DSP First - Chapter 4 - Sampling and Aliasing

Sinusoids, or sinusoidal signals, representing the cosine or sine signals/waves, are the most basic signals in the theory of signals and systems. This chapter introduces the basic sinusoid concepts and operations.

Review of sine and cosine functions

• Basic trigonometric functions.

Properties

• Equivalence: $\sin\theta = \cos(\theta-\pi/2)$ or $\cos\theta=\sin(\theta+\pi/2)$; the sine function is just a cosine function that is shifted to the right by $\pi/2$,
• Periodicity: $\cos(\theta + 2\pi k) = \cos\theta$, where $k\in \mathbb{Z}$,
• Evenness of cosine: $\cos(-\theta) = \cos\theta$,
• Oddness of sine: $\sin(-\theta) = -\sin\theta$,
• Zeros of sine: $\sin(\pi k) = 0$, for $k\in\mathbb{Z}$,
• Ones of sine: $\cos(2\pi k) = 1$, for $k\in\mathbb{Z}$,
• Minus ones of cosine: $\cos[2\pi(k+\dfrac{1}{2})]=-1$, for $k\in\mathbb{Z}$,x
• Derivatives: $\dfrac{d \sin\theta}{d \theta} = \cos\theta$ and $\dfrac{d \cos\theta}{d \theta} = -\sin\theta$.

Trigonometric identities

• $\sin^2\theta + \cos^2\theta = 1$,
• $\cos^2\theta = \cos^2\theta - \sin^2\theta$,
• $\sin^2\theta = 2\sin\theta\cos\theta$,
• $\sin(\alpha\pm\beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$,
• $\cos(\alpha\pm\beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$,
• $\cos^2\theta = \frac{1}{2}(1+\cos 2\theta)$,
• $\sin^2\theta = \frac{1}{2}(1-\cos 2\theta)$.

Sinusoidal signals

The general mathematical formula for a cosine signal is

$$\begin{equation} x(t) = A\cos(\omega_0 t + \phi) = A\cos(2\pi f_0 t + \phi), \end{equation}\label{cos}$$ where - $A$ is the amplitude, - $\omega_0$ is the radian frequency (rad/sec), - $\phi$ represents the radian phase-shift (rads), - $f_0 = \omega_0/2\pi$, the cyclic frequency (sec$^{-1}$), represents the number of periods (cycles) per second, - $T_0 = \dfrac{1}{f_0} = \dfrac{2\pi}{\omega_0}$, the period (sec), is the one cycle length of the sinusoid in time.

Phase shift and time shift

• Having $x_1(t) = x(t-t_1)$, we say $x(t)$ is a time-shifted version of $s(t)$
  • if $t_1 > 0$ (positive), shifted to the right = delayed,
  • if $t_1 < 0$ (negative), shifted to the left = advanced.
• Taking the sinusoid as the form in Eq. $\eqref{cos}$,
  • convert time shift to a phase shift: $x(t-t_1) = A\cos(\omega_0(t-t_1)+\phi) = A\cos(\omega_0t+\phi+\phi_1)$, where $\phi_1 = -\omega_0t_1$ is the phase shift.
  • $t_1 = -\dfrac{\phi}{\omega_0} = -\dfrac{\phi}{2\pi f_0}$,
  • $\phi_1 = -2\pi f_0 t_1 = -2\pi\dfrac{t_1}{T_0}$.
• Based on the definition of the time shift and the phase shift, they have the opposite direction, e.g., if the time shift is positive (delay), the phase shift would be negative.
• modulo reduction and principal value of the phase.

Sampling and plotting sinusoids

• Be careful of the use of $n$ and $t$, meaning one can use either $x(nT_s)$ or $x(t)$ but never $x(tT_s)$.

Complex exponentials and phasors

• Complex exponentials signals provide an alternative representation for the real cosine signal and might make some manipulation or analysis easier.

Review of complex numbers

• Real part and imaginary part.
• Cartesian form or polar form.
• Magnitude and argument
• Euler's formula: $e^{j\theta} = \cos\theta + j\sin\theta$

Complex exponentials signal

• $\bar{x}(t) = Ae^{j(\omega_0t + \phi)}$
• $x(t) = \Re{\{Ae^{j(\omega_0t+\phi)}\}} = A\cos(\omega_0t+\phi)$

The rotating phasor interpretation

• The complex exponential signal could be expressed as $\bar{x}(t)=Xe^{j\omega_0t}$, i.e., the product of the complex amplitude $X=Ae^{j\phi}$ and the complex-valued function $e^{j\omega_0t}$.
• The complex amplitude $X$ is also called the phasor (vs. vector) (相量 vs. 向量).
• $\bar{x}(t)=Xe^{j\omega_0t}=Ae^{j\theta(t)}$, where $\theta(t) = \omega_0t + \phi$.
• In the complex plane, $\bar{x}(t)$ is simply a rotating vector at a constant rate $\omega_0$ with initial phase $\phi$ ($t=0$). So a complex exponential signal is a rotating phasor.
  • $\omega_0 > 0$: rotating counterclockwise,
  • $\omega_0 < 0$: rotating clockwise.

Inverse Euler formulas

• Applying the inverse Euler's formula, the real cosine signal with radian frequency $\omega_0$ is composed of two conjugated complex exponential signals with frequencies of $\omega_0$ and $-\omega_0$, and also complex amplitudes of $\frac{1}{2}Ae^{j\phi}$ and $-\frac{1}{2}Ae^{j\phi}$, respectively. $$x(t) = A\cos(\omega_0t+\phi) = \frac{1}{2}\bar{x}(t) + \frac{1}{2}\bar{x}^*(t) = \Re{\{\bar{x}(t)\}}$$

Phasor Addition

• Additions of sinusoids with the same frequency but different amplitudes and phases

Addition of complex numbers

• $z_1+z_2= (x_1+x_2)+j(y_1+y_2)$.

Phasor addition rule

• The summation of sinusoids with the same frequency is a sinusoid with the identical frequency with the amplitude and phase of a certain phasor calculated by the summation of the phasors of each sinusoid.
• Summation of phasors is also a phasor: $$\begin{equation} \sum_{k=1}^N A_ke^{j\phi_k} = Ae^{j\phi} \end{equation} \label{phasor_sum}$$
• Finally, lead us to: $$\sum_{k=1}^N A_k\cos(\omega_0t + \phi_k) = A\cos(\omega_0t + \phi)$$ which could be proved either by
  • trigonometric identities, or
  • summation of phasors following the steps:
    1. Get the phasors $X_k = A_ke^{j\phi_k}$ of each individual cosine signals,
    2. Add phasors using Eq. $\eqref{phasor_sum}$, employing polar-to-Cartesian-to-polar conversion,
    3. Multiply the resulting phasor $X=Ae^{j\phi}$ with the rotating function $e^{j\omega_0t}$ and get $\bar{x}(t)$,
    4. Take the real part and get $x(t) = \bar{x}(t)$.

Tuning fork and its physics

• higher-frequency "ting" and the lower-frequency "hum", where the "ting" comes from the transient.