which fades out this range. I.e. the faded out signal fw(t) then

results in

fw(t) = w(t) ⋅f(t) .

(5.64)

In general, w(t) is a function that is symmetric around time t0, e.g. a square or Gaus-

sian function. In the frequency domain, the spectrum must then be convolved with

the frequency transformed function W(t) of the window function w(t); i.e.

Fw(f) = W(f) ∗F(f) .

(5.65)

The spectrum now created by convolution naturally also has a width that is influenced

by the width of the spectrum of the window function W(f). The spectrum of the win-

dow function W(f) can have a similar shape as the window function w(t) in the time

domain. If, for example, w(t) can be described by a Gaussian pulse, this can also be

done for the associated spectrum W(f) – a Gaussian pulse i(t) in the time domain also

produces a Gaussian pulse I(f) in the frequency domain. Unfortunately, the width of

the spectrum of the window function in the frequency domain is large when the width

of the window function w(t) in the time domain is small, and vice versa. This can be

shown by examining a pulse width in the time and frequency domains. If the pulse

width is defined according to the width of an equal-area rectangular pulse (cf. Fig-

ure 5.25), it follows:

T =

1

i(0) ^⋅

∞

∫

−∞

i(t) dt ,

(5.66)

B =

1

I(0) ^⋅

∞

∫

−∞

I(f) df.

(5.67)