Continuous time and frequency

Periodic signals

Periodic signals can be composed of sine waves :

  $$x(t) = \sum_{k = -\infty}^{\infty} c_{k} e^{j2\pi k F_1 t}$$ (27)

where $F_1$ is the principle or fundamental frequency and $k \neq 1$ are the harmonics with strength $c_k$. Usually $c_1$ is set to $1$. This is a Fourier series with $c_{k}$ as coefficients. For example an ECG has a fundamental frequency of about 1Hz (60 beats per minute). However, the harmonics give the ECG its characteristic peak like shape.

How do we find the coefficients $c_k$?

  $$c_{k} = \frac{1}{T_p} \int_{T_p} x(t) e^{-j2 \pi k F_1 t} dt$$ (28)

For simple cases there are analytical solutions for $c_k$, for example for square waves, triangle wave, etc.

What are the properties of $c_{k}$?

  $$c_{k} = c_{- k}^{*} \qquad \Leftrightarrow \qquad x(t)\mbox{ is real}$$ (29)

or

$$c_{k} = \mid c_k \mid e^{j \theta_k}$$ (30)

$$c_{-k} = \mid c_k \mid e^{-j \theta_k}$$ (31)

Proof: with the handy equation...

  $$\cos z = \frac{1}{2} \left(e^{zi} + e^{-zi}\right)$$ (32)

we get

$$x(t) = c_0 + \sum_{k = 1}^{\infty} \mid c_k \mid e^{j \theta_k} e^{j 2\p... ...1 t} + \sum_{k = 1}^{\infty} \mid c_{k}\mid e^{-j \theta_k} e^{-j 2\pi k F_1 t}$$ (33)

$$= c_0 + 2 \sum_{k = 1}^{\infty} \mid c_k \mid \cos(2 \pi k F_1 t + \theta_{k})$$ (34)

How are the frequencies distributed? Let's have a look at the frequency spectrum of a periodic signal: $P_{k} = \mid c_{k} \mid^{2}$

• There are discrete frequency peaks
• Spacing of the peaks is $\frac{1}{T_1} = F_1$
• Only the positive frequencies are needed: $c_{-k} = c_{k}^{*}$

A-periodic signals

In case nothing is known about $X(t)$ we need to integrate over all frequencies instead of just the discrete frequencies.

  $$X(F) = \int_{-\infty}^{\infty} x(t) e^{-j 2 \pi F t} dt$$ (35)

Consequently, the frequency spectrum $X(F)$ is continuous.

  $$x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(F) e^{j 2 \pi F t} dF$$ (36)

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