2. SPECTRA OF CONTINUOUS AND DISCRETE SIGNALS 
The spectral density function (spectrum) of a given 
continuous or discrete process is defined as the Fourier 
transform of the autocorrelation function of this process. 
The spectrum is a function that measures the average rates 
of fluctuation of a given set of signals. It represents, 
therefore , the distribution of power with respect to 
frequency . The spectrum of a continuous process can be 
determined as follows : 
P(e) = j| ga) ej?* ak 
z 5 R,(k) { cos wk - j sin wk Ï dk (1) 
Where : 
Fat?) is the spectral density at frequency co 
R (X) is the autocorrelation function at lag k. 
As the autocorrelation function is always an even function, 
the product R (k) . sin «w k integrates to zero , therefore, 
PL.) is alwäys real(Schwartz,et 81,1975) 
Fl) a em Re Ck) cos Wk dk (2) 
For O < w ££ © , the inverse Fourier transform will give : 
à 
Ry) = = ) mq) coesk aw qosi ta) 
If the continuous process is sampled at intervals ^x, 
then the spectrum of the resulting discrete data will be : 
| od 
"iE chi 
Pal«w) Ryj(k) © (4) 
Ka SO 
The inverse Fourier transform will give : 
RU 
\ 
Ra (&) = de Fa(w) cos Wk dw (5) 
- 307 =