Periodic signals#

A continuous-time signal $x(t)$ is said to be periodic with period $T$ if there is a positive nonzero value of $T$ for which

An example of such a signal is given in Fig. 17.

We can use the periodicity to synthesize a periodic signal such as that shown in Fig. 17.

Let’s first define the signal over one period. We will use MATLAB and the symbolic math toolbox for this example:

Let one period of the periodic signal be defined by

We can use the Heaviside function (unit step) (MATLAB function heaviside: see The Unit Step Function) to sythesise this signal.

Define a $t$ as a symbolic variable t

syms t

Define the period $T$ of the periodic signal (you might want to play with this value)

T = 1; % period of periodic signal

Now define the signal using the Heaviside function to limit the range of the signal.

x(t) = t * (heaviside(t) - heaviside(t-T));

Now plot one period of the signal $x(t)$

fplot(x(t)),ylim([0 1.2]),grid,title('A Single period of x(t)'),xlabel('t')

One period earlier:

signal1 = x(t + T)

fplot(signal1),ylim([0 1.2]),grid,title('A Single period of x(t+T)')

Two periods later:

signal2 = x(t-2*T)

fplot(signal2),ylim([0 1.2]),grid,title('A Single period of x(t)')

It follows that

for all $t$ and any integer $m$.

Now we use a loop and the definition of periodic function to repeat this signal multiple times

periodic_signal = 0;
for m = 5:-1:-5
    periodic_signal = periodic_signal + x(t + m*T);
end
periodic_signal

Now we plot the result

fplot(periodic_signal,'g-',"LineWidth",2),...
    grid,ylabel('x(t)'),xlabel('t'),title('T = 1')
xlim([-3.00 3.00])
ylim([0 1.2])

Fundamental period#

The fundamental period $T_0$ of $x(t)$ is the smallest value of $T$ for which $x(t+mT)=x(t)$ holds.

For the previous example this was $T_0=1$.

DC signals#

Note that the definition of the fundamental period does not hold for a constant signal $x(t)$ (known as a DC signal).

For a constant signal $x(t)=c$ the fundamental period is undefined since $x(t)$ is periodic for any choice of $T$ (and so there is no smallest postive value). See Fig. 18.

Nonperiodic signals#

Any continuous-time signal which is not periodic is called a nonperiodic (or aperiodic) signal. For example see Fig. 19

Normalised energy content of a signal#

For an arbitrary continuous-time signal $x(t)$, the normalised energy content $E$ of $x(t)$ is defined as

Normalised average power of a signal#

The normalised average power $P$ of $x(t)$ is defined as

Energy and power signals#

Based on the previous definitions, the following classes of signals can be defined:

• $x(t)$ is said to be an energy signal if and only if $0<E<\mathrm{\infty }$, and so $P=0$.

• $x(t)$ is said to be an power signal if and only if $0<P<\mathrm{\infty }$, thus implying that $E=\mathrm{\infty }$.

• Signals that satisfy neither property are referred to as neither energy signals nor power signals.

Note that a periodic signal is a power signal if its energy content per period is finite, and then the average power of this signal need only be calulated over a period (ex:1.18).

Mean value#

For periodic signals with fundamental period $T_0$

The mean value is also known as the dc value.

Observations

• the mean value corresponds to the arithmetic average

• the signal $x(t)-M_x$ has zero mean

Measures of spread#

Example 2.1: Sum of two periodic signals#

Let $x_1(t)$ and $x_2(t)$ be periodic signals with fundamental periods $T_1$ and $T_2$ respectively. Under which conditions is the sum $x(t)=x_1(t)+x_2(t)$ periodic, and what is the fundamental period of $x(t)$ if it is periodic?

For the answer, refer to the lecture recording or see solved problem 1.14 in [Hsu, 2020].

Exercise 2.2: Periodic signals#

Determine whether or not each of the following signals is periodic. If a signal is periodic, determine its fundamental period.

a). $x(t)=\cos (t+\frac{\pi }{4})$;

b). $x(t)=\sin \frac{2\pi }{3}t$;

c). $x(t)=\cos \frac{\pi }{3}t+\sin \frac{\pi }{4}t$;

d). $x(t)=\cos t+\sin \sqrt{2}t$;

e). $x(t)={\sin }^{2}t$;

f). $x(t)=e^{j[(\pi /2)t-1]}$;

For the answers, refer to the lecture recording or see solved problem 1.16 in [Hsu, 2020].

Exercise 2.3: Integral properties of periodic signals#

Show that if $x(t+T)=x(t)$, then

for any real $\alpha$, $\beta$ and $a$.

For the answer, refer to the lecture recording or see solved problem 1.17 in [Hsu, 2020].

Exercise 2.4: Power in a periodic signal#

Show that if $x(t)$ is periodic with fundamental period $T_0$, then the normalized average power $P$ of $x(t)$ defined by

is the same as the average power $x(t)$ over any interval of length $T_0$, that is,

For the answer, refer to the lecture recording or see solved problem 1.18 in [Hsu, 2020].

Exercise 2.5: Power and energy signals#

Determine whether the following signals are energy signals, power signals, or neither.

a). $x(t)=e^{-at}{u}_0(t)$, $a>0$;

b). $x(t)=A\cos ({\omega }_{0}t+\theta )$

c). $x(t)=t{u}_0(t)$

Note $u_0(t)$ is the unit step (or Heaviside) function formally introduced in the next lecture. For the answer, refer to the lecture recording or see solved problem 1.20 in [Hsu, 2020].

Exercise 2.6: Power in domestic mains electricty#

Domestic mains power in the UK is delivered as a sinusoidal signal $x(t)=A\cos ({\omega }_{0}t+\theta )$ with frequency of $50\mathrm{Hz}$ and RMS value of $240\mathrm{V}$.

Calculate the fundamental period $T_0$, fundamental angular frequency ${\omega }_0$, average power $P$, peak voltage $A=x_{\mathrm{peak}}$, crest factor (CF) and peak-to-average power ratio (PAPR) of the power signal provided to the home.

Express CF and PAPR in dB.

Unit 2.2: Take aways#

• A signal is periodic, with period $T$ is $x(t+T)=x(t)\mathrm{all}t$

• Signal energy for a signal $x(t)$:

• Signal power for a signal $x(t)$:

• $x(t)$ is said to be an energy signal if and only if $0<E<\mathrm{\infty }$, and so $P=0$.

• $x(t)$ is said to be an power signal if and only if $0<P<\mathrm{\infty }$, thus implying that $E=\mathrm{\infty }$.

• Signals that satisfy neither property are referred to as neither energy signals nor power signals.

• Signal mean (average or DC value):

• Root mean square (RMS): $\sqrt{P_x}$;

• Peak value: ${\left|x\right|}_{\mathrm{peak}}=\underset{t}{max}|x(t)|$;

• Crest factor:

• Peak-to-average power ratio (PAPR):

• Value of $y$ in decibel (dB) is $20{\log }_{10}y$.