In this section we will show that any periodic signal with a, so called, Fundamental period T0 can be composed as the sum of sinusoidal signals from which each of these frequencies is related to the, so called, Fundamental frequency F0=1/T0 as an integer multiple. In order to show this Fig. 1 depicts a signal y2(t) which is composed of the sum of two sinusoidal signals x1(t) of frequency f1=6 [Hz] and x2(t) of frequency f2=2 [Hz]. From this figure it follows that y2(t) is periodic and we can measure that the period equals T0=1/F0=0.5 [sec]. The reason for this periodicity of signal y2(t) is that exactly three periods of x1(t) and one period of x2(t) fit into one period T0 of signal y2(t). Thus after each T0=0.5 [sec] the same composition of the signals x1(t) and x2(t) appears in the signal y2(t).
Signals of various frequencies with there cumulative signals.
The figure also shows another signal y3(t) which is composed of the sum three sinusoidal signals. The same sinusoidal signals x1(t), x2(t) and a new sinusoidal signal x3(t), with frequency f3=1.2 [Hz]. Again it follows from the figure that y3(t) is periodic but now we can measure that the period equals T0=1/F0=2.5 [sec]. This is because of the fact that exactly fifteen periods of x1(t), fi ve periods of x2(t) and three periods of x3(t) fit into one period T0 of signal y3(t). Mathematically we can calculate F0=1/T0 of y3(t) as the Greatest Common Divisor (gcd) of the frequencies from the individual sinusoidal components, since F0=gcd{f1,f2,f3}=gcd{6,2,1.2}=0.4 [Hz]. The greatest common divisor of a set of numbers is the largest number that all numbers of that set can be fully divided by. From this it follows that the Fundamental period of y3(t) equals T0=1/F0=2.5 [sec]. This leads to the following general statement:
A real signal x(t), which consists of a DC component and a sum of N different frequencies f1,f2,…,fN, with f1<f2<…<fN, is a periodic signal with Fundamental period T0=1/F0 and has a Fundamental frequency F0=gcd{f1,f2,…,fN}.
In the case that this general statement is true, all N different frequencies of signal x(t) are related by an integer number times the Fundamental frequency F0. Thus we can define the largest frequency by fN=M⋅F0 with integer M≥N. With this definition we can write such a periodic signal x(t) as a weighted sum of phasor components as
x(t)=M∑k=−Mαkej2πkF0t,
in which the complex weights αk represent the amplitude and phase of the frequency components at frequency k⋅F0. Finally note that possible α0 and definitely only N out of M complex weights αk are not equal to zero.
Example
x(t)=2+2cos(102πt−π8)−sin(238πt+π3)+3cos(340πt+π2)
Determine if the above signal is periodic and if so calculate the Fundamental frequency F0=1/T0 and give the general expression for x(t) as in the above equation and make a spectral plot of x(t).
The signal consists out of a DC component and a sum of 3 sinusoids with frequencies f1=51, f2=119 and f3=170 [Hz]. The Fundamental frequency can be determined as F0=gcd{51,119,170}=17 [Hz]. Therefore x(t) is a periodic signal with Fundamental period T0=1/F0=1/17 [sec]. The relationships between the three frequencies and the Fundamental frequencies are f1=51=3⋅F0, f2=119=7⋅F0 and f3=170=10⋅F0 [Hz].
The highest frequency contains 10 times the Fundamental frequency and therefore x(t) can be written as
x(t)=10∑k=−10αkej2πkF0t.
By using Euler we can write x(t) as
x(t)=32e−jπ2e−j2π170t+12je−jπ3e−j2π119t+ejπ8e−j2π51t+2+e−jπ8ej2π51t+−12jejπ3ej2π119t+32ejπ2ej2π170t.
When comparing the above two equations, the values of αk can be determined as
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⎪⎩32ejπ2,for k=10−12jejπ3,for k=7e−jπ8,for k=32,for k=0ejπ8=α∗3,for k=−312je−jπ3=α∗7,for k=−732e−jπ2=α∗10,for k=−100.otherwise
The spectral plot of x(t) is given in Fig. 2. The x-axis contains both the frequency as the integer values of k, which are the frequency divided by the Fundamental frequency.
