Week 4

Periodic Pulses

A periodic pulse has the general equation

\[x(t+T)\]

Sinusoidal Signals

\[sin(\omega_0t)=sin(2\pi f_0t)=sin(2\pi\frac{t}{T_0})\]

where $\omega=2\pi f_0=2\pi\frac{t}{T_0}$ is the angular velocity, and $f_0$ is the linear frequency.

A sinusoidal signal may be expressed in terms of cosine such that

\[cos(\omega_0t)=cos(2\pi f_0t)=cos(2\pi\frac{t}{T_0})\]

whose graph is shifted to the left/right of sin by $\frac{\pi}{2}$ (sin has a peak at $\frac{\pi}{2}$, cos has a peak at $0$).

Phase Shifted Cosine

\[x=A\,cos(\omega_0+\phi)\]

where $T_0=\frac{2\pi}{\omega_0}$

Truncated Signal

A signal may be truncated by multiplying it by a pulse. The resulting signal is defined within the window bounded by the pulse.

Sinc Function

The $sinc()$ function is defined as

\[sinc(t)=\frac{sin(t)}{t}\]

It is important to note that $sinc(0)=1$, for

\[\lim_{t\to 0} sin(t)=t\]

therefore

\[\lim_{t\to 0} \frac{sin(t)}{t}=\lim_{t\to 0}\frac{t}{t}=1\]

• We shall later see that $sinc()$ and the rectangular pulse are related through the Fourier transformation.
• Note that sinc is 0 at $t=n$-th values of $\pi$.

Sketching $sin(\omega_0(t-t_0))$

• First crossing at $t=t_0$
• For $sin(\omega_0 t-\omega_0 t_0)$, $w_0 t_0$ is constant and the phase shift.
• $w_0 t$=$n\pi$, thus $t=\frac{n\pi}{\omega_0}$. The spacing between crossings therefore is $\frac{\pi}{\omega_0}$.

Exponentials

Exponential signals take the form

\[x(t)=Ae^{\beta t}\]

• When $\beta>0$, the signal is increasing
• When $\beta<0$, the signal is decreasing
• When $t=0$, we have our y-intercept at $y=A$.

Multiplying Signals with Exponentials

• Growing sinusoid: $x(t)=Ae^{\Beta t}cos(\omega_0 t),\, \beta >0$
• Decaying sinusoid: $x(t)=Ae^{\Beta t}cos(\omega_0 t),\, \beta <0$

Complex Exponentials / Complex Sinusoids

From Euler’s formula, we have

\[re^{j\omega_0 t}=r[cos(\omega_0 t)+jsin(\omega_0 t)]\\ Re[re^{j\omega_0 t}=rcos(\omega_0 t)\\ Im[re^{j\omega_0 t}=rsin(\omega_0 t)]\]

Magnitude and phase: \(|re^{j\omega_0 t}|=|r|\sqrt{cos(\omega_0 t)^2+sin(\omega_0 t)^2}=r\\ \angle{re^{j\omega_0 t}}=arctan(\frac{rsin(\omega_0 t)}{rcos(\omega_0 t)})=\omega_0 t\)

Operations on Complex Sinusoids

The conjugate of a complex sinusoid is:

\[(e^{j\omega_0 t})^*=e^{-j\omega_0 t}=cos(\omega_0 t)-jsin(\omega_0 t)\\ Re[conj]=cos(\omega_0 t)\\ Im[conj]=-sin(\omega_0 t)\\ |conj| = 1\\ \angle{conj}=-\omega_0 t\]

Rewriting Sinusoids

Any sinusoid can be written in terms of complex exponentials

\[A \cos(\omega_0 t+\phi)=\frac{A}{2}e^{j\phi}e^{j\omega_0 t}+\frac{A}{2}e^{-j\phi}e^{-j\omega_0 t}\\ A \sin(\omega_0 t+\phi)=\frac{A}{j2}e^{j\phi}e^{j\omega_0 t}+\frac{A}{j2}e^{-j\phi}e^{-j\omega_0 t}\]

Alternately,

\[A\cos(\omega_0 t+\phi)=A Re[e^{j(\omega_0 t+\phi)}]\\ A\sin(\omega_0 t+\phi)=A Im[e^{j(\omega_0 t+\phi)}]\]