Fourier analysis

 

Principle

 

In medicine, in all kinds of ways and for many aims signals are measured from the human body. Signals can be represented as a function of area or space (2-D or 3-D), for example with CT and MRI scans and as a function of the time, as with ECG, EEG (Electroencephalography) and Magnetoencephalography (MEG), but also as function of time and space as with fMRI (see MRI: functional MRI).

We will limit our description to analogue, ongoing time signals. These can be periodic (for example a cosine or ECG) and non-periodic. Non-periodic signals are for example a once occurring pulse or a noisy signal, such as the EEG. This description is limited to the real notation. For the description in complex notation see Fourier transforms.

 

Often it is important to know which frequencies occur in the signal, e.g. in communication technology but also in medicine. Then the signal is no longer represented in the time domain but in the so-called frequency domain. The theory on which this is based, is the theory of Fourier, which says that each (random) signal can be described by the sum of a series of sine or cosine functions, everyone with its own amplitude and phase (or as the sum of a series of sinus and cosine functions, everyone with their its amplitude).

The frequencies of the sine and cosine waves have the ratio 1, 2, 3, 4 etc. and they are called the harmonics. The first harmonic (with ratio 1) is the ground or fundamental frequency. Since any signal comprises this harmonic series from frequencies Fourier analysis is also called frequency analysis.

The term harmonic comes from the term harmonic oscillator. An example is the harmonic movement performed by a weight suspended from a feather. The excursion of the weight as a function of time is described by a sine wave.

Fig. 1 shows an analysis of the blood flow signal.

 

 

Fig.   Fourier analysis of blood flow in a dog lung artery. The upper panel gives the measured signal, indicated by a dotted line and the drawn curve is the mean signal plus the first 4 harmonics.

 

Fourier analysis and synthesis

Like already said, an important description method is the approach of a signal by the sum of a number of sine and cosine functions. A single sine function is described by its amplitude C, angular frequency ω (ω = 2πf with f the frequency in Hz) and its phase φ (in radians) with regard to another signal of the same frequency (or position in time):

  x(t) = Csin(2πft + φ).                                          (1)

In Fig. 2 is demonstrated how a number of harmonic sine waves can describe some signal, here a square wave. The fluently drawn line in the four panels is the sum of the solid curve in the previous  panel (e.g. in panel a) plus the respective dotted harmonic (in b). When this procedure is continued, then a sum signal with the shape of a square wave arises, such as indicated in panel d. The more harmonics are used, the better the synthesis approaches the square signal. The square wave is a special case of the general function:

,                   (2)

where x(t) is the sum of an infinite number of harmonics, each with its own amplitude C, frequency f and phase angle φ. The amplitudes Ck together constitute the amplitude spectrum (with amplitude versus k) and similarly φk yields the phase spectrum.

The above makes clear that any signal can be developed in a series of Fourier terms and that a signal can be synthesized with such a series: Fourier synthesis.

 

Fig. 2  Fourier analysis and synthesis of a square wave. a. signal fitted with the 1st harmonic with amplitude 1. b. With the first 3 harmonics. The third contributes with amplitude 1/3. The sum signal is the drawn curve, dashed curves are the harmonics. c. The same for the first 5 harmonics. d. For the first 7 harmonics.

 

The average value and the Fourier components

 

The average value of a signal x(t) concerning a duration T has been defined as:

                                       (3)

Averaged over one period, sine and cosine waves have an average value of zero. This means that the average of a signal described with equation (2) has also as an average value of zero. (The average of a sum = the sum of the averages). For a random signal the average value xmean is generally not zero but has some value, C0. This C0 must be added to (2), yielding:

 

 .       (4)

Equation 4 describes any periodic signal or function. More Info describes how Ck is calculated.

 

 

Application

 

Fourier-analysis, in its digital version, is performed by many PC software packages (MatLab etc.) and spread sheets.

 

 

More info

 

By defining that φ0 = π/2 (4) becomes the general equation:

    , or          (5a)

      (5b),

(with the goniometric rule sin(a+b) = sina∙cosb + cosa∙sinb) where C0 adopts the character of the amplitude of a sinusoid with frequency zero.

Now, the sine of (5a) with phase φ and amplitude C has been written as the sum of a cosine and sin of 2πkft. With Csinφ = A and Ccosφ = B, (5b) becomes:

  .    (6)

Ck (the amplitude) and φk of equation (5) are calculated from:

   Ck = √(Ak2 + Bk2)   and                                   (7a)                        

   φk = arctan Bk/Ak.                                              (7b)

 

It is general practice to present Ak along the x-axis and Bk along the y-axis. After applying the theorem of Pythagoras the resulting Ck makes the angle of φk with the x-axis. With a vector representation, Ck is absolute value and φk the argument.

The coefficients Ak and Bk are calculated as follows:

  , and  

             (8)

The integral only gives an output Ak or Bk > 0 when x(t) comprises a cosine or sine of the same frequency kf. Ak and Bk are independent since the integral of sinjacoskb gives zero (j and k are the harmonic rank numbers). This property is a condition for so called orthogonality of two functions, here sine and cosine.

 The elegance of the Fourier development is that its coefficients agree exactly with the least square fit of the signal and this applies even to each component separately.

 

When x(t) is written as a cosine series, then formula (5) becomes:

      (9)

 

Self-evident, the power of the signal in the time domain and in the frequency domain should be the same. This is the theorema of Parseval;

 

                            (10)

 

If Bk = 0 for each k, then we call the signal even because the signal can be mirrored with respect to the y-as (amplitude) or for the line parallel with the y-as on t = kT. As Ak = 0 (apart from A0) for each k, then the signal is called odd because there is a so-called point-symmetry at the origin. This holds also for the time instants kT on the x-as. One speaks more commonly of even when f(t) = f(─t), and odd functions when f(t) = ─f (t). Polynomials with exclusively terms with even exponents are even functions. Likewise polynomials with exclusively odd exponents are odd functions. However, an arbitrarily signal or function is nor even nor odd. A sine is odd and a cosine even. Also these signals can be written as a as polynomials (with infinite terms). The step function (see Linear first order system) on t=0 is odd (sum of sinuses) and the delta function (sum of cosines on t = 0) is an even function. (The delta function is everywhere 0, except on t = 0 the amplitude is 4, see Linear first order system).

Looking back once more to the synthesis of the block-signal (Fig. 2), then we see that the number of maxima is equal to the rank number of the highest term that is used for the synthesis. Additionally, the ripples at the sides are larger than those in the middle of the plateau, the phenomenon of Gibbs.

 

The elegance of the Fourier development is that its coefficients agree exactly with the least square fit of the signal and this applies even to each component separately.